Abstract and Applied Analysis Function Spaces, Compact Operators, and Their Applications

Guest Editors: S. A. Mohiuddine, M. Mursaleen, Adem Kiliçman, and Abdullah Alotaibi Function Spaces, Compact Operators, and Their Applications Abstract and Applied Analysis

Function Spaces, Compact Operators, and Their Applications

Guest Editors: S. A. Mohiuddine, M. Mursaleen, Adem Kilic¸man, and Abdullah Alotaibi Copyright © 2014 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Ravi P. Agarwal, USA Juan C. Corts, Spain Luca Guerrini, Italy Bashir Ahmad, Saudi Arabia Graziano Crasta, Italy Yuxia Guo, China M. O. Ahmedou, Germany Zhihua Cui, China Qian Guo, China Nicholas D. Alikakos, Greece Bernard Dacorogna, Switzerland Chaitan P. Gupta, USA Debora Amadori, Italy Vladimir Danilov, Russia Uno Hmarik, Estonia Douglas R. Anderson, USA Mohammad T. Darvishi, Iran Maoan Han, China Jan Andres, Czech Republic L. F. Pinheiro de Castro, Portugal Ferenc Hartung, Hungary Giovanni Anello, Italy Toka Diagana, USA Jiaxin Hu, China Stanislav Antontsev, Portugal Jess I. Daz, Spain Zhongyi Huang, China Mohamed K. Aouf, Egypt Josef Diblk, Czech Republic Chengming Huang, China Narcisa C. Apreutesei, Romania Fasma Diele, Italy Gennaro Infante, Italy Natig M. Atakishiyev, Mexico Tomas Dominguez, Spain Ivan Ivanov, Bulgaria Ferhan M. 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Peariffj, Croatia Wenyu Sun, China Agacik Zafer, Turkey Shuangjie Peng, China Robert Szalai, UK Jianming Zhan, China Antonio M. Peralta, Spain Sanyi Tang, China Weinian Zhang, China Sergei V.Pereverzyev, Austria Chun-Lei Tang, China Chengjian Zhang, China Allan Peterson, USA Gabriella Tarantello, Italy Zengqin Zhao, China Andrew Pickering, Spain Nasser-Eddine Tatar, Saudi Arabia Sining Zheng, China Cristina Pignotti, Italy Gerd Teschke, Germany Yong Zhou, China Somyot Plubtieng, Thailand Bevan Thompson, Australia Tianshou Zhou, China Milan Pokorny, Czech Republic Sergey Tikhonov, Spain Chun-Gang Zhu, China Sergio Polidoro, Italy Claudia Timofte, Romania Qiji J. Zhu, USA Ziemowit Popowicz, Poland Thanh Tran, Australia Malisa R. Zizovic, Serbia Maria M. Porzio, Italy Juan J. Trujillo, Spain Wenming Zou, China Enrico Priola, Italy Gabriel Turinici, France Contents

Function Spaces, Compact Operators, and Their Applications, S. A. Mohiuddine, M. Mursaleen, Adem Kilic¸man, and Abdullah Alotaibi Volume 2014, Article ID 342194, 1 pages

On Analog of Fourier Transform in Interior of the Light Cone, Tatyana Shtepina Volume 2014, Article ID 685794, 7 pages

Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces,MirnaDzamonjaˇ Volume2014,ArticleID184071,11pages

Multipliers of Modules of Continuous Vector-Valued Functions, Liaqat Ali Khan and Saud M. Alsulami Volume2014,ArticleID397376,6pages

Lipschitz Spaces and Fractional Integral Operators Associated with Nonhomogeneous Metric Measure Spaces, Jiang Zhou and Dinghuai Wang Volume2014,ArticleID174010,8pages

Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers,AhmetFarukC¸akmak and Feyzi Bas¸ar Volume2014,ArticleID236124,12pages

A 𝑘-Dimensional System of Fractional Neutral Functional Differential Equations with Bounded Delay, Dumitru Baleanu, Sayyedeh Zahra Nazemi, and Shahram Rezapour Volume2014,ArticleID524761,6pages

Integral Equation-Wavelet Collocation Method for Geometric Transformation and Application to Image Processing, Lina Yang, Yuan Yan Tang, Xiang Chu Feng, and Lu Sun Volume 2014, Article ID 798080, 17 pages

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The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations, Stefan Balint and Agneta M. Balint Volume 2014, Article ID 872548, 10 pages

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Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces, Vatan Karakaya, Kadri Dogan,ˇ Faik Gursoy,¨ and Muzeyyen¨ Erturk¨ Volume 2013, Article ID 560258, 9 pages

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Applications of Hankel and Regular Matrices in Fourier Series, Abdullah Alotaibi and M. Mursaleen Volume 2013, Article ID 947492, 3 pages

Statistical Summability of Double Sequences through de la Vallee-Poussin´ Mean in Probabilistic Normed Spaces,S.A.MohiuddineandAbdullahAlotaibi Volume 2013, Article ID 215612, 5 pages Existence of Periodic Solutions to Multidelay Functional Differential Equations of Second Order, Cemil Tunc¸ and Ramazan Yazgan Volume 2013, Article ID 968541, 5 pages

Sequence Spaces Defined by Musielak-Orlicz Function over 𝑛-Normed Spaces, M. Mursaleen, Sunil K. Sharma, and A. Kılıc¸man Volume 2013, Article ID 364743, 10 pages

Strongly Almost Lacunary 𝐼-Convergent Sequences, Adem Kılıc¸man and Stuti Borgohain Volume 2013, Article ID 642535, 5 pages

The Existence and Attractivity of Solutions of an Urysohn Integral Equation on an Unbounded Interval, Mohamed Abdalla Darwish, Jozef´ Bana´s, and Ebraheem O. Alzahrani Volume 2013, Article ID 147409, 9 pages

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations,M.DelaSen Volume 2013, Article ID 890657, 14 pages Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 342194, 1 page http://dx.doi.org/10.1155/2014/342194

Editorial Function Spaces, Compact Operators, and Their Applications

S. A. Mohiuddine,1 M. Mursaleen,2 Adem Kiliçman,3 and Abdullah Alotaibi1

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India 3 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to S. A. Mohiuddine; [email protected]

Received 7 July 2014; Accepted 7 July 2014; Published 14 September 2014

Copyright © 2014 S. A. Mohiuddine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of this special issue is to focus on the latest Acknowledgments developments and achievements of the theory of compact operators on function spaces and their applications in dif- The editors thank all the contributors and colleagues who did ferential, functional, and integral equations. The concept of the refereeing work very sincerely. the compactness plays a fundamental role in creating the S. A. Mohiuddine basis of several investigations conducted in nonlinear analysis M. Mursaleen andisveryusefulinseveraltopicsofappliedmathematics, Adem Kilic¸man engineering, mathematical physics, numerical analysis, and Abdullah Alotaibi so on. The compactness is very often used in fixed point theory and its applications to the theories of functional, differential, and integral equations of various types. On the other hand, the sequence spaces offer relevant tools for illustrating abstract results and properties in functional analysis. The research papers in this special issue cover various topics like function spaces and compact operators on them, sequence spaces and their topological and geometric prop- erties, paranormed Norlund¨ sequence spaces of nonabsolute type,spacesoffunctionsoverthefieldofnon-Newtonian complex numbers, statistical summability methods and their application to Fourier series, applications of Hankel and regu- lar matrices in Fourier series, statistical approximation results for Kantorovich-type operators, Mellin transform and Kratzel transform, fixed point theory and its applications, Fourier transform, convergence methods of iterative algorithm, iso- morphic universality, Sobolev type spaces, fractional integral operators, well-posedness and stability for Euler equations, integral equation-wavelet collocation method, functional differential equations, Urysohn integral equation, difference equations, and Lipschitz spaces and integral operators. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 685794, 7 pages http://dx.doi.org/10.1155/2014/685794

Research Article On Analog of Fourier Transform in Interior of the Light Cone

Tatyana Shtepina

Donetsk Institute of Municipal Economy, Bulavina Street 1, Donetsk 83053, Ukraine

Correspondence should be addressed to Tatyana Shtepina; [email protected]

Received 7 November 2013; Revised 6 May 2014; Accepted 7 May 2014; Published 6 August 2014

Academic Editor: Adem Kilicman

Copyright © 2014 Tatyana Shtepina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝜌 We introduce an analog of Fourier transform F in interior of light cone that commutes with the action of the Lorentz group. We 𝜌 ℎ describe some properties of F , namely, its action on pseudoradial functions and functions being products of pseudoradial function ℎ 𝜌 and space hyperbolic harmonics. We prove that Fℎ-transform gives a one-to-one correspondence on each of the irreducible components of quasiregular representation. We calculate the inverse transform too.

1. Introduction Our analog of Fourier transform is an intertwining oper- ator of quasiregular representation of Lie group 𝐺,soitmaps One of the most valuable integral transforms used in many- each of the irreducible components of decomposition in itself. dimensional analysis is the classical Fourier transform. It Following Stein and Weiss in Euclidean space [1], we describe 𝜌 is caused by the fact that this transform has a very simple action of F on pseudoradial functions and functions that transformation law at tensions and commutes with action of ℎ 𝑛 represent a product of pseudoradial function and space Lie group 𝑆𝑂(𝑛) in R .Asaconsequenceoftheseproper- 2 𝑛 hyperbolic harmonics. The obtained formulas allow us to ties, the 𝐿 (R ) decomposes in a direct sum of irreducible 𝜌 −1 writetheinversetransform(F ) with ease. These results subspaces that are invariant under rotations (e.g., [1,chapter ℎ 4] and [2, chapter 9]). This decomposition is an analog may be applicable to constructing an equivariant extension 2 of wave operator in interior of the light cone. For Laplace of decomposition 𝐿 (R) in a direct integral of irreducible representations: operator it was completed in [4]. ∞ 2 𝑖𝑥𝜉 𝐿 (R) = ∫ C ⋅𝑒 𝑑𝜉, (1) −∞ 2. Spherical Harmonics and Classical Funk-Hecke Theorem which act in one-dimensional subspaces invariant under 𝑛,𝑙 translations. There exists a general theorem that guarantees Let R be the space of homogeneous harmonic polynomials 𝑛,𝑙 the existence of a direct integral decomposition into irre- of degree 𝑙 in 𝑛 variables. If 𝑓(𝑥) belongs to R then ducible subrepresentations: it suffices that the topological 𝑛−1 its restriction to sphere 𝑆 is called the surface spherical group have a countable dense subset. 𝑛−1 harmonics of degree 𝑙 and is denoted by 𝑓(𝜉), 𝜉∈𝑆 .The The goal of this paper is to introduce an analog of Fourier 𝜌 relation between 𝑓(𝑥) and 𝑓(𝜉) follows from homogeneity transform Fℎ in the interior of the light cone on which Lie 1 condition: group 𝐺=𝑆𝑂0(𝑛−1, 1) acts. Suppose 𝐿loc(𝑆𝐻(𝑅)) is a space of locally integrable functions on pseudosphere 𝑆𝐻(𝑅) of radius 𝑙 𝑓 (𝑥) =𝑓(𝑟⋅𝜉) =𝑟𝑓 (𝜉) ,𝑟=|𝑥| . (2) 𝑅, so this space allows a direct integral decomposition into irreducible subspaces invariant under action of Lie group 𝐺 𝐿2(R𝑛) 𝐿2(R) Surface spherical harmonics of degree 𝑙 form a linear space that is similar to decompositions of and in 𝑛,𝑙 classical case. Actually, this decomposition was obtained by over C too,andwedenoteitbyR .Itisquiteevidentthat,for 𝑛,𝑙 2 𝑛−1 Gel’fand et al. in [3] in the sixties of the last century. any 𝑙 = 0, 1, 2, . ., the inclusion R ⊂𝐿(𝑆 ) is valid. But in 2 Abstract and Applied Analysis thesamespaceactstheso-calledquasiregularrepresentation Proposition 3 (see [1, chapter IV, Theorems 3.3, 3.10]). (a) of Lie group 𝑆𝑂(𝑛), defined by the next equality: Let function 𝑓(𝑥) be a product of radial function and space 𝑙 −1 𝑛−1 spherical harmonics of degree : [𝑇 (𝑔) 𝑓] (𝜉) =𝑓(𝑔 𝜉) , 𝜉 ∈𝑆 ,𝑔∈𝑆𝑂(𝑛) . (3) 𝑙 𝑓 (𝑥) =𝑓0 (|𝑥|) 𝑆𝐾 (𝑥) , (7) This representation is unitary with respect to the standard 2 𝑛−1 1 𝑛 2 𝑛 inner product in 𝐿 (𝑆 ). The next theorem is widely known where 𝑓0(𝑟) is such that 𝑓(𝑥) ∈𝐿 (R )∩𝐿(R ).Thenits (see, e.g., [2, 5, 6]). Fourier transform has a form:

2 𝑛−1 𝑙 Theorem 1. (a) The next decomposition is valid 𝐿 (𝑆 )= [F𝑓] (𝑥) =𝐹𝑙 (|𝑥|) 𝑆𝐾 (𝑥) , (8) ⨁∞ R𝑛,𝑙 𝑙=0 (decomposition of Hilbert space into orthogonal where direct sum). 𝑛,𝑙 2𝜋 (b) The subspaces R consisting of the space harmonics of 𝐹𝑙 (𝑟) = degree 𝑙 are invariant under Fourier transform F. 𝑖𝑙𝑟(𝑛−2)/2+𝑙 𝑛,𝑙 R +∞ (9) (c) Each of the subspaces is invariant with respect to 𝑙+𝑛/2 the quasiregular representation 𝑇 of Lie group 𝑆𝑂(𝑛) and is iso- × ∫ 𝑓0 (𝑠) 𝑠 𝐽(𝑛+2𝑙−2)/2 (2𝜋𝑟𝑠) 𝑑𝑠. 0 morphic to the irreducible representation 𝑇(𝑙,0,...,0) with a highest weight (𝑙,0,...,0). (b) In particular, Fourier transform for radial function 𝑓(𝑥) = 2 𝑛−1 (d) The quasiregular representation of 𝑆𝑂(𝑛) in 𝐿 (𝑆 ) 𝑓0(|𝑥|) is also radial: has a simple spectrum. 𝑛,𝑙 [F𝑓] (𝑥) =𝐹0 (|𝑥|) (10) (e) The space R has a dimension (𝑛+𝑙−3)!(𝑛+2𝑙− 2)/𝑙!(𝑛−2)!and an orthogonal basis consisting of the next sur- (one sets 𝑙=0in the above formula). 𝑙 face spherical harmonics 𝑆𝐾(𝜃): As Proposition 3 implies, the infinite-dimensional sub- 𝑛−3 𝑛/2−𝑗/2−1+𝑚 spaces 𝑆𝑙 (𝜃) =(∏𝐶 𝑗+1 ( 𝜃 ) 𝐾 𝑚𝑗−𝑚𝑗+1 cos 𝑛−𝑗−1 𝑗=0 𝑙 H𝑙 = span {𝑓 (|𝑥|) 𝑆𝐾 (𝑥)} , (11) (4) 𝑓(𝑟) 𝑚𝑗+1 where runs over the set of radial functions satisfying × sin (𝜃𝑛−𝑗−1)) 𝑙 conditions of Proposition 3 and 𝑆𝐾(𝑥) runs over the set of space spherical harmonics of degree 𝑙, are invariant under ±𝑖𝑚 𝜃 2 𝑛 ×𝑒 𝑛−2 1 , Fourier transform in 𝐿 (R ). Ontheotherhand,ifwefixthefunction𝑓(|𝑥|) we 𝑛−1 where 𝜃1,𝜃2,...,𝜃𝑛−1 are Euler angles on sphere 𝑆 ; 𝑙=𝑚0; get a subspace in H𝑙,whichisinvariantwithrespectto 2 𝑛 and 𝐾 is multi-index 𝐾=(𝑚1,...,𝑚𝑛−3;±𝑚𝑛−2) such that quasiregular representation of group 𝑆𝑂(𝑛) in 𝐿 (R ).Itcan 𝑚0 ≥𝑚1 ≥⋅⋅⋅≥𝑚𝑛−3 ≥𝑚𝑛−2 ≥0, 𝑚𝑖 ∈ Z. be easily verified that it is isomorphic to the irreducible representation 𝑇(𝑙,0,...,0) with a highest weight (𝑙,0,...,0). The known Funk-Hecke theorem states that for integral Since spaces of irreducible nonisomorphic unitary represen- operators whose kernels depend only on the distance 𝜌 (in tations of compact group are mutually orthogonal (H. Weyl’s spherical geometry) between points 𝜉 and 𝜂 where 𝜉, 𝜂 ∈ 𝑛−1 theorem),wehaveonemoreimportantconsequenceofthe 𝑆 everysurfacesphericalharmonicsisaneigenvector.We classical Funk-Hecke theorem. give a contemporary formulation of the Funk-Hecke theorem following the monograph [7]ofErdelyi.´ Corollary 4. The next decomposition into orthogonal direct sum is valid: Theorem 2 (Funk, Hecke). Let 𝐹(𝑥) be a function of a real ∞ 𝑥 [−1, 1] 2 𝑛 variable which is absolutely Lebesgue integrable on 𝐿 (R )=⨁ H𝑙. (12) together with its square. Then, for any unit vector 𝜂, 𝑙=0

𝑙 𝑙 We will try to extend the classical theorem of Funk and ∫ 𝐹[(𝜉,𝜂)]𝑆𝐾 (𝜉) 𝑑𝜉 =𝜆𝑛,𝑙𝑆𝐾 (𝜂) , (5) 𝑛−1,1 𝑆𝑛−1 Hecke and its corollaries on the hyperbolic space R with where indefinite inner product. ∞ 𝑙 𝑛/2 (2−𝑛)/2 𝜆𝑛,𝑙 =𝑖(2𝜋) ∫ 𝑡 𝐽𝑙+𝑛/2−1 (𝑡) 𝑓 (𝑡) 𝑑𝑡, 3. Hyperbolic Harmonics and Generalized −∞ Funk-Hecke Theorem 1 (6) 1 −𝑖𝑥𝑡 𝑛−1,1 𝑓 (𝑡) = ∫ 𝑒 𝐹 (𝑥) 𝑑𝑥. Let R be the pseudo-Euclidean space with the indefinite 2𝜋 −1 inner product The simple consequences of the Funk-Hecke theorem are [𝑥,] 𝑦 =−𝑥𝑦 −⋅⋅⋅−𝑥 𝑦 +𝑥 𝑦 . the following two propositions. 1 1 𝑛−1 𝑛−1 𝑛 𝑛 (13) Abstract and Applied Analysis 3

𝑛,𝜎 This inner product may be used for definition of a distance harmonics.Thismeansthatwemaycall𝐻𝐿 (𝜃) surface 𝑛−1,1 𝑟(𝑥, 𝑦) between two points 𝑥, 𝑦 ∈ R that do not belong hyperbolic harmonics and consider them analogs of surface 𝑙 to the light cone [𝑧, 𝑧]. =0 We assume, for such two points, spherical harmonics 𝑆𝐾(𝜃). 𝑛𝜎 𝑛,𝜎 The relation between 𝐻 and 𝐻̃ follows from homo- [𝑥, 𝑦] 𝐿 𝐿 𝑟(𝑥,𝑦)= . geneous condition: cosh (14) √[𝑥, 𝑥] ⋅ [𝑦, 𝑦] ̃𝑛,𝜎 ̃𝑛,𝜎 𝜎 ̃𝑛,𝜎 𝐻𝐿 (𝑢) = 𝐻𝐿 (𝑟⋅𝑥) =𝑟 𝐻𝐿 (𝑥) ,𝑟=|𝑥| . (21) Such distance may take either real nonnegative or pure 𝑛,𝜎 Suppose G is the minimal closed subspace in imaginary values. However, if we restrict ourselves by the 1 𝐿loc(𝑆𝐻,𝑑𝜉) containing all surface hyperbolic harmonics interior 𝑈 of the light cone’s upper sheet 𝑛,𝜎 𝑛,𝜎 𝐻𝐿 (𝜃). Similarly to Euclidean case, denote by G the 1 𝑛−1,1 minimal closed subspace in 𝐿loc(𝑈, 𝑑𝑢) containing all space 𝑈={𝑥∈R | [𝑥, 𝑥] >0,𝑥𝑛 >0}, (15) hyperbolic harmonics. It is obvious, from what is stated 𝐻𝑛,𝜎(𝜃) then, for all 𝑥, 𝑦 ∈𝑈,wehave𝑟(𝑥, 𝑦). ⩾0 above, that the 𝐿 are linearly independent for different 2 𝜎 andallofthemaresubspacesinthespaceofwaveequation Letuscallthesetofallpoints𝑥 of 𝑈,forwhich[𝑥, 𝑥] =𝑅 solutions. holds, pseudosphere of radius 𝑅.Wewillusethedesignation 𝑛,𝜎 Basic properties of G areprovedin[6]. We formulate 𝑆𝐻(𝑅) for pseudosphere of radius 𝑅 and 𝑆𝐻 for pseudosphere 𝑛−1,1 them in a compact form now. of radius 1 in R .Recallthat𝑆𝐻 is a manifold of a R𝑛−1,1 constant negative curvature in on the one hand and Theorem 5 (analog of Theorem 1). (a) The next decomposi- a homogeneous symmetric space with respect to the action 1 ∞ 𝑛,𝜎 tion is valid 𝐿𝑙𝑜𝑐(𝑆𝐻,𝑑𝜉)=∫ G 𝑑𝜌 (the decomposition into of Lie group 𝐺=𝑆𝑂0(𝑛 − 1, 1) on the other hand, because 0 continuous direct sum). G𝑛,𝜎 𝑆𝐻 ≅𝑆𝑂0 (𝑛−1,1) /𝑆𝑂 (𝑛−1) . (16) (b) Each of the subspaces is invariant with respect to the quasiregular representation 𝑅 of Lie group 𝑆𝑂0(𝑛 − 1, 1). 𝑛,𝜎 It follows from here that 𝑆𝐻 possesses the unique up to (c) The representations of 𝑆𝑂0(𝑛 − 1, 1) in G are irredu- constant multiplier left-invariant with respect to 𝐺 measure cible and mutually nonisomorphic. 𝑑𝜉 1 : (d) The quasiregular representation 𝑅 of 𝐺 in 𝐿𝑙𝑜𝑐(𝑆𝐻,𝑑𝜉) has a simple spectrum. 𝑑𝜉 = 𝑛−2𝜃 𝑛−3𝜃 ⋅...⋅ 𝜃 𝑑𝜃 𝑑𝜃 ⋅⋅⋅𝑑𝜃 . 𝑛,𝜎 sinh 𝑛−1sin 𝑛−2 sin 2 1 2 𝑛−1 (e) The space G is infinite-dimensional and has a basis 𝑛,𝜎 (17) generated by functions of the form 𝐻𝐿 (𝜃). 1 We denote by 𝐿loc(𝑆𝐻,𝑑𝜉) the space of complex-valued The following theorem generalizes the classical Funk- functions on 𝑆𝐻 locally integrable in measure 𝑑𝜉.In Hecke theorem to the case of hyperbolic space. This theorem, 1 𝑛=2 𝑛=4 𝐿loc(𝑆𝐻,𝑑𝜉) acts the quasiregular representation 𝑅 of Lie for cases and ,wasprovedin[8]. But the general group 𝐺, defined by case was published in [6]. −1 Theorem 6. 𝐹(𝑥) 𝑥 (𝑅 (𝑔) 𝑓) (𝑥) =𝑓(𝑔 𝑥) , 𝑥𝐻 ∈𝑆 ,𝑔∈𝑆𝑂0 (𝑛−1,1) . Suppose is a function of a real variable (18) such that 1 2 (a) 𝐹(𝑥) ∈𝐿 (−∞, +∞) ∩ 𝐿 (−∞, +∞); We need the notion of space and surface hyperbolic harmon- ics to decompose the representation 𝑅 into irreducible ones. (b) 𝐹(𝑥) can be continued analytically to a function 𝐹(𝛼) 1 𝛼=𝑥+𝑖𝑦 We will consider in 𝐿loc(𝑆𝐻,𝑑𝜉)functions of the complex variable that is bounded and analytic in the lower half-plane 𝑦⩽0; (3−𝑛)/2−𝑚 𝐻𝑛,𝜎 (𝜃) = (3−𝑛)/2𝜃 P 1 ( 𝜃 ) 1 𝐿 sinh 𝑛−1 𝜎+(𝑛−3)/2 cosh 𝑛−1 (c) 𝐹(𝑥) has Fourier preimage 𝑓(𝑡) ∈𝐿 (0, +∞). (19) 𝑚 𝑛,𝜎 1 𝐻 ×𝑆𝐾 (𝜃1,𝜃2,...,𝜃𝑛−2), Let 𝐿 be an arbitrary surface hyperbolic harmonic of homogeneity degree 𝜎.Then,foranyvector𝜂∈𝑆𝐻,the 𝜇 where P] (𝑥) are adjoined Legendre functions of genus one, following equality holds: 𝐿=(𝑘0,𝐾), 𝐾=(𝑘1,...,𝑘𝑛−4,±𝑘𝑛−3),with𝑘0 ⩾𝑘1 ⩾⋅⋅⋅⩾ 𝑛,𝜎 𝑛,𝜎 𝑘𝑛−2 ⩾0, and all parameters 𝑘𝑖 are integers. ∫ 𝐹 ([𝜉, 𝜂]) 𝐻 (𝜉) 𝑑𝜉 =𝜆 𝐻 (𝜂) , 𝑛,𝜎 𝐿 𝑛,𝜎 𝐿 (22) It is easy to see that if we extend functions 𝐻𝐿 (𝜃) from 𝑆𝐻 pseudosphere 𝑆𝐻 to the interior 𝑈 of the light cone’s upper where the eigenvalue 𝜆𝑛,𝜎 does not depend on index 𝐿 of sheet “by homogeneity” with the degree 𝑛,𝜎 harmonics 𝐻𝐿 and equals 𝑛−2 𝜎=− +𝑖𝜌, 𝜌∈[0, +∞) , (20) 𝑛/2 (𝑛−2)/2 2 𝜆𝑛,𝜎 =2 (−𝜋𝑖) 𝐻̃𝑛,𝜎 𝑈 +∞ (23) then obtained functions 𝐿 on are solutions of the × ∫ 𝛼2−𝑛/2𝐾 (𝑖𝛼) 𝑓 (𝛼) 𝑑𝛼, ̃𝑛,𝜎 𝜎+𝑛/2−1 wave equation ◻𝐻𝐿 =0;thatis,theyarespace hyperbolic 0 4 Abstract and Applied Analysis

퐾 (푧) 儨 儨1+푖휌 −푖[푥,휉] −1−푖휌 where ] is the McDonald function and = 儨휉儨 ∫ 푒 |푥| (푅 (푔) 푓) (푥) 푑푥 푈 1 +∞ 푓 (훼) = ∫ 푒푖훼푡퐹 (푡) 푑푡. 儨 儨1+푖휌 −푖[푥,휉] −1−푖휌 −1 √ (24) = 儨휉儨 ∫ 푒 |푥| 푓(푔 푥) 푑푥. 2휋 −∞ 푈 (27) The idea of the proof lies in using of intertwining operators theory. Namely, let us define an operator A in On the other hand, 1 퐿loc(푆퐻) by the equality 휌 푅[(푔)∘Fℎ푓] (휉) =푅(푔)(F휌푓) (휉) [A푓] (휂) = ∫ 퐹 [(휉, 휂)] 푓 (휉) 푑휉. (25) ℎ 푆𝐻 儨 儨1+푖휌 =푅(푔)(儨휉儨 ∫ 푒−푖[푥,휉]|푥|−1−푖휌푓 (푥) 푑푥) A 儨 儨 Itcanbeeasilyseenthat is an intertwining operator of 푈 (28) quasiregular representation 푅.Becausethespectrumof푅 is 푛,휎 儨 儨1+푖휌 −푖[푥,푔−1휉] −1−푖휌 simple, A can map the invariant subspace G only to itself. = 儨휉儨 ∫ 푒 |푥| 푓 (푥) 푑푥 푈 Schur’s lemma implies that A|G𝑛,𝜎 =휆푛,휎 ⋅퐸,where휆푛,휎 does not depend on the multi-index 퐿 and 퐸 is identity operator. 儨 儨1+푖휌 = 儨휉儨 ∫ 푒−푖[푔푥,휉]|푥|−1−푖휌푓 (푥) 푑푥. Thus one can assume that 퐿 = (0,0,...,0);thatis, 儨 儨 푛휎 푈 examine the zonal hyperbolic harmonics 퐻푂 (휉) instead of 퐻푛휎(휉) −1 arbitrary spherical harmonics 퐿 . Change variable 푔푥 = 푡.Then,푥=푔 푡 and 푑푥 = 푑푡 since the The nontrivial part of the proof lies in calculation of measure on 푈 is invariant under action of Lie group 푆푂0(푛 − eigenvalue 휆푛,휎 rather than in verifying if surface hyperbolic 1, 1).Wehave A harmonics are eigenvectors for . 휌 [푅 (푔) ∘ Fℎ푓] (휉)

儨 儨−1−푖휌 4. The Hyperbolic Fourier Transform and 儨 儨1+푖휌 −푖[푡,휉]儨 −1 儨 −1 = 儨휉儨 ∫ 푒 儨푔 푡儨 푓(푔 푡) 푑푡 Some of Its Properties 푈 (29)

儨 儨1+푖휌 To obtain analogs of Proposition 3 we need some integral = 儨휉儨 ∫ 푒−푖[푡,휉]|푡|−1−푖휌푓(푔−1푡) 푑푡. 휌 1 儨 儨 푈 transform Fℎ in 퐿loc(푈) similar to the Fourier transform in 2 푛 퐿 (R ). 휌 휌 휌 So, Fℎ ∘ 푅(푔) = 푅(푔) ∘ Fℎ;thatis,Fℎ is an intertwining Definition 7. The hyperbolic Fourier transform in space operator. 퐿1 (푈) F휌 loc is a transform ℎ, defined by (this integral should Proposition 9. Let 푓(푥) be a pseudoradial function belonging be understood in a regularized value sense) 1 1/2 to the space 퐿푙표푐(푈);thatis,푓(푥) = 푓0([푥, 푥] ) for almost all 휌 푥∈푈. Then its integral transform F is pseudoradial for all 휌 儨 儨1+푖휌 −푖[휉,푥] −1−푖휌 ℎ [Fℎ푓] (휉) = 儨휉儨 ∫ 푒 |푥| 푓 (푥) 푑푥, (26) 휉∈푈: 푈 휌 1/2 [F 푓] (휉) =퐹0 ([휉, 휉] ), (30) where 푑푥 = 푑푟푑푠 and 푑푠 is an invariant measure on ℎ 푆 (푟) hyperboloid 퐻 . where 휌 F (푛−2)/2 푛/2 −(푛−4)/2+푖휌 Note that hyperbolic Fourier transform ℎ is dependent 퐹0 (푠) = (−휋푖) ⋅2 ⋅푠 on 휌; the reason is that this transform acts on its “own” 푛휎 +∞ (31) component G (i.e., for 휎=−(푛−2)/2+푖휌) simply (푛−2)/2−푖휌 × ∫ 푓0 (푟) 푟 퐾(푛−2)/2 (푖푟푠) 푑푟. as a scalar operator (see Corollary 11 from Proposition 10). 0 Perhaps, uniform integral operator, acting on all subspaces G푛휎 푅(푔) ̂ 휌 as a scalar and invariant under , does not exist. Proof. Let 푓(휉) = (Fℎ푓)(휉).Then,takinguseof 휌 Proposition 8, Proposition 8. The hyperbolic Fourier transform Fℎ is an intertwining operator of quasiregular representation 푅 in [푅 (푔) 푓]̂ (휉) =(푅(푔)∘F휌푓) (휉) 1 ℎ 퐿푙표푐(푈). 휌 =(Fℎ ∘푅(푔)푓)(휉) (32) Proof. By definition of quasiregular representation and 휌 ̂ hyperbolic Fourier transform, we have =(Fℎ푓) (휉) = 푓 (휉) . 휌 푅(푔)푓 = 푓 푓 [Fℎ ∘푅(푔)푓](휉) We take into account that because is pseu- doradial. 휌 = Fℎ (푅 (푔) 푓) (휉) This proves the first part of the proposition. Abstract and Applied Analysis 5

We fix now √[푥, 푥] = 푟 and √[휉, 휉] = 푠.Introducethe We calculate the inner integral on sphere in a standard way: 耠耠 耠耠 Euler coordinates on hyperboloids 푆퐻(푟) and 푆퐻(푠): first we integrate on a parallel ⟨푥 ,휉 ⟩=cos 푎,orthogonal 耠耠 to vector 휉 ;thenweintegrateby푎 theobtainedfunctionin 푥1 =푟sinh 휃푛−1 sin 휃푛−2 ⋅...⋅cos 휃1, variable 푎, 0⩽푎⩽휋:

푥2 =푟sinh 휃푛−1 sin 휃푛−2 ⋅...⋅sin 휃1, 푖푟푠 sinh 휃 sinh 휑 cos 푎 ∫ 푒 𝑛−1 𝑛−1 푑푆 . 푆𝑛−2 . 휋 (푛−2)/2 푖푟푠 sinh 휃 sinh 휑 cos 푎 2휋 푛−3 = ∫ 푒 𝑛−1 𝑛−1 ( ) ( 푎) 푑푎, 푥푛−1 =푟sinh 휃푛−1 cos 휃푛−2, sin 0 Γ ((푛−2) /2)

푥푛 =푟cosh 휃푛−1, (37) (33) 휉 =푠 휑 휑 ⋅...⋅ 휑 , 1 sinh 푛−1 sin 푛−2 cos 1 (푛−2)/2 푛−3 where (2휋 /Γ((푛 − 2)/2))(sin 푎) is area of surface of 휉2 =푠sinh 휑푛−1 sin 휑푛−2 ⋅...⋅sin 휑1, (푛 − 3)-dimensional sphere with radius sin 푎. After changing variables cos 푎=푡,sin푎=√1−푡2,weget . . 푖푟푠 sinh 휃 sinh 휑 cos 푎 ∫ 푒 𝑛−1 𝑛−1 푑푆 휉푛−1 =푠sinh 휑푛−1 cos 휑푛−2, 푆𝑛−2

휉푛 =푠cosh 휑푛−1. (푛−2)/2 1 2휋 푖푟푠 sinh 휃 sinh 휑 ⋅푡 = ∫ 푒 𝑛−1 𝑛−1 By definition of hyperbolic Fourier transform, Γ ((푛−2) /2) −1 (푛−4)/2 휌 ×(1−푡2) 푑푡 [Fℎ푓] (휉) +∞ (푛−2)/2 儨 儨1+푖휌 −푖[푥,휉] −1−푖휌 √ 2휋 Γ ((푛−2) /2) Γ (1/2) = 儨휉儨 ∫ ∫ 푒 |푥| 푓0 ( [푥, 푥])푑푟푑푠 = ⋅ 0 푆 (푟) Γ ((푛−2) /2) (푛−3)/2 𝐻 (−푟푠 sinh 휃푛−1 sinh 휑푛−1/2) +∞ 儨 儨1+푖휌 푛−1 −1−푖휌 (푛−3)/2 = 儨휉儨 ∫ 푓0 (푟) 푟 푟 푑푟 (−푟푠 sinh 휃푛−1 sinh 휑푛−1/2) 0 ⋅ Γ ((푛−2) /2) Γ (1/2) (38) −푖[푟푥󸀠,푠휉󸀠] 푛−2 ⋅ ∫ 푒 sinh 휃푛−1 1 (2((푛−3)/2)−1)/2 푖(푟푠 sinh 휃𝑛−1 sinh 휑𝑛−1)푡 2 푆𝐻(푟) ⋅ ∫ 푒 (1 − 푡 ) 푑푡 −1 푛−3 ⋅ sin 휃푛−2 ⋅...⋅sin 휃2푑휃푛−1푑휃푛−2 ⋅⋅⋅푑휃1, (2휋)(푛−2)/2 Γ ((푛−2) /2) Γ (1/2) (34) = ⋅ Γ ((푛−2) /2) (푛−3)/2 (−푟푠 휃푛−1 휑푛−1) 耠 耠 sinh sinh where 푥 ,휉 ∈푆퐻. Consider the inner integral 퐼 on hyperboloidinmoredetail: ⋅퐽(푛−3)/2 (−푟푠 sinh 휃푛−1 sinh 휑푛−1)

∞ (푛−1)/2 −푖푟푠 cosh 휃 cosh 휑 (2휋) 퐼=∫ ∫ 푒 𝑛−1 𝑛−1 = 𝑛−2 (푛−3)/2 0 푆 (−푟푠 sinh 휃푛−1 sinh 휑푛−1) 푖푟푠 sinh 휃 sinh 휑 ⟨푥󸀠󸀠,휉󸀠󸀠⟩ ×푒 𝑛−1 𝑛−1 ×퐽 (−푟푠 휃 휑 ). (35) (푛−3)/2 sinh 푛−1 sinh 푛−1 푛−2 푛−3 ⋅ sinh 휃푛−1sin 휃푛−2 Putting the found value of integral on sphere into the ⋅...⋅sin 휃2푑휃푛−1 ⋅⋅⋅푑휃1, expression of hyperbolic Fourier transform, we get:

耠耠 耠耠 where 푥 and 휉 belong to spheres, which are intersections (2휋)(푛−1)/2푠1+푖휌 +∞ 푓 (푟) 푟푛−1 2 2 [F휌푓] (휉) = ∫ 0 푟−1−푖휌 [푥, 푥] = 푟 [휉, 휉] = 푠 ℎ (푛−3)/2 (푛−3)/2 of hyperboloids and by hyperplanes 0 푟 (−푠 sinh 휑푛−1) 푥푛 =푟cosh 휃푛−1 and 휉푛 =푠cosh 휑푛−1 correspondingly: +∞ ∞ −푖푟푠 cosh 휃𝑛−1 cosh 휑𝑛−1 −푖푟푠 cosh 휃𝑛−1 cosh 휑𝑛−1 푛−2 ⋅ ∫ 푒 퐼= ∫ 푒 sinh 휃푛−1푑휃푛−1 0 0 (푛−1)/2 󸀠󸀠 󸀠󸀠 × 휃푛−1 푖푟푠 sinh 휃𝑛−1 sinh 휑𝑛−1⟨푥 ,휉 ⟩ 푛−3 (36) sinh ⋅ ∫ 푒 sin 휃푛−2 푆𝑛−2 ×퐽(푛−3)/2 (−푟푠 sinh 휃푛−1 sinh 휑푛−1)푑휃푛−1푑푟. ⋅...⋅sin 휃2푑휃푛−2 ⋅⋅⋅푑휃1. (39) 6 Abstract and Applied Analysis

√ 2 Change variables 푧=sinh 휃푛−1,cosh휃푛−1 = 1+푧 ,and Proof. We have, by definition of hyperbolic Fourier trans- √ 2 푑푧 = 1+푧 푑휃푛−1,sowehave form, 휌 儨 儨1+푖휌 −푖[푥,휉] −1−푖휌 푛휎 +∞ 儨 儨 √ ̃ [Fℎ푓] (휉) = 儨휉儨 ∫ 푒 |푥| 푓0 ( [푥, 푥]) 퐻퐿 (푥) 푑푥 −푖푟푠 cosh 휃𝑛−1 cosh 휑𝑛−1 (푛−1)/2 푈 ∫ 푒 sinh 휃푛−1 0 +∞ 儨 儨1+푖휌 −1−푖휌 ×퐽 (−푟푠 휃 휑 )푑휃 = 儨휉儨 ∫ 푓0 (푟) 푟 (푛−3)/2 sinh 푛−1 sinh 푛−1 푛−1 0 (40) −푖푟푠 cosh 휑 √1+푧2 (푛−1)/2 +∞ 푒 𝑛−1 푧 −푖[푥,휉] ̃푛휎 = ∫ ×(∫ 푒 퐻퐿 (푥) 푑푥) 푑푟. 푆 (푟) 0 √1+푧2 𝐻 (46) ⋅퐽 (−푟푠 휑 푧) 푑푧. (푛−3)/2 sinh 푛−1 耠 耠 耠 耠 ̃푛휎 Let 푥=푟푥, 휉=푠휉,where푥 ,휉 ∈푆퐻.Because퐻퐿 is a 휎 Now we apply integral from [9], Section 2.12.10,formula(10), homogeneous function of homogeneity degree , 휌=푖푟푠 휑 푧=1 푥=푧 푐=−푟푠 휑 푛휎 푛휎 耠 휎 푛휎 耠 and set cosh 푛−1, , , sinh 푛−1,and 퐻̃ (푥) = 퐻̃ (푟푥 )=푟 퐻 (푥 ). (47) ] =(푛−3)/2.Finallywehave 퐿 퐿 퐿 Hence, −푖푟푠 cosh 휑 √1+푧2 (푛−1)/2 +∞ 푒 𝑛−1 푧 [F휌푓] (휉) ∫ ℎ 0 √1+푧2 儨 儨1+푖휌 −푖[푥,휉] −1−푖휌 √ ̃푛휎 = 儨휉儨 ∫ 푒 |푥| 푓0 ( [푥, 푥]) 퐻퐿 (푥) 푑푥 ⋅퐽(푛−3)/2 (−푟푠 sinh 휑푛−1푧) 푑푧 푈 2 ( 푥=푟푥耠, 푑푥=푟푛−1푑푥耠) √ (푛−3)/2 −(푛−2)/2 we change variable = (−푟푠 sinh 휑푛−1) (푖푟푠) 퐾(푛−2)/2 (푖푟푠) . 휋 +∞ (41) 儨 儨1+푖휌 −1−푖휌 = 儨휉儨 ∫ 푓0 (푟) 푟 0 Hence, −푖푟푠[푥󸀠,휉󸀠] 휎 푛휎 耠 푛−1 耠 ×(∫ 푒 푟 퐻퐿 (푥 )푟 푑푥 )푑푟 휌 푛/2 (푛−2)/2 푆𝐻 [Fℎ푓] (휉) =2 휋 +∞ +∞ 儨 儨1+푖휌 휎+푛−1 −1−푖휌 1+푖휌 푛−1 −1−푖휌 = 儨휉儨 ∫ 푓0 (푟) 푟 푟 ×푠 ∫ 푓0 (푟) 푟 푟 0 0 󸀠 󸀠 −(푛−2)/2 ×(∫ 푒−푖푟푠[푥 ,휉 ]퐻푛휎 (푥耠)푑푥耠)푑푟. ⋅ (푖푟푠) 퐾(푛−2)/2 (푖푟푠) 푑푟 (42) 퐿 푆𝐻 = (−휋푖)(푛−2)/22푛/2푠−(푛−4)/2+푖휌 (48)

∞ We make use of formula (32) from [6]tocalculatetheintegral (푛−2)/2−푖휌 푆 휉耠 ∈푆 × ∫ 푓0 (푟) 푟 퐾(푛−2)/2 (푖푟푠) 푑푟. on 퐻. Namely, for each 퐻,theequalitytakesplace: 0 −푖푟푠[푥󸀠,휉󸀠] 푛휎 耠 耠 ∫ 푒 퐻퐿 (푥 )푑푥 Proposition 9 is proved. 푆𝐻 (49) 1 휋푖 (푛−2)/2 Proposition 10. Suppose function 푓(푥) ∈ 퐿푙표푐(푈) is a product 푛/2 푛휎 耠 =2 (− ) 퐾휎+(푛−2)/2 (푖푟푠) 퐻퐿 (휉 ). of pseudoradial function and space hyperbolic harmonic of 푟푠 휎 homogeneity degree : Now we have 휌 푛/2 1+푖휌 푛휎 耠 1/2 ̃푛,휎 [Fℎ푓] (휉) =2 푠 퐻퐿 (휉 ) 푓 (푥) =푓0 ([푥, 푥] ) 퐻퐿 (푥) , (43) +∞ 휌 휎+푛−1 −1−푖휌 F ⋅ ∫ 푓0 (푟) 푟 푟 and then its ℎ-transform is 0

(푛−2)/2 [F휌푓] (휉) =퐹푛,휎 ([휉, 휉]1/2) 퐻̃푛,휎 (휉) , 휋푖 ℎ 퐿 (44) ×(− ) 퐾휎+(푛−2)/2 (푖푟푠) 푑푟 푟푠 (50) 푛/2 ̃푛휎 (푛−2)/2 where =2 퐻퐿 (휉) 푠⋅(−휋푖) 퐹푛,휎 (푠) =2푛/2(−휋푖)(푛−2)/2 +∞ × ∫ 푓0 (푟) 퐾휎+(푛−2)/2 (푖푟푠) 푑푟 +∞ (45) 0 ⋅푠∫ 푓0 (푟) 퐾휎+(푛−2)/2 (푖푟푠) 푑푟. 푛휎 ̃푛휎 0 =퐹 퐻퐿 (휉) , Abstract and Applied Analysis 7

where any function 휑(푥) definedonaLobachevskyspace푆퐻 could be raised to function 휑̃ on 퐺=푆푂0(푛 − 1, 1) that is constant 퐹푛휎 (푠) =2푛/2퐻̃푛휎 (휉) 푠⋅(−휋푖)(푛−2)/2 퐿 on the left cosets under subgroup 푆푂(푛−1). An analog of this 1 2 +∞ (51) theorem in 퐿 ∩퐿 (푆퐻) is valid as it was shown in the author’s × ∫ 푓0 (푟) 퐾휎+(푛−2)/2 (푖푟푠) 푑푟. paper [10]. Our proof method uses Fourier transform and an 0 ordinary convolution of functions on 퐺: Proposition 10 is proved. −1 [휑∗̃ 휓̃] (푥) = ∫ 휑̃ (푔) 휓̃ (푔 푥) 푑푔. (58) Corollary 11. If 휎=−(푛−2)/2+푖휌,thenhyperbolic 퐺 휌 푛휎 Fourier transform F acts on the space G of space hyperbolic ℎ We hope that the technique developed in this work (including harmonics as a scalar operator: the hyperbolic Fourier transform) will be able to prove 휌 ̃푛휎 ̃푛휎 that intertwining operators of quasiregular representation of [F 퐻 ] (휉) =휆푛휎퐻 (휉) , (52) ℎ 퐿 퐿 Lorentz group are also involutions in interior of the light cone. where Conflict of Interests (−2휋푖)푛/2 휆푛휎 = . (53) 2 cosh (휋휌/2) The author declares that there is no conflict of interests regarding the publication of this paper. 휌 ̃푛휎 Proof. Consider [Fℎ퐻 ](휉). Let us use Proposition 10. 1/2 Because 푓0([휉, 휉] )≡1,wehave References

휌 ̃푛휎 푛/2 (푛−2)/2 ̃푛휎 [1] E. M. Stein and G. Weiss, Introduction to Fourier analysis on [Fℎ퐻 ] (휉) =2 (−휋푖) 푠퐻 (휉) Euclidean Spaces, Princeton University Press, 1971. +∞ (54) [2] N. Y. Vilenkin, Special Functions and Representations of Groups, × ∫ 퐾휎+(푛−2)/2 (푖푟푠) 푑푟. 0 Nauka, Moscow, Russia, 2nd edition, 1991. [3] I. M. Gel’fand, M. I. Graev, and N. Y. Vilenkin, Integral Geom- We apply the integral from [9], Section 2.16.2,formula(1), etry and Representation Theory, Generalized Functions,vol.5, and set ] =푖휌and 푐=푖푠. Finally we have (this integral should Academic Press, New York, NY, USA, 1966. be understood in a regularized value sense too) the following: [4] V. P. Burskii and T. V. Shtepina, “On the spectrum of an equiv- ariant extension of the Laplace operator in a ball,” Ukrainian 휌 ̃푛휎 [Fℎ퐻 ] (휉) Mathematical Journal,vol.52,no.11,pp.1679–1690,2000. [5] S. Helgason, Groups and Geometric Aanalysis,vol.113,Academic 휋 1 =2푛/2(−휋푖)(푛−2)/2 ⋅푠⋅ ⋅ 퐻̃푛휎 (휉) Press, 1984. 2푖푠 cos (푖휋휌/2) (55) [6] V.V.Shtepin and T. V.Shtepina, “An application of intertwining operators in functional analysis,” Izvestiya Mathematics,vol.73, (−2휋푖)푛/2 = 퐻̃푛휎 (휉) . no. 6, pp. 1265–1288, 2009. 2 cosh (휋휌/2) [7] A. Erdelyi,´ “Die Funksche Integralgleichung der Kugelflachen-¨ funktionen und ihre Ubertragung¨ auf die Uberkugel,”¨ Mathe- Corollary 11 is proved. matische Annalen,vol.115,no.1,pp.456–465,1938. [8] T. V. Shtepina, “A generalization of the Funk-Hecke theorem to Corollary 12. The inverse hyperbolic Fourier transform thecaseofhyperbolicspace,”Izvestiya: Mathematics,vol.68,no. 휌 −1 푛휎 (Fℎ) on each of the spaces G has the next form: 5, pp. 1051–1061, 2004. [9] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals 휌 −1 and Series: Special Functions,vol.2,Gordon&Breach,New [(Fℎ) 푓] (휉) York, NY, USA, 1990. (56) 2 cosh (휋휌/2)儨 儨1+푖휌 푖[휉,푥] −1−푖휌 [10] T. V.Shtepina, “About representation as convolution of the oper- = 儨휉儨 ∫ 푒 |푥| 푓 (푥) 푑푥. 푛/2 儨 儨 ator, permutable with the operator quasiregular representations (−2휋푖) 푈 of group of Lorentz,” Trudy Instituta Prikladnoj Matematiki i Mekhaniki,vol.7,pp.225–228,2002. The proof evidently follows from Corollary 11.

Remark 13. A well-known theorem asserts that any inter- twining operator of the quasiregular representation of a compact group is a convolution [5,chapterV,Section2, Theorem 2.3]. However, the question whether this theorem is 1 true for representation 푅 of Lie group 푆푂0(푛 − 1, 1) in 퐿loc(푈) is still open. Due to the exact sequence

1㨀→푆푂(푛−1) 㨀→ 푆 푂 0 (푛−1,1) 㨀→ 푆 퐻 㨀→ 1 , (57) Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 184071, 11 pages http://dx.doi.org/10.1155/2014/184071

Research Article Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces

Mirna DDamonja

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

Correspondence should be addressed to Mirna Dzamonja;ˇ [email protected]

Received 7 December 2013; Revised 22 March 2014; Accepted 30 March 2014; Published 20 May 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 Mirna Dzamonja.ˇ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding between 𝐶(𝐾) and 𝐶 (𝐿) for 0-dimensional 𝐾 and 𝐿 such that the following requirement holds for all ℎ =0̸ and 𝑓≥0in 𝐶(𝐾):if0≤𝑇ℎ≤𝑇𝑓, then there are constants 𝑎 =0̸ and 𝑏 with 0≤𝑎⋅ℎ+𝑏≤𝑓and 𝑎⋅ℎ+𝑏=0̸ .

1. Introduction not only by set-theoretic methods such as forcing, but also by the methods coming from classification theory in model Inthispaperwejoinarecenttrendthathasseenacombina- theory and pcf theory. These ideas were explored in the tion of model theory and set theory address questions coming context of isometric embeddings in [2], but the methods from analysis and topology. Examples are the spectacular applicable to isometries do not at all apply in the context of p = t proof by Malliaris and Shelah of the in [1]or isomorphicembeddings;hencewehaveneededtoconstructa more directly connected to this paper, work by Shelah and new framework. Using this framework we are able to provide UsvyatsovontheisometricuniversalityofBanachspaces[2], atemplate(Theorem 16(1)) of results which state that, for which will be mentioned in more detail below. We are specif- an uncountable cardinal number 𝜆 under certain cardinal ically interested in the isomorphic embeddings of Banach arithmetic assumptions, there is no universal Banach space spaces,inparticularintheuniversality number of this class. of density 𝜆, under certain kinds of isomorphic embeddings. This is the minimal number of the Banach spaces of a given The general kind of embeddings considered in the template density which isomorphically embed all the other spaces is called very positive embeddings and it includes many of the same density, allowing, depending on the context, natural examples of embeddings. One is presented in the the embeddings or the spaces to have extra properties. In following theorem which easily follows from Theorem 16(1) many contexts, for example, when working just with plain (see Section 5.3). isomorphic embeddings, it suffices to work with spaces of the form 𝐶(𝐾) andeventoassumethat𝐾 is 0-dimensional and Theorem 1. Suppose that 𝜃 and 𝜆 are two regular cardinals so of the form St(A) where A isaBooleanalgebraandSt(A) with itsStonespace.ThisisbecauseeveryBanachspaceofagiven + 𝜃 density embeds isometrically into one of the form 𝐶(St(A)) ℵ2 ≤𝜃<𝜃 <𝜆<2, (1) with the same density. In the main body of this paper we ℵ concentrate on the spaces of the form 𝐶(St(A)). and that (∀𝜅 < 𝜆)𝜅 0 <𝜆. The topic of universality of Banach spaces has already Thentheminimalnumberofspacesoftheform𝐶(St(A)) received a significant input from set theory, notably in the of density 𝜆 needed to embed all Banach spaces of the form work of Brech and Koszmider [3, 4], which will be reviewed 𝐶(St(B)) of density 𝜆 by an embeddings 𝑇 satisfying the 𝜃 below. The new element we bring is the study of the topic following conditions (i)–(iii) is 2 ,where 2 Abstract and Applied Analysis

(i) 𝑇 is positive; that is, 𝑓≤𝑔⇒𝑇𝑓≤𝑇𝑔; 2. Background For a quasi-ordered class (M,≤), the universality number is (ii) 𝑇 is onto; defined as the smallest size of N ⊆ M such that for every 𝑀∈ M there is 𝑁∈N such that 𝑀≤𝑁.InBanachspacetheory 0≤𝑇ℎ≤𝑇𝑓ℎ =0̸ 𝑓≥0 (iii) if , ,and , then there are we find many examples of classes whose universality numbers 𝑎 =0̸ 𝑏 0≤𝑎⋅ℎ+𝑏≤𝑓 constants , such that and have been studied, with respect to isomorphic, isometric, and 𝑎⋅ℎ+𝑏=0̸ . other kinds of embeddings. A classical result by Banach [10, page 185] shows that 𝐶([0, 1]) is isometrically universal for all In particular there is no surjectively universal space separable Banach spaces. 𝐶(St(A)) of density 𝜆, under the embeddings satisfying (i)–(iii). For the nonseparable case, the situation is more com- plex. The cardinal arithmetic assumption GCH automatically It would of course be desirable to weaken the condition of gives one universal model for each uncountable density, as very positivity. In recent work, Shelah [5]introducesamodel- explained below. Specific models of the failure of GCH were theoretic property called the olive property which generalizes studied by Brech and Koszmider [3] who considered Banach the model-theoretic tools that can be used to apply the spacesofdensitythecontinuumandprovedthatintheCohen ℵ method of invariants which were used in the context of linear model for 2 manyCohenrealsthereisanisomorphically c =ℵ orders in [6] and which are necessary for [2]andforthis universal Banach space of density 2.In[9]there work. The restriction to very positivity in our work basically arenegativeuniversalityresultsinCohenandCohen-like comes from the connection of the invariants to linear orders. models; for example, the isomorphic universality number ℵ ℵ Therefore it is a promising direction for future work to finda for Banach spaces of density 1 is 2 in the Cohen model Banach space isomorphism context where one could use the andmoreoveroneCohenrealaddsaw.c.g.Banachspaceof ℵ olivepropertyinplaceoftheorder. density 1 whichdoesnotembedintoanyBanachspacewith ℵ An additional consideration of the paper is the subject a dense set of size 1 in the ground model. In [4], which of the number of pairwise nonisomorphic Banach spaces studies w.c.g. Banach spaces, it is stated (page 1268), without of a given density. For example, the celebrated Kaplansky proof, that Koszmider and Thompson noted that a version theorem [7]showsthatif𝐶(𝐾) and 𝐶(𝐿) admit a bijective of the proof from [3]givesamodelwherethereisno ℵ isomorphism which also preserves pointwise order, then 𝐾 isomorphically universal Banach space of density 1.Letus and 𝐿 are homeomorphic. Coupled with model-theoretic briefly explain the known positive universality results in the 𝜅 𝜅 results which show that for every uncountable 𝜅 there are 2 context of GCH. Throughout, stands for an infinite cardinal. pairwise nonisomorphic Boolean algebras of size 𝜅 and with By combining the Stone duality theorem, the fact that any 𝜅 𝑋 𝐶(𝐵 ∗ ) the Stone representation theorem, this gives that there are 2 Banach space is isometric to a subspace of 𝑋 and that 𝐵 ∗ pairwise non-order-isomorphic Banach spaces of density 𝜅. 𝑋 has a totally disconnected continuous preimage, Brech Itfollowsfromourwork,asweshowinTheorem 16(2),that and Koszmider proved the following. thesameistrueundertheweakerassumptionofverypositive (1) embeddings (that the assumption of very positivity is strictly Fact 1 (see [3], Fact 1.1). The universality number of the class of Boolean algebras of size 𝜅 isgreaterorequaltothe weaker than the assumption of preserving order follows 𝜅 from our examples in Section 5). A very general model- universality number of the class of Banach spaces of density theoretic study of the properties which lead to a large number with isometric embeddings, which is greater or equal than the of pairwise nonisomorphic models in metric structures is universality number of the class of Banach spaces of density 𝜅 with isomorphic embeddings. undertakenbyShelahandUsvyatsovin[8]andinthefutureit (2) 𝐶( (A)) A may yield more general results about nonisomorphic Banach Theclassofspacesoftheform St for a Boolean algebra of size 𝜅 is isometrically universal for the spaces. 𝜅 We should finish this introduction by mentioning that class of Banach spaces of density , and in particular its uni- for uncountable 𝜆 the successor of a regular it is not difficult versality number with either isometric or isomorphic embed- to construct specific models of set theory in which there dings is the same as the universality number of the whole 𝜅 are no isomorphically universal Banach spaces of density classofBanachspacesofdensity . 𝜆; for example, the classical Cohen model will do (paper Fact 1 is only interesting in the context of uncountable [9] addresses this and finer versions of it). The point of 𝜅 𝜅=ℵ Theorem 16 is that it is not a result which is true just in ,sincefor 0 we have a universal Boolean algebra as some specifically constructed model; it is a result which well as an isometrically universal Banach space, as explained holdsassoonascertaincardinalarithmeticassumptions above. On the other hand, it is known from the classical are fulfilled. Another remark is that on the basis of what model theory (see [11]forsaturatedandspecialmodels)that isknownintheliteratureandwhatweobtainhere,no inthepresenceofGCHthereisauniversalBooleanalgebraat known result differentiates between the universality number every uncountable cardinal, so the questions of universality of Banach spaces of a given density under isometries or for the above classes are interesting in the context of the under isomorphisms. Furthermore, it is not known how to failure of the relevant instances of GCH. Negative universality differentiate them from the universality number of Boolean resultsforBooleanalgebrasareknowntoholdwhenGCH algebras. failssufficientlybytheworkofKojmanandShelah[6]and Abstract and Applied Analysis 3 in Cohen-like extensions by the work of Shelah (see [6] Remark 5. Every isometry is an isomorphism. An isomor- foraproof).ShelahandUsvyatsovprovedin[2] that, in phism is in particular an injective continuous function, and the models where the negative universality results that were in fact, a linear map 𝑇 is an isomorphism if and only if both −1 obtained for Boolean algebras in [6]hold,thesamenegative 𝑇 and 𝑇 are linear and continuous. def universality results hold for Banach spaces under isometric For 𝑇 an isomorphism, we define ‖𝑇‖ = sup{‖𝑇(𝑓)‖ : embeddings. The smallest cardinal at which these results can ‖𝑓‖ = 1}. apply is ℵ2. For example, if 𝜆 is a regular cardinal greater than ℵ0 ℵ0 A B ℵ1 but smaller than 2 (so 2 ≥ℵ3),thereisnouniversal Throughout the paper letters and will be used for under isometries Banach space of density 𝜆. Boolean algebras, 𝜅, 𝜆 for infinite cardinals, and 𝐾 and 𝐿 for compact spaces. The space 𝐶(𝐾) is the space of all continuous Conjecture 2. The universality number of the class of Banach real-valued functions on 𝐾 with the topology given by the 𝜅 def spaces of density with isomorphic embeddings is the same supremum norm ‖𝑓‖ = sup{𝑓(𝑥) : 𝑥.Wewillwrite ∈𝐾} as the universality number of the class of Boolean algebras of St(A) for the Stone space of a Boolean algebra A,whichis 𝜅 size . defined as the space of all ultrafilters 𝑢 on A with the topology [𝑎] def= {𝑢:𝑎∈𝑢} It follows from the above discussion that Conjecture 2 generated by sets as a clopen basis. Let us would improve Fact 1(1) and it would imply the negative note that Fact 1 implies. universalityresultsofShelahandUsvyatsov.Forallweknow Observation 1. The universality number of Banach spaces of at this point Conjecture 2 could be a theorem of ZFC; that is, + density 𝜅, under any kind of embeddings, is either 1 or ≥𝜅 . it is not known to fail at any 𝜅 even consistently. A particular case of Conjecture 2 is the following Conjecture 3,which ∗ + This is so because if for any 𝛼 ∈[1,𝜅 ),wehadthat{𝑋𝛼 : ∗ summarizes the most interesting case from the point of view 𝛼<𝛼} were a universal family of Banach spaces of density of Banach space theory. 𝜅,thenwecouldassumethateach𝑋𝛼 =𝐶(St(A𝛼)) for some Boolean algebras A𝛼 of size 𝜅.Thereforewecouldfindasingle Conjecture 3. The universality number of the class of Banach algebra A of size 𝜅 such that all A𝛼 embed into it (simply by spaces of density 𝜅 with isomorphic embeddings is the same as freely generating an algebra by a disjoint union of all A𝛼)and the universality number of the class of Banach spaces of density 𝐶( (A)) 𝜅 hence St wouldbeasingleuniversalBanachspaceof with isometric embeddings. density 𝜅. There is a considerable amount of study of other kinds of embeddings of Banach spaces, but isometries and isomor- 3. Natural Spaces of Functions phisms and our work will fit into that area. Sticking to the spaces of the form 𝐶(𝐾), among the classically studied iso- Our methods will involve a combination of model theory, set morphic embeddings are those that preserve multiplication theory, and Banach space theory. In this section we introduce or the ones that preserve the pointwise order of functions. It a simple model-theoretic structure which will be used to is known for either one of them (Gelfand and Kolmogorov achieve that mixture of methods. A [12] for the former and Kaplansky [7] for the latter) that Suppose that is a Boolean algebra. We will associate to it a structure whose role is to represent the space 𝐶(𝐾),where if they are onto, they actually characterize the topological 𝐾 A 𝐾= (A) structure of the space; that is, if 𝑇 : 𝐶(𝐾) → 𝐶(𝐿) is an is the Stone space of , St .Theideaisasfollows. onto embeddings which either preserves multiplication or We are interested in the set of all simple functions with rational coefficients defined on 𝐾,sofunctionsofthetype the pointwise order, then 𝐾 and 𝐿 are homeomorphic. We Σ𝑖≤𝑛𝑞𝑖𝜒[𝑎 ],whereeach𝑞𝑖 is rational, 𝑎𝑖 ∈ A,and[𝑎𝑖] denotes will show that, in moving from the order preserving onto 𝑖 the basic clopen set in 𝐾 determined by 𝑎𝑖. Every element of assumption just a small bit, we no longer have the preserva- 𝐶(𝐾) is the limit of a sequence of such functions, since the tion of the homeomorphic structure, but under the assump- limits of such sequences form exactly the class of Lebesgue tionthatGCHfailssufficiently,wedohavealargenumber integrable functions, which of course includes 𝐶(𝐾).Letus of pairwise nonisomorphic spaces and a large universality then consider the vector space freely generated by A over number. Q,callit𝑉=𝑉(A) (this vector space figures in3 [ ]with We now finish the introduction by giving some back- the notation 𝐶Q(A) and is considered in a different context). ground information for the readers less familiar with Banach Hence every simple function on 𝐾 with rational coefficients space theory. corresponds uniquely to an element of 𝑉, via an identification of each 𝑎∈A with 𝜒[𝑎]. Using coordinatwise addition and 𝜔 Definition 4. A Banach space is a normed vector space com- scalar multiplication, the product 𝑊=𝑉 becomes a vector plete in the metric induced by the norm. A linear embeddings space. Any function 𝑓 in 𝐶(𝐾) can be identified with an 𝑇:𝑋 →𝑌 between Banach spaces is an isometry if for every element of this vector space, namely, a sequence of simple 𝑥∈𝑋 ‖𝑥‖ = ‖𝑇𝑥‖ 𝑇𝑥 ,wehave ,whereweuse to denote rational functions whose limit is 𝑓, and hence 𝐶(𝐾) can be 𝑇(𝑥). A linear embeddings 𝑇:𝑋between →𝑌 Banach identified with a subset of 𝑊. spaces is an isomorphism if there is a constant 𝐷>0such Toencapsulatethisdiscussionwewillworkwithvector that, for every 𝑥∈𝑋,wehave(1/𝐷)‖𝑥‖ ≤ ‖𝑇𝑥‖ ≤𝐷‖𝑥‖. spaces with rational coefficients and with two distinguished 4 Abstract and Applied Analysis

unary predicates 𝐶, 𝐶0 satisfying 𝐶0 ⊆𝐶.Withour of coefficients in 𝑓𝑘(𝑛) is ≤𝑛+1,and𝑔𝑛 =𝑔𝑛−1 otherwise. motivation in mind, we will call them function spaces.Ifsuch Hence ⟨𝑔𝑛 :1≤𝑛<𝜔⟩is the sequence ⟨𝑓𝑛 :1≤𝑛<𝜔⟩with aspace(𝑉,𝐶,0 𝐶 ) is the space of sequences of simple rational possible repetitions of each element finitely many times and functions over a Stone space 𝐾=St(A) and 𝐶, 𝐶0 correspond, so lim𝑛→∞𝑔𝑛 =𝑓. respectively, to the set of such sequences which converge or converge to 0, then we call (𝑉,𝐶,0 𝐶 ) a natural space and we Definition 7. Suppose that 𝑓=⟨𝑓𝑛 :𝑛>𝜔⟩is a sequence in 󸀠 denote it by 𝑁(A).Inspacesoftheform𝑁(A) for an element 𝑁(A) 𝑓 =⟨𝑓󸀠 :𝑛>𝜔⟩ 𝑁(A) and suppose that 𝑛 was obtained by 𝑓 of 𝐶 we define ‖𝑓‖ as the norm in 𝐶(St(A)) of the limit 𝑓 Σ 𝑞 𝜒 first replacing each 𝑛 with an equivalent function 𝑖≤𝑛 𝑖 [𝑎𝑖] 𝑓 of 𝑓.If𝜙 is an embeddings between 𝑁(A) and 𝑁(B),we with disjoint 𝑎𝑖s and then replacing the coefficients 𝑞𝑖 by 𝑠𝑖 will say that 𝐷>0is a constant of the embeddings if for every using the procedure described in the proof of Lemma 6.We 𝑁(A) 󸀠 𝑓 of 𝐶 ,wehavethat(1/𝐷) ⋅ ‖𝑓‖ ≤ ‖𝜙(𝑓)‖ ≤ 𝐷𝑓‖ ⋅‖ .Not say that 𝑓 is a top-up of 𝑓. every embeddings has such a constant, but we will only work with the ones which do. Corollary 8. Suppose that A and B are Boolean algebras, 𝑁(A) We will mostly be interested in a specific case of the and let A denote the linear subspace of 𝐶 spanned by representation of continuous functions as limits of simple the functions whose rational coefficients are in [0, 1].Thenfor functions, given by the following observation. 𝑁(A) every 𝑓=⟨𝑓𝑛 :𝑛<𝜔⟩in 𝐶 ,thereis𝑔∈A with lim𝑛𝑓𝑛 = lim𝑛𝑔𝑛. Lemma 6. Suppose that 𝐾=S𝑡(A) is the Stone space of a Boolean algebra A and let 𝑓≥0be a function in 𝐶(𝐾) with ∗ ∗ Proof. Let 𝑓=lim𝑛𝑓𝑛;hence𝑓 can be written as 𝑓= ‖𝑓‖ ≤ 𝐷 𝐷 >0 ⟨𝑓 :𝑛< + − + − for some . Then there is a sequence 𝑛 𝑓 −𝑓 where 𝑓 = max{𝑓,0} and 𝑓 = min{𝑓,0} are both 𝜔⟩ of simple functions, where each 𝑓𝑛 is of the form Σ𝑖≤𝑛𝑞𝑖𝜒[𝑎 ], 𝑖 continuous and positive. Therefore, by the closure of A under 𝑞 (0, 𝐷∗] 𝑎 ∈ A 𝑓= with each 𝑖 rational in and 𝑖 ,suchthat linear combinations, it suffices to prove the corollary in the 𝑓 ∗ lim 𝑛. case of 𝑓≥0.Let𝐷 =‖𝑓‖, and we now apply Lemma 6. Moreover, we can assume that for every 𝑛, 𝑓𝑛 is the sum of at most 𝑛+1functions of the form 𝑠⋅𝜒[𝑎]. Thepointofthesedefinitionsistheconnectionbetween Proof. By multiplying by a constant if necessary, we can ∗ the embeddability in the class of spaces of the form 𝐶(St(A)) assume that 𝐷 =1. Functions of the form Σ𝑖≤𝑛𝑟𝑖𝜒[𝑎 ] with 𝑟 𝑖 and the class of function spaces. Namely, we have the each 𝑖 real, contain the constant function 1, form an algebra, following. and separate the points of 𝐾; hence by the Stone-Weierstrass 𝐶(𝐾) theorem, they form a dense subset of .Noticethat Theorem 9. Suppose that A and B are Boolean algebras, and every function Σ𝑖≤𝑛𝑟𝑖𝜒[𝑎 ] can be, by changing the coefficients 𝑁(A) 𝑖 let A denotethelinearsubspaceof𝐶 spanned by the family and the sets 𝑎𝑖 if necessary, represented in the form where 󸀠 A of sequences ⟨𝑓𝑛 :𝑛<𝜔⟩of simple functions whose all [𝑎𝑖]s are pairwise disjoint, so we can without loss of coefficients are rationals in [0, 1] and which satisfy that each 𝑓𝑛 generality work only with such functions. Given 𝜀>0and 𝑛+1 Σ 𝑟 𝜒 𝑎 𝑖≤𝑛 has at most many elements. Then if there is an isomorphic 𝑖≤𝑛 𝑖 [𝑎𝑖] with 𝑖s disjoint, we can find for rational 𝑇 𝐶( 𝑡(A)) 𝐶( (B)) 𝑞 |𝑞 −𝑟|<𝜀 Σ 𝑟 𝜒 embeddings from S to St , then there is an numbers 𝑖 with 𝑖 𝑖 ;hencethefunction 𝑖≤𝑛 𝑖 [𝑎𝑖] 𝜙 A 𝑁(B) 𝜀 Σ 𝑞 𝜒 isomorphic embeddings from to satisfying that for is approximated within by 𝑖≤𝑛 𝑖 [𝑎𝑖],showingthatalso every 𝑓=⟨𝑓𝑛 :𝑛<𝜔⟩in A,if𝑓=lim𝑛∈𝜔𝑓𝑛,then functions with rational coefficients and disjoint 𝑎𝑖s are dense. 𝜙(𝑓) = 𝑇(𝑓) Now given 𝑓≥0afunctionin𝐶(𝐾) with ‖𝑓‖ ≤ 1 and lim𝑛∈𝜔 𝑛 . 𝜀>0,letΣ𝑖≤𝑛𝑞𝑖𝜒[𝑎 ] be a function with rational coefficients 𝑖 Proof. Let 𝑇:𝐶(St(A)) → 𝐶(St(B)) be an isomorphic and disjoint 𝑎𝑖ssatisfying‖𝑓 −𝑖≤𝑛 Σ 𝑞𝑖𝜒[𝑎 ]‖<𝜀,recallingthat 𝑖 ‖𝑇‖ < ∞ the ‖‖ in 𝐶(𝐾) is the supremum norm. Define now embeddings, so . We intend to define an isomorphic embeddings 𝜙 from A to 𝑁(B). By linearity it is sufficient to 𝑞 , 𝑞 ∈ [0, 1] , A󸀠 A 𝜋 { 𝑖 if 𝑖 work with the basis of . Let us use the notation 𝑛 for the 𝑠 = 0, 𝑞 <0, projection on the 𝑛th coordinate. First we define the action of 𝑖 { if 𝑖 (2) 󸀠 { 𝜙 on those 𝑓∈A which have the property that there is at {1, if 𝑞𝑖 >1, most one 𝑛 such that 𝜋𝑛(𝑓) is not the identity zero function, 𝜋 (𝑓) 𝜒 𝑎∈A andconsiderthefunctionΣ𝑖≤𝑛𝑠𝑖𝜒[𝑎 ].Weclaimthat‖𝑓 − and then 𝑛 is a function of the form [𝑎] for some .If 𝑖 𝑛 𝑎 =0̸ 𝜙(𝑓) Σ𝑖≤𝑛𝑠𝑖𝜒[𝑎 ]‖<𝜀. By the assumption that 𝑎𝑖s are disjoint, for there is no such with ,thenwelet be the element 𝑖 𝑁(B) 𝜋 𝑛 any 𝑥,thereisatmostone𝑖=𝑖(𝑥)such that 𝑥∈[𝑎𝑖].If of whose all projections 𝑛 are zero. Otherwise, let 𝑥∉⋃ [𝑎 ] 𝑠 =𝑞 |𝑓(𝑥) −Σ 𝑠 𝜒 (𝑥)| = 𝜋𝑛(𝑓) =0̸ 𝑇(𝜋𝑛(𝑓)) 𝑖≤𝑛 𝑖 or 𝑖(𝑥) 𝑖(𝑥),then 𝑖≤𝑛 𝑖 [𝑎𝑖] be such that and consider ,whichiswell |𝑓(𝑥) −Σ 𝑞 𝜒 (𝑥)| < 𝜀 𝑥∈[𝑎] 𝑞 <0 𝑇(𝜋𝑛(𝑓)) 𝑖≤𝑛 𝑖 [𝑎𝑖] .If 𝑖 and 𝑖 ,then defined. We have no reason to believe that is a |𝑓(𝑥)𝑖 −𝑠 | = 𝑓(𝑥) < 𝑓(𝑥)𝑖 −𝑞 <𝜀as 𝑓(𝑥).If ≥0 𝑥∈[𝑎𝑖] and simple function with rational coefficients. However, there isa 𝐹((𝜋 (𝑓))) 𝑞𝑖 >1,then|𝑓(𝑥)−𝑠𝑖|=1−𝑓(𝑥)<𝑞𝑖 −𝑓(𝑥) <𝜀 as 𝑓(𝑥). ≤1 function 𝑛 which is a simple function with rational To finish the proof, we observe that given a sequence of coefficients and whose distance to 𝑇(𝜋𝑛(𝑓)) in 𝐶(St(B)) is 𝑛+1 functions ⟨𝑓𝑛 :𝑛<𝜔⟩of the above form which converges to less than 1/2 . We define 𝜙(𝑓) tobetheuniqueelement𝑔 of afunction𝑓, we can define by induction on 𝑛, 𝑔0 =0, 𝑘(𝑛) = 𝑁(B) such that the only 𝜋𝑚(𝑔) which is not identically zero min{𝑘 :𝑘 𝑓 ∉{𝑔0,...,𝑔𝑛−1}} and 𝑔𝑛 =𝑓𝑘(𝑛) if the number is 𝜋𝑛(𝑔) and 𝜋𝑛(𝑔) = 𝐹((𝜋𝑛(𝑓))). Abstract and Applied Analysis 5

󸀠 𝜆 Now suppose that 𝑓∈A is such that for exactly one 𝑛, (3) A club guessing sequence on 𝑆𝜃 is a sequence ⟨𝐶𝛿 :𝛿∈ 𝜋 (𝑓) 𝜋 (𝑓) = ∑𝑚 𝑞 𝜒 𝑆𝜆⟩ 𝐶 𝛿 𝐸⊆𝜆 𝑛 is not identity zero, and 𝑛 𝑖=0 𝑖 [𝑎𝑖] for some 𝜃 such that each 𝛿 is a club in ,andforeveryclub , 𝑎0,...,𝑎𝑚 ∈ A and some rational 𝑞0,...,𝑞𝑚 in [0, 1].For there is 𝛿 such that 𝐶𝛿 ⊆𝐸. 𝑖≤𝑚,let𝑓𝑖 be the element of 𝑁(A) whose 𝑛th projection 𝜒 Observation 2. Suppose that 𝜃>ℵ1 and there is a club is [𝑎𝑖] andallotherprojectionsareidentityzero.Hencewe 𝜆 guessing sequence ⟨𝐶𝛿 :𝛿∈𝑆𝜃 ⟩. Then there is a club guessing have already defined 𝜙(𝑓 ),andwelet𝜙(𝑓) = ∑ 𝑞𝑖𝜙(𝑓 ). 𝑖 𝑖≤𝑚 𝑖 ⟨𝐷 :𝛿∈𝑆𝜆⟩ 𝑖<𝜃 Finally suppose that 𝑓=⟨𝑓𝑛 :𝑛<𝜔⟩is any element sequence 𝛿 𝜃 such that, for all , 󸀠 of A . Therefore for every 𝑛,wehavealreadydefined𝑔𝑛 = (𝑖) =𝜔󳨐⇏ (𝛼𝛿) =𝜔,̸ 𝜙(⟨0,...,𝑓𝑛,0,...⟩),where𝑓𝑛 is on the 𝑛th coordinate. Let cf cf 𝑖 (4) 𝜙(𝑓) =𝑛 ⟨𝑔 :𝑛<𝜔⟩. Hence we have defined a linear 󸀠 ⟨𝛼𝛿 :𝑖<𝜃⟩ 𝐷 embeddings of A to 𝑁(B). We extend this embeddings to A where 𝑖 is the increasing enumeration of 𝛿,for by linearity. We need to check that this embeddings preserves each 𝛿. 󸀠 𝐶 and 𝐶0,andagainitsufficestoconcentrateonthebasisA . 𝑓=⟨𝑓 :𝑛<𝜔⟩ 𝐶𝑁(A) ∩ A󸀠 Proof. First of all notice that by passing to subsets if necessary So suppose that 𝑛 is in ,and 𝐶 𝑔=⟨𝑔 :𝑛<𝜔⟩ 𝜙 we can without loss of generality assume that each 𝛿 has let 𝑛 be its image under .Wewillshow 𝜃 𝛿 𝐶󸀠 𝐶 𝑔∈𝐶𝑁(B) order type .Given ,let 𝛿 consist of the points of 𝛿 of that byshowingthatitisaCauchysequence.Let >𝜔 𝐷 𝐶󸀠 𝛿 𝜃> 𝑛, 𝑚 <𝜔 ‖𝑔 −𝑔 ‖ 𝑓 = ∑ 𝑞 ⋅𝜒 cofinality ,andlet 𝛿 be the closure of 𝛿 in .Since ; we will consider 𝑛 𝑚 .Let 𝑛 𝑖≤𝑘 𝑖 [𝑎𝑖] 󸀠 ℵ1,wehavethat𝐶𝛿 is unbounded in 𝛿,soitisclearthat𝐷𝛿 and 𝑓𝑚 = ∑𝑗≤𝑙 𝑟𝑗 ⋅𝜒[𝑏 ].Wehave 𝑗 is a club of 𝛿, and since we have 𝐷𝛿 ⊆𝐶𝛿,weobtainthatthe 𝜆 󵄩 󵄩 󵄩 󵄩 resulting sequence is a club guessing sequence on 𝑆𝜃 .Italso 󵄩𝑔𝑛 −𝑔𝑚󵄩 = 󵄩𝜙(𝑓𝑛)−𝜙(𝑓𝑚)󵄩 follows that otp(𝐷𝛿)=𝜃, so the increasing enumeration as 󵄩 󵄩 󵄩 󵄩 claimed exists. ≤ 󵄩𝜙(𝑓𝑛)−𝑇(𝑓𝑛)󵄩 + 󵄩𝑇(𝑓𝑛)−𝑇(𝑓𝑚)󵄩 󵄩 󵄩 The main definition we need is the definition ofthe + 󵄩𝑇(𝑓𝑚)−𝜙(𝑓𝑚)󵄩 invariant.Letussupposethat𝜃>ℵ1 is regular and that 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 ⟨𝐷 :𝛿∈𝑆𝜆⟩ ≤ ∑ 󵄨𝑞 󵄨 󵄩𝜙(𝜒 )−𝑇(𝜒 )󵄩 + 󵄩𝑇(𝑓 −𝑓 )󵄩 𝛿 𝜃 is a club guessing sequence with an increasing 󵄨 𝑖󵄨 󵄩 [𝑎𝑖] [𝑎𝑖] 󵄩 󵄩 𝑛 𝑚 󵄩 𝛿 𝑖≤𝑘 enumeration ⟨𝛼𝑖 :𝑖<𝜃⟩of 𝐷𝛿,foreach𝛿 and satisfying 󵄩 󵄩 the requirement (4).Thissequencewillbefixedthroughout. 󵄨 󵄨 󵄩 󵄩 + ∑ 󵄨𝑟𝑗󵄨 󵄩𝜙(𝜒[𝑏 ])−𝑇(𝜒[𝑏 ])󵄩 The existence of such a sequence will be discussed at the 󵄨 󵄨 󵄩 𝑗 𝑗 󵄩 𝑗≤𝑙 end of the section, but for the moment let us say that Shelah (see Theorem 15) proved that such a sequence exist in many (𝑘+1) 󵄩 󵄩 (𝑙+1) 𝜆 ≥𝜃++ ≤ + ‖𝑇‖ ⋅ 󵄩𝑓 −𝑓 󵄩 + , circumstances, notably for any regular . 2𝑛+1 󵄩 𝑛 𝑚󵄩 2𝑚+1 (3) Definition 11. Suppose that A isaBooleanalgebraofsize𝜆, 𝜆 A =⟨A𝛼 :𝛼<𝜆⟩a filtration of A,that𝛿∈𝑆𝜃 and that 𝑓∈ which goes to 0 as 𝑛, 𝑚. →∞ 𝐶𝑁(A) \𝑁(A ) 𝑖∈𝑆𝜃 𝑁(A) 󸀠 𝛿 .Anordinal =𝜔̸ is an element of the invari- 𝑓=⟨𝑓 :𝑛<𝜔⟩ 𝐶 ∩A 󸀠 𝑁(A ) At the end suppose that 𝑛 is in 0 , 𝛼𝛿 ant inv (𝑓) if and only if there is 𝑓 ∈𝐶 𝑖+1 such that, and let 𝑔=⟨𝑔𝑛 :𝑛<𝜔⟩be its image under 𝜙.Bythe A,𝛿 𝑁(A ) 𝑛+1 𝛼𝛿 definition of 𝜙,wehavethat‖𝑔𝑛‖ ≤ ‖𝑇(𝑓𝑛)‖ + (𝑛 + 1)/2 for every 𝑔 in 𝐶 𝑖 ,wehave A󸀠 (this was the point of requiring the members of to be 󵄨 󵄨 sequences whose 𝑛th element has ≤𝑛+1coefficients). Since 0≤ 𝑔 ≤ 󵄨 𝑓 − 𝑓󸀠󵄨 󳨐⇒ 𝑔∈𝐶. lim𝑛 𝑛 󵄨lim𝑛 𝑛 lim𝑛 𝑛󵄨 0 (5) ‖𝑇‖ < ∞,wehavelim𝑛→∞‖𝑇(𝑓𝑛)‖ = 0,soinconclusion, lim𝑛→∞‖𝑔𝑛‖=0. We will be interested in the kind of embeddings between Banach spaces which will allow us to define appropri- 4. Invariants for the Natural Spaces and ate 𝜙 which preserve the invariants; see the Preservation Very Positive Embeddings Lemma 13. We have succeeded to do this in the case of a special kind of positive embeddings, as defined in the We will now adapt the Kojman-Shelah method of invariants following definition. [6],tothenaturalspacesandaspecifickindofisomorphic 𝑇: embeddings between Banach spaces, which we call very Definition 12. We say that an isomorphic embeddings 𝐶(𝐾) → 𝐶(𝐿) positive embeddings (see Definition 12). From this point on is very positive if the following requirements we assume that 𝜆 is a regular uncountable cardinal. hold: (i) 𝑔≥0⇒𝑇𝑔≥0(positivity), Definition 10. (1) Suppose that 𝑀 is a model of size 𝜆.A 𝑔∈𝐶(𝐿)\{0} 0≤𝑔 ℎ filtration of 𝑀 is a continuous increasing sequence ⟨𝑀𝛼 :𝛼< (ii) for every with ,thereis with 𝜆⟩ of elementary submodels of 𝑀, each of size <𝜆. 0≤𝑇ℎ≤𝑔and ℎ =0̸ , 𝜆 (2) Foraregularcardinal𝜃<𝜆,weusethenotation𝑆𝜃 (iii) if 0≤𝑇ℎ≤𝑇𝑓, ℎ, 𝑓 =0̸ ,and𝑓≥0, then there is 𝑠 =0̸ for {𝛼 < 𝜆 : cf(𝛼) = 𝜃}. definable from ℎ with 0≤𝑠≤𝑓. 6 Abstract and Applied Analysis

󸀠 󸀠 󸀠 We do not know if very positive embeddings was studied have that lim 𝜙(𝑓 )=𝑇(𝑓) and similarly 0≤lim 𝜙(𝑓 ).By in the literature, but clearly, one kind of embeddings that the fact that 𝐷𝛿 ⊆𝐸and since 𝜙 is an isomorphism, we have 󸀠 𝑁(B ) 󸀠 is very positive is an order preserving onto embeddings. In 𝛼𝛿 that 𝜙(𝑓 )∈𝐶 𝑖+1 . We would like to use 𝜙(𝑓 ) to witness this case we have Kaplansky’s theorem [7]mentionedabove, 𝑖∈ (𝜙(𝑓)) which shows that in the presence of such an embeddings from that invB,𝛿 ,soletustry.Wehavealreadyestablished 𝐶(𝐾) 𝐶(𝐿) 𝐾 𝐿 󸀠 󸀠 󸀠 onto we have that and are homeomorphic. that lim 𝜙(𝑓 )=𝑇(𝑓) and 0≤lim 𝜙(𝑓 ).Itremainstocheck 󸀠 We show in the example in Section 5 that the analogue is not 𝜙(𝑓 ) true for very positive embeddings. In particular the question the property (5)of . Suppose for a contradiction that there is 𝑔∈𝑁(B𝛼𝛿 ) such of the number of pairwise nonisomorphic by very positive 𝑖 𝐶( (A)) def 󸀠 embeddings spaces of the form St does not reduce that 0≤𝑔 = lim𝑛𝑔𝑛 ≤|𝑇𝑓−𝑇𝑓| but that 𝑔∉𝐶0. Applying to the well-studied and understood question of the number (ii) we can find ℎ with 0≤𝑇ℎ≤𝑔and ℎ =0̸ .ByCorollary 8, of pairwise nonisomorphic Boolean algebras of a given 𝜅 we can assume that there is ℎ∈A with ℎ=lim𝑛ℎ𝑛,and cardinality (which for any infinite 𝜅 is always equal to 2 ,see hence 𝑇ℎ = lim𝑛𝜙(ℎ). Translating the properties guaranteed Shelah’s [13]). 𝜆 by (ii) into the terms of 𝜙 and applying the elementarity of Let us now make a further assumption on : 𝛿 𝑀↾𝛼 ℎ∈𝑁(A 𝛿 ) 𝑖 ,wecanassumethat 𝛼𝑖 . Now we apply ℵ 𝜅<𝜆󳨐⇒𝜅 0 <𝜆. (6) (iii) to find 𝑠≥0, 𝑠 =0̸ definable from ℎ and satisfying 𝑠≤ 󸀠 |𝑓 − 𝑓 |. Being definable from ℎ, 𝑠 has an approximation 𝑠 Lemma 13 A B (Preservation Lemma). Let and be Boolean with 𝑠 definable from ℎ;hence𝑠∈𝑁(A𝛼𝛿 ).Bytoppingupif 𝜆 𝑇:𝐶( (A))→𝐶( (B)) 𝑖 algebras of size and suppose that St St necessary as in Lemma 6 and in the above paragraph, we can A B is a very positive embeddings. Let and be any filtrations assume that every element 𝑠𝑛 in 𝑠 satisfies 𝑠𝑛 ≥0; therefore 𝑠 of A and B,respectively,andletA denote the linear subspace 󸀠 𝑁(A) 󸀠 contradicts the choice of 𝑓 . of 𝐶 spanned by the set A of sequences of functions whose rational coefficients are in [0, 1] and that satisfy that the 𝑛th Now let us prove the other direction of the desired 𝑖∈ (𝜙(𝑓)) 𝑔∈ coordinate has ≤𝑛+1nonzero coefficients. equality. Let invB,𝛿 as exemplified by some 𝜙:A →𝑁(B) 𝑁(B 𝛿 ) If is an isomorphic embeddings satisfying 𝛼𝑖+1 . As in the previous paragraphs, we can assume that 𝜙(𝑓) = 𝑇( (𝑓)) 𝑓∈A def that lim lim for every , then there is a 0≤𝑔 = lim𝑛𝑔𝑛, and hence by (ii) we can assume that for 𝐸 𝜆 𝛿 𝐷 ⊆𝐸 󸀠 󸀠 club of such that, for every with 𝛿 and for every some 𝑓 which is not 0 we have 0≤𝑇𝑓 ≤𝑔and by the 󸀠 󸀠 𝑓∈A \𝑁(A𝛿) with 0≤lim𝑛𝑓𝑛 and ‖lim𝑛𝑓𝑛‖=1,wehave same argument as above we can assume that there is 𝑓 = that 󸀠 󸀠 󸀠 ⟨𝑓 :𝑛<𝜔⟩∈𝑁(A 𝛿 )∩A 𝑓 =𝑓 𝑛 𝛼𝑖+1 such that lim𝑛 𝑛 . 𝑖𝑛V (𝑓) = 𝑖𝑛V (𝜙𝑓)) ( . 󸀠 A,𝛿 B,𝛿 (7) Now we claim that 𝑓 exemplifies that 𝑖∈invA,𝛿(𝑓).Suppose ⟨ℎ :𝑛∈𝜔⟩∈𝑁(A 𝛿 )\𝐶 for a contradiction that 𝑛 𝛼𝑖 0 and A B 󸀠 Proof. We may assume that the underlying set of and 0≤lim ℎ𝑛 ≤|lim 𝑓𝑛 − lim 𝑓 |.Letℎ=lim ℎ𝑛.Asbefore, 𝜆 𝑀 𝑛 is the ordinal . Let us define a model with the universe ℎ∈A ℎ ≥0 𝜔 we can assume that and each 𝑛 .Sobythe two disjoint copies of the -sequences of the simple functions 0 ≤ 𝑇ℎ ≤ 𝑇(|𝑓 −𝑓󸀠|) = |𝑇𝑓 − 𝑇𝑓󸀠| 𝜆 positivity we have ,by on with rational coefficients, interpreted as the elements of 𝜙 𝜙(ℎ) = 𝑇ℎ 𝑁(A) and 𝑁(B),allthesymbolsof𝑁(A) and 𝑁(B) with the choice of we have that , and by elementarity 𝜙(ℎ) ∈ 𝑁(B 𝛿 )\𝐶 𝜙(ℎ) interpretations induced from these models, and the symbols we have 𝛼𝑖 0.Itfollowsthat contradicts A A󸀠 𝜙 𝐸 𝜆 , ,and . By the assumption (6), there is a club of 𝑖∈invB,𝛿(𝜙(𝑓)). such that, for every 𝛿∈𝐸of cofinality not 𝜔,wehavethat 𝑀 restricted to the sequences whose ordinal coefficients are <𝛿is an elementary submodel of 𝑀 andthatithasuniverse corresponding to 𝑁(A𝛿)∪𝑁(B𝛿). Let us denote the latter The next task is to construct lots of Boolean algebras model by 𝑀↾𝛿. A with different invariants for 𝑁(A) and then to us the 𝑁(A) 󸀠 𝑁(B) Suppose now that 𝐷𝛿 ⊆𝐸.Choose𝑓∈𝐶 ∩A \𝑁(A𝛿) Preservation Lemma to show that no fixed can embed with 0≤lim𝑛𝑓𝑛 and ‖lim𝑛𝑓𝑛‖=1.Let𝑓=lim 𝑓𝑛.Bythe them all. 𝑁(B) choice of 𝛿,wehavethat𝜙(𝑓) ∈ 𝐶 \𝑁(B𝛿).Supposefirst + 󸀠 𝑁(A ) Lemma 14 𝜃 <𝜆 𝛼𝛿 (Construction Lemma). Suppose that . that 𝑖∈invA,𝛿(𝑓) and let 𝑓 ∈𝐶 𝑖+1 demonstrate this. Let 𝜆 Then the club guessing sequence ⟨𝐷𝛿 :𝛿∈𝑆𝜃 ⟩ can be chosen so 󸀠 def 󸀠 󸀠 𝑁(A) 𝐴⊆𝜃 𝑓 = lim 𝑓𝑛,whichiswelldefinedas𝑓 ∈𝐶 .Noticethat that for any which is a closed set of limit ordinals, there 󸀠 A = A[𝐴] A A the requirement (5)willholdifwereplace𝑓 by any top-up is a Boolean algebra ,afiltration of and a club 󸀠󸀠 𝐸 𝜆 𝛿∈𝐸 𝑓∈A󸀠 \𝑁(A ) 𝑓 (see Definition)as 7 0≤𝑓≤1, and hence for all 𝑥,we of such that for every there is 𝛿 with 󸀠󸀠 󸀠 (𝑓) = 𝐴𝜃 ∩𝑆 ‖ 𝑓 ‖=1 have |𝑓(𝑥) −𝑓 (𝑥)| ≤ |𝑓(𝑥) −𝑓 (𝑥)|.Sincethetoppingup invA,𝛿 =𝜔̸ and lim 𝑛 . 𝛿 procedure is definable in 𝑀↾𝛼𝑖+1, we may assume that 0≤ 󸀠 󸀠 The proof of this lemma is presented in Section 6.The 𝑓𝑛 ≤1for all 𝑛 and 0≤𝑓 ≤1.ByLemma 6 applied within 󸀠 following theorem of Shelah will be used in the proof of the 𝑀 𝛿 𝑓 ∈ A 𝜙 𝛼𝑖+1 , we can assume that . By the choice of ,we Construction Lemma as well as in the proof of Theorem 16. Abstract and Applied Analysis 7

휃 Theorem 15 (Shelah, [14, Claim 1.4]). Let 𝜃<𝜆be two regular but then 𝜙(𝑓) also has invariant 𝐴∩𝑆=휔̸ , by the choice of 𝐸1, + 휆 𝐴 cardinals with 𝜃 <𝜆. Then there is a stationary set 𝑆⊆𝑆휃 and andhencewehaveacontradictionwiththechoiceof . (2) G ={𝐶((A[𝐴])) : sequences ⟨𝑐훿 :𝛿∈𝑆⟩, ⟨P훼 :𝛼<𝜆⟩such that Consider the family St 𝐴 a closed set of limit ordinals in 𝜃}. By the argument in 1 (i) otp(𝑐훿)=𝜃and sup(𝑐훿)=𝛿; for every 𝐴,theset{𝐵 a closed set of limit ordinals in 𝜃: 𝐶( (A[𝐵])) 𝐶( (A[𝐴]))} 𝐸 𝜆 𝛿∈𝑆 𝑐훿 ⊆𝐸 St that embeds very positively into St (ii) for every club of ,thereis with ; 휃 has size <2,soclearlyevery𝐶(St(A[𝐴])) is very posi- (iii) P훼 ⊆ P(𝛼) and |P훼|<𝜆; 휃 tively isomorphic with <2 many 𝐶(St(A[𝐵])).Hencewe 𝛼 𝑐 𝑐 ∩𝛼 ∈ (iv) if is a nonaccumulation point of 훿,then 훿 (2휃) ⋃ P耠 can choose cf pairwise non-very positively isomorphic 훼耠<훼 훼; elements of G by a simple induction. (v) the nonaccumulation points of every 𝑐훿 are successor ordinals. 5. Examples Claim 1.4. in [14] does not state property (c) explicitly, but 5.1. Cardinal Arithmetic. An example of circumstances when it follows from the first line of the proof of that Claim. Theorem 16 applies is when Nowwepresentthemaintheoremofthepaper. ℵ0 ℵ1 𝜃=ℵ2,𝜆=ℵ4,2=ℵ1 but 2 ≥ℵ5. (8) Theorem 16. Suppose that 𝜃 and 𝜆 are two regular cardinals + 휃 ℵ0 with ℵ2 ≤𝜃<𝜃 <𝜆<2 and that (∀𝜅 < 𝜆) 𝜅 <𝜆. 5.2.VeryPositiveEmbeddingsontoDoNotGiveRiseto Then Homeomorphism. We give an example of two 0-dimensional 𝐾 𝐿 (1) the minimal number of spaces of the form 𝐶(St(A)) of spaces and which are not homeomorphic; yet they admit density 𝜆 needed to embed all Banach spaces of the form a very-positive isomorphism onto. The example itself was 휃 𝐶(St(B)) of density 𝜆 very positively is 2 .Inparticular constructedbyPlebanekin[15,Example5.3],whenconsid- 𝐶( (A)) ering positive onto isomorphisms. there is no very-positively universal space St of 𝐾 ⟨𝑥 : density 𝜆; Let consist of two disjoint convergent sequences 푛 𝑛<𝜔⟩with lim푛𝑥푛 =𝑥and ⟨𝑦푛 :𝑛<𝜔⟩with lim푛𝑦푛 =𝑦=𝑥̸ , (2휃) (2) there are at least cf pairwise non-very positively and let 𝐿 consist of a single convergent sequence ⟨𝑧푛 :𝑛<𝜔⟩ 𝐶(𝐾) 𝜆 isomorphic Banach spaces of density . with lim푛𝑧푛 =𝑧.Define𝑇 : 𝐶(𝐾) → 𝐶(𝐿) by letting for all 𝑛

휆 Proof. Fix sequences ⟨𝑐훿 :𝛿∈𝑆⊆𝑆휃 ⟩ and ⟨P훼 :𝛼<𝜆⟩as 𝑇𝑓 (𝑧 0)=𝑓(𝑦), guaranteed by Theorem 15.Noticethat⟨𝑐훿 :𝛿∈𝑆⟩satisfies 훿 𝑓(𝑥푛)+𝑓(𝑦) that, with ⟨𝛼푖 :𝑖<𝜃⟩being the increasing enumeration of 𝑐훿, 𝑇𝑓 (𝑧 )= , 훿 2푛−1 2 (9) we have that cf(𝑖) =𝜔⇏ cf(𝛼푖 ) =𝜔̸ .Henceletting𝐷훿 =𝑐훿 for 𝛿∈𝑆and 𝐷훿 an arbitrary club of 𝛿 of order type 𝜃 satisfying 𝑓 (𝑥) +𝑓(𝑦 ) 훿 휆 푛 (𝑖) =𝜔̸ ⇒ (𝛼 ) =𝜔̸ 𝛿∈𝑆 \𝑆 𝑇𝑓 (𝑧 2푛+2)= . cf cf 푖 for 휃 ,thesequencecanbe 2 used in the context of the Preservation Lemma 13.Itcanalso be used in the context of the Construction Lemma 14.Letus Plebanek shows that 𝑇 is a positive isomorphism onto 𝐶(𝐿) −1 therefore find the Boolean algebras A[𝐴] as described in the andmoreoverhecalculatestheinverse𝑆=𝑇 which is given statement of the Construction Lemma. Notice that there are by 휃 휃 2 many different choices for 𝐴∩𝑆 . =휔̸ 𝑆ℎ (𝑦) = ℎ(𝑧 ), 𝑆ℎ 𝑥 =2ℎ 𝑧 −ℎ(𝑧 ), (1) Suppose for a contradiction that there is a family 0 ( ) ( ) 0 {𝐶( (A )):𝛼<𝛼∗} 𝛼∗ <2휃 A St 훼 for some for some algebras 훼 𝑆ℎ (𝑥 )=2ℎ(𝑧 )−ℎ(𝑧 ), of size 𝜆 which is very positively universal for all 𝐶(St(A)) for 푛 2푛−1 0 A 𝜆 (10) Boolean algebras of size . Notice that the assumptions we 𝑆ℎ 푛(𝑦 )=2ℎ(𝑧2푛)−2ℎ(𝑧) +ℎ(𝑧0) ℵ0 have made on 𝜆 imply that 𝜆 =𝜆,sothesizeofeach𝑁(A훼) is 𝜆.LetF be the family of all subsets of 𝜃 that appear as for 𝑛≥1. invariants of elements of ⋃훼<훼∗ 𝑁(A훼);hencethesizeofF is 휃 휃 𝑇 <2 (since we have assumed 𝜆<2), and in particular there We will show that is a very positive embeddings. Consider- 휃 ing property (ii) of Definition 12,supposethat𝑔∈𝐶(𝐿)\{0} is 𝐴⊆𝜃a closed set of limit ordinals such that 𝐴∩𝑆 ∉ F. =휔̸ with 0≤𝑔; we need to find ℎ with 0≤𝑇ℎ≤𝑔and ℎ =0̸ . Let A = A[𝐴].Supposethat𝑇 is a very positive embeddings Since 𝑇 is onto, there is ℎ such that 𝑇ℎ = 𝑔 and since 𝑇 is an of 𝐶(St(A)) into some 𝐶(St(A훼)),andlet𝜙 be an embeddings isomorphism and 𝑔 =0̸ ,wealsohaveℎ =0̸ . 𝑁(A) 𝑁(A ) 𝑓∈A ∩𝐶푁(A) of into 훼 satisfying that for every , For the property (iii), we will have an existential proof we have 𝜙(𝑓) = 𝑇(𝑓), which exists by Theorem 9.Let𝐸0 be of the existence of the 𝑠 as required, given ℎ and 𝑓 as in the aclubof𝜆 as guaranteed by the Preservation Lemma, let 𝐸1 assumptions. First let us deal with the case that ℎ≥0.LetS be a club of 𝜆 as guaranteed by the Construction Lemma, let be the family of all non-negative functions in 𝐶(𝐾) for which 𝐸=𝐸0 ∩𝐸1,andsupposethat𝛿 is such that 𝐷훿 ⊆𝐸.Thenby there is exactly one point with non-zero value, and on that 휃 the choice of 𝐸0 there is 𝑓 in 𝑁(A) whose invariant is 𝐴∩𝑆=휔̸ , point the value is equal to that of ℎ. Each element of S is 8 Abstract and Applied Analysis clearly definable from ℎ.Weclaimthatsome𝑠∈S can be 𝛿∈𝑆.Forallthedefinitionsofinvariantsweusehere,the chosen to demonstrate (iii). Namely, since we do not have value of the invariant is the same with respect to ⟨𝑐훿 :𝛿∈𝑆⟩ 𝑇𝑓 ≤ 𝑇ℎ 𝑓≤ℎ 휆 ,wecannothave by positivity. Hence there as it is with respect to ⟨𝐷훿 :𝛿∈𝑆휃 ⟩,sowewillnotmake is some value 𝑤 with ℎ(𝑤) < 𝑓(𝑤),andbythecontinuityof a difference between the two. We start with a construction the functions ℎ and 𝑓, there must be some such 𝑤∈{𝑥푛,𝑦푛 : lemma for a certain family of linear orders, as obtained by 𝑛∈𝜔}.Thenletting𝑠(𝑤) = ℎ(𝑤) and 𝑠(V)=0for V =𝑤̸ gives Kojman and Shelah in [6]. Let us give their definition of the afunctioninS,andwehave0≤𝑠≤𝑓and 0 =𝑠̸.Letusnow invariants of linear orders. deal with the general case. Since 0≤𝑇ℎ≤𝑇𝑓,itfollowsthatℎ(𝑦) ≥.Suppose 0 Definition 18. Suppose that 𝐿 is a linear order with the first that ℎ(𝑦) >; 0 hence certainly 𝑓(𝑦).Let >0 𝜀>0be universe 𝜆 and L =⟨𝐿훿 :𝛿<𝜆⟩is a filtration of 𝐿.Then such that 𝜀 ⋅ ℎ(𝑦) < 𝑓(𝑦) (so we can take 𝜀=1if ℎ(𝑦) < 𝑓(𝑦) for every 𝛿∈𝑆such that the universe of 𝐿훿 is 𝛿, we define and 𝜀=1/2if ℎ(𝑦) = 𝑓(𝑦)). We are going to define a function def 耠 훿 훿 𝑠푛 ∈ 𝐶(𝐾) for 𝑛<𝜔by letting 𝑠푛(𝑦) = 𝜀ℎ(𝑦), 𝑠푛(𝑦푚)=𝜀ℎ(𝑦푚) (𝛿) ={𝑖<𝜃:(∃𝛿∈(𝛼 ,𝛼 ]) (∀𝑥 ∈ 𝐿 훿 )𝑥≤ 𝛿 invL,훿 푖 푖+1 훼푖 퐿 for 𝑚≥𝑛and 𝑠푛 =0otherwise. Since ℎ is continuous and ℎ(𝑦) > 0 𝑛 ℎ(𝑦 )≥0 耠 ,wehave,thatforlargeenough , 푛 ,and ⇐⇒ 𝑥 ≤ 퐿 𝛿 }. hence 𝑠푛 ≥0and 𝑠푛 =0̸ .Since𝜀 ⋅ ℎ(𝑦) < 𝑓(𝑦) and both ℎ and 𝑓 are continuous, we must have that, for large enough 𝑛, 𝜀ℎ(𝑦푛)<𝑓(𝑦푛), and therefore for large enough 𝑛 we have (11) 𝑠푛 ≤𝑓.So,some𝑠푛 will work to exemplify (iii). Suppose now that ℎ(𝑦) =.If 0 ℎ(𝑥) >,bycontinuity 0 Lemma 3.7 in [6]provesthat,undertheassumptionswehave ℎ(𝑥 ) >0 𝑓(𝑥) >0 𝜀 stated, for every closed set 𝐴 of limit ordinals in 𝜃,thereis 푛 is eventually .If ,wecanchoose as in the 𝐿[𝐴] 𝜆 L[𝐴] = previous paragraph, so we are done by a similar argument. a linear order with universe and a filtration ⟨𝐿훿[𝐴] : 𝛿 < 𝜆⟩ of 𝐿[𝐴] such that for every 𝛿∈𝑆with Otherwisewehavethat𝑓(𝑥) =0 so by looking at 𝑇ℎ(𝑧2푛+2) 𝐿훿[𝐴] = 𝛿 we have invL[퐴],훿(𝛿) = 𝐴 (Lemma 3.7 in [6]also and 𝑇𝑓 (𝑧 2푛+2),weobtainℎ(𝑥) ≤ 𝑓(𝑦).Alsowehavethat 2휃 <𝜆 ℎ(𝑥푛) is eventually >0.Choose𝜀>0such that 𝜀⋅ℎ(𝑥) < 𝑓(𝑦) states the assumption , but this assumption is not used and define 𝑠푛 by letting 𝜀⋅𝑠푛(𝑦) = ℎ(𝑥) and for 𝑚≥𝑛, in the proof of the lemma, only in the proof of the final result). 𝑠푛(𝑦푚)=𝜀⋅ℎ(𝑥푚),weseethatforlargeenough𝑛,wehavethat 0≤𝑠푛 ≤𝑓. The idea of our proof is to transform the Kojman-Shelah If ℎ(𝑥) < 0 then eventually 𝑇ℎ(𝑧2푛+2)<0,whichisacon- construction first into a construction of a family of Boolean 𝜆 tradiction and so ℎ(𝑥) ≥.If 0 ℎ(𝑥) = 0 then we conclude from algebras of size andthentousetheseBooleanalgebrasto the definition of 𝑇ℎ that, for every 𝑛, ℎ(𝑥푛)=2⋅𝑇ℎ(𝑧2푛−1)≥ define natural spaces of functions with appropriate invariants. 0 ℎ(𝑦 )=2⋅𝑇ℎ(𝑧 )≥0 and similarly 푛 2푛+2 , therefore A ℎ≥0, and we can use the very first argument. Definition 19. Suppose that is a Boolean algebra with the set of generators {𝑎훼 :𝛼<𝜆}and A =⟨A훿 :𝛿<𝜆⟩is a filtration of A,while𝛿∈𝑆is such that A훿 is generated by 5.3. Specific Very Positive Embeddings {𝑎훼 :𝛼<𝛿}. We define

Theorem 17. Theorem 16(1) implies Theorem 1. def 耠 훿 훿 훿 invA,훿 (𝑎훿) = {𝑖 < 𝜃 : (∃𝛿 ∈(𝛼푖 ,𝛼푖+1]) (∀𝛼 <푖 𝛼 )

Proof. We just need to show that the assumptions of 耠 𝑎훼 ∩𝑎훿 =𝑎훼 ∩𝑎훿 mod A훼훿 , (12) Theorem 1 imply those of Theorem 16(1).Thecardinalarith- 푖 metic assumption and the requirement of positivity are the 푐 푐 耠 𝑎훼 ∩𝑎훿 =𝑎훼 ∩𝑎훿 mod A훼훿 }, same in both theorems, so we proceed to show that any 𝑇 푖 as in the assumptions of Theorem 1 satisfies the requirements where 𝑎=𝑏 mod A훼훿 means that for any element 𝑤 of A훼훿 (ii) and (iii) of the definition of very positivity. Requirement 푖 푖 𝑤≤𝑎 𝑤≤𝑏 (ii) follows easily by the surjectivity of 𝑇. Finally (iii), letting we have if and only if . 𝑎 𝑏 𝑠=𝑎⋅ℎ+𝑏 , be as given by (iii) in Theorem 1 and suffices 𝐿 for (iii) of the very positivity. Definition 20. Suppose that is a linear order with universe 𝜆. We define a Boolean algebra A[𝐿] as being generated by {𝑎훼 :𝛼<𝜆}freely except for the equations 5.4. Not Every Positive onto Embeddings Satisfies the Require- ments of Theorem 1. Note that it is a consequence of the 𝑎훿 ≤𝑎휀 ⇐⇒ 𝛿 ≤ 퐿 𝜀. (13) assumptions of Theorem 1 that if 𝑇ℎ ≤ 𝑇𝑓,thenforevery𝑥, 𝑓(𝑥)=0⇒ℎ(𝑥)≤−𝑏/𝑎. Since the equations in (13) are finitely consistent with the axioms of a Boolean algebra, it follows from the compactness theoremthatthealgebraA[𝐿] is well defined. Now we will 6. Proof of the Construction Lemma see a translation between the calculation of the invariants of 휆 the linear orders and the associated Boolean algebras. We present a proof of Lemma 14.Let𝑆⊆𝑆휃 and sequences ⟨𝑐훿 :𝛿∈𝑆⟩, ⟨P훼 :𝛼<𝜆⟩be as in the statement of Sublemma 14.1.Let𝐿 be a linear order on 𝜆,andletA[𝐿] be 휆 Theorem 15, while ⟨𝐷훿 :𝛿∈𝑆휃 ⟩ is such that 𝐷훿 =𝑐훿 for the algebra associated to 𝐿 as per Definition 20.Let𝐿 and A be Abstract and Applied Analysis 9

𝐿 A[𝐿] 𝛽 < 𝛽 𝛽 > 𝛿 𝑎푐 ≤𝑎푐 any filtrations of and , respectively. Then there is a club we conclude 0 퐿 1.Also,if 0 퐿 ,thenwehave 훽0 훿, 𝐸 such that for every 𝛿∈𝑆∩𝐸one has 𝑎푐 ∩𝑎 >0 𝛽 < 𝛿 contradicting that 훽0 훿 .Hence 0 퐿 . 耠 耠 푐 If 𝛽1 <퐿 𝛿,then𝛽1 <퐿 𝛿 ,so𝑎훽 ≤𝑎훿, and hence 𝑎훽 ∩𝑎훽 ≤ 𝑖𝑛V (𝛿) =𝑖𝑛V (𝑎 ) 1 0 1 퐿,훿 A,훿 훿 (14) 耠 𝑎훿, as required. So assume that 𝛿<퐿 𝛽1.Hence𝑎훽 >𝑎훿 and so 푐 푐 푐 1 耠 𝑎훽 ∩𝑎훽 >𝑎훽 ∩𝑎훿 ≥𝑎훽 ∩𝑎훽 , a contradiction. This finishes the and moreover, for any 𝑖∈𝑖𝑛V퐿,훿(𝛿),thisisexemplifiedby𝛿 if 0 1 0 0 1 𝑖∈𝑖𝑛V (𝑎 ) 𝑎 耠 proofoftheforwarddirection,andtheotherdirectionfollows and only if A,훿 훿 is exemplified by 훿 . 耠 from the symmetry of the roles of 𝛿 and 𝛿 in the proof. 耠 Now suppose that 𝑖∈invA,훿(𝑎훿) as exemplified by 𝑎훿. Proof. Let 𝐸 be the club of 𝛿 such that the universe of 𝐿훿 is 𝛼<𝛼훿 𝛼< 𝛿⇔𝛼<𝛿耠 𝛼< 𝛿 𝛿, 𝐿훿 is an elementary submodel of 𝐿,andA[𝐿]훿 is generated Let 푖 ; we need to prove 퐿 퐿 .If 퐿 , 𝑎 <𝑎 𝑎 <𝑎耠 by {𝑎훼 :𝛼<𝛿}and is an elementary submodel of A.Suppose then 훼 훿,hence 훼 훿 by the assumption, and hence 𝛼< 𝛿耠 A[𝐿] that 𝛿∈𝐸∩𝑆. 퐿 by the definition of . The other direction follows 耠 First suppose that 𝑖∈invL,훿(𝛿) as exemplified by 𝛿 .Let by symmetry. 훿 耠 푐 𝛼<𝛼 𝑎 ∩𝑎 =𝑎 ∩𝑎 A 훿 𝑎 ∩ 푖 ,weneedtoprove 훼 훿 훼 훿 mod 훼푖 and 훼 푐 耠 The next step is going from the invariants of Boolean 𝑎 =𝑎 ∩𝑎 A 훿 훿 훼 훿 mod 훼푖 . algebras to the invariants of natural spaces. 耠 耠 Case 1 (𝛼<퐿 𝛿). Hence by the choice of 𝛿 ,wehave𝛼<퐿 𝛿 and Sublemma 14.2.LetA[𝐿] be one of the algebras described in 耠 耠 𝑎훼 ≤𝑎훿, 𝑎훼 ≤𝑎훿. Therefore 𝑎훼 ∩𝑎훿 =𝑎훼 ∩𝑎훿 =𝑎훼. the above, and let A be its filtration. Then there is a club 𝐸 of 푐 𝑧>0 A 훿 𝑧≤𝑎 ∩𝑎 𝜆 𝛿∈𝑆 𝐷 ⊆𝐸 Suppose that is in 훼푖 and satisfies 훼 훿.By such that, for every with 훿 , one has that the Disjunctive Normal Form for Boolean algebras, we can 푙(푖,푗) 𝑧=⋁ ⋀ 𝑎 𝑙(𝑖, 𝑗) ∈ {0, 1} 𝑖𝑛VA,훿 (𝜒[푎 ])=𝑖𝑛VA,훿 (𝑎훿) (15) assume that 푖≤푛 푗≤푘푖 훽(푖,푗) for some 훿 𝛽(𝑖, 𝑗) <𝛼훿 𝑖 and 푖 . It suffices to prove that for every we have 𝑖∈𝑖𝑛V (𝜒 ) 푙(푖,푗) 푐 耠 and moreover, for any A,훿 [푎훿] ,thisisexemplifiedby ⋀푗≤푘 𝑎 ≤𝑎 ∩𝑎 .Fixan𝑖 and without loss of generality 푖 훽(푖,푗) 훼 훿 𝜒[푎耠 ] if and only if 𝑖∈𝑖𝑛VA,훿(𝑎훿) is exemplified by 𝑎훿耠 . Here, 푙(푖,푗) 훿 assume that ⋀푗≤푘 𝑎훽(푖,푗) >0, as otherwise the conclusion is the invariant on the left refers to the invariant in the natural 푖 𝑁(A) trivial. space and the invariant on the right to the invariant A 𝜒 Let 𝐴푙 ={𝑗≤𝑘푖 :𝛽(𝑖,𝑗)=𝑙},for𝑙∈{0,1}.For in the algebra .Thenotation [푎] is used for the sequence 푁(A) simplicity assume that both of these sets are nonempty, as ⟨𝜒[푎],𝜒[푎],𝜒[푎],...⟩in 𝐶 . otherwise the proof is easier. Let 𝛽1 be the 𝐿-minimal element ∗ 𝐴 ⋀ 𝑎푙(푖,푗) =𝑎 𝛽 𝐿 Proof. Let M be a model consisting of 𝐿, A,twodisjoint of 1;hence 푗∈퐴1 훽(푖,푗) 훽1 .Let 0 be the -maximal 푐 푐 푐 copies of the 𝜔-sequences of the simple functions on 𝜆 with element of 𝐴0;hence⋀ 𝑎 =(⋁ 𝑎훽(푖,푗)) =𝑎 .In 푗∈퐴0 훽(푖,푗) 푗∈퐴0 훽0 𝑁(A) 푙(푖,푗) rational coefficients, interpreted as the elements of and ⋀ 𝑎 =𝑎푐 ∩𝑎 𝑁(A) conclusion, 푗≤푘푖 훽(푖,푗) 훽0 훽1 . Since we have assumed all the symbols of with induced interpretations induced ⋀ 𝑎푙(푖,푗) >0 𝑎 ≤𝑎 from these models. Recall the assumption that for all 𝜅<𝜆, that 푗≤푘 훽(푖,푗) ,wecannothave 훽1 훽0 ,equivalently ℵ 푖 𝜅 0 <𝜆 𝛽 ≤ 𝛽 𝛽 < 𝛽 𝑎 ∩𝑎푐 >0 we have andnoticethatitimpliesthatthereisaclub 1 퐿 0.Hencewehave 0 퐿 1. Similarly, since 훽1 훼 𝐸0 of 𝜆 such that, for every 𝛿∈𝐸0 of cofinality >ℵ0,the we can conclude that 𝛼<퐿 𝛽1.Finally,ifwehad𝛽0 >퐿 𝛿,then ∗ ∗ 푐 푐 model M ↾𝛿is ℵ1-saturated in M ; that is, it realizes all we would obtain 𝑎훽 ≤𝑎훿, in contradiction with 0<𝑎훽푐 ∩ ∗ 0 0 the types with countably many parameters in M ↾𝛿which 𝑎훽 ≤𝑎훿 𝛽0 <퐿 𝛿 ∗ 1 , and therefore . M 𝐿 𝐿 𝐸⊆𝐸 𝛿< 𝛽 𝑎 ∩𝑎푐 =0̸ are realized in .Let be any filtration of ,let 0 be Suppose now that 퐿 1. Therefore 훽1 훿 .Onthe 𝛿∈𝑆 푐 푐 a club witnessing Sublemma 14.1, and let be such that 𝑎 ∩𝑎 ∩𝑎 ≤𝑎∩𝑎 푐 ∩𝑎 푐 =0 other hand, 훽 훽1 훿 훿 훼 훿 ,andhencewemust 𝐷 ⊆𝐸 0 푐 훿 . 𝑎 ∩𝑎 ∩𝑎 >0 𝛽 < 𝛽 𝑖∈ (𝑎 ) 𝑎 耠 𝑔≥0 have 훽0 훽1 훿 ,which,takingintoaccount 0 퐿 1, Suppose invA,훿 훿 as exemplified by 훿 but 푐 푁(A ) gives that 𝑎훽 ∩𝑎훿 >0, and hence 𝛿<퐿 𝛽0, a contradiction. 훼훿 0 with 𝑔∈𝐶 푖 \𝐶0 and the limit 𝑔 of 𝑔 satisfies 𝑔≤ 𝛽 < 𝛿 𝛿耠 𝛽 < 𝛿耠 Hence we have 1 퐿 . By the choice of ,wehave 1 퐿 , |𝜒[푎 ] −𝜒[푎耠 ]|=𝜒[푎 Δ푎 ] 푐 훿 훿 훿 훿耠 .Bytoppingupifnecessary(see 𝑎 ≤𝑎耠 𝑎 ∩𝑎 ≤𝑎耠 and hence 훽1 훿 and in particular 훽0 훽1 훿 , 𝑔 ≥0 耠 Definition), 7 we may assume that each 푛 ,andby as required. Since the roles of 𝛿 and 𝛿 in this proof were throwing away unnecessary elements of 𝑔, we may assume symmetric, we can prove in the same way that for any 𝑧>0 𝑔 =0̸ 𝑛 푐 耠 푐 that every 푛 . We can then assume that for each there A 훿 𝑧≤𝑎 ∩𝑎 𝑧≤𝑎 ∩𝑎 푛 푘푛 푛 in 훼푖 which satisfies 훼 훿 we also have 훼 훿. {𝑏 ,...,𝑏 }∈A 훿 𝑞 (𝑖 ≤ 𝑘 )∈ are pairwise disjoint 0 푛 훼푖 and 푖 푛 + 푛 耠 耠 푐 Q 𝑔 =Σ 𝑞 𝜒 푛 ‖𝑔 −𝜒 ‖→ 𝛼> 𝛿 𝛼> 𝛿 𝛿 𝑎 ≥ such that 푛 푖≤푘푛 푖 [푏 ].Since 푛 [푎훿Δ푎 耠 ] Case 2 ( 퐿 ,so 퐿 bythechoiceof ). We have 훿 푛 푖 훿 푐 푐 푐 耠 0,therehastobea[𝑏푖 ] with a nonempty intersection 𝑎훼,so𝑎훼 ∩𝑎훿 =0,andsimilarly𝑎훼 ∩𝑎훿 =0.Wealsohave 耠 with [𝑎훿Δ𝑎훿耠 ].ByapplyingtheDisjunctiveNormalForm, 𝑎훼 ∩𝑎훿 =𝑎훿 and similarly for 𝛿 ; hence we need to prove that 𝑏푛 = ⋁ ⋀ 𝑎푙(푚,표) 𝑎 =𝑎耠 A 훿 we can assume that 푖 푗≤푚 표≤표푚 훽(푚,표) for some 훿 훿 mod 훼푖 .AsinCase1,itsufficestoshowthat,for 훿 푐 푐 𝑙(𝑚, 𝑜) ∈ {0, 1} 𝛽(𝑚, 𝑜) <𝛼훿 𝑗≤𝑚 every 𝛽0,𝛽1 <𝛼 with 0<𝑎 ∩𝑎훽 <𝑎훿,wehave𝑎 ∩ and 푖 . Therefore there is 푖 훽0 1 훽0 푙(푚,표) 耠 such that ⋀표≤표 [𝑎 ]∩[𝑎훿Δ𝑎훿耠 ] =0̸ .Thenwehavethat 𝑎훽 ≤𝑎훿 (the equality cannot occur), and vice versa. Let us 푚 훽(푚,표) 1 푐 푙(푚,표) 푙(푚,표) 耠 0<𝑎 ∩𝑎 ⋀ [𝑎 ]∩[𝑎훿] =⋀̸ [𝑎 ]∩[𝑎 ] start with the forward direction. As before, from 훽0 훽1 , 표≤표푚 훽(푚,표) 표≤표푚 훽(푚,표) 훿 , and hence there 10 Abstract and Applied Analysis

푙(푚,표) 푙(푚,표) 耠 has to be 𝑜≤𝑝푚 such that [𝑎훽(푚,표)]∩[𝑎훿] =[𝑎̸ 훽(푚,표)]∩[𝑎훿]. Let 𝜀∈[0,1]be such that 𝑓 is constantly 𝜀 on [𝑎훿 \𝑎훿耠 ].Say 1−푙(푚,표) 1−푙(푚,표) 耠 𝜀≤1/2as the other case is symmetric. Hence max{𝑓,1 − 𝑓} It follows that [𝑎훽(푚,표) ]∩[𝑎훿] =[𝑎̸ 훽(푚,표) ]∩[𝑎훿].Let𝛽= on 𝑎훿 \𝑎훿耠 is 1−𝑓,inparticularwehave 𝛽(𝑚, 0). From the choice of 𝐸,usingSublemma14.1,wehave 耠 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 that for 𝑅∈{<퐿,>퐿}, 𝛽𝑅𝛿 if and only if 𝛽𝑅𝛿 .Wegothrough 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 0≤𝑔≤󵄨𝜒 −𝑓󵄨 + 󵄨𝑓−𝜒 耠 󵄨 ≤2⋅󵄨𝜒 −𝑓󵄨 󵄨 [푎훿] 󵄨 󵄨 [푎 ]󵄨 󵄨 [푎훿] 󵄨 (17) a case analysis like in the proof of Sublemma 14.1. If 𝛽<퐿 𝛿, 훿 耠 then we have 𝛽<퐿 𝛿 ,so[𝑎훽]∩[𝑎훿]=[𝑎훽]=[𝑎훽]∩[𝑎훿耠 ],a 푐 푐 1/2 ⋅ 𝑔 𝑓 𝛽> 𝛿 [𝑎 ]∩[𝑎 ]=0=[𝑎]∩[𝑎 耠 ] and hence is a function which contradicts that contradiction. If 퐿 ,then 훽 훿 훽 훿 ,a 𝑖∈ (𝜒 ) exemplifies that invA,훿 [푎훿] . A contradiction and hence contradiction. Therefore, 𝑖∈invA,훿(𝜒[푎 ]). 훿 𝜒[푎耠 ] 𝑖∈ (𝜒[푎 ]) 훿 exemplifies that invA,훿 훿 as required. 𝑖∈ (𝜒 ) Claim 1. Suppose that invA,훿 [푎훿] as exemplified by some 𝑖∈ (𝜒 ) Now suppose that invA,훿 [푎훿] and assume without 𝑓. Without loss of generality, we can assume that 𝑓=𝜒[푎耠 ] for 耠 훿 𝑓=𝜒[푎耠 ] 𝛿 耠 loss of generality by Claim 1 that 훿 for some .We some 𝛿 . need to prove that 𝑖∈invA,훿(𝑎훿) as exemplified by 𝑎훿耠 .But 𝛽<𝛼훿 𝑙<2 𝑎푙 ∩𝑎 =𝑎̸ ∩ Proof of the Claim. First let us notice that if 𝑓=lim푛𝑓푛,then indeed, if for some 푖 and ,wehavethat 훽 훿 훽 耠 耠 훿 훿 𝑓 = {𝑓,1} |𝜒 −𝑓 |≤|𝜒 −𝑓| 훼푖 훼푖 for min ,wehave [푎훿] [푎훿] ,sowecan 𝑎훿耠 mod A , then there is 𝑤>0in A with 𝑤≤𝑎훿Δ𝑎훿耠 𝑓≤1 0 without loss of generality assume that .Similarlywecan and then clearly the sequence ⟨𝜒[푤],𝜒[푤],...⟩is not in 𝐶 and 𝑓≥0 assume that , and then by applying a similar logic, we ⟨𝜒[푎 Δ푎 ],𝜒[푎 Δ푎 ],...⟩ is below 훿 훿耠 훿 훿耠 , a contradiction. can also assume that 0≤𝑓푛 ≤1for all 𝑛 and that 𝑓푛 =0̸ for all 𝑛. Each 𝑓푛 is a simple function with rational coefficients With this, putting together all the sublemmas and the defined on (without loss of generality) disjoint basic clopen Kojman-Shelah result, we finish the proof of the Construction 푙 훽 훿 Lemma. sets of the form [𝑎훽 ] where 𝛽<𝛼푖+1 and 𝑙훽 <2.Let{𝛽푛 :𝑛< 𝜔} enumerate all the relevant 𝛽.Foreach𝑛 and 𝑅∈{<퐿,>퐿}, 𝑗푛 𝑎 𝑅𝑎 let 푅 be the truth value of “ 훽푛 훿.” Consider the following Conflict of Interests 훿 sentence with parameters 𝑓, 𝛼푖 and the elements of {𝛽푛 :𝑛< 𝜔} 𝛽 The author declares that there is no conflict of interests ;thereis such that regarding the publication of this paper. 𝑔∈𝑁(A 훿 ) 0≤ 𝑔 ≤| 𝑓 −𝜒 | (i) for all 훼푖 if lim푛 푛 lim 푛 [푎훽] ,we 0 have 𝑔∈𝐶; Acknowledgments 𝑛 𝑅∈{< ,> } 𝑎 𝑅𝑎 (ii) for all and 퐿 퐿 ,wehave 훽푛 훽 if and only 𝑗푛 =1 The author thanks EPSRC for the Grants EP/G068720 and if 푅 . EP/I00498 which supported this research and the University Thissentenceistrueasexemplifiedby𝛿,sobythechoice of Wroclaw in Poland for their invitation in October 2010, ∗ 훿 耠 of 𝐸0,itistrueinM ↾𝛼푖+1; say as exemplified by 𝛿 .Letus when some of the preliminary results were presented. The 耠 耠 author especially thanks Grzegorz Plebanek for the many note that in 𝐿,wehave𝛿<퐿 𝛿 or 𝛿 <퐿 𝛿, and let us assume 耠 productive conversations during the development of this that 𝛿 <퐿 𝛿, as the other case is symmetric. We claim that paper. 𝜒[푎耠 ] 𝑖∈ (𝜒[푎 ]) 훿 exemplifies that invA,훿 훿 .Ifnot,wecanfind 푁(A ) def 훼훿 0 𝑔∈𝐶 푖 \𝐶 with 0≤𝑔 and 𝑔 = lim푛𝑔푛 ≤|𝜒[푎 ] −𝜒[푎耠 ]|= 훿 훿 References 𝜒[푎 Δ푎耠 ] ≤1 훿 훿 . By the triangle inequality it follows that 󵄨 󵄨 󵄨 󵄨 [1] M. Malliaris and S. Shelah, “General topology meets model 𝑔≤󵄨𝜒[푎 ] −𝑓󵄨 + 󵄨𝑓−𝜒[푎耠 ]󵄨 . 𝑝 𝑡 󵄨 훿 󵄨 󵄨 훿 󵄨 (16) theory, on and ,” Proceedings of the National Academy of Sci- ences of the United States of America,vol.110,no.33,pp.13300– 耠 We have that for every 𝑥 both |𝜒[푎 ] −𝑓|(𝑥)and |𝜒[푎 ] −𝑓 |(𝑥) 13305, 2013. 푐 훿 훿 are equal to 𝑓(𝑥) if 𝑥∈[𝑎훿] and 1−𝑓(𝑥)if 𝑥∈[𝑎훿耠 ].The [2] S. Shelah and A. Usvyatsov, “Banach spaces and groups—order possible difference is on [𝑎훿 \𝑎훿耠 ],wheretheformerfunction properties and universal models,” IsraelJournalofMathematics, is equal to 1−𝑓(𝑥)and the latter to 𝑓(𝑥).Wenowclaimthat vol. 152, pp. 245–270, 2006. 𝑓 is constant on [𝑎훿 \𝑎훿耠 ]. [3] C. Brech and P.Koszmider, “On universal Banach spaces of den- Clearly, it suffices to show that each 𝑓푛 is constant on [𝑎훿 \ sity continuum,” Israel Journal of Mathematics,vol.190,pp.93– 𝑎훿耠 ],andbythechoiceof{𝛽푛 :𝑛<𝜔},itsufficestoshowthat, 110, 2012. for each 𝑙<2and each 𝑛, 𝜒[푎푙 ] is constant on [𝑎훿\𝑎훿耠 ].Let𝛽= [4] C. Brech and P. Koszmider, “On universal spaces for the class 훽푛 耠 of Banach spaces whose dual balls are uniform Eberlein com- 𝛽 𝑛 𝛽≤ 𝛿 𝑎 ≤𝑎耠 𝜒 푛 for some .If 퐿 ,then 훽 훿 so [푎훽] is constantly 0 pacts,” Proceedings of the American Mathematical Society,vol. 푐 푐 耠 [𝑎훿 \𝑎훿耠 ] 𝑎 ≥𝑎耠 ≥(𝑎훿 \𝑎 ) 𝜒[푎푐 ] 141, no. 4, pp. 1267–1280, 2013. on . In addition, we have 훽 훿 훿 ,so 훽 耠 [5] S. Shelah, “No universal group in a cardinal,” http://xxx.tau is constantly 1 on [𝑎훿 \𝑎훿耠 ].If𝛽≥퐿 𝛿 ,then𝛽≥퐿𝛿 by the choice 耠 .ac.il/abs/1311.4997v1 . 𝛿 𝑎 ≥𝑎 𝜒 [𝑎 \𝑎 耠 ] of ,so 훽 훿, and hence [푎훽] is constantly 1 on 훿 훿 . 푐 푐 [6] M. Kojman and S. Shelah, “Nonexistence of universal orders in In addition, 𝑎훽 ≤𝑎훿 so 𝜒[푎푐 ] is constantly 0 on [𝑎훿 \𝑎훿耠 ],and 훽 many cardinals,” TheJournalofSymbolicLogic,vol.57,no.3,pp. the statement is proved. 875–891, 1992. Abstract and Applied Analysis 11

[7] I. Kaplansky, “Lattices of continuous functions,” Bulletin of the American Mathematical Society,vol.53,pp.617–623,1947. [8] S. Shelah and A. Usvyatsov, “Unstable classes of metricstruc- tures,” http://arxiv.org/abs/0810.0734 . [9] M. Dzamonja,ˇ “Banach spaces from one Cohen real”. [10] S. Banach, Theorie Des Operations Lineaires, Volume 1 of Monografie Matem-Atyczne, IMPAN, Warszawa, Poland, 1932. [11] C. C. Chang and H. J. Keisler, Model Theory,vol.73ofStudies in Logic and the Foundations of Mathematics,North-Holland Publishing, Amsterdam, The Netherlands, 3rd edition, 1990. [12] I. Gelfand and A. Kolmogorov, “On rings of continuous func- tions on topo- logical spaces,” Soviet Mathematics Doklady,vol. 22, pp. 11–15, 1939 (Russian). [13] S. Shelah, Classification Theory and the Number of Nonisomor- phic Models,vol.92ofStudies in Logic and the Foundations of Mathematics, North-Holland Publishing, Amsterdam, The Netherlands, 2nd edition, 1990. [14] S. Shelah, “Advances in cardinal arithmetic,” in Finite and Infinite Combinatorics in Sets and Logic,N.W.Sauer,R.E. Woodrow, and B. Sands, Eds., vol. 411 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pp. 355–383, Kluwer Academic, Dordrecht, The Netherlands, 1993. [15] G. Plebanek, “On positive embeddings of 𝐶(𝐾) spaces,” Studia Mathematica,vol.216,no.2,pp.179–192,2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 397376, 6 pages http://dx.doi.org/10.1155/2014/397376

Research Article Multipliers of Modules of Continuous Vector-Valued Functions

Liaqat Ali Khan and Saud M. Alsulami

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Liaqat Ali Khan; [email protected]

Received 8 November 2013; Accepted 11 January 2014; Published 12 May 2014

Academic Editor: M. Mursaleen

Copyright © 2014 L. A. Khan and S. M. Alsulami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

∗ In 1961, Wang showed that if 𝐴 is the commutative 𝐶 -algebra 𝐶0(𝑋) with 𝑋 a locally compact Hausdorff space, then 𝑀(𝐶0(𝑋)) ≅ 𝐶𝑏(𝑋). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these (𝐶 (𝑋, 𝐸), 𝐶 (𝑋, 𝐹)) ≃𝐶 (𝑋, (𝐸, 𝐹)), 𝐸 𝐹 𝑝 results by showing that Hom𝐶0(𝑋,𝐴) 0 0 𝑠,𝑏 Hom𝐴 where and are -normed spaces which are also essential isometric left 𝐴-modules with 𝐴 being a certain commutative 𝐹-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

1. Introduction 𝐶0(𝑋, 𝐴) is isometrically isomorphic to the completed tensor product 𝐶0(𝑋)⊗𝜆𝐴 with respect to the smallest cross norm 𝜆 Characterizations of multipliers on algebras and modules of (see [2–5]).Wewillavoidtheuseofthistechniqueasitneed continuous functions with values in a commutative Banach not work in our case. In fact, when 𝐴 is not locally convex, 𝐶∗ 𝐴 or -algebra have been obtained by several authors. In ⊗𝜆 is no longer appropriate; even for 𝐴 acomplete𝑝-normed 𝐴 1961, Wang [1]showedthatif is taken as the commutative space, many complications arise (see [7,Section10.4];[8,p. 𝐶∗ 𝐶 (𝑋) 𝑋 -algebra 0 with being a locally compact Hausdorff 100]). space, then 𝑀(𝐶0(𝑋)) ≅𝑏 𝐶 (𝑋). This result has also been generalized to vector-valued functions by several authors (see, e.g., [2–6]). In 1985, Lai [6]showedthatif𝑋 is a 2. Preliminaries locally compact abelian group and 𝐴 is a commutative In this section, we include some basic definitions and study Banach algebra with a bounded approximate identity, then various classes of topological algebras considered in this 𝑀(𝐶0(𝑋, 𝐴))𝑏 ≅𝐶 (𝑋, 𝑀(𝐴)𝑢). In 1992, Candeal Haro and paper. Lai [3]hadobtained 𝐸 Hom𝐶 (𝑋,𝐴) (𝐶0 (𝑋, 𝐸) ,𝐶0 (𝑋, 𝐹)) ≃𝐶𝑠,𝑏 (𝑋, Hom𝐴 (𝐸,) 𝐹 ) , Definition 1 (see [9, 10]). Let be a vector space over the field 0 K ∈{R C} (1) , . in the case when 𝐴 isacommutativeBanachalgebraand𝐸 (a) A function 𝑞:𝐸 → R is called an 𝐹-seminorm on 𝐸 and 𝐹 are left Banach 𝐴-modules. if it satisfies the following: A natural question arises is to investigate the extent to which these characterizations can be made beyond Banach (F1) 𝑞(𝑢) ≥0 for all 𝑢∈𝐸; modules. We will focus mainly on the nonlocally convex case (F2) 𝑞(𝑢) =0 if 𝑢=0; by considering 𝐴 a commutative complete 𝑝-normed algebra, (F3) 𝑞(𝛼𝑢) ≤ 𝑞(𝑢) for all 𝑢∈𝐸and 𝛼∈K with 0<𝑝≤1, having a minimal approximate identity and 𝐸 and |𝛼| ≤ 1 𝐹 𝐹 𝐴 ; being -spaces which are also left -modules. 𝑞(𝑢 + V) ≤ 𝑞(𝑢) +V 𝑞( ) 𝑢, V ∈𝐸 We mention that the arguments of earlier authors relied (F4) for all ; ) 𝛼 →0 K 𝑞(𝛼 𝑢) → 0 𝑢∈𝐸 heavily on the fact that, in the case of 𝐴,aBanachalgebra, (F5 if 𝑛 in ,then 𝑛 for all . 2 Abstract and Applied Analysis

(b) An 𝐹-seminorm 𝑞 on 𝐸 is called an 𝐹-norm if, for any where 𝐷 is a bounded (resp., finite) subset of 𝐸 and 𝑊 is a 𝑢∈𝐸, 𝑞(𝑢) =0 implies 𝑢=0. neighborhood of 0 in 𝐹. Clearly, 𝑠≤𝜎.Inparticular,if(𝐴,𝐴 𝑞 ) 𝑝 𝜎 𝐶𝐿(𝐴) (c) An 𝐹-seminorm (or 𝐹-norm) 𝑞 on 𝐸 is called a 𝑝- is a -normed algebra, then the -topology on is the one given by the 𝑝-norm ‖⋅‖𝐴 . In this setting, the strong seminorm (resp., 𝑝-norm), 0<𝑝≤1, if it also satisfies 𝑝 operator topology 𝑠 on 𝐶𝐿(𝐴) is given by the family of {𝑃𝑎 : 𝑝 𝑞 (𝛼𝑢) = |𝛼| 𝑞 (𝑢) ∀𝑢∈𝐸,𝛼∈K.(𝑝-homogeneous). 𝑎∈𝐴}of 𝐹-seminorms, where (2) 𝑃𝑎 (𝑇) =𝑞𝐴 (𝑇 (𝑎)) ,𝑇∈𝐶𝐿(𝐴) . (9) 𝑞 𝐹 𝑝 (d) If is an -norm (resp., a -norm) on a vector space (𝐸, 𝑞 ) 𝐹 ||𝑇|| 𝐸 (𝐸, 𝑞) 𝐹 Remark 5. If 𝐸 is a general -algebra, then 𝑞𝐸 need , then the pair is called an -normed (resp., a {𝑢 ∈ 𝐸 :𝑞 (𝑢) ≤ 1} 𝑝-normed) space. not exist since the set 𝐸 may not be bounded (see ([10,p.8];[11, 12]) for counterexamples). (e) An 𝐹-norm (or a 𝑝-norm) 𝑞 on an algebra 𝐴 is called submultiplicative if Definition 6. Let 𝑋 be a Hausdorff topological space and 𝐸 a Hausdorff topological vector space over the field K (= R or 𝑞 (𝑎𝑏) ≤𝑞(𝑎) 𝑞 (𝑏) ∀𝑎, 𝑏 ∈ 𝐴. (3) C)withabaseW of neighborhoods of 0 in 𝐸.Afunction𝑓: 𝑋→𝐸is said to vanish at infinity if, for each neighborhood 𝐴 𝐹 𝑝 An algebra with a submultiplicative -norm (resp., - 𝑊 of 0 in 𝐸, there exists a compact set 𝐾=𝐾𝑊 ⊆𝑋such that norm) 𝑞 is called an 𝐹-normed (resp., 𝑝-normed) algebra. 𝑓 (𝑥) ∈𝑊 ∀𝑥∈𝑋\𝐾. (10) Definition 2. (1) Anet{𝑒𝜆 :𝜆∈𝐼}in a topological algebra 𝐴 is called an approximate identity if We will denote by 𝐶𝑏(𝑋, 𝐸) the vector space of all continuous bounded 𝐸-valued functions on 𝑋 and by 𝐶0(𝑋, 𝐸) the lim 𝑒𝜆𝑎=lim 𝑎𝑒𝜆 =𝑎 ∀𝑎∈𝐴. 𝐶 (𝑋, 𝐸) 𝜆 𝜆 (4) subspace of 𝑏 consisting of those functions which vanish at infinity. When 𝐸=K (= R or C),thesespaceswill (2) An approximate identity {𝑒𝜆 :𝜆∈𝐼}in an 𝐹-normed be denoted by 𝐶𝑏(𝑋) and 𝐶0(𝑋).Let𝐶𝑏(𝑋) ⊗ 𝐸 denote the algebra (𝐴, 𝑞) is said to be minimal if 𝑞(𝑒𝜆)≤1for all 𝜆∈𝐼. vector subspace of 𝐶𝑏(𝑋, 𝐸) spannedbythesetofallfunctions of the form 𝜑⊗𝑢,where𝜑∈𝐶𝑏(𝑋), 𝑢∈𝐸,and If 𝐸 and 𝐹 are topological vector spaces over the field K ∈ {R or C}, then the set of all continuous linear mappings 𝑇: (𝜑⊗𝑢) (𝑥) =𝜑(𝑥) 𝑢, 𝑥 ∈ 𝑋. (11) 𝐸→𝐹is denoted by 𝐶𝐿(𝐸,. 𝐹) Clearly, 𝐶𝐿(𝐸, 𝐹) is a vector 𝑋 𝐶 (𝑋, 𝐸) space over K with the usual pointwise operations. Further, if We mention that, if is not locally compact, then 0 𝐹 = 𝐸, 𝐶𝐿(𝐸) = 𝐶𝐿(𝐸,𝐸) is an algebra under composition may be the trivial vector space {0}. For example, if 𝑋=Q,the 𝐸=R 𝐶 (Q, R)={0} (i.e., (𝑆𝑇)(𝑢) = 𝑆(𝑇(𝑢)), 𝑢∈𝐸) and has the identity 𝐼:𝐸 → space of rationals, and ,then 0 . 𝐸 given by 𝐼(𝑢) = 𝑢 (𝑢. ∈𝐸) Remarks 7. (i) If 𝐸=𝐴is an algebra, then 𝐶𝑏(𝑋, 𝐴) is also an Definition 3. Let (𝐸,𝐸 𝑞 ) and (𝐹,𝐹 𝑞 ) be 𝑝-normed spaces. For algebra with respect to the pointwise multiplication defined any linear map 𝑇:𝐸, →𝐹 define by ‖𝑇‖ = {𝑞 (𝑇𝑢) :𝑢∈𝐸,𝑞 (𝑢) ≤1}. (𝑓𝑔) (𝑥) =𝑓(𝑥) 𝑔 (𝑥) ,𝑥∈𝑋. 𝑞𝐸,𝑞𝐹 sup 𝐹 𝐸 (5) (12)

Then, by 10([ ,p.101-102]),𝑇 ∈ 𝐶𝐿(𝐸, 𝐹) if and only if (ii) If 𝐸=𝐴is a commutative algebra, then 𝐶𝑏(𝑋, 𝐴) ‖𝑇‖ <∞ ‖⋅‖ 𝐹 𝐶𝐿(𝐸, 𝐹) 𝐶𝑏(𝑋) 𝑞𝐸,𝑞𝐹 .Further, 𝑞𝐸,𝑞𝐹 is an -norm on is also commutative; in particular, is a commutative and, for any 𝑇 ∈ 𝐶𝐿(𝐸,, 𝐹) algebra. (iii) If 𝐸 is only a vector space, then 𝐶𝑏(𝑋, 𝐸) is a 𝑞 (𝑇𝑢) ≤ ‖𝑇‖ ⋅𝑞 (𝑢) ∀𝑢∈𝐸. 𝐹 𝑞𝐸,𝑞𝐹 𝐸 (6) 𝐶𝑏(𝑋)-bimodule with respect to the module multiplications (𝜑, 𝑓) → 𝜑 ⋅𝑓 and (𝑓,𝜑) → 𝑓 ⋅𝜑 defined by In particular, if 𝑇 ∈ 𝐶𝐿(𝐸) = 𝐶𝐿(𝐸,, 𝐸) we denote (𝜑 ⋅ 𝑓) (𝑥) =𝜑(𝑥) 𝑓 (𝑥) =(𝑓⋅𝜑)(𝑥) ,𝑥∈𝑋. ‖𝑇‖ := {𝑞 (𝑇 (𝑢)) :𝑢∈𝐸,𝑞 (𝑢) ≤1} . (13) 𝑞𝐸 sup 𝐸 𝐸 (7) (iv) If 𝐸 is a vector space and 𝐴 is algebra, then 𝐶𝑏(𝑋, 𝐸) In this case, for any 𝑆, 𝑇 ∈ 𝐶𝐿(𝐸), ||𝑆𝑇||𝑞 ≤ ||𝑆||𝑞 ||𝑇||𝑞 ; 𝐸 𝐸 𝐸 is a left 𝐴-module with respect to the module multiplication hence (𝐶𝐿(𝐸),𝑞 ‖⋅ ) is a 𝑝-normed algebra. 𝐸 (𝑎, 𝑓) → 𝑎 ⋅𝑓 as pointwise action: Definition 4. Let 𝐸 and 𝐹 be topological vector spaces. (𝑎⋅𝑓) (𝑥) =𝑎𝑓(𝑥) , 𝑎∈𝐴,𝑓∈𝐶𝑏 (𝑋, 𝐴) ,𝑥∈𝑋. (14) The uniform operator topology 𝜎 (resp., the strong operator 𝑠 𝐶𝐿(𝐸, 𝐹) topology )on is defined as the linear topology In particular, 𝐶0(𝑋, 𝐸) is a left 𝐴-module. which has a base of neighborhoods of 0 consisting of all the sets of the form Definition 8. Let 𝑋 be a Hausdorff space and 𝐸 aHausdorff topological vector space (TVS) over K (= R or C). The 𝑁 (𝐷,) 𝑊 = {𝑇∈𝐶𝐿(𝐴) :𝑇(𝐷) ⊆𝑊} , (8) uniform topology 𝑢 on 𝐶𝑏(𝑋, 𝐸) is the linear topology which Abstract and Applied Analysis 3 has a base of neighborhoods of 0 consisting of all sets of the Proof. Let V ∈𝐹.Then form 󵄩 󵄩 󵄩𝐿V󵄩𝑞 ,𝑞 = sup {𝑞𝐹 (𝐿V (𝑎)):𝑞𝐴 (𝑎) ≤1} 𝑁 (𝑋, 𝐺) = {𝑓∈𝐶𝑏 (𝑋, 𝐸) :𝑓(𝑋) ⊆𝑊} , (15) 𝐴 𝐹 where 𝑊 is a neighborhood of 0 in 𝐸.Inparticular,if𝐸= = sup {𝑞𝐹 (V ⋅𝑎) :𝑞𝐴 (𝑎) ≤1} (23) (𝐸,𝐸 𝑞 ) is an 𝐹-normed space, the 𝑢-topology on 𝐶𝑏(𝑋, 𝐸) is ≤ {𝑞 (𝑎) 𝑞 (V) :𝑞 (𝑎) ≤1}=𝑞 (V) . given by the 𝐹-norm sup 𝐴 𝐹 𝐴 𝐹 󵄩 󵄩 󵄩𝑓󵄩 = 𝑞 (𝑓 (𝑥)), 𝑓∈𝐶 (𝑋, 𝐸) . 󵄩 󵄩𝑞 ,∞ sup 𝐸 𝑏 On the other hand, 𝐸 𝑥∈𝑋 (16) 󵄩 󵄩 󵄩𝐿V󵄩 = {𝑞𝐹 (V ⋅𝑎) :𝑞𝐴 (𝑎) ≤1} 󵄩 󵄩𝑞𝐴,𝑞𝐹 sup 3. Main Results (24) ≥𝑞𝐹 (V ⋅𝑒𝜆)∀𝜆∈𝐼, In this section we extend some results of [2–6] from Banach modules to the more general setting of topological modules. so

Definition 9 (cf. [13, 14]). Let (𝐴,𝐴 𝑞 ) be a commutative 𝑝- 󵄩 󵄩 󵄩𝐿V󵄩𝑞 ,𝑞 ≥ lim 𝑞𝐹 (V ⋅𝑒𝜆)=𝑞𝐹 (limV ⋅𝑒𝜆)=𝑞𝐹 (V) . (25) normed algebra, and let (𝐸,𝐸 𝑞 ) be a 𝑝-normed space which 𝐴 𝐹 𝜆 𝜆 is also an 𝐴-module in the usual algebraic sense. Then 𝐸 is 𝐴 ‖𝐿 ‖ =𝑞 (V) ||𝑅 || =𝑞 (V) called an isometric -module if Hence V 𝑞𝐴,𝑞𝐹 𝐹 .Similarly, V 𝑞𝐸 𝐸 .

𝑞𝐹 (𝑎𝑢) ≤𝑞𝐴 (𝑎) 𝑞𝐹 (𝑢) for any 𝑎∈𝐴,𝑢∈𝐸. (17) Lemma 12. Let (𝐴,𝐴 𝑞 ) a commutative 𝑝-normed algebra, and (𝐹, 𝑞 ) 𝐴 𝐴 If (𝐴,𝐴 𝑞 ) has a minimal approximate identity {𝑒𝜆 :𝜆∈𝐼}, let 𝐹 be an essential isometric -bimodule. If has an 𝑒 (𝐴, 𝐹) ≅𝐹 𝑀(𝐴) ≅𝐴 then 𝐸 is called an essential 𝐴-module if lim𝜆𝑒𝜆𝑢=lim𝜆𝑢𝑒𝜆 = identity ,thenHom𝐴 and . 𝑢 for all 𝑢∈𝐸. Proof. We claim that Definition 10. Let (𝐴,𝐴 𝑞 ) be a commutative 𝑝-normed alge- (𝐴, 𝐹) ≅{𝐿 :𝑇∈ (𝐴, 𝐹)} bra, and let 𝐸=(𝐸,𝑞𝐸) and 𝐹=(𝐹,𝑞𝐹) be 𝑝-normed spaces Hom𝐴 𝑇(𝑒) Hom𝐴 which are also 𝐴-modules. One writes (26) ={𝐿V : V ∈𝐹}≅𝐹. Hom𝐴 (𝐸,) 𝐹 ={𝑇∈𝐶𝐿(𝐸,) 𝐹 : Clearly, 𝑇 (𝑎⋅𝑢)=𝑎⋅𝑇(𝑢) for any 𝑎∈𝐴,𝑢∈𝐸}. (18) {𝐿𝑇(𝑒) :𝑇∈Hom𝐴 (𝐴, 𝐹)} ⊆ {𝐿V : V ∈𝐹} ⊆ Hom𝐴 (𝐴, 𝐹) . If 𝐸 is an 𝐴-bimodule, then defining 𝑎∗𝑇by (27) (𝑎∗𝑇)(𝑢) =𝑇(𝑢⋅𝑎)(𝑎∈𝐴,𝑢∈𝐸) , (19) On the other hand, if 𝑇∈Hom𝐴(𝐴, 𝐹),then,forany𝑎∈𝐴, Hom𝐴(𝐸, 𝐹) becomes a left 𝐴-module. In fact, for any 𝑎, 𝑏 ∈ 𝑇 (𝑎) =𝑇(𝑒𝑎) =𝑇(𝑒) ⋅𝑎=𝐿 (𝑎) . 𝐴, 𝑢∈𝐸, 𝑇(𝑒) (28) (𝑎∗𝑇)(𝑏⋅𝑢) =𝑇((𝑏⋅𝑢) ⋅𝑎) =𝑇(𝑏⋅(𝑢⋅𝑎)) 𝑇=𝐿 ‖𝐿 ‖ = Hence 𝑇(𝑒).Further,byLemma 11, 𝑇(𝑒) 𝑞 ,𝑞 (20) 𝐴 𝐹 =𝑏⋅𝑇(𝑢⋅𝑎) =𝑏⋅(𝑎∗𝑇)(𝑢) . 𝑞𝐹(𝑇(𝑒)).ThusHom𝐴(𝐴, 𝐹) ≅𝐹.Inparticular,𝑀(𝐴) ≅ 𝐴. In particular, Hom𝐴(𝐴, 𝐹) is a left 𝐴-module. If 𝐸=𝐹=𝐴, then Hom𝐴(𝐴, 𝐴) = 𝑀(𝐴) is the usual multiplier algebra of 𝐴 : Density Assumption. In the sequel, we will always assume 𝑀 (𝐴) = {𝑇∈𝐶𝐿(𝐴,) 𝐴 :𝑇(𝑎𝑏) =𝑎𝑇(𝑏) =𝑇(𝑎) 𝑏 that, for 𝑋 a locally compact Hausdorff space and 𝐸 a (21) topological vector space, 𝐶0(𝑋) ⊗ 𝐸 is 𝑢-dense in 𝐶0(𝑋, 𝐸). ∀𝑎, 𝑏} ∈𝐴 , This assumption is crucial for the proof of our main results. which is a commutative algebra (without 𝐴 being commuta- For its justification, we mention that as a consequence of the tive) and has the identity 𝐼:𝐴, →𝐴 𝐼(𝑥) = 𝑥 (𝑥 ∈𝐴). vector-valued versions of Stone-Weierstrass theorem [8, 12, 15], 𝐶0(𝑋) ⊗ 𝐸 is 𝑢-dense in 𝐶0(𝑋, 𝐸) in each of the following Lemma 11. Let (𝐴,𝐴 𝑞 ) a commutative 𝑝-normed algebra cases. having a minimal approximate identity, and let (𝐹,𝐹 𝑞 ) be 𝐸 𝑝-normed space which is an essential isometric 𝐴-bimodule. (a) is locally convex. V ∈𝐹 Then, for any , (b) Every compact subset of 𝑋 has a finite covering 󵄩 󵄩 󵄩 󵄩 󵄩𝐿 󵄩 = 󵄩𝑅 󵄩 =𝑞 V , dimension and 𝐸 is any topological vector space. 󵄩 V󵄩𝑞 󵄩 V󵄩𝑞 𝐹 ( ) (22) 𝐹 𝐹 𝑝 (c) 𝐸 is an 𝐹-space with a basis (e.g., 𝐸=ℓ for 𝑝>0). where 𝐿V,𝑅V :𝐴 →are 𝐹 the maps given by 𝐿V(𝑎) = V ⋅𝑎and 𝑅V(𝑎) = 𝑎 ⋅ V,𝑎∈𝐴. (d) 𝐸 has the approximation property. 4 Abstract and Applied Analysis

Recall that if 𝑇∈𝑀(𝐶0(𝑋, 𝐴)),then𝑇(𝑎 ⋅ 𝑓) =𝑎⋅ By 𝑇 being linear and 𝐶0(𝑋)⊗𝐸 being assumed to be 𝑢-dense 𝑇(𝑓) for 𝑓∈𝐶0(𝑋, 𝐴) and 𝑎∈𝐴([16,Lemma4.5]). in 𝐶0(𝑋, 𝐸),itfollowsthat𝑇(𝑎 ⋅ 𝑓) = 𝑎 ⋅𝑇(𝑓) holds for all We also mention that if (𝐴,𝐴 𝑞 ) is an 𝑝-normed algebra 𝑓∈𝐶0(𝑋, 𝐴) and 𝑎∈𝐴. having a minimal approximate identity, then, by ([16,Lemma (b) Similar to the above part. 4.4]), 𝐶0(𝑋, 𝐴) has an approximate identity and hence it is afaithfultopological𝐴-module. Consequently, for any 𝑇∈ We now give the following characterization in the pseu- 𝐶0(𝑋) 𝐶0(𝑋, 𝐹) 𝑀(𝐶0(𝑋, 𝐴)), 𝑇(𝑓𝑔) = 𝑓𝑇(𝑔) =𝑇(𝑓)𝑔 for all 𝑓, 𝑔 ∈ doscaler case by considering both and as 𝐶0(𝑋) 𝐶0(𝑋, 𝐴);wewillwrite -modules. 󵄩 󵄩 Theorem 15. 𝑋 ‖𝑇‖𝑞 := sup {𝑞𝐴 (𝑇 (𝑓)) :0 𝑓∈𝐶 (𝑋, 𝐴) , 󵄩𝑓󵄩𝑞 ,∞ ≤1}. Let be a locally compact Hausdorff space and 𝐴 𝐴 𝐹=(𝐹,𝑞 ) 𝑝 (29) 𝐹 a -normed space. Then (𝐶 (𝑋) ,𝐶 (𝑋, 𝐹))≅𝐶 (𝑋, 𝐹) . 𝑇∈ (𝐶 (𝑋, 𝐸), 𝐶 (𝑋, 𝐹)) Hom𝐶0(𝑋) 0 0 𝑏 (35) If Hom𝐶0(𝑋,𝐴) 0 0 ,welet 󵄩 󵄩 󵄩 󵄩 𝑇∈ 𝐶 (𝑋)(𝐶0(𝑋),0 𝐶 (𝑋, 𝐹)) 𝑥∈𝑋 ‖𝑇‖𝑞 ,𝑞 := {𝑞𝐹 (𝑇 (𝑓)) :0 𝑓∈𝐶 (𝑋, 𝐸) , 󵄩𝑓󵄩 ≤1}. Proof. Let Hom 0 and .If 𝐸 𝐹 sup 󵄩 󵄩𝑞𝐸,∞ 𝜑, 𝜓0 ∈𝐶 (𝑋) with 𝜑(𝑥) =0̸ and 𝜓(𝑥) =0̸ , then there is a (30) neighborhood 𝑁(𝑥) of 𝑥 in 𝑋 such that Definition 13. Now, let 𝐸=(𝐸,𝑞𝐸) and 𝐹=(𝐹,𝑞𝐹) be 𝐹- 𝜑 (𝑡) =0,̸ 𝜓(𝑡) =0̸ for any 𝑡∈𝑁(𝑥) . (36) normed spaces. For any closed subspace 𝑈=𝑈𝑠(𝐸, 𝐹) of 𝐶𝐿(𝐸, 𝐹) endowed with the strong operator topology 𝑠,we Since 𝐶0(𝑋) is commutative and 𝐶0(𝑋, 𝐹) is a 𝐶0(𝑋)-module, define following as in ([1, p. 1135]), we have 𝐶𝑠,𝑏 (𝑋, 𝑈) ={𝐺:𝑋󳨀→𝑈: 𝜓 (𝑡) (𝑇𝜑) (𝑡) =𝑇(𝜓⋅𝜑)(𝑡) =𝑇(𝜑⋅𝜓)(𝑡) 𝐺 is strongly continuous and bounded}. (37) =𝜑(𝑡) (𝑇𝜓) (𝑡) (31)

We now define an 𝐹-norm on 𝐶𝑠,𝑏(𝑋, 𝑈) by and then ‖𝐺‖ = ‖𝐺 (𝑥)‖ = 𝑞 (𝐺 (𝑥)(𝑢)) . 𝑇(𝜓)(𝑡) (𝑇𝜑) (𝑡) 𝐶𝑠,𝑏 sup 𝑞𝐸,𝑞𝐹 sup sup 𝐹 𝑥∈𝑋 𝑥∈𝑋 𝑢∈𝐸,𝑞 (𝑢)≤1 = for any 𝑡∈𝑁(𝑥) . (38) 𝐸 𝜓 (𝑡) 𝜑 (𝑡) (32) Now, for each 𝑥∈𝑋with 𝜑(𝑥) =0̸ , define 𝑔𝑇 :𝑋 →by 𝐹 Then 𝐶𝑠,𝑏(𝑋, 𝑈) is a complete 𝑝-normed space under the 𝑝- norm ‖⋅‖𝑞,∞ definedin(24). (𝑇𝜑) (𝑥) 𝑔 (𝑥) = . 𝑇 𝜑 (𝑥) (39) Recall that a left 𝐴-module 𝐸 is called faithful (or without order)if,forany𝑢∈𝐸, 𝑎⋅𝑢 =0for all 𝑎∈𝐴implies that Bytheaboveargument,thefunction𝑔𝑇(𝑥) defined in this way 𝑥=0(cf. [13, 14]). is independent of the choice of 𝜑∈𝐶0(𝑋);hence𝑔𝑇 is well- defined. Lemma 14. Let 𝐴=(𝐴,𝑞𝐴) be a commutative complete 𝑝- Clearly if 𝜑(𝑥) =0̸ ,then(𝑇𝜑)(𝑥)𝑇 =𝑔 (𝑥)𝜑(𝑥).The normed algebra, and let 𝐸 and 𝐹 be 𝐴-modules. Then, for any equality also holds when 𝜑(𝑥). =0 [To see this, choose 𝑇∈Hom𝐶 (𝑋,𝐴)(𝐶0(𝑋, 𝐸),0 𝐶 (𝑋, 𝐹)), 0 𝜓∈𝐶0(𝑋) such that 𝜓(𝑥) =0̸ .Then (a) 𝑇(𝑎 ⋅ 𝑓) = 𝑎 ⋅𝑇(𝑓) for 𝑎∈𝐴and 𝑓∈𝐶0(𝑋, 𝐸), 𝜓 (𝑥) (𝑇𝜑) (𝑥) =𝑇(𝜓𝜑) (𝑥) =𝜑(𝑥) (𝑇𝜓) (𝑥) =0, (40) (b) 𝑇(𝜑 ⋅ 𝑓) = 𝜑 ⋅𝑇(𝑓) for 𝜑∈𝐶0(𝑋) and 𝑓∈𝐶0(𝑋, 𝐸). and so 𝑇𝜑(𝑥).] =0 Proof. (a) We first note that 𝐶0(𝑋) is a Banach algebra with a Next, 𝑔𝑇 ∈𝐶𝑏(𝑋, 𝐹), as follows. For any 𝑥∈𝑋with bounded approximate identity, {𝜓𝛼} (say). Then, for any 𝑎∈ 𝜑(𝑥) =0̸ , by Urysohn’s lemma, we can choose a 𝜑∈𝐶0(𝑋) 𝐴,𝑢∈𝐸,and𝜑∈𝐶0(𝑋), such that ‖𝜑‖∞ =|𝜑(𝑥)|.So [(𝜓 ⊗𝑎)⋅(𝜑⊗𝑢)]= (𝜓 𝜑⊗𝑎⋅𝑢) lim𝛼 𝛼 lim𝛼 𝛼 󵄩 󵄩 𝑞 [𝑇𝜑 (𝑥)] ‖𝑇‖𝑞 󵄩𝜑󵄩 (33) 𝑞 [𝑔 (𝑥)]= 𝐹 ≤ 𝐹 ∞ = ‖𝑇‖ 𝐹 𝑇 󵄨 󵄨 󵄨 󵄨 𝑞𝐹 (41) =𝜑⊗𝑎⋅𝑢=𝑎(𝜑⊗𝑢). 󵄨𝜑 (𝑥)󵄨 󵄨𝜑 (𝑥)󵄨 𝑇∈ (𝐶 (𝑋, 𝐸), 𝐶 (𝑋, 𝐹)) 𝜓 ⊗𝑎 ∈ 𝑥∈𝑋 ‖𝑔 ‖ ≤‖𝑇‖ 𝑔 ∈𝐶(𝑋, 𝐹) Since Hom𝐶0(𝑋,𝐴) 0 0 and 𝛼 for all .Hence 𝑇 𝑞,∞ 𝑞 ,andso 𝑇 𝑏 . 𝐶 (𝑋, 𝐴) 𝜑⊗𝑢∈𝐶(𝑋, 𝐸) 𝐹 0 , 0 ,wehave On the other hand, since 󵄩 󵄩 󵄩 󵄩 𝑇 (𝑎 ⋅ (𝜑 ⊗ 𝑢))= lim𝑇[(𝜓𝛼 ⊗𝑎)⋅(𝜑⊗𝑢)] 𝑞 [(𝑇𝜑) 𝑥 ]=𝑞 [𝑔 𝑥 𝜑 𝑥 ]≤󵄩𝑔 󵄩 󵄩𝜑󵄩 , 𝛼 𝐹 ( ) 𝐹 𝑇 ( ) ( ) 󵄩 𝑇󵄩𝑞,∞󵄩 󵄩∞ (42)

= lim (𝜓𝛼 ⊗𝑎)⋅𝑇(𝜑⊗𝑢) (34) ‖𝑇‖ ≤‖𝑔‖ ‖𝑔 ‖ =‖𝑇‖ 𝛼 we have 𝑞𝐹 𝑇 𝑞,∞.Consequently 𝑇 𝑞𝐹,∞ 𝑞𝐹 . This shows that Hom𝐶 (𝑋)(𝐶0(𝑋),0 𝐶 (𝑋, 𝐹)) is isometrically =𝑎⋅𝑇(𝜑⊗𝑢). 0 embedded in 𝐶𝑏(𝑋, 𝐹). Abstract and Applied Analysis 5

Conversely, for any 𝑔∈𝐶𝑏(𝑋, 𝐹), we define 𝑇𝑔 : or 𝐶0(𝑋) →0 𝐶 (𝑋, 𝐹) by 𝐺 (𝑥)(𝑎⋅𝑢) =𝑎⋅𝐺(𝑥)(𝑢) . (52) 𝑇𝑔 (𝜑) =𝑔⋅𝜑,𝜑∈𝐶0 (𝑋) . (43) This implies that 𝐺(𝑥) ∈ Hom 𝐴(𝐸, 𝐹), and hence 𝐺∈ Then one can easily show that 𝑇𝑔 is a multiplier from 𝐶𝑠,𝑏(𝑋, Hom 𝐴(𝐸, 𝐹)). Next we establish isometry between 𝑇 𝐶0(𝑋) to 𝐶0(𝑋, 𝐹) and that ‖𝑔‖𝑞,∞ =‖𝑇𝑔‖ . and 𝐺.For𝑥∈𝑋and 𝜑⊗𝑢0 ∈𝐶 (𝑋)⊗𝐸 with ‖𝜑 ⊗ 𝑢‖𝑞 ,∞ ≤1, 𝑞𝐹 𝐸 ‖𝐺(𝑥)‖ = 𝑞 [𝐺 (𝑥)(𝑢)] = 𝑞 [𝑔 (𝑥)] Now we can establish the main theorem by considering 𝑞𝐸,𝑞𝐹 sup 𝐹 sup 𝐹 𝑢 𝑞 (𝑢)≤1 𝑞 (𝑢)≤1 both 𝐶0(𝑋, 𝐸) and 𝐶0(𝑋, 𝐹) as 𝐶0(𝑋, 𝐴)-modules. 𝐸 𝐸 󵄩 󵄩 󵄩 󵄩 Theorem 16. 𝐴=(𝐴,𝑞 ) 𝑝 ≤ sup 󵄩𝑔𝑢󵄩𝑞 ,∞ = sup 󵄩𝑔𝑢 ⋅𝜑󵄩𝑞 ,∞ Let 𝐴 be a commutative complete - 𝑞 (𝑢)≤1 𝐹 𝑞 (𝑢)≤1 𝐹 𝐸=(𝐸,𝑞) 𝐹=(𝐹,𝑞) 𝑝 𝐸 𝐸 (53) normed algebra, and let 𝐸 and 𝐹 be - ‖𝜑‖∞≤1 normed spaces which are also essential isometric 𝐴-modules. 󵄩 󵄩 = 󵄩𝑇(𝜑⊗𝑢)󵄩 = ‖𝑇‖𝑞 ,𝑞 , Then sup 󵄩 󵄩𝑞𝐹,∞ 𝐸 𝐹 ‖𝜑⊗𝑢‖ ≤1 (𝐶 (𝑋, 𝐸) ,𝐶 (𝑋, 𝐹))≅𝐶 (𝑋, (𝐸,) 𝐹 ) . 𝑞𝐸,∞ Hom 𝐶0(𝑋,𝐴) 0 0 𝑠,𝑏 Hom 𝐴 𝐶 (𝑋) ⊗ 𝐸 𝑢 𝐶 (𝑋, 𝐸) ‖𝐺‖ ≤‖𝑇‖ (44) since 0 is -dense in 0 .So 𝐶𝑠,𝑏 𝑞𝐸,𝑞𝐹 . But 𝑇 The correspondence between the multiplier and the function 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑇(𝜑⊗𝑢)󵄩 = 󵄩𝑔𝑢 ⋅𝜑󵄩 ≤ 󵄩𝑔𝑢󵄩 󵄩𝜑󵄩 𝐺 is given by the following relation: 󵄩 󵄩𝑞𝐹,∞ 󵄩 󵄩𝑞𝐹,∞ 󵄩 󵄩𝑞𝐹,∞󵄩 󵄩∞ 󵄩 󵄩 󵄩 󵄩 (𝑇𝑓) (𝑥) =𝐺(𝑥) ⋅𝑓(𝑥) ≤ ‖𝐺‖𝐶 ‖𝑢‖ 󵄩𝜑󵄩 = ‖𝐺‖𝐶 󵄩𝜑⊗𝑢󵄩 𝑠,𝑏 󵄩 󵄩∞ 𝑠,𝑏 󵄩 󵄩𝑞𝐸,∞ (45) (54) 𝑓𝑜𝑟 𝑥 ∈ 𝑋 𝑎𝑛𝑑 𝑎𝑛𝑦𝑓∈𝐶 0 (𝑋, 𝐸) . 𝜑⊗𝑢∈𝐶 (𝑋) ⊗ 𝐸 ‖𝑇‖ ≤‖𝐺‖ for all 0 .Consequently, 𝑞𝐸,𝑞𝐹 𝐶𝑠,𝑏 . Proof. Let 𝑇∈Hom𝐶 (𝑋,𝐴)(𝐶0(𝑋, 𝐸),0 𝐶 (𝑋, 𝐹)).Thenwecan 0 Conversely, let 𝐺∈𝐶𝑠,𝑏(𝑋, Hom 𝐴(𝐸, 𝐹)) and 𝜑∈𝐶0(𝑋). define a map Ψ𝑇 :𝐸 → Hom𝐶 (𝑋) (𝐶0(𝑋),0 𝐶 (𝑋, 𝐹)) by 0 Then 𝐺⋅𝜑is a continuous function on 𝑋 given by Ψ (𝑢) (𝜑) =𝑇(𝜑⊗𝑢) 𝑢∈𝐸,𝜑∈𝐶 (𝑋) . 𝑇 for 0 (46) (𝐺⋅𝜑) (𝑥)(𝑢) = (𝐺 (𝑥) 𝑢) 𝜑 (𝑥) ,𝑥∈𝑋,𝑢∈𝐸. (55) Ψ (𝑢)(𝜑) ∈ To see that this map is well-defined, first note that 𝑇 It is easy to see that 𝐺⋅𝜑vanishes at infinity, and so 𝐺⋅𝜑∈ 𝐶0(𝑋, 𝐹).Forafixed𝑢∈𝐸,theoperatorΦ𝑇(𝑢) defines a 𝐶0(𝑋, Hom 𝐴(𝐸, 𝐹)).Forany𝑢∈𝐸and 𝜑∈𝐶0(𝑋), 𝐺 bounded linear operator from 𝐶0(𝑋) into 𝐶0(𝑋, 𝐹),sinceby determines a bounded linear operator 𝑇 from 𝐶0(𝑋, 𝐸) to (46), 𝐶0(𝑋, 𝐹) given by 󵄩 󵄩 󵄩 󵄩 󵄩Ψ𝑇 (𝑢) (𝜑)󵄩𝑞 ,∞ = 󵄩𝑇(𝜑⊗𝑢)󵄩𝑞 ,∞ 𝐸 𝐸 𝑇 (𝜑⊗𝑢) (𝑥) = (𝐺 (𝑥) 𝑢) 𝜑 (𝑥) . (56) (47) 󵄩 󵄩 ≤ ‖𝑇‖𝑞 ⋅ 󵄩𝜑⊗𝑎󵄩 ; 𝐸 󵄩 󵄩𝑞𝐸,∞ Again, since 𝐶0(𝑋) ⊗ 𝐸 is 𝑢-dense in 𝐶0(𝑋, 𝐸),itfollowsthat ‖𝑇‖ =‖𝐺‖ 𝑞𝐸,𝑞𝐹 𝐶𝑠,𝑏 . further, it is a multiplier since, for any 𝜑, 𝜓0 ∈𝐶 (𝑋), Since 𝐸 and 𝐹 are 𝐴-modules, for any ℎ⊗𝑎∈𝐶0(𝑋) ⊗ 𝐴 and 𝜑⊗𝑢∈𝐶0(𝑋) ⊗ 𝐸, Ψ𝑇 (𝑢) (𝜑𝜓) =𝑇(𝜑𝜓 ⊗𝑢) =𝜑⋅𝑇(𝜓⊗𝑢) . (48) 𝑇((ℎ⊗𝑎) ⋅ (𝜑 ⊗ 𝑢)) = 𝑇 (ℎ𝜑 ⊗𝑎𝑢) Hence Ψ𝑇(𝑢) ∈ Hom 𝐶 (𝑋)(𝐶0(𝑋),0 𝐶 (𝑋, 𝐹)).ByTheorem 15, 0 𝑔 𝐶 (𝑋, 𝐹) there exists an element, say 𝑢,in 𝑏 such that =𝐺(⋅)(𝑎⋅𝑢) (ℎ𝜑) (⋅) Ψ 𝑢 (𝜑) = 𝑔 ⋅𝜑, 𝑢∈𝐸,𝜑∈𝐶 𝑋 . (57) 𝑇 ( ) 𝑢 for 0 ( ) (49) =𝑎⋅ℎ(⋅) 𝐺 (⋅)(𝑢) 𝜑 (⋅) 𝐺:𝑋 → (𝐸, 𝐹) Now, we can define a map Hom 𝐴 by = (ℎ⊗𝑎) ⋅𝑇(𝜑⊗𝑢).

𝐺 (𝑥)(𝑢) =𝑔𝑢 (𝑥) for 𝑥∈𝑋,𝑢∈𝐸. (50) Hence 𝑇 is a multiplier on 𝐶0(𝑋, 𝐸) since 𝐶0(𝑋)⊗𝐸 is 𝑢-dense 𝐶 (𝑋, 𝐸) 𝐺 𝑇 To see that this map is well-defined, first note that, for a fixed in 0 . The isometry between and now implies that 𝑥∈𝑋, 𝐺(𝑥) is a linear operator from 𝐸 into 𝐹.Moreover,for (𝐶 (𝑋, 𝐸) ,𝐶 (𝑋, 𝐹)) ≅𝐶 (𝑋, (𝐸,) 𝐹 ) . Hom𝐶0(𝑋,𝐴) 0 0 𝑠,𝑏 Hom𝐴 𝑎∈𝐴and 𝜑∈𝐶0(𝑋),wehave (58)

𝐺 (𝑥)(𝑎⋅𝑢) ⋅𝜑(𝑢) =𝑔𝑎𝑢 (𝑥) 𝜑 (𝑥) =𝑇(𝜑⊗𝑎⋅𝑢)(𝑥) =𝑎⋅𝑇(𝜑⊗𝑢)(𝑥) =𝑎⋅𝑔 (𝑥) 𝜑 (𝑥) 𝑢 4. Applications =𝑎⋅𝐺(𝑥)(𝑢) 𝜑 (𝑥) , As an application of the above results, in particular of (51) Theorem 16, we can deduce several known results, as follows. 6 Abstract and Applied Analysis

Corollary 17 (see [3]). Let 𝑋 be a locally compact Hausdorff References space and 𝐴 = (𝐴, || ⋅ ||) a commutative Banach algebra, and let 𝐸 and 𝐹 be Banach 𝐴-modules. Then [1] J.-K. Wang, “Multipliers of commutative Banach algebras,” Pacific Journal of Mathematics, vol. 11, pp. 1131–1149, 1961. Hom 𝐶 (𝑋,𝐴)(𝐶0 (𝑋, 𝐸) ,𝐶0 (𝑋, 𝐹))≅𝐶𝑠,𝑏 (𝑋, Hom 𝐴 (𝐸,) 𝐹 ) . [2] C. A. Akemann, G. K. Pedersen, and J. Tomiyama, “Multipliers 0 𝐶∗ (59) of -algebras,” Journal of Functional Analysis,vol.13,pp.277– 301, 1973. Corollary 18 (see [3, 5]). Let 𝑋 be a locally compact Hausdorff [3] J. C. Candeal Haro and H. C. Lai, “Multipliers in continuous space and 𝐴 = (𝐴, ‖ ⋅‖) be a commutative Banach algebra with vector-valued function spaces,” Bulletin of the Australian Math- identity of norm 1,andlet𝐸 be a Banach 𝐴-module. Then ematical Society,vol.46,no.2,pp.199–204,1992. [4] H. C. Lai, “Multipliers of a Banach algebra in the second (𝐶 (𝑋, 𝐸) ,𝐶 (𝑋, 𝐸))≅𝐶 (𝑋, 𝐸) . conjugate algebra as an idealizer,” The Tohoku Mathematical Hom 𝐶0(𝑋,𝐴) 0 0 𝑏 (60) Journal,vol.26,pp.431–452,1974. Corollary 19 (see [16]). Let 𝑋 be a locally compact Hausdorff [5] H. C. Lai, “Multipliers for some spaces of Banach algebra valued space and 𝐴 = (𝐴, 𝑞) a commutative complete 𝑝-normed functions,” The Rocky Mountain Journal of Mathematics,vol.15, algebra with a minimal approximate identity. Then no. 1, pp. 157–166, 1985. [6] H. C. Lai, “Multipliers of Banach valued function spaces,” 𝑀(𝐶0 (𝑋, 𝐴))≅𝐶𝑠,𝑏 (𝑋,( 𝑀 𝐴)𝑢). (61) Australian Mathematical Society A,vol.39,no.1,pp.51–62,1985. [7] A. Mallios, Topological Algebras-Selected Topics,vol.124of Proof. This follows from the fact that Hom 𝐴(𝐴, 𝐴) = 𝑀(𝐴). North-Holland Mathematics Studies,North-Holland,NewYork, NY, USA, 1986. [8] A. H. Shuchat, “Approximation of vector-valued continuous Corollary 20 𝑋 (see [1]). Let be a locally compact Hausdorff functions,” Proceedings of the American Mathematical Society, space. Then vol.31,pp.97–103,1972. 𝑀 (𝐶 (𝑋)) ≅𝐶 (𝑋) . [9] G. Kothe,¨ Topological Vector Spaces I, Springer, New York, NY, 0 𝑏 (62) USA, 1969. [10] S. Rolewicz, Metric Linear Spaces,D.Reidel,Dordrecht,The Proof. ThisfollowsfromthefactthatHom𝐶 (𝑋)(𝐶0(𝑋), 0 Netherlands, 1985. 𝐶0(𝑋)) ≅𝑏 𝐶 (𝑋). [11] M. Adib, A. H. Riazi, and L. A. Khan, “Quasimultipliers on F- algebras,” Abstract and Applied Analysis, vol. 2011, Article ID Example 21. Let 𝐴𝑝,0<𝑝≤1, denote the algebra of all 235273, 30 pages, 2011. holomorphic functions in the unit disc 𝐷={𝑧∈C :|𝑧|≤1}: [12] L. A. Khan, Linear Topological Spaces of Continuous Vector- ∞ Valued Functions, Academic Publications, 2013. 𝑛 𝜑 (𝑧) = ∑𝑎𝑛𝑧 ,𝑧∈𝐷, (63) [13] M. A. Rieffel, “Induced Banach representations of Banach 𝑛=0 algebras and locally compact groups,” Journal of Functional Analysis,vol.1,pp.443–491,1967. for which [14] L. A. Khan, “Topological modules of continuous homomor- ∞ phisms,” Journal of Mathematical Analysis and Applications,vol. 󵄩 󵄩 󵄨 󵄨𝑝 󵄩𝜑󵄩𝑝 = ∑󵄨𝑎𝑛󵄨 <∞. (64) 343, no. 1, pp. 141–150, 2008. 𝑛=0 [15] M. Abel, “Topological bimodule-algebras,” in Proceedings of 𝑝 the 3rd International Conference on Topological Algebra and This is a commutative complete -normed algebra with the Applications,pp.25–42,Oulu,Finland,2004. pointwise multiplication and has an identity ([7,p.135];[17, [16] L. A. Khan and S. M. Alsulami, “Multipliers of commutative F- p. 8]). In this case, algebras ofcontinuous vector-valued functions,” Bulletin of the Malaysian Mathematical Sciences Society.Inpress. 𝑀 (𝐶 (𝑋, 𝐴 )) ≃𝐶 (𝑋, 𝑀(𝐴 ) ) ≃𝐶 (𝑋, 𝐴 ) . 0 𝑝 𝑏 𝑝 𝑠 𝑏 𝑝 (65) [17] W. Zelazko,˙ “Metric generalizations of Banach algebras,” Disser- tations Mathematics,vol.47,70pages,1965. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under the Project no. 3-059/429. The authors therefore acknowledge with thanks DSR technical and financial support. The authors are also grateful to the referee for his many useful suggestions to improve the paper. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 174010, 8 pages http://dx.doi.org/10.1155/2014/174010

Research Article Lipschitz Spaces and Fractional Integral Operators Associated with Nonhomogeneous Metric Measure Spaces

Jiang Zhou and Dinghuai Wang

Department of Mathematics, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Jiang Zhou; [email protected]

Received 1 December 2013; Revised 3 April 2014; Accepted 5 April 2014; Published 17 April 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 J. Zhou and D. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑝 The fractional operator on nonhomogeneous metric measure spaces is introduced, which is a bounded operator from 𝐿 (𝜇) into 𝑞,∞ the space 𝐿 (𝜇). Moreover, the Lipschitz spaces on nonhomogeneous metric measure spaces are also introduced, which contain the classical Lipschitz spaces. The authors establish some equivalent characterizations for the Lipschitz spaces, and some results of the boundedness of fractional operator in Lipschitz spaces are also presented.

1. Introduction Let (X,𝑑,𝜇)be a nonhomogeneous metric measure space in the sense of Hytonen¨ [14]. In this paper, we establish As we know, the theory on spaces of homogeneous type is the definition of fractional operator on nonhomogeneous needed to assume that measure 𝜇 of metric spaces (X,𝑑,𝜇) metric measure spaces, which contains the classical fractional satisfies the doubling measure condition, which means that integral operator introduced by Garc´ıa-Cuerva and Gatto [7], there exists a constant 𝐶,suchthat,foreveryball𝐵(𝑥, 𝑟) of 𝑥 𝑟 𝜇(𝐵(𝑥, 2𝑟)) ≤ 𝐶𝜇(𝐵(𝑥, 𝑟)) and similar to the definition introduced by Fu et al.11 [ ], center and radius , .Inrecent (𝐿𝑝(𝜇),𝑞,∞ 𝐿 (𝜇)) years, many classical theories have been proved still valid then we get the -boundedness for frac- without the assumption of doubling measure condition; see tional integral operator on nonhomogeneous metric measure 𝑑 spaces. In Section 3, we also establish the definition of Lip- [1–12]. Recall that a Radon measure 𝜇 on 𝑅 is said to only schitz spaces on nonhomogeneous metric measure spaces, satisfy the polynomial growth condition, if there exists a 𝑑 which contains the classical Lipschitz spaces. We establish positive constant 𝑐 such that, for all 𝑥∈R and 𝑟>0, 𝑛 some equivalent characterizations for the Lipschitz spaces. 𝜇(𝐵(𝑥, 𝑟)) ≤𝑐𝑟 ,where𝑛 is some fixed number in (0, 𝑑] and In Section 4, we present some results of the boundedness of 𝐵(𝑥, 𝑟) = {𝑦∈ R𝑑 : |𝑥−𝑦|< 𝑟} .Theanalysisassociated fractional operator in Lipschitz spaces. with such nondoubling measures 𝜇 is proved to play a striking To state the main results of this paper, we first recall some role in solving the long-standing open Painleve’s´ problem necessary notions and remarks. by Tolsa [13]. Obviously, the nondoubling measure 𝜇 with the polynomial growth condition may not satisfy the well- (X,𝑑,𝜇) known doubling condition, which is a key assumption in Definition 1 (see [15]). A metric space is said to be 𝑁 ∈ N harmonic analysis on spaces of homogeneous type. In 2010, geometrically doubling if there exists some 0 such that, 𝐵(𝑥, 𝑟) ⊂ X Hytonen¨ [14] introduced a new class of metric measure spaces for any ball , there exist a finite ball covering {𝐵(𝑥 , 𝑟/2)} 𝐵(𝑥, 𝑟) satisfying both the so-called geometrically doubling and the 𝑖 𝑖 of such that the cardinality of this cov- 𝑁 upper doubling conditions (see the definition below), which ering is at most 0. are called nonhomogeneous spaces. Recently, many classical results have been proved still valid if the underlying spaces Definition 2 (see [14]). A metric measure space (X,𝑑,𝜇) is are replaced by the nonhomogeneous spaces of Hytonen¨ (see said to be upper doubling if 𝜇 is a Borel measure on X and [4–6, 9–12]). there exist a dominating function 𝜆:X × (0, ∞) → (0, ∞) 2 Abstract and Applied Analysis

and a positive constant 𝑐𝜆 such that, for each 𝑥∈X, 𝑟→ Definition 7 (see [14]). For any two balls 𝐵⊂𝑆, define 𝜆(𝑥, 𝑟) is nondecreasing and 1 𝑟 𝐾𝐵,𝑆 =1+∫ 𝑑𝜇 (𝑥) , (6) 𝜇 (𝐵 (𝑥,)) 𝑟 ≤𝜆(𝑥,) 𝑟 ≤𝑐𝜆(𝑥, )∀𝑥∈X,𝑟>0. 2𝑆\𝐵 𝜆(𝑐𝐵,𝑑(𝑥,𝑐𝐵)) 𝜆 2 (1) where 𝑐𝐵 is the center of the ball 𝐵. A metric measure space (X,𝑑,𝜇) is called a nonhomo- ̃ Remark 8. The following discrete version, 𝐾𝐵,𝑆,of𝐾𝐵,𝑆 geneous metric measure space if (X,𝑑,𝜇) is geometrically defined in Definition,wasfirstintroducedbyBuiand 7 doubling and (X,𝑑,𝜇)is upper doubling. Duong [4] in nonhomogeneous metric measure spaces, 𝐾 Remark 3. (i) Obviously, a space of homogeneous type is which is more close to the quantity 𝑄,𝑅 introduced by Tolsa a special case of upper doubling spaces, where we take the [1] in the setting of nondoubling measures. For any two balls 𝐵⊂𝑆 𝐾̃ dominating function 𝜆(𝑥, 𝑟) := 𝜇(𝐵(𝑥,. 𝑟)) On the other hand, ,let 𝐵,𝑆 be defined by R𝑑 𝜇 the Euclidean space with any Radon measure as in (1) 𝑁 𝑘 𝐵,𝑆 𝜇(6 𝐵) is also an upper doubling space by taking the dominating 𝐾̃ := 1 + ∑ , (7) 𝜆(𝑥, 𝑟) :=𝑘 𝐶𝑟 𝐵,𝑆 𝑘 function . 𝑘=1 𝜆(𝑐𝐵,6𝑟 𝑟𝐵) (ii) Let (X,𝑑,𝜇) be upper doubling with 𝜆 being the X ×(0,∞) dominating function on as in Definition.Itwas 2 where 𝑟𝐵 and 𝑟𝑆, respectively, denote the radii of the balls ̃ proved in [6] that there exists another dominating function 𝜆 𝐵 and 𝑆,and𝑁𝐵,𝑆 denotes the smallest integer satisfying ̃ 𝑁𝐵,𝑆 ̃ such that 𝜆≤𝜆and, for all 𝑥, 𝑦 ∈ X with 𝑑(𝑥, 𝑦), ≤𝑟 6 𝑟𝐵 ≥𝑟𝑆.Obviously,𝐾𝐵,𝑆 ≤𝐶𝐾𝐵,𝑆.AswaspointedbyBui 𝐾 ∼𝐶𝐾̃ ̃ ̃ and Duong [4], in general, it is not true that 𝐵,𝑆 𝐵,𝑆. 𝜆 (𝑥,) 𝑟 ≤𝐶𝜆̃𝜆(𝑦,𝑟). (2) 1 Definition 9 (see [14]). Let 𝜌 ∈ (1, ∞).Afunction𝑓∈𝐿 (𝜇) Thus, in this paper, we always suppose that 𝜆 satisfies (2). loc is said to be in the space RBMO(𝜇) if there exist a positive constant 𝐶,andforanyball𝐵⊂X,anumber𝑓𝐵 such that Definition 4 (see [14]). Let 𝛼,𝛼 𝛽 ∈(1,∞).Aball𝐵⊂𝑋is (𝛼, 𝛽) 𝜇(𝛼𝐵) ≤𝛽 𝜇(𝐵) called -doubling if 𝛼 . 1 󵄨 󵄨 ∫ 󵄨𝑓 (𝑥) −𝑓󵄨 𝑑𝜇 (𝑥) ≤𝐶, As stated in lemma of [4], there exist plenty of doubling 𝜇(𝜌𝐵) 󵄨 𝐵󵄨 (8) balls with small radii and with large radii. In the rest of the 𝐵 paper, unless 𝛼 and 𝛽𝛼 are specified otherwise, by an (𝛼,𝛼 𝛽 )- for any two balls 𝐵 and 𝐵1 such that 𝐵⊂𝐵1, doubling ball we mean a (6, 𝛽6)-doubling with a fixed number 3log 6 2 𝑛 󵄨 󵄨 𝛽6 > max{𝐶𝜆 ,6 },where𝑛=log2𝑁0 is viewed as a geo- 󵄨 󵄨 󵄨𝑓𝐵 −𝑓𝐵 󵄨 ≤𝐶𝐾𝐵,𝐵 . (9) metric dimension of the spaces. 󵄨 1 󵄨 1 𝐶 Definition 5 (see [11]). Let 𝜖 ∈ (0, ∞). A dominating function The infimum of the positive constants satisfying above 𝜆 𝜖 two inequalities is defined to be the RBMO(𝜇) norm of 𝑓 and is satisfying the -weak reverse doubling condition if, for ‖𝑓‖ all 𝑟∈(0,2diam(X)) and 𝑎∈(1,2diam(X)/𝑟), there exists denoted by RBMO(𝜇). (𝜇) anumber𝐶(𝑎) ∈ [1, ∞), depending only on 𝑎 and X,such From [14, Lemma 4.6], it follows that the space RBMO 𝜌 ∈ (1, ∞) that, for all 𝑥∈X, is independent of . In this paper, we consider a variant of the fractional 𝜆 (𝑥, )𝑎𝑟 ≥𝐶(𝑎) 𝜆 (𝑥,) 𝑟 (3) integrals from [7, Definition 4.1]. and, moreover, Definition 10. Let 0<𝛼<𝑛and 0<𝛿≤1.Afunction𝐾𝛼 ∈ 1 𝐿 (X × X \{(𝑥,𝑦):𝑥=𝑦})is said to be a fractional kernel ∞ 1 loc ∑ <∞. of order 𝛼 and regularity 𝛿 if it satisfies the following two 𝑘 𝜖 (4) 𝑘=1 [𝐶(𝑎 )] conditions: 𝑥, 𝑦 ∈ X 𝑥 =𝑦̸ Remark 6. (i) It is easy to see that if 𝜖1 <𝜖2 and 𝜆 satisfies the (i) for all with , 𝜖1-weak reverse doubling condition, then 𝜆 also satisfies the 󵄨 󵄨 1 𝜖2-weak reverse doubling condition. 󵄨𝐾 (𝑥, 𝑦)󵄨 ≤𝐶 ; 󵄨 𝛼 󵄨 1−𝛼/𝑛 (10) (ii) Assume that diam(X)=∞.Foranyfixed𝑥∈X,we [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] know that 𝜆 (𝑥,) 𝑟 =0, 𝜆 (𝑥,) 𝑟 =∞. (ii) for all 𝑥, 𝑥,̃ 𝑦 ∈ X with 𝜆(𝑥, 𝑑(𝑥, 𝑦)) ≥ 2𝜆(𝑥, 𝑑(𝑥, 𝑥))̃ , 𝑟→0lim 𝑟→∞lim (5) 󵄨 󵄨 󵄨 󵄨 󵄨𝐾 (𝑥, 𝑦) −𝐾 (𝑥,̃ 󵄨𝑦) + 󵄨𝐾 (𝑦, 𝑥) −𝐾 (𝑦, 𝑥)̃ 󵄨 (ii) It is easy to see that the 𝜖-weak reverse doubling con- 󵄨 𝛼 𝛼 󵄨 󵄨 𝛼 𝛼 󵄨 dition is much weaker than the assumption introduced by Bui [𝜆 (𝑥, 𝑑 (𝑥, 𝑥̃))]𝛿/𝑛 (11) and Duong in [4,Subsection7.3]:thereexists𝑚 ∈ (0, ∞) such ≤𝐶 . 𝑚 1−𝛼/𝑛+𝛿/𝑛 that, for all 𝑥∈X and 𝑎, 𝑟 ∈ (0,, ∞) 𝜆(𝑥, 𝑎𝑟) =𝑎 𝜆(𝑥,. 𝑟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] Abstract and Applied Analysis 3

A linear operator 𝐼𝛼 is called fractional integral operator The second main result of this paper is the following some with 𝐾𝛼 satisfying (10)and(11), equivalent characterizations for the Lipschitz spaces.

1 Theorem 16. For a function 𝑓∈𝐿𝑙𝑜𝑐(𝜇), the conditions (A), 𝐼𝛼𝑓 (𝑥) := ∫ 𝐾𝛼 (𝑥, 𝑦) 𝑓 (𝑦) 𝑑𝜇 (𝑦). (12) X (B),and(C) are equivalent as follows. 𝑛 𝐶 Remark 11. By taking 𝜆(𝑥, 𝑑(𝑥, 𝑦)) = 𝑑(𝑥,𝑦) ,itiseasyto (A) Thereexistsomeconstant 1 and a collection of num- 𝑓 𝐵 see that Definition 10 in this paper contains Definition 4.1 bers of 𝐵,oneforeach , such that these two properties 𝐵 𝑟 introduced by Garc´ıa-Cuerva and Gatto in [7], and Defini- hold: for any all with radius tion 10 is similar to Definition 1.9 introduced by Fu et al. in 1 󵄨 󵄨 𝛽/𝑛 ∫ 󵄨𝑓 (𝑥) −𝑓𝐵󵄨 𝑑𝜇 (𝑥) ≤𝐶1𝜆(𝑥,) 𝑟 , (16) [11]. 𝜇 (2𝐵) 𝐵

Definition 12. Let 𝐾𝛼 be a fractional kernel of order 𝛼 and and for any ball 𝑈 such that 𝐵∈𝑈and radius (𝑈) ≤ 𝑝 regularity 𝛿, 𝑓∈𝐿 (𝜇) and 0<𝛼−𝑛/𝑝<𝛿. We define 2𝑟, 󵄨 󵄨 󵄨𝑓 −𝑓 󵄨 ≤𝐶𝜆(𝑥,) 𝑟 𝛽/𝑛. ̃ 󵄨 𝐵 𝑈󵄨 1 (17) 𝐼𝛼𝑓 (𝑥) =∫ {𝐾𝛼 (𝑥,) 𝑦 −𝐾𝛼 (𝑥0,𝑦)} 𝑓 (𝑦) 𝑑𝜇 (𝑦) , X (B) There is a constant 𝐶2 such that (13) 󵄨 󵄨 𝛽/𝑛 󵄨𝑓 (𝑥) −𝑓(𝑦)󵄨 ≤𝐶2𝜆(𝑥, 𝑑(𝑥, 𝑦)) , (18) where 𝑥0 is some fixed point of X. We observe that the integral in (13)convergesboth for 𝜇-almost every 𝑥 and 𝑦 in the support of 𝜇. locally and at ∞ as a consequence of (10), (11), and Holder’s¨ (C) For any given 𝑝, 1≤𝑝≤∞,thereisaconstant𝐶(𝑝), inequality. Of course the function just defined depends on the such that for every ball 𝐵 of radius 𝑟,onehas election of 𝑥0, but the difference between any two functions 1 1/𝑝 obtained in (13) for different elections of 𝑥0 is just a constant. 󵄨 󵄨𝑝 𝛽/𝑛 ( ∫ 󵄨𝑓 (𝑥) −𝑚𝐵 (𝑓)󵄨 𝑑𝜇 (𝑥)) ≤𝐶(𝑝)𝜆(𝑥,) 𝑟 , 𝜇 (𝐵) 𝐵 𝜇(X)=∞ From now on, we will assume that .Theresults (19) below are also true when 𝜇(X)<∞. 𝑚 = (1/𝜇(𝐵)) ∫ 𝑓(𝑦)𝑑𝜇(𝑦) Now we state the first main theorem of this paper. where 𝐵 𝐵 and also for any ball 𝑈 such that 𝐵⊂𝑈and radius (𝑈) ≤ 2𝑟 Theorem 13. Let 1≤𝑝<𝑛/𝛼and 1/𝑞 = 1/𝑝 − 𝛼/𝑛. 󵄨 󵄨 󵄨𝑚 (𝑓) − 𝑚 (𝑓)󵄨 ≤𝐶(𝑝)𝜆(𝑥,) 𝑟 𝛽/𝑛. If 𝜆 satisfy the 𝜖-weak reverse doubling condition with 𝜖∈ 󵄨 𝐵 𝑈 󵄨 (20) 󸀠 (0, min{𝛼/𝑛, (1/𝑝 − 𝛼/𝑛)𝑝 }),then In addition, the quantities inf 𝐶1, inf 𝐶2,andinf 𝐶(𝑝) with 󵄩 󵄩 𝑞 𝑝 𝐶󵄩𝑓󵄩 afixed are equivalent. 󵄨 󵄨 󵄩 󵄩𝐿𝑝(𝜇) 𝜇 ({𝑥∈X : 󵄨𝐼𝛼𝑓 (𝑥)󵄨 > ]}) ≤ ( ) ; (14) ] Now we state the third main result of this paper.

𝑝 Theorem 17. 𝐾 𝑛/𝛼 < 𝑝 ≤∞ that is, 𝐼𝛼 is a bounded operator from 𝐿 (𝜇) into the space Let 𝛼 be a fractional kernel. and 𝑞,∞ 𝐿 (𝜇). 𝛼−𝑛/𝑝<𝛿.If𝜆 satisfy the 𝜖-weak reverse doubling condition 󸀠 󸀠 ̃ with 𝜖 ∈ (0, min{(1 − ((𝛼 − 𝛿)/𝑛))𝑝 ,(𝛼/𝑛−1/𝑝)𝑝 }),then𝐼𝛼 𝑝 Next, let us introduce Lipschitz spaces on nonhomoge- maps 𝐿 (𝜇) boundedly into Lip(𝛼 − 𝑛/𝑝). neous metric measure spaces. Theorem 18. Let 𝐾𝛼 be a fractional kernel and 𝛼+𝛽; <𝛿 Definition 14. Given that 𝛽 ∈ (0, 1], we say that the function if 𝜆 satisfy the 𝜖-weak reverse doubling condition with 𝜖∈ 𝑓:X → C 𝛽 ̃ satisfies a Lipschitz condition of order (0, min{(𝛼+𝛽)/𝑛, (𝛿−𝛼−𝛽)/𝑛}),then𝐼𝛼 maps Lip(𝛽) bound- ̃ provided that edly into Lip(𝛼 + 𝛽) if and only if 𝐼𝛼(1) = 0. 󵄨 󵄨 𝛽/𝑛 󵄨𝑓 (𝑥) −𝑓(𝑦)󵄨 ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] for every 𝑥, 𝑦 ∈ X Finallywepresentaresultwhichcanbeviewedeither (15) as an extension of the case 𝑝=∞of the Theorem 17 or as extension of the case 𝛽=0of Theorem 18. and the smallest constant in inequality (15) will be denoted by ‖𝑓‖Lip(𝛽). It is easy to see that the linear space with the norm Theorem 19. Let 𝐾𝛼 be a fractional kernel and 0<𝛼<𝛿. ‖⋅‖Lip(𝛽) is a Banach space, and we will call it Lip(𝛽). If 𝜆 satisfy the 𝜖-weak reverse doubling condition with 𝜖∈ ̃ (0, min{𝛼/𝑛, (𝛿 − 𝛼)/𝑛}),then𝐼𝛼 maps RBMO(𝜇) boundedly ̃ Remark 15. Lipschitz condition can also be defined by into Lip(𝛼) if and only if 𝐼𝛼(1) = 0. 󵄨 󵄨 𝛽/𝑛 󵄨𝑓 (𝑥) −𝑓(𝑦)󵄨 ≤𝐶[𝜆(𝑦,𝑑(𝑥,𝑦))] for every 𝑥, 𝑦 ∈ X; Finally, we make some conventions on notation. 󸀠 (15) Throughout the whole paper, 𝐶 stands for a positive con- stant, which is independent of the main parameters, but it 󸀠 by (2), it is easy to see that (15)and(15) are equivalent. mayvaryfromlinetoline. 4 Abstract and Applied Analysis

2. Proof of Theorem 13 which holds even for 𝑝=1. We can and assume that ‖𝑓‖퐿𝑝(휇) =1.By(5), we can choose 𝑟 ∈ (0, ∞) such that Proof of Theorem 13. We are going to adapt to our context of 𝐶[𝜆(𝑥,훼/푛−1/푝 𝑟)] = ]/2 the proof given by Garc´ıa-Cuerva and Gatto [7]. Consider .Then ] 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑓(𝑦)󵄨 {𝑥 ∈ X : 󵄨𝐼훼𝑓 (𝑥)󵄨 > ]}⊂{𝑥∈X : 󵄨𝐼1󵄨 > } 󵄨𝐼 𝑓 (𝑥)󵄨 ≤ ∫ 󵄨 󵄨 𝑑𝜇 (𝑦) 2 󵄨 훼 󵄨 1−훼/푛 X [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] (23) 󵄨 󵄨 ] 󵄨 󵄨 ∪{𝑥∈X : 󵄨𝐼2󵄨 > }. 󵄨𝑓(𝑦)󵄨 2 ≤ ∫ 󵄨 󵄨 𝑑𝜇 (𝑦) 1−훼/푛 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] (21) By the relation between 𝑟 and ], 𝜇({𝑥 ∈ X :|𝐼2|>]/2}) = 0. 󵄨 󵄨 󵄨𝑓(𝑦)󵄨 We use Holder’s¨ inequality once more to obtain + ∫ 󵄨 󵄨 𝑑𝜇 (𝑦) 1−훼/푛 X\퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 1/푝󸀠 󵄨 󵄨 𝑑𝜇 (𝑦) 󵄨𝐼 󵄨 ≤(∫ ) 󵄨 1󵄨 1−훼/푛 := 𝐼1 +𝐼2. 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))]

耠 󵄨 󵄨푝 1/푝 By Holder’s¨ inequality, if 𝑝>1,then1 − (1 − 𝛼/𝑛)𝑝 <0; 󵄨𝑓(𝑦)󵄨 ×(∫ 󵄨 󵄨 𝑑𝜇 (𝑦)) 1−훼/푛 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄨 󵄨 󵄩 󵄩 󵄨𝐼2󵄨 ≤ 󵄩𝑓󵄩퐿𝑝(휇) 1/푝󸀠 ∞ 𝜇(𝐵(𝑥,2−푗𝑟)) 1/푝󸀠 ≤𝐶(∑ ) 1 −푗−1 1−훼/푛 푗=0 [𝜆 (𝑥, 2 𝑟)] ×(∫ 󸀠 𝑑𝜇 (𝑦)) X\퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))](1−훼/푛)푝 󵄨 󵄨푝 1/푝 󵄩 󵄩 󵄨𝑓(𝑦)󵄨 ≤ 󵄩𝑓󵄩 ×(∫ 󵄨 󵄨 𝑑𝜇 (𝑦)) 󵄩 󵄩퐿𝑝(휇) 1−훼/푛 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 1/푝󸀠 ∞ 1 1/푝󸀠 ∞ 𝜆(𝑥,2푗𝑟) ×(∑ ∫ 󸀠 𝑑𝜇 (𝑦)) 2𝑗퐵\2𝑗−1퐵 [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))](1−훼/푛)푝 ≤𝐶(∑ ) 푗=1 −푗−1 1−훼/푛 푗=0 [𝜆 (𝑥, 2 𝑟)] 󵄩 󵄩 ≤ 󵄩𝑓󵄩 𝑝 퐿 (휇) 󵄨 󵄨푝 1/푝 󵄨𝑓(𝑦)󵄨 󸀠 ×(∫ 𝑑𝜇 (𝑦)) 푗 1/푝 1−훼/푛 ∞ 𝜇(𝐵(𝑥,2 𝑟)) 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] (24) ×(∑ ) (1−훼/푛)푝󸀠 푗=1 [𝜆 (𝑥,푗−1 2 𝑟)] 1/푝󸀠 ∞ −푗−1 훼/푛 󵄩 󵄩 ≤𝐶(∑[𝜆 (𝑥, 2 𝑟)] ) ≤ 󵄩𝑓󵄩퐿𝑝(휇) 푗=0

1/푝󸀠 󵄨 󵄨푝 1/푝 ∞ 𝜆(𝑥,2푗𝑟) 󵄨𝑓(𝑦)󵄨 ×(∫ 󵄨 󵄨 𝑑𝜇 (𝑦)) ×(∑ ) 1−훼/푛 (1−훼/푛)푝󸀠 퐵(푥,푟) 푗=1 [𝜆 (𝑥,푗−1 2 𝑟)] [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))]

󸀠 1/푝󸀠 1/푝 ∞ ∞ 󵄩 󵄩 1−(1−훼/푛)푝󸀠 1 훼/푛 ≤𝐶󵄩𝑓󵄩 (∑[𝜆 (𝑥,푗−1 2 𝑟)] ) ≤𝐶(∑ [𝜆 (𝑥,) 𝑟 ] ) 󵄩 󵄩퐿𝑝(휇) 푗 훼/푛 푗=1 푗=0 [𝐶 (2 )]

󸀠 󵄨 󵄨푝 1/푝 1/푝 󵄨𝑓(𝑦)󵄨 󵄩 󵄩 ∞ 1 ×(∫ 󵄨 󵄨 𝑑𝜇 (𝑦)) ≤𝐶󵄩𝑓󵄩 (∑ ) 1−훼/푛 󵄩 󵄩퐿𝑝(휇) (1−훼/푛)푝󸀠−1 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 푗=1 [𝐶 푗(2 )]

훼/푝󸀠푛 × [𝜆 (𝑥,) 𝑟 ]훼/푛−1/푝 ≤𝐶[𝜆 (𝑥,) 𝑟 ]

󵄩 󵄩 󵄨 󵄨푝 1/푝 ≤𝐶󵄩𝑓󵄩 [𝜆 (𝑥,) 𝑟 ]훼/푛−1/푝, 󵄨𝑓(𝑦)󵄨 󵄩 󵄩퐿𝑝(휇) ×(∫ 󵄨 󵄨 𝑑𝜇 (𝑦)) . 1−훼/푛 (22) 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] Abstract and Applied Analysis 5

Then, by applying Tchebichev’s inequality, we have and by (5), Lemma 21,weobtainthat 󵄨 󵄨 󵄨 󵄨 lim 𝑓퐵 =𝑓(𝑥) . 𝜇 ({𝑥 ∈ X : 󵄨𝐼훼𝑓 (𝑥)󵄨 > ]}) 푗→∞ 𝑗 (29)

󵄨 󵄨 ] Let 𝑥 and 𝑦 be two points as in the lemma; take 𝐵=𝐵(𝑥,𝑟) ≤𝜇({𝑥∈X : 󵄨𝐼1󵄨 > }) 2 anyballwith𝑟≤𝑑(𝑥,𝑦)and let 𝑈 = 𝐵(𝑥, 2𝑑(𝑥,.Now 𝑦)) 푘 ̃ ̃ 훼푝/푝󸀠푛 −푝 define 𝐵푘 =𝐵(𝑥,2𝑟),for0≤𝑘≤𝑘,where𝑘 is the first inte- ≤ 𝐶[𝜆(𝑥, 𝑟)] ] ̃푘 ger such that 2 𝑟≥𝑑(𝑥,𝑦).Then 󵄨 󵄨푝 󵄨𝑓(𝑦)󵄨 × ∫ ∫ 󵄨 󵄨 𝑑𝜇 (𝑦) 𝑑𝜇 (𝑥) ̃푘−1 1−훼/푛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 X 퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄨𝑓 −𝑓 󵄨 ≤ ∑ 󵄨𝑓 −𝑓 󵄨 + 󵄨𝑓 −𝑓 󵄨 󵄨 퐵 푈󵄨 󵄨 퐵𝑘 퐵𝑘+1 󵄨 󵄨 퐵̃𝑘 푈󵄨 푘=0 󸀠 =𝐶[𝜆 (𝑥,) 𝑟 ]훼푝/푝 푛]−푝 ̃푘 훽/푛 ≤𝐶 ∑[𝜆 (𝑥,푘 2 𝑟)] 𝑑𝜇 (𝑥) 󵄨 󵄨푝 1 × ∫ ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝜇 (𝑦) 푘=0 1−훼/푛 󵄨 󵄨 (30) X 퐵(푦,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] ̃푘 훽푘/푛 훽/푛 훼푝/푝󸀠푛 −푝 훼/푛󵄩 󵄩 󵄩 󵄩 ≤𝐶1 ∑𝑐휆 [𝜆 (𝑥,) 𝑟 ] ≤ 𝐶[𝜆(𝑥, 𝑟)] ] [𝜆(𝑥, 𝑟)] 󵄩𝑓󵄩퐿𝑝(휇) 푘=0 −푞 =𝐶[𝜆 (𝑥,) 𝑟 ] =𝐶] . 耠 훽/푛 ≤𝐶𝐶1𝜆(𝑥,𝑑(𝑥,𝑦)) , (25) 耠 where 𝐶 is independent of 𝑥 and 𝑑(𝑥,. 𝑦) This completes the proof of Theorem 13. A similar argument can be made for the point 𝑦 with any 耠 ball 𝐵 = 𝐵(𝑦, 𝑠) such that 𝑠≤𝑑(𝑥,𝑦)and 𝑉=𝐵(𝑦,3𝑑(𝑥,𝑦)). Corollary 20. Let 1<𝑝<𝑛/𝛼and 1/𝑞 = 1/𝑝 − 𝛼/𝑛.If 𝜆 𝜖 𝜖∈ Therefore satisfy the -weak reverse doubling condition with 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 耠 󵄨𝑓 −𝑓 󸀠 󵄨 ≤ 󵄨𝑓 −𝑓 󵄨 + 󵄨𝑓 −𝑓 󵄨 + 󵄨𝑓 −𝑓 󸀠 󵄨 (0, min{𝛼/𝑛, (1/𝑝 − 𝛼/𝑛)𝑝 }),then 󵄨 퐵 퐵 󵄨 󵄨 퐵 푈󵄨 󵄨 푈 푉󵄨 󵄨 푉 퐵 󵄨 (31) 󵄩 󵄩 󵄩 󵄩 耠耠 훽/푛 󵄩𝐼 𝑓󵄩 ≤𝐶󵄩𝑓󵄩 . ≤𝐶 𝐶 𝜆(𝑥, 𝑑 (𝑥, 𝑦)) . 󵄩 훼 󵄩퐿𝑞(휇) 󵄩 󵄩퐿𝑝(휇) (26) 1

Take two sequences of (2, 𝛽2)-doubling balls 𝐵푗 =𝐵(𝑥,𝑟푗) Proof. It suffices to apply Marcinkiewicz’s interpolation the- 𝐵耠 =𝐵(𝑦,𝑠) 𝑟 →0 𝑠 →0 orem with indices slightly bigger and slightly smaller than and 푗 with 푗 and 푗 .Wehave 𝑝 󵄨 󵄨 . 󵄨 󵄨 󵄨 󵄨 耠耠 훽/푛 󵄨𝑓 (𝑥) −𝑓(𝑦)󵄨 = lim 󵄨𝑓퐵 −𝑓퐵󸀠 󵄨 ≤𝐶 𝐶1𝜆(𝑥, 𝑑 (𝑥, 𝑦)) . 󵄨 󵄨 푗→0󵄨 𝑗 𝑗 󵄨 3. Proof of Theorem 16 (32) ( )⇒( ) 𝑥 ∈ 𝐵 = 𝐵(𝑥, 𝑟) Before we give the proof of Theorem 16,wefirstintroducea B C .For 0 , by the properties of 𝜆 technical lemma from [8, Lemma 3.2]. function and Holder’s¨ inequality, we obtain 1/푝 Lemma 21. 𝑓∈𝐿1 (𝜇) 𝛽 >2푑 1 󵄨 󵄨푝 Let 푙표푐 .If 2 ,then,foralmostevery ( ∫ 󵄨𝑓 (𝑥) −𝑚퐵 (𝑓)󵄨 𝑑𝜇 (𝑥)) 𝜇 (𝐵) 퐵 𝑥 with respect to 𝜇, there exists a sequence of (2, 𝛽2)-doubling 𝐵 =𝐵(𝑥,𝑟) 𝑟 →0 balls 푗 푗 with 푗 ,suchthat 󵄨 󵄨푝 1/푝 1 󵄨 1 󵄨 ≤( ∫ 󵄨 ∫ (𝑓 (𝑥) − 𝑓 (𝑦)) 𝑑𝜇 (𝑦)󵄨 𝑑𝜇 (𝑥)) 1 𝜇 (𝐵) 퐵 󵄨𝜇 (𝐵) 퐵 󵄨 lim ∫ 𝑓(𝑦)𝑑𝜇(𝑦)=𝑓(𝑥) . 푗→∞ (27) 𝜇(𝐵푗) 퐵𝑗 1/푝 1 훽푝/푛 ≤( ∫ [𝐶2𝜆 (𝑥,) 𝑟 ] 𝑑𝜇 (𝑥)) Proof of Theorem 16. (A)⇒(B). Consider 𝑥 as in the lemma 𝜇 (𝐵) 퐵 and let 𝐵푗 =𝐵(𝑥,𝑟푗), 𝑗≥1,asequenceof(2, 𝛽2)-doubling 1/푝 𝑟 →0 1 훽푝/푛 balls with 푗 . Consider ≤( ∫ [𝐶2𝐶휆𝜆(𝑥0,𝑟)] 𝑑𝜇 (𝑥)) 𝜇 (𝐵) 퐵 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 1 󵄨 󵄨 󵄨𝑚퐵 (𝑓) −퐵 𝑓 󵄨 ≤ ∫ 󵄨𝑓(𝑦)−𝑓퐵 󵄨 𝑑𝜇 (𝑦) 훽/푛 2 훽/푛 󵄨 𝑗 𝑗 󵄨 󵄨 𝑗 󵄨 ≤𝐶2𝐶휆[𝜆 0(𝑥 ,𝑟)] ≤𝐶2𝐶휆[𝜆 (𝑥,) 𝑟 ] . 𝜇(𝐵푗) 퐵𝑗 (33) 𝜇 (2𝐵 ) 󵄨 󵄨 푗 1 󵄨 󵄨 By the similar argument, for any ball 𝑈 such that 𝐵⊂𝑈and ≤ ∫ 󵄨𝑓(𝑦)−𝑓퐵 󵄨 𝑑𝜇 (𝑦) 󵄨 𝑗 󵄨 𝑈≤2𝑟 𝜇(𝐵푗) 𝜇(2𝐵푗) 퐵𝑗 radius , 󵄨 󵄨 2 훽/푛 훽/푛 󵄨𝑚퐵 (𝑓) −푈 𝑚 (𝑓)󵄨 ≤𝐶2𝐶휆[𝜆 (𝑥, )2𝑟 ] ≤𝛽𝐶1[𝜆 (𝑥,푗 𝑟 )] , (34) 훽/푛 2 훽/푛 (28) ≤𝑐휆 𝐶2𝐶휆[𝜆 (𝑥,) 𝑟 ] . 6 Abstract and Applied Analysis

( )⇒( ) 𝑓 =𝑚 (𝑓) 1/푝󸀠 C A .Definefirst 퐵 퐵 .Then(17) is exactly 󵄩 󵄩 ∞ 1 ≤𝐶󵄩𝑓󵄩 (∑ ) (20). To prove (16), we write 󵄩 󵄩퐿𝑝(휇) (훼/푛−1/푝)푝󸀠 푗=0 [𝐶 푗(2 )] 1 󵄨 󵄨 ∫ 󵄨𝑓 (𝑥) −𝑓퐵󵄨 𝑑𝜇 (𝑥) 훼/푛−1/푝 𝜇 (2𝐵) 퐵 × [𝜆 (𝑥,) 𝑟 ]

1/푝 󵄩 󵄩 훼/푛−1/푝 1 󵄨 󵄨푝 1/푝󸀠 󵄩 󵄩 󵄨 󵄨 ≤𝐶󵄩𝑓󵄩퐿𝑝(휇)[𝜆 (𝑥,) 𝑟 ] . ≤ (∫ 󵄨𝑓 (𝑥) −𝑓퐵󵄨 𝑑𝜇 (𝑥)) 𝜇(𝐵) 𝜇 (2𝐵) 퐵 (35) (37) 1/푝 𝜇 (𝐵) 1 󵄨 󵄨푝 ≤ ( ∫ 󵄨𝑓 (𝑥) −𝑓퐵󵄨 𝑑𝜇 (𝑥)) 𝜇 (2𝐵) 𝜇 (𝐵) 퐵 The second term is estimated in a similar way after noting that 2𝐵 ⊂ 𝐵(𝑦, 3𝑟). 훽/푛 ≤𝐶(𝑝)𝜆(𝑥,) 𝑟 . Next, by using (2), Holder’s¨ inequality, and 𝛼−𝑛/𝑝<𝛿, we get This concludes the proof of the theorem. 훿/푛 Remark 22. Theorem 16 is also true if the number 2 in con- [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄨 󵄨 𝐽 ≤𝐶∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝜇 (𝑧) ( ) 𝜌>1 3 1−훼/푛+훿/푛 󵄨 󵄨 dition A is replaced by any fixed .Inthatcase,the X\2퐵 [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] proof uses (𝜌,휌 𝛽 )-doubling balls, that is, balls satisfying 훿/푛󵄩 󵄩 𝜇(𝜌𝐵)휌 ≤𝛽 𝜇(𝐵). 󵄩 󵄩 ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄩𝑓󵄩퐿𝑝(휇)

󸀠 4. Proofs of Theorems 17–19 1 1/푝 ×(∫ 𝑑𝜇 (𝑧)) (1−(훼−훿)/푛)푝󸀠 Proof of Theorem 17. Without loss of generality, we assume X\2퐵 [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] 𝑝<∞ 𝑥 =𝑦̸ 𝐵 that . Consider and let be the ball with center 󵄩 󵄩 훿/푛 ≤𝐶󵄩𝑓󵄩 [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 𝑥 and radius 𝑟=𝑑(𝑥,𝑦).Then,wehave 󵄩 󵄩퐿𝑝(휇) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 1/푝󸀠 󵄨𝐼̃ 𝑓 (𝑥) − 𝐼̃ 𝑓(𝑦)󵄨 ≤ ∫ 󵄨𝐾 (𝑥,) 𝑧 󵄨 󵄨𝑓 (𝑧)󵄨 𝑑𝜇 (𝑧) 1 󵄨 훼 훼 󵄨 󵄨 훼 󵄨 󵄨 󵄨 ×(∫ 𝑑𝜇 (𝑧)) 2퐵 󸀠 X\2퐵 [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ](1−(훼−훿)/푛)푝 󵄨 󵄨 󵄨 󵄨 + ∫ 󵄨𝐾훼 (𝑦, 𝑧)󵄨 󵄨𝑓 (𝑧)󵄨 𝑑𝜇 (𝑧) 󵄩 󵄩 훿/푛 (훼−훿)/푛−1/푝 2퐵 ≤𝐶󵄩𝑓󵄩퐿𝑝(휇)[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄨 󵄨 + ∫ 󵄨𝐾 (𝑥,) 𝑧 −𝐾 (𝑦, 𝑧)󵄨 󵄩 󵄩 훼/푛−1/푝 󵄨 훼 훼 󵄨 ≤𝐶󵄩𝑓󵄩퐿𝑝(휇)[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] . X\2퐵 (38) 󵄨 󵄨 × 󵄨𝑓 (𝑧)󵄨 𝑑𝜇 (𝑧)

:= 𝐽1 +𝐽2 +𝐽3. Putting together the three estimates, (36) 󵄨 󵄨 󵄨̃ ̃ 󵄨 󵄩 󵄩 훼/푛−1/푝 耠 󵄨𝐼훼𝑓 (𝑥) − 𝐼훼𝑓 (𝑦)󵄨 ≤𝐶󵄩𝑓󵄩 𝑝 [𝜆 (𝑥, 𝑑 (𝑥,))] 𝑦 , For the first term, by 𝑛/𝛼,then <𝑝 1 − (1 − 𝛼/𝑛)𝑝 >0, 󵄨 󵄨 퐿 (휇) (39) 󵄨 󵄨 󵄨𝑓 (𝑧)󵄨 𝐽 ≤ ∫ 󵄨 󵄨 𝑑𝜇 (𝑧) 1 1−훼/푛 2퐵 ̃ 푝 [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] then 𝐼훼 maps 𝐿 (𝜇) boundedly into Lip(𝛼 − 𝑛/𝑝).

1/푝󸀠 󵄩 󵄩 𝑑𝜇 (𝑧) ≤ 󵄩𝑓󵄩 (∫ ) 󵄩 󵄩퐿𝑝(휇) 󸀠 2퐵 [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))](1−훼/푛)푝 ̃ Proof of Theorem 18. If 𝐼훼 ∈ Lip(𝛽), then by the continuity of 𝐼̃ 𝐼̃ (1) 1/푝󸀠 the operator 훼 implies that 훼 must be constant; that is, ∞ 𝜇(𝐵(𝑥,2−푗+1𝑟)) ̃ ̃ 󵄩 󵄩 𝐼훼(1)(𝑥) = 𝐼훼(1)(𝑥0)=0. ≤ 󵄩𝑓󵄩 (∑ ) 󵄩 󵄩퐿𝑝(휇) (1−훼/푛)푝󸀠 On the other hand, we can observe that 푗=0 [𝜆 (𝑥, 𝑑 (𝑥,2−푗𝑟))]

󸀠 −푗+1 1/푝 𝐼̃ (1) =0⇐⇒𝐼̃ (1)(𝑥) − 𝐼̃ (1) (𝑦) = 0; 󵄩 󵄩 ∞ 𝜆(𝑥,2 𝑟) 훼 훼 훼 (40) ≤ 󵄩𝑓󵄩 (∑ ) 󵄩 󵄩퐿𝑝(휇) (1−훼/푛)푝󸀠 푗=0 [𝜆 (𝑥, 𝑑 (𝑥,2−푗𝑟))] this implies that 1/푝󸀠 ∞ 󸀠 󵄩 󵄩 −푗 1−(1−훼/푛)푝 ≤𝐶󵄩𝑓󵄩퐿𝑝(휇)(∑[𝜆 (𝑥, 𝑑 (𝑥,2 𝑟))] ) ∫ {𝐾훼 (𝑥,) 𝑧 −𝐾훼 (𝑦, 𝑧)} 𝑑𝜇 (𝑧) =0. (41) 푗=0 X Abstract and Applied Analysis 7

훼/푛 훽/푛 Thus we can write ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] [𝜆 (𝑦, 𝑑 (𝑥, 𝑦))]

̃ ̃ (훼+훽)/푛 𝐼훼 (𝑓) (𝑥) − 𝐼훼 (𝑓) (𝑦) ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] . (44) = ∫ {𝐾훼 (𝑥,) 𝑧 −𝐾훼 (𝑦, 𝑧)} (𝑓 (𝑧) −𝑓(𝑥))𝑑𝜇(𝑧) X In order to estimate 𝑀3,weuse(2)toobtain = ∫ 𝐾훼 (𝑥,) 𝑧 (𝑓 (𝑧) −𝑓(𝑥))𝑑𝜇(𝑧) 2퐵(푥,푟) 󵄨 󵄨 󵄨𝑀3󵄨 + ∫ −𝐾 (𝑦, 𝑧) (𝑓 (𝑧) −𝑓(𝑥))𝑑𝜇(𝑧) 훼 훿/푛 훽/푛 2퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] ≤𝐶∫ 𝑑𝜇 (𝑧) 1−(훼−훿)/푛 X\2퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] + ∫ {𝐾훼 (𝑥,) 𝑧 −𝐾훼 (𝑦, 𝑧)} X\2퐵(푥,푟) ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))]훿/푛 ×(𝑓(𝑧) −𝑓(𝑥))𝑑𝜇(𝑧) 1 × ∫ 𝑑𝜇 (𝑧) := 𝑀 +𝑀 +𝑀, 1−(훼+훽−훿)/푛 1 2 3 X\2퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] (42) ≤ 𝐶[𝜆(𝑥, 𝑑(𝑥,훿/푛 𝑦))] [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))](훼+훽−훿)/푛 where 𝑟=𝑑(𝑥,𝑦).For𝑀1, (훼+훽)/푛 󵄨 󵄨 ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] . 󵄨 󵄨 󵄨𝑓 (𝑧) −𝑓(𝑥)󵄨 󵄨𝑀 󵄨 ≤ ∫ 󵄨 󵄨 𝑑𝜇 (𝑧) (45) 󵄨 1󵄨 1−훼/푛 2퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ]

[𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ]훽/푛 Combining the estimates for 𝑀1, 𝑀2,and𝑀3,weobtain ≤𝐶∫ 𝑑𝜇 (𝑧) (43) 1−훼/푛 2퐵(푥,푟) [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] 󵄨 󵄨 󵄨̃ ̃ 󵄨 (훼+훽)/푛 󵄨𝐼훼𝑓 (𝑥) − 𝐼훼𝑓(𝑦)󵄨 ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] ; (46) ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥,(훼+훽)/푛 𝑦))] .

this finishes the proof. Similarly, we know that 2𝐵(𝑥, 𝑟) ⊂ 3𝐵(𝑦,;usingH 𝑟) older’s¨ 耠 inequality (1/𝑡 + 1/𝑡 =1),andletting𝑡>𝑛/𝛼,thenwehave 󵄨 󵄨 ̃ ̃ 󵄨𝑀2󵄨 Proof of Theorem 19. 𝐼훼(𝑓) ∈ Lip(𝛼) implies that 𝐼훼(1) = 0 󵄨 󵄨 and, equivalently, that 󵄨𝑓 (𝑧) −𝑓(𝑥)󵄨 ≤ ∫ 󵄨 󵄨 𝑑𝜇 (𝑧) 1−훼/푛 2퐵(푥,푟) [𝜆 (𝑦, 𝑑 (𝑦, 𝑧))] ∫ (𝐾훼 (𝑥, 𝑦)훼 −𝐾 (𝑥0, 𝑦)) 𝑑𝜇 (𝑦) =0 (47) X [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ]훽/푛 ≤𝐶∫ 𝑑𝜇 (𝑧) 1−훼/푛 2퐵(푥,푟) [𝜆 (𝑦, 𝑑 (𝑦, 𝑧))] for all 𝑥. 𝑥 =𝑦̸ 𝐵=𝐵(𝑥,𝑟) 𝑟=𝑑(𝑥,𝑦) 1/푡 Take two points and let with . ≤𝐶(∫ [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ]훽푡/푛𝑑𝜇 (𝑧)) Then 2퐵(푥,푟) 󵄨 󵄨 󸀠 󵄨𝐼̃ (𝑓) (𝑥) − 𝐼̃ (𝑓) (𝑦)󵄨 1/푡 󵄨 훼 훼 󵄨 1 ×(∫ 𝑑𝜇 (𝑧)) 󵄨 󵄨 (1−훼/푛)푡󸀠 󵄨 󵄨 2퐵(푥,푟) [𝜆 (𝑦, 𝑑 (𝑦, 𝑧))] = 󵄨∫ {𝐾훼 (𝑥,) 𝑧 −𝐾훼 (𝑦, 𝑧)} (𝑓 (𝑧) −𝑓2퐵)𝑑𝜇(𝑧)󵄨 󵄨 X 󵄨 ∞ 󵄨 󵄨 󵄨 󵄨 ≤𝐶(∑ ∫ [𝜆 (𝑥, ≤ 󵄨∫ 𝐾훼 (𝑥,) 𝑧 (𝑓 (𝑧) −𝑓2퐵)𝑑𝜇(𝑧)󵄨 −𝑗+1 −𝑗 󵄨 2퐵 󵄨 푗=0 2 퐵(푥,푟)\2 퐵(푥,푟) 󵄨 󵄨 󵄨 󵄨 1/푡 + 󵄨∫ 𝐾훼 (𝑦, 𝑧) (𝑓 (𝑧) −𝑓2퐵)𝑑𝜇(𝑧)󵄨 훽푡/푛 󵄨 2퐵 󵄨 𝑑(𝑥,2−푗+1𝑟))] 𝑑𝜇(𝑧)) 󵄨 󵄨 󵄨 󵄨 + 󵄨∫ {𝐾훼 (𝑥,) 𝑧 −𝐾훼 (𝑦, 𝑧)} (𝑓 (𝑧) −𝑓2퐵)𝑑𝜇(𝑧)󵄨 󵄨 X\2퐵 󵄨 1/푡󸀠 1 ×(∫ 𝑑𝜇 (𝑧)) := 𝑁1 +𝑁2 +𝑁3. (1−훼/푛)푡󸀠 3퐵(푦,푟) [𝜆 (𝑦, 𝑑 (𝑦, 𝑧))] (48) 8 Abstract and Applied Analysis

For the first term, by Holder’s¨ inequality with some 𝑝>𝑛/𝛼, Acknowledgments

1 󵄨 󵄨 Jiang Zhou is supported by the National Science Foundation 𝑁 ≤ ∫ 󵄨𝑓 (𝑧) −𝑓 󵄨 𝑑𝜇 (𝑧) 1 1−훼/푛 󵄨 2퐵󵄨 of China (Grants nos. 11261055 and 11161044) and by the 2퐵 𝜆 𝑥, 𝑑 𝑥, 𝑧 [ ( ( ))] National Natural Science Foundation of Xinjiang (Grants 󸀠 nos. 2011211A005 and BS120104). 𝑑𝜇 (𝑧) 1/푝 ≤(∫ ) (1−훼/푛)푝󸀠 2퐵 [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] References

1/푝 (49) 1 󵄨 󵄨푝 [1] X. Tolsa, “BMO, 𝐻 ,andCalderon-Zygmund´ operators for non ×(∫ 󵄨𝑓 (𝑧) −𝑓2퐵󵄨 𝑑𝜇 (𝑧)) 2퐵 doubling measures,” Mathematische Annalen,vol.319,no.1,pp. 󵄩 󵄩 89–149, 2001. ≤𝐶[𝜆 (𝑥,) 𝑟 ]훼/푛−1/푝𝜇(𝜌𝐵)1/푝󵄩𝑓󵄩 󵄩 󵄩RBMO(휇) [2]X.Tolsa,“Littlewood-Paleytheoryandthe𝑇(1) theorem with non-doubling measures,” Advances in Mathematics,vol.164,no. 훼/푛󵄩 󵄩 ≤𝐶[𝜆 (𝑥,) 𝑟 ] 󵄩𝑓󵄩 (휇). 1, pp. 57–116, 2001. RBMO 1 [3]X.Tolsa,“Thespace𝐻 for nondoubling measures in terms of 2𝐵 ⊂ 𝐵(𝑦, 3𝑟) a grand maximal operator,” Transactions of the American Math- Using ,thesecondtermcanbedealtwithin ematical Society,vol.355,no.1,pp.315–348,2003. 𝑁 𝑁 ≤ 𝐶[𝜆(𝑥,훼/푛 𝑟)] ‖𝑓‖ thesamewayasthe 1;then 2 RBMO(휇). [4] T. A. Bui and X. T. Duong, “Hardy spaces, regularized BMO |𝑓 𝑘+1 −𝑓 | ≤ ‖𝑓‖ 𝐾 𝑘+1 ≤ It is easy to see that 2 퐵 2퐵 RBMO(휇) 2퐵,2 퐵 spaces and the boundedness of Calderon-Zygmund´ operators 𝑘‖𝑓‖ on non-homogeneous spaces,” Journal of Geometric Analysis, RBMO(휇);then vol.23,no.2,pp.895–932,2013.

훿/푛 [5] T. Hytonen,S.Liu,D.Yang,andD.Yang,“BoundednessofCal-¨ [𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄨 󵄨 𝑁 ≤ ∫ 󵄨𝑓 (𝑧) −𝑓 󵄨 𝑑𝜇 (𝑧) deron-Zygmund´ operators on non-homogeneous metric mea- 3 1−훼/푛+훿/푛 󵄨 2퐵󵄨 X\2퐵 [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] sure spaces,” Canadian Journal of Mathematics,vol.64,no.4, pp. 892–923, 2012. 1 ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥,훿/푛 𝑦))] [6] T.Hytonen,¨ D. Yang, and D. Yang, “The Hardy space 𝐻 on non- homogeneous metric spaces,” Mathematical Proceedings of the ∞ 󵄨 󵄨 󵄨𝑓 (𝑧) −𝑓𝑘+1 󵄨 Cambridge Philosophical Society,vol.153,no.1,pp.9–31,2012. × ∑ (∫ 󵄨 2 퐵󵄨 𝑑𝜇 (𝑧) 1−훼/푛+훿/푛 [7] J. Garc´ıa-Cuerva and A. E. Gatto, “Boundedness properties of 2𝑘+1퐵\2𝑘퐵 [𝜆 (𝑥, 𝑑 (𝑥,)) 𝑧 ] 푘=1 fractional integral operators associated to non-doubling mea- 󵄨 󵄨 sures,” Studia Mathematica,vol.162,no.3,pp.245–261,2004. 󵄨𝑓 𝑘+1 −𝑓 󵄨 + 󵄨 2 퐵 2퐵󵄨 𝜇(2푘+1𝐵)) [8] J. Garc´ıa-Cuerva and A. E. Gatto, “Lipschitz spaces and Cal- [𝜆 (𝑥, 𝑑푘 (𝑥,2 𝑟))]1−훼/푛+훿/푛 deron-Zygmund´ operators associated to non-doubling mea- sures,” Publicacions Matematiques` ,vol.49,no.2,pp.285–296, 훿/푛󵄩 󵄩 2005. ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄩𝑓󵄩 (휇) RBMO [9] Y. Cao and J. Zhou, “Morrey spaces for non-homogeneous ∞ 𝜇(𝜌2푘+1𝐵) metric measure spaces,” Abstract and Applied Analysis,vol.2013, ×(∑ ArticleID196459,8pages,2013. 푘 1−훼/푛+훿/푛 푘=1 [𝜆 (𝑥, 𝑑 (𝑥,2 𝑟))] [10]X.Fu,D.Yang,andW.Yuan,“Boundednessofmultilinear commutators of Calderon-Zygmund´ operators on Orlicz spaces ∞ 𝜇(2푘+1𝐵) over non-homogeneous spaces,” Taiwanese Journal of Mathe- + ∑𝑘 ) matics, vol. 16, no. 6, pp. 2203–2238, 2012. [𝜆(𝑥,𝑑(𝑥,2푘𝑟))]1−훼/푛+훿/푛 푘=1 [11] X. Fu, D. Yang, and W. Yuan, “Generalized fractional integrals 󵄩 󵄩 and their commutators over non-homogeneous spaces,” Tai- ≤ 𝐶[𝜆(𝑥, 𝑑(𝑥,훿/푛 𝑦))] 󵄩𝑓󵄩 󵄩 󵄩RBMO(휇) wanese Journal of Mathematics,vol.18,no.2,pp.509–557,2014.

∞ [12] H. Lin and D. Yang, “Equivalent boundedness of Marcinkiewicz 훼/푛−훿/푛 ×(∑ (𝑘+1) [𝜆(𝑥,𝑑(𝑥,2푘𝑟))] ) integrals on non-homogeneous metric measure spaces,” Science China. Mathematics,vol.57,no.1,pp.123–144,2014. 푘=0 [13]X.Tolsa,“Painleve’s´ problem and the semiadditivity of analytic 훼/푛󵄩 󵄩 ≤ 𝐶[𝜆 (𝑥, 𝑑 (𝑥, 𝑦))] 󵄩𝑓󵄩 . capacity,” Acta Mathematica,vol.190,no.1,pp.105–149,2003. 󵄩 󵄩RBMO(휇) [14] T. Hytonen,¨ “A framework for non-homogeneous analysis on (50) metric spaces, and the RBMO space of Tolsa,” Publicacions Matematiques` ,vol.54,no.2,pp.485–504,2010. Thus the proof of Theorem 19 is completed. [15] R. R. Coifman and G. Weiss, Analyse Harmonique Non-Com- mutative sur Certains Espaces Homogenes` ,vol.242ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1971. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 236124, 12 pages http://dx.doi.org/10.1155/2014/236124

Research Article Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers

Ahmet Faruk Çakmak1 and Feyzi BaGar2

1 Department of Mathematical Engineering, Yıldız Technical University, Davutpas¸a Campus, Esenler, 80750 Istanbul, Turkey 2 Department of Mathematics, Faculty of Arts and Sciences, Fatih University, Hadimkoy¨ Campus, Buy¨ ukc¨ ¸ekmece, 34500 Istanbul, Turkey

Correspondence should be addressed to Feyzi Bas¸ar; [email protected]

Received 3 December 2013; Accepted 22 January 2014; Published 15 April 2014

Academic Editor: S.A. Mohiuddine

Copyright © 2014 A. F. C¸akmakandF.Bas¸ar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct ∗ the field C of ∗-complex numbers and the concept of ∗-metric. Also, we give the definitions and the basic important properties of ∗-boundedness and ∗-continuity. Later, we define the space 𝐶∗(Ω) of ∗-continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that 𝐶∗(Ω) is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship 󸀠 between 𝐶∗(Ω) and the set of 𝐶∗(Ω)∗-differentiable functions.

1. Preliminaries, Background and Notations multiplication. Further, the non-Newtonian exponent, surd, and absolute value are defined and some of their properties As a popular non-Newtonian calculus, multiplicative calculus are given. They also proved that the spaces of all bounded, was studied by Stanley [1] in a brief overview. Bashirov et al. convergent, null and absolutely p-summable sequences of [2] have recently emphasized on the multiplicative calculus the non-Newtonian real numbers are the complete metric and gave the results with applications corresponding to the spaces. Quite recently, Tekin and Bas¸ar have [7]proved well-known properties of derivative and integral in the clas- that the corresponding classical sequence spaces are Banach sical calculus. Recently, in [3], the multiplicative calculus has spaces over the non-Newtonian complex field. Quite recently, extended to the complex valued functions and interested in C¸akir[8] has defined the sets 𝐵(𝐴) and 𝐶(𝐴) of geometric the statements of some fundamental theorems and concepts complexvaluedboundedandcontinuousfunctionsand of multiplicative complex calculus and demonstrated some showed that 𝐵(𝐴) and 𝐶(𝐴) form a vector space with respect analogies between the multiplicative complex calculus and totheadditionandscalarmultiplicationinthesenseof classical calculus by theoretical and numerical examples. multiplicative calculus and are complete metric spaces, where Bashirov and Riza [4]havestudiedonthemultiplicative 𝐴 denotes the compact subset of the complex plane C.Quite differentiation for complex valued functions and established recently, Uzer [9] has investigated the waves near the edge of the multiplicative Cauchy-Riemann conditions. Bashirov et a conducting half plane. Firstly, the series is converted into al. [5]haveinvestigatedvariousproblemsfromdifferentfields contour integrals in a complex plane and then some contour which can be modeled more efficiently using multiplicative deformations are made. After that, the resultant integrals are calculus, in place of Newtonian calculus. Quite recently, converted back into the series forms, which are seen to be C¸akmakandBas¸ar [6]haveshowedthatnon-Newtonianreal rapidlyconvergentnearthereflection/shadowboundariesof numbers form a field with the binary operations addition and the conducting half plane. In the second part, a multiplicative 2 Abstract and Applied Analysis calculus is employed for evaluating the relevant integrals, the operations are defined as follows: for 𝑥, 𝑦 ∈𝐴 and any approximately. By the way, he derives a simple expression, generator 𝛼, which can be used whenever the series derived in the first part −1 −1 of the paper is not rapidly convergent. 𝛼-addition 𝑥+𝑦̇ = 𝛼 [𝛼 (𝑥) +𝛼 (𝑦)] , Non-Newtonian calculus is an alternative to the usual ̇ −1 −1 calculus of Newton and Leibniz. It provides differentiation 𝛼-subtraction 𝑥−𝑦 = 𝛼 [𝛼 (𝑥) −𝛼 (𝑦)] , and integration tools based on non-Newtonian operations ̇ −1 −1 instead of classical operations. Every property in classical 𝛼-multiplication 𝑥×𝑦 = 𝛼 [𝛼 (𝑥) ×𝛼 (𝑦)] , (3) calculus has an analogue in non-Newtonian calculus. Gen- ̇ −1 −1 erally speaking, non-Newtonian calculus is a methodology 𝛼-division 𝑥/𝑦 = 𝛼 [𝛼 (𝑥) ÷𝛼 (𝑦)], that allows one to have a different look at problems which ̇ −1 −1 can be investigated via calculus. In some cases, for exam- 𝛼-order 𝑥<𝑦 ⇐⇒ 𝛼 (𝑥) <𝛼 (𝑦) . ple, for wage-rate- (in dollars, euro, etc.) related problems, 𝐼 the use of bigeometric calculus which is a kind of non- Particularly, if we choose , the identity function, as an 𝛼 𝛼(𝑥) =𝑥 𝛼−1(𝑥) = 𝑥 𝑥∈𝐴 Newtonian calculus is advocated instead of a traditional -generator, then and for all 𝛼 Newtonian one. and therefore -arithmetic obviously turns out to the classical Throughout this paper, non-Newtonian calculus is arithmetic. Consider −1 −1 denoted by (NC), and classical calculus is denoted by (CC). 𝛼-addition 𝑥⊕𝑦=𝛼[𝛼 (𝑥) +𝛼 (𝑦)] = 𝛼 (𝑥 +𝑦) Also for short we use ∗-continuity for non-Newtonian continuity. A generator is a one-to-one function whose =𝑥+𝑦classical addition, domain is R and whose range is a subset of R. Each −1 −1 generator generates exactly one type of arithmetic, and 𝛼-subtraction 𝑥⊖𝑦=𝛼[𝛼 (𝑥) −𝛼 (𝑦)] = 𝛼 (𝑥 −𝑦) conversely each type of arithmetic is generated by exactly one generator. As a generator, we choose the function exp =𝑥−𝑦classical subtraction, + from R to the set R that is to say that −1 −1 𝛼-multiplication 𝑥⊙𝑦=𝛼[𝛼 (𝑥) ×𝛼 (𝑦)] + 𝛼:R 󳨀→ R =𝛼(𝑥×𝑦)=𝑥×𝑦 𝑥 𝑥 󳨃󳨀→ 𝛼 (𝑥) =𝑒 =𝑦, classical multiplication, (1) −1 + −1 −1 𝛼 : R 󳨀→ R 𝛼-division 𝑥⊘𝑦=𝛼[𝛼 (𝑥) ÷𝛼 (𝑦)] = 𝛼 (𝑥 ÷𝑦) −1 𝑦 󳨃󳨀→ 𝛼 (𝑦) = ln 𝑦=𝑥. =𝑥÷𝑦,=0 𝑦 ̸ classical division. (4)

In the special cases 𝛼=𝐼and 𝛼=exp, 𝛼 generates 𝑧 If we choose exp as an 𝛼-generator defined by 𝛼(𝑧) =𝑒 for the classical and geometric arithmetic, respectively, where 𝐼 −1 ∗ 𝑧∈C 𝛼 (𝑧) = 𝑧 𝛼 denotes the identity function whoseinverseisitself.ThesetR ,then ln and -arithmetic turns out to ∗ of non-Newtonian real numbers are defined by R := {𝛼(𝑥) : geometric arithmetic. Consider 𝑥∈R}. −1 −1 (ln 𝑥+ln 𝑦) 𝛼-addition 𝑥⊕𝑦=𝛼[𝛼 (𝑥) +𝛼 (𝑦)] = 𝑒 Following Bashirov et al. [2]andUzer[3], the main 𝐶 (Ω) purpose of this paper is to construct the space ∗ of =𝑥⋅𝑦geometric addition, non-Newtonian complex valued continuous functions which −1 −1 (ln 𝑥−ln 𝑦) forms a Banach space with the norm defined on it. Finally, 𝛼-subtraction 𝑥⊖𝑦=𝛼[𝛼 (𝑥) −𝛼 (𝑦)] = 𝑒 we give some applications to seek how (NC) can be applied to the classical Functional Analysis problems such as approx- =𝑥÷𝑦,=0 𝑦 ̸ imation and inner product properties. geometric subtraction, We should know that all concepts in classical 𝛼 −1 −1 (ln 𝑥×ln 𝑦) arithmetic have natural counterparts in -arithmetic. 𝛼-multiplication 𝑥⊙𝑦=𝛼[𝛼 (𝑥) ×𝛼 (𝑦)] = 𝑒 For instance, 𝛼-zero and 𝛼-one turn out to be 𝛼(0) and 𝛼(1). 𝛼 Similarly, the -integers turn out to =𝑥ln 𝑦 =𝑦ln 𝑥

...,𝛼(−3) ,𝛼(−2) ,𝛼(−1) ,𝛼(0) ,𝛼(1) ,𝛼(2) ,𝛼(3) ,... geometric multiplication, (2) −1 −1 (ln 𝑥÷ln 𝑤) 𝛼-division 𝑥⊘𝑦=𝛼[𝛼 (𝑥) ÷𝛼 (𝑦)] = 𝑒

1/ ln 𝑦 Consider any generator 𝛼 with range 𝐴⊆R.By𝛼- =𝑥 ,𝑦=1̸ geometric division. arithmetic, we mean the arithmetic whose domain is 𝐴 and (5) Abstract and Applied Analysis 3

Arithmetic is any system that satisfies the whole of the Table 1: Notation in 𝛼-arithmetic and 𝛽-arithmetic. R ordered field axioms whose domain is a subset of .There 𝛼 𝛽 are infinitely many types of arithmetic, all of which are -arithmetic -arithmetic isomorphic, that is, structurally equivalent. Nevertheless, the Realm 𝐴𝐵 fact that two systems are isomorphic does not exclude their Addition +̇ +̈ separate usage. In [2], it is shown that each ordered pair of Subtraction −̇ −̈ arithmetic give rise to a calculus by a sensible use of the first Multiplication ×̇ ×̈ ̇ ̈ arithmetic or function arguments and the second arithmetic Division / / for function values. Ordering ≤̇ ≤̈ Let 𝛼 and 𝛽 be arbitrarily selected generators and (𝛼 ,𝛽 ) -arithmetic -arithmetic is the ordered pair of arithmetic. −1 3 Table 1 may be useful for the notation used in 𝛼-arithmetic =𝛼{[𝛼 (𝑥)] } and 𝛽-arithmetic. 𝛼 𝛽 . Definitions for -arithmetic are also valid for -arith- . metic. For example, 𝛽-convergence is defined by means of 𝛽- 𝑝 intervals and their 𝛽-interiors. 𝑥𝑝̇=𝑥𝑝−1̇ ×𝑥=𝛼{[𝛼̇ −1 (𝑥)] } In the (NC), 𝛼-arithmetic is used for arguments and 𝛽-arithmeticisusedforranges;inparticular,changes . in arguments and ranges are measured by 𝛼-differences . and 𝛽-differences, respectively. The operators of the (NC) (7) are applied only to functions with arguments in 𝐴 and ∗ 𝑞̇ values in 𝐵. and √𝑞̇𝑥=𝑦provided there exists an 𝑦∈R such that 𝑦 =𝑥. 𝛼 𝛽 The 𝛼-absolute value of a number 𝑥 in 𝐴⊂R is defined The isomorphism from -arithmetic to -arithmetic is −1 ̇̇ the unique function 𝜄 (iota) which has the following three as 𝛼(|𝛼 (𝑥)|) and is denoted by |𝑥|. ⋅ properties: For each 𝛼-nonnegative number 𝑥,thesymbol√𝑥 will be used to denote 𝛼{√𝛼−1(𝑥)} which is the unique 𝛼-non- 𝑦 𝛼 𝑥 𝜄 negative number whose -square is equal to .Foreach (i) is one to one; number 𝑥 in 𝐴,

⋅ 󵄨 󵄨 (ii) 𝜄 is on 𝐴 and onto 𝐵; √ 2̇ ̇ ̇ 󵄨 −1 󵄨 𝑥 = |𝑥| =𝛼(󵄨𝛼 (𝑥)󵄨), (8) ̇̇ (iii) for any numbers 𝑢 and V in 𝐴, where the absolute value |𝑥| of 𝑥∈R(𝑁) is defined by

{𝑥, 𝑥 >𝛼̇ (0) , 𝜄(𝑢 +̇V)=𝜄(𝑢) +𝜄̈(V) , { |𝑥̇|̇= 𝛼 (0) , 𝑥=𝛼(0) , { (9) 𝜄(𝑢 −̇V)=𝜄(𝑢) −𝜄̈(V) , {𝛼 (0) −𝑥,̇ <𝛼 𝑥 ̇ (0) .

𝜄(𝑢 ×̇V)=𝜄(𝑢) ×𝜄̈(V) , The ∗-distance between two points 𝑥1 and 𝑥2 is defined (6) ⋅ ⋅ |𝑥 −𝑥̇ | 𝜄(𝑢 /V̇)=𝜄(𝑢) /𝜄̈(V) , V ≠0,̇ by 1 2 andhasthesymmetryproperty,since 󵄨 −1 −1 󵄨 |𝑥̇ −𝑥̇ |̇=𝛼[󵄨𝛼 (𝑥 )−𝛼 (𝑥 )󵄨] 𝑢 ≤̇V ⇐⇒ 𝜄 (𝑢) ≤𝜄̈(V) . 1 2 󵄨 1 2 󵄨 󵄨 󵄨 󵄨 −1 −1 󵄨 =𝛼[󵄨𝛼 (𝑥2)−𝛼 (𝑥1)󵄨] (10) = |𝑥̇ −𝑥̇ |.̇ −1 2 1 It turns out that 𝜄(𝑥) = 𝛽{𝛼 (𝑥)} for all 𝑥 in 𝐴 and that 𝜄(𝑛)̇ = ∗ 𝑛̈for every integer 𝑛. Let any 𝑧∈R be given. Then, 𝑧 is called a positive −1 𝑧>𝛼̇ (0) 𝑧 Since, for example, 𝑢 +̇V =𝜄 {𝜄(𝑢) +𝜄(̈ V)},itshould non-Newtonian real number if , is called a non- 𝑧<𝛼̇ (0) 𝑧 be clear that any statement in 𝛼-arithmetic can readily be Newtonian negative real number if ,and,finally, is called an unsigned non-Newtonian real number if 𝑧=𝛼(0). transformed into a statement in 𝛽-arithmetic. ∗ ∗ By R and R ,wedenotethesetsofnon-Newtonianpositive 𝑝̇ ∗ 𝑥𝑝̇ + − Throughout this paper, we define the -th -exponent and negative real numbers, respectively. 𝑞̇ ∗ √𝑞̇𝑥 𝑥∈R∗ and the -th -root of by In (CC), we have |𝑥𝑦| = |𝑥||𝑦| and |𝑥 + 𝑦| ≤ |𝑥| +|𝑦| for 𝑥, 𝑦 ∈ R.Thefollowinglemmasshowthatthecorresponding ̇ 2 resultsalsoholdinnon-Newtoniancalculus. 𝑥2 =𝑥×𝑥=𝛼{𝛼̇ −1 (𝑥) ×𝛼−1 (𝑥)}=𝛼{[𝛼−1 (𝑥)] } ∗ ̇ ̇ Lemma 1 ([6, Proposition 2.2]). For any 𝑥, 𝑦 ∈ R , |𝑥 ×𝑦̇|= 3̇ 2̇ −1 −1 −1 −1 ̇̇ ̇̇ 𝑥 =𝑥 ×𝑥=𝛼{𝛼̇ {𝛼 [𝛼 (𝑥) ×𝛼 (𝑥)]}×𝛼 (𝑥)} |𝑥| ×̇|𝑦|. 4 Abstract and Applied Analysis

∗ Lemma 2 (∗-Triangle inequality see [6, Lemma 3.1]). Let The classical derivatives [𝐷𝑓](𝑎) and [𝐷 𝑓](𝑎) do not ∗ 𝑥, 𝑦 ∈ R .Then, necessarily coexist and are seldom equal; however, if the following exist, |𝑥̇+𝑦̇ |̇≤ |𝑥̇|+̇̇|𝑦̇|.̇ (11) [𝐷 (𝛼−1)] (𝑎) , [𝐷𝛼] (𝛼−1 (𝑎)), Let (𝑢𝑛) be an infinite sequence of the elements in 𝐴 𝑢 𝐴 (16) . Then, there is at most one element in such that [𝐷 (𝛽−1)] (𝑓 (𝑎)), [𝐷𝛽]{𝛽−1 (𝑓 (𝑎))} , every𝛼-interval with 𝑢 in its 𝛼-interior contains all but finitely many terms of (𝑢𝑛).Ifthereissuchanumber𝑢,then(𝑢𝑛) is ∗ then both [𝐷𝑓](𝑎) and [𝐷 𝑓](𝑎) exist. said to be 𝛼-convergent to 𝑢,whichiscalledthe𝛼-limit of Wedenote the sets of ∗-bounded functions, ∗-continuous (𝑢𝑛).Inotherwords, functions and ∗-differentiable functions in the ∗-closed [𝑎,̇ ] 𝑏 ̇ 𝐵 [𝑎,̇ ] 𝑏 ̇𝐶 [𝑎,̇ ] 𝑏 ̇ 𝐶󸀠 [𝑎,̇ ] 𝑏 ̇ ̇̇ interval by ∗ , ∗ ,and ∗ ,respectively. 𝑢𝑛 󳨀→ 𝑢 (𝛼 -convergent)⇐⇒∀𝜀> 0, ∗ ∗ 𝑓 [𝑎,̇ ] 𝑏 ̇ (12) The -average of a -continuous function on ̇ ̇̇ ∗𝑏 ∃𝑛0 ∈ N ∋ |𝑢𝑛 −𝑢̇ |<𝜀 ∀𝑛≥𝑛0 and some 𝑢∈𝐴. is denoted by 𝑀𝑎 𝑓 anddefinedtobethe𝛽-limitofthe 𝛽-convergent sequence whose 𝑛th term is the 𝛽-average of The ∗-limit of a function 𝑓 at an element 𝑎 in 𝐴 is, if it 𝑓(𝑎1),...,𝑓(𝑎𝑛),where𝑎1,...,𝑎𝑛 is the 𝑛-fold 𝛼-partition of ̇ ̇ ̇ ̇ exists, the unique number 𝑏 in 𝐵 such that, for every infinite [𝑎,] 𝑏 .The∗-average of a ∗-uniform function on [𝑎, 𝑏] is equal sequence (𝑎𝑛) of arguments of 𝑓 distinct from 𝑎,if(𝑎𝑛) to the 𝛽-average of its values at 𝑎 and 𝑏 andisequaltoitsvalue is 𝛼-convergent to 𝑎,then{𝑓(𝑎𝑛)} 𝛽-converges to 𝑏 and is at the 𝛼-average of 𝑎 and 𝑏. ∗ 𝑓(𝑥) =𝑏 ∗𝑏 denoted by -lim𝑥→𝑎 .Thatis, ∗ ∫ 𝑓(𝑥)𝑑𝑥 ∗ 𝑓 The -integral 𝑎 of a -continuous function ̇ ̇ ∗𝑓 (𝑥) =𝑏⇐⇒∀𝜖>̈0,̈ on [𝑎,] 𝑏 ,isthefollowingnumberin𝐵: lim𝑥→𝑎 (13) ̈ ̈ ̇ ̇ [𝜄 (𝑏) −𝜄̈(𝑎)] ×𝑀̈ ∗𝑏𝑓. ∃𝛿 >̇0∋̇ |𝑓 (𝑥) −𝑏̈|<𝜖̈ ∀𝑥∈𝐴 with |𝑥 −𝑎̇|<𝛿.̇ 𝑎 (17) It is trivial that Afunction𝑓 is ∗-continuous at a point 𝑎 in 𝐴 if and only 𝑎 𝑓 ∗ 𝑓(𝑥) = 𝑓(𝑎) ∗𝑎 if is an argument of and -lim𝑥→𝑎 .When ̈ 𝛼 𝛽 𝐼 ∗ ∫ 𝑓 (𝑥) 𝑑𝑥 = 0. (18) and are the identity function ,theconceptsof -limit 𝑎 and ∗-continuity are identical with those of classical limit and classical continuity. Since ̇ ̇ The 𝛽-change of a function 𝑓 over an interval [𝑎, 𝑏] is ∗𝑏 𝑓(𝑏)−𝑓(𝑎)̈ ∗ ∗𝑏 ̈ ̈ the number .A -uniform function is a function ∫ 𝑓 (𝑥) 𝑑𝑥=𝑀𝑎 {[𝜄 (𝑏) −𝜄(𝑎)] ×𝑓} , (19) in 𝐴,is∗-continuous,andhasthesame𝛽-change over any 𝑎 𝛼 𝛼 ∗ two -interval of equal -extent. The -uniform functions the ∗-integral is a weighted ∗-average. arethoseexpressibleintheform𝜄{(𝑚×𝑥)̇ +𝑐}̇,where𝑚 and ∗𝑏 ∫ 𝑓(𝑥)𝑑𝑥 𝛽 𝑐 are constants in 𝐴 and 𝑥 is unrestricted in 𝐴.Bychoosing Furthermore, 𝑎 equals to the -limitofthe 𝛽 𝑛 𝑚=1̇and 𝑐=0̇,weseethat𝜄 is ∗-uniform. It is characteristic -convergent sequence whose th term is ∗ 𝛼 of a -uniform function that, for each -progression of [𝜄 (𝑘 ) ×𝑓̈ (𝑎 )] +⋅⋅⋅̈ +̈[𝜄 (𝑘 ) ×𝑓̈ (𝑎 )] , arguments, the corresponding sequence of values is a 𝛼- 𝑛 1 𝑛 𝑛−1 (20) progression. The ∗-slope of a ∗-uniform function is its 𝛽- ̇ ̇ 𝛼 𝛼 1̇ where 𝑎1,...,𝑎𝑛 is the 𝑛-fold partition of 𝛼-partition of [𝑎,] 𝑏 change over any interval of -extent . For example, the 𝑘 𝑎 −𝑎̇ ,...,𝑎 −𝑎̇ ∗-slope of the function 𝜄{(𝑚×𝑥)̇ +𝑐}̇ turns out to be 𝜄(𝑚).In and 𝑛 isthecommonvalueof 2 1 𝑛 𝑛−1. ∗ 𝜄 1̈ ∗ If 𝛼 is classically continuous function and 𝛽 = 𝐼(𝑥), =𝑥 particular, the -slope of equals ,andthe -slope of a ∗ constant function on 𝐴 equals 0̈. then the -integral is a Stieltjes integral. ∗ 𝑓 [𝑎,̇ ] 𝑏 ̇ ∗ The -gradient of a function over is the -slope Theorem 3. The ∗-derivative and ∗-integral are inversely of the ∗-uniform function containing (𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏)) ∗𝑏 related in the sense indicated by the following two statements. showed as 𝐺𝑎 and turns out to be ̇ ̇ ∗𝑥 (i) If 𝑓 is ∗-continuous on [𝑎,] 𝑏 and 𝑔(𝑥) =∫ 𝑓(𝑡)𝑑𝑡 ∗𝑏 𝑎 𝐺 =[𝑓(𝑏) −𝑓̈ (𝑎)] /[𝜄̈ (𝑏) −𝜄̈(𝑎)]. ̇ ̇ ∗ ̇ ̇ 𝑎 (14) for every 𝑥∈[𝑎,] 𝑏 ,then𝐷 𝑔=𝑓on [𝑎,] 𝑏 . ∗ ̇ ̇ ∗𝑏 ∗ If the following ∗-limit exists, the ∗-derivative of (ii) If 𝐷 ℎ is ∗-continuous on [𝑎,] 𝑏 ,then∫ 𝐷 ℎ(𝑥)𝑑𝑥 = 𝑓[𝐷∗𝑓](𝑎) 𝑎 𝑓 ∗ 𝑎 𝑎 at ,andsaythat is -differentiable at , ℎ(𝑏)−ℎ(𝑎)̈ . [𝐷∗𝑓] (𝑎) =∗ {[𝑓 (𝑏) −𝑓̈ (𝑎)] /[̈𝜄 (𝑏) −𝜄̈(𝑎)]} . -lim𝑥→𝑎 (15) It is convenient to indicate the uniform relationships between the corresponding notions of the ∗-calculus and ∗ If it exists, [𝐷 𝑓](𝑎) is necessarily in 𝐵. classical calculus. ∗ −1 The ∗-derivative 𝐷 𝑓 of 𝑓 is the function that assigns to For each number 𝑎∈𝐴,let𝑎=𝛼̄ (𝑎).Let𝑓 be ∗ ̄ each number in 𝐴 the number [𝐷 𝑓](𝑡), if it exists. afunctionfrom𝐴 into 𝐵,andset𝑓(𝑡) = 𝛽{𝑓(𝛼(𝑡))}. Abstract and Applied Analysis 5

̄ Then, both ∗-lim𝑥→𝑎𝑓(𝑥) and lim𝑡→𝑎̄𝑓(𝑡) exist and follows: ⊕:C∗ × C∗ 󳨀→ C∗ ̄ ∗-lim𝑓 (𝑥) =𝛽[lim 𝑓 (𝑡)]. (21) (24) 𝑥→𝑎 𝑡→𝑎̄ ∗ ∗ ∗ ∗ ̇ ̈ ̈̈ (𝑧1 ,𝑧2 ) 󳨃󳨀→ 𝑧 1 ⊕𝑧2 =(𝑎1̇+𝑎2̇, 𝑏1+𝑏2),

̄ ⊙:C∗ × C∗ 󳨀→ C∗ Furthermore, 𝑓 is ∗-continuous at 𝑎 if and only if 𝑓 is classically continuous at 𝑎.̄ (𝑧∗,𝑧∗) 󳨃󳨀→ 𝑧 ∗ ⊙𝑧∗ ∗𝑏 ̇ ̇ ∗𝑏 1 2 1 2 (25) If 𝐺𝑎 𝑓 is the ∗-gradient of 𝑓 over [𝑎,] 𝑏 ,then𝐺𝑎 𝑓= 𝛽{𝐺𝑏̄}𝑓̄ 𝐺𝑏̄𝑓̄ 𝑓̄ [𝑎,̇ ] 𝑏 ̇ 𝑎̄ ,where 𝑎̄ is the classical gradient of over . =(𝛼(𝑎̇𝑎̇− 𝑏̈𝑏̈),𝛽(𝑎̇𝑏̈+ 𝑏̈𝑎̇)) , ∗ ̄ 1 2 1 2 1 2 1 2 If both [𝐷 𝑓](𝑎) and 𝐷𝑓(𝑎)̄ exist, then we have [𝐷∗𝑓](𝑎) =𝑓( 𝛽[𝐷 ̄𝑎)]̄ 𝑓 ∗ [𝑎,̇ 𝑏]̇ .If is -continuous on ,then ̇ ̇ ̈ ̈ ∗𝑏 𝑏̄ ̄ where 𝑎1, 𝑎2 ∈𝐴and 𝑏1, 𝑏2 ∈𝐵with 𝑀𝑎 𝑓 = 𝛽{𝑀𝑎̄}𝑓 and −1 −1 𝑎1̇=𝛼 (𝑎1̇)=𝛼 (𝛼 1(𝑎 )) = 𝑎1 ∈ R, ∗𝑏 ∗𝑏̄ (26) ̈ −1 ̈ −1 ∫ 𝑓=𝛽(∫ 𝑓)̄ 𝑏1 =𝛽 (𝑏1)=𝛽 (𝛽 1(𝑏 )) = 𝑏1 ∈ R. 𝑎 𝑎̄ ∗ (22) Then, Tekin and Bas¸ar [7, Theorem 2.1] proved that (C ,⊕,⊙) 𝛼−1(𝑏) =𝛽{∫ 𝛽−1 [𝑓 (𝛼 (𝑥))]𝑑𝑥}. is a field. −1 ∗ ∗ ̈ 𝛼 (𝑎) The ∗-distance 𝑑 between any two elements 𝑧1 =(𝑎1̇, 𝑏1) ∗ ̈ ∗ and 𝑧2 =(𝑎2̇, 𝑏2) of the set C is defined by

The rest of the paper is organized as follows. ∗ ∗ ∗ 󸀠 ∗ 𝑑 : C × C 󳨀→ [0,̈̈∞̈)=𝐵̈ ⊂𝐵 In Section 2, it is shown that the set C of non-Newtonian complex (∗-complex) numbers forms a field with the binary (𝑧∗,𝑧∗)󳨀→𝑑∗ (𝑧∗,𝑧∗) operations addition (+)̈and multiplication (×)̈.Further,some 1 2 1 2 basic properties and inequalities which play the basic role ̈ (27) in ∗-convergence and ∗-continuity are proved. Section 3 is √⋅⋅ 2̈ ̈ ̈2 = [𝜄( 𝑎1̇−̇𝑎2̇)] +(̈𝑏1−̈𝑏2) devoted to the space 𝐶∗(Ω) of ∗-continuous functions of a ∗-complex variable. We prove that 𝐶∗(Ω) is a complete metric space with the natural metric and is a Banach √ 2 2 ∗ =𝛽[ (𝑎1 −𝑎2) +(𝑏1 −𝑏2) ]. space with the natural norm and the space 𝐵 (𝐴; 𝐸) of all ∗-bounded mappings from 𝐴 into 𝐸 is a Banach space. As C∗ an application part, in Section 4, we try to create the ∗-inner Here and after, we know that is a field and the distance C∗ product space specifically for (MC) and give an inclusion rela- between two points in is computed by the relation 𝑑∗ 𝑑∗ tion between 𝐶∗(Ω) and the set of ∗-differentiable functions. .Now,wewillseewhetherthisrelation is metric C∗ ∗ In the final section of the paper, we note the significance of over or not, define -norm, and try to obtain some the (NC) and record some further suggestions. required inequalities in the sense of non-Newtonian complex calculus. ∗ ∗ ∗ ∗ ∗ 𝑑 (𝑧 ,𝜃 ) is called ∗-norm of 𝑧 ∈ C and is denoted by ∗ ∗ ̈∗ ̈ 2. -Complex Field and -Inequalities |𝑧 |;thatis,

In this section, following Tekin and Bas,ar [7], we give some ∗ ⋅⋅ 2̈ 2̈ knowledge on the -complex field and some concerning |𝑧̈∗ |̈=𝑑∗ (𝑧∗,𝜃∗)=√[𝜄( 𝑎−̇̇0)]̇ +(̈𝑏̈−̈0)̈ =𝛽(√𝑎2 +𝑏2), inequalities. ̇ ̈ ̈ Let 𝑎̇ ∈ (𝐴, +,̇−,̇×,̇/, ≤)̇and 𝑏 ∈ (𝐵, +,̈−,̈×,̈/, ≤)̈be (28) arbitrarily chosen elements from corresponding arithmetic. ̈ 𝑧∗ =(𝑎,̇𝑏)̈ 𝜃∗ =(0,̇0)̈ Then, the ordered pair (𝑎,̇𝑏) is called as a ∗-point. The set where and . of all ∗-points is called the set ∗-complex numbers and is ∗ ∗ ∗ C Lemma 4 (∗-triangle inequality [7, Lemma 2.3]). Let 𝑧1 ,𝑧2 ∈ denoted by ,thatis, ∗ C .Then, ∗ ∗ ̈ ̈ C := {𝑧 =(𝑎,̇𝑏) : 𝑎∈𝐴⊆̇ R, 𝑏∈𝐵⊆R}. (23) ̈∗ ∗ ̈̈̈∗ ̈̈̈∗ ̈ |𝑧1 ⊕𝑧2 | ≤ |𝑧1 | + |𝑧2 |. (29)

Lemma 5 𝑧∗,𝑧∗ ∈ C∗ |𝑧̈∗⊙𝑧∗|=̈ Define the binary operations addition (⊕) and multiplication ([7,Lemma2.4]). Let 1 2 .Then, 1 2 ∗ ̈ ∗ ̈ |𝑧̈∗|̈×̈|𝑧̈∗|̈ (⊙) of ∗-complex numbers 𝑧1 =(𝑎1̇, 𝑏1) and 𝑧2 =(𝑎2̇, 𝑏2) as 1 2 . 6 Abstract and Applied Analysis

̈̈ ∗ Lemma 6 (∗-Minkowski inequality [7,Lemma2.5]). Let 𝑝≥1 Lemma 9. (𝐶∗(Ω), 𝑑 ) is a metric space. ∗ ∗ ∗ and 𝑧𝑘 ,𝑡𝑘 ∈ C for all 𝑘∈{1,2,3,...,𝑛}.Then, Proof. Let 𝛼 and 𝛽 be the generators on the sets of arguments ̈ ̈ 𝑛 1/𝑝 𝑛 1/𝑝 ⋅⋅ ⋅⋅ and values, respectively. ̈∗ ∗ 𝑝̈ ̈ ̈∗ 𝑝̈ (∑|𝑧𝑘 ⊕𝑡𝑘 | ) ≤(∑|𝑧𝑘 | ) 𝑘=1 𝑘=1 (i) For every 𝑓, 𝑔 ∈𝐶 ∗(Ω) and for every 𝑧∈Ω,wehave (30) ̈ 𝑛 1/𝑝 ∗ ⋅⋅ 𝑑 (𝑓,𝑔) ̈ ̈∗ 𝑝̈ +(∑|𝑡𝑘 | ) . ̈ ̈ 󵄨 −1 󵄨 𝑘=1 = max |𝑓 (𝑧) −𝑔̈(𝑧) | = max 𝛽{󵄨𝛽 [𝑓 (𝑧) −𝑔̈(𝑧)]󵄨} 𝑧∈Ω 𝑧∈Ω 󵄨 󵄨 Theorem 7 (C∗,𝑑∗) (see [7, Theorem 2.6]). is a complete 󵄨 −1 −1 −1 󵄨 ∗ = max 𝛽{󵄨𝛽 (𝛽 {𝛽 [𝑓 (𝑧)]−𝛽 [𝑔 (𝑧)]})󵄨} metric space, where 𝑑 is defined by (27). 𝑧∈Ω 󵄨 󵄨 󵄨 󵄨 󵄨 −1 −1 󵄨 Inthispaper,wemainlyfocusonthesupmetriconthe = max 𝛽{󵄨𝛽 [𝑓 (𝑧)]−𝛽 [𝑔 (𝑧)]󵄨} ∗-complex numbers because ∗-continuity always required to 𝑧∈Ω use that metric relation. Therefore, we present the complete- ̈ ∗ = 0⇐⇒𝑓=𝑔; ness of the set C with respect to the sup metric. (34) ∗ ̈ ̈ Theorem 8. C is a Banach space with the norm ‖⋅‖ defined by that is, (M1) holds.

̈ 2̈ ‖̈𝑧∗ ‖̈= √⋅⋅ [𝜄( 𝑎−̇̇0)]̇2+(̈𝑏̈−̈0)̈, (31) (ii) One can easily see for every 𝑓, 𝑔 ∈𝐶 ∗(Ω) that

∗ ̈ ∗ where 𝑧 =(𝑎,̇𝑏) and 𝜃 =(0,̇0)̈. 𝑑∗ (𝑓,𝑔)

󵄨 −1 −1 󵄨 = max 𝛽{󵄨𝛽 [𝑓 (𝑧)]−𝛽 [𝑔 (𝑧)]󵄨} 3. Continuous Function Space over 𝑧∈Ω 󵄨 󵄨 ∗ the Field C 󵄨 󵄨 (35) 󵄨 −1 −1 󵄨 = max 𝛽{󵄨𝛽 [𝑔 (𝑧)]−𝛽 [𝑓 (𝑧)]󵄨} In this section, we construct the space of continuous func- 𝑧∈Ω ∗ C ̈ ̈ ∗ tions over the field and show that this space is a complete = max|𝑔 (𝑧) −𝑓̈ (𝑧) | =𝑑 (𝑔, 𝑓) , metric space with max metric such that 𝑧∈Ω 𝑑∗ (𝑓,𝑔) = |𝑓̈(𝑧) −𝑔̈(𝑧) |.̈ max∗ (32) 𝑧∈C which shows that the symmetry axiom (M2) also Itwouldnotbetoohardtofindoutthatthespaceof∗- holds. continuous functions creates a normed space with the norm (iii) By a routine verification for every 𝑓, 𝑔, ℎ ∈𝐶 ∗(Ω),if −1 ̈ ̈ reduced from the sup metric. Finally, we investigate the we apply 𝛽 to |𝑓(𝑧)−𝑔(𝑧)̈ | completeness property of the spaces of ∗-bounded and ∗- continuous functions. ∗ ̈ ̈ ∗ 𝑑 (𝑓,𝑔) = max|𝑓 (𝑧) −𝑔̈(𝑧) | Let Ω⊂C be compact. Then, by 𝐶∗(Ω),wedenotethe 𝑧∈Ω space of ∗-continuous functions defined on the set Ω.One 󵄨 −1 −1 󵄨 can easily see that the set 𝐶∗(Ω) forms a vector space over = max𝛽{󵄨𝛽 [𝑓 (𝑧)]−𝛽 [𝑔 (𝑧)]󵄨} (36) ∗ 𝑧∈Ω 󵄨 󵄨 C with respect to the algebraic operations addition (+) and 󵄨 󵄨 scalar multiplication (×) defined on 𝐶∗(Ω) as follows: ̈ ̈ ̈ 󵄨 −1 −1 󵄨 𝑢= |𝑓 (𝑧) −𝑔 (𝑧) | =𝛽{󵄨𝛽 [𝑓 (𝑧)]−𝛽 [𝑔 (𝑧)]󵄨}, +:𝐶∗ (Ω) ×𝐶∗ (Ω) 󳨀→ 𝐶 ∗ (Ω) (𝑓,𝑔) 󳨃󳨀→ ( 𝑓 + 𝑔) (𝑧) =𝑓(𝑧) +𝑔̈(𝑧) , then we obtain that

−1 𝑓=𝑓(𝑧) , 𝑔=𝑔(𝑧) ∈𝐶∗ (Ω) , 𝛽 (𝑢) ∗ (33) 󵄨 −1 −1 󵄨 ×:C ×𝐶∗ (Ω) 󳨀→ 𝐶 ∗ (Ω) 󵄨 󵄨 = 󵄨𝛽 [𝑓 (𝑧)]−𝛽 [𝑔 (𝑧)]󵄨 ̈ 󵄨 󵄨 (𝛼, 𝑓 (𝑧)) 󳨃󳨀→ ( 𝛼 × 𝑓) (𝑧) =𝛼×𝑓(𝑧) , 󵄨 −1 −1 −1 −1 󵄨 = 󵄨𝛽 [𝑓 (𝑧)]−𝛽 [ℎ (𝑧)] +𝛽 [ℎ (𝑧)] −𝛽 [𝑔 (𝑧)]󵄨 𝑧∗ =(𝑧∗)∈𝜔∗,𝛼∈C∗. 󵄨 󵄨 𝑘 −1 󵄨 −1 −1 󵄨 ≤(𝛽∘𝛽 ) 󵄨𝛽 [𝑓 (𝑧)]−𝛽 [ℎ (𝑧)]󵄨 In order to show that 𝐶∗(Ω) is a metric space with the 󵄨 󵄨 ∗ −1 󵄨 −1 −1 󵄨 metric 𝑑 defined by (32), we give the following lemma. +(𝛽∘𝛽 ) 󵄨𝛽 [ℎ (𝑧)] −𝛽 [𝑔 (𝑧)]󵄨 Abstract and Applied Analysis 7 󵄨 󵄨 −1 󵄨 −1 −1 󵄨 =𝛽 (𝛽 {󵄨𝛽 [𝑓 (𝑧)]−𝛽 [ℎ (𝑧)󵄨}) in some cases, but not every time when we want. In [3], 󵄨 󵄨 Uzer showed by using multiplicative calculus which is a kind −1 󵄨 −1 −1 󵄨 +𝛽 (𝛽 {󵄨𝛽 [ℎ (𝑧)] −𝛽 [𝑔 (𝑧)󵄨}) of non-Newtonian calculus that it is more flexible than the classical calculus for Bessel functions in a special domain. −1 ̈ ̈ =𝛽 (|𝑓 (𝑧) −ℎ̈ (𝑧) |) We can reproduce more examples for the same situation but we mainly focused on the theoretical properties of the space −1 ̈ ̈ ̈ +𝛽 (|ℎ (𝑧) −𝑔 (𝑧) |), 𝐶∗(Ω). (37) Theorem 11. 𝐶∗(Ω) is a normed space with the norm given by which yields by applying 𝛽 that 󵄩 󵄩∗ ̈ ̈ 󵄩𝑓󵄩 = max|𝑓 (𝑧) |;𝑓=𝑓(𝑧) ∈𝐶∗ (Ω) . 󵄩 󵄩 𝑧∈Ω (41) 𝑢 ≤𝛽{𝛽̈ −1 (|𝑓̈(𝑧) −ℎ̈ (𝑧) |)+𝛽̈ −1 (|ℎ̈(𝑧) −𝑔̈(𝑧) |)}̈ 𝑓, 𝑔 ∈𝐶 (Ω) 𝜆∈C (38) Proof. Let ∗ and .Then,thefollowinghold. ̈ ̈ ̈̈̈ ̈ ̈ = |𝑓 (𝑧) −ℎ(𝑧) |+|ℎ (𝑧) −𝑔 (𝑧) |. (i) One can easily show that 󵄩 󵄩∗ ̈ ̈ ̈ ̈ Therefore, by taking maximum, one can derive that 󵄩𝑓󵄩 = 0⇐⇒max |𝑓 (𝑧) | = 0 ∗ ̈ ̈ ⇐⇒ |𝑓̈(𝑧) |̈= 0∀𝑧∈Ω̈ 𝑑 (𝑓,𝑔) = max |𝑓 (𝑧) −𝑔̈ (𝑧) | 𝑧∈Ω 󵄨 󵄨 󵄨 −1 󵄨 ̈ ̈ ̈ ̈ ̈ ⇐⇒ 𝛽 { 󵄨𝛽 [𝑓 (𝑧)]󵄨}=0∀𝑧∈Ω ≤̈max|𝑓 (𝑧) −ℎ̈ (𝑧) | +̈max|ℎ (𝑧) −𝑔̈(𝑧) | (39) 󵄨 󵄨 𝑧∈Ω 𝑧∈Ω (42) 󵄨 −1 󵄨 ⇐⇒ 󵄨𝛽 [𝑓 (𝑧)]󵄨 =0 ∀𝑧∈Ω =𝑑∗ (𝑓,ℎ) +𝑑̈∗ (ℎ, 𝑔) . 󵄨 󵄨 ⇐⇒ 𝛽 −1 [𝑓 (𝑧)]=0 ∀𝑧∈Ω This means that the triangle inequality (M3) also holds. ∗ Therefore, since (i)–(iii) are satisfied, 𝑑 is a metric on ⇐⇒ 𝑓 (𝑧) = 0∀𝑧∈Ω.̈ 𝐶∗(Ω).Thiscompletestheproof. That is to say that the axiom (N1) holds. Definition 10. A ∗-norm is a nonnegative ∗-real valued ∗ function on Ω⊂C whose value at an 𝑥∈Ωis denoted (ii) From the property of vector space axioms of the space ∗ ∗ 𝐶 (Ω) by ‖𝑥‖ ,thatis,‖⋅‖ :Ω → R, and satisfies the following ∗ ,itisimmediatethat conditions: 󵄩 󵄩∗ ̈ ̈ 󵄩𝜆 ×𝑓̈ 󵄩 = max |𝜆 ×𝑓̈ (𝑧) | ∗ 𝑧∈Ω (N1) ‖𝑥‖ = 0⇔𝑥=̈ 0̈, ∗ ∗ ̈ ̈ ̈ ̈ ‖𝜆 ×𝑥‖̈ = |𝜆̈|̈×‖𝑥‖̇ = |𝜆| ×̇max |𝑓 (𝑧) | (43) (N2) (absolute homogeneity), 𝑧∈Ω ‖𝑥+𝑦‖̈ ∗≤‖𝑥‖̈ ∗+‖𝑦‖̈ ∗ (N3) (triangle inequality), ̈ ̈ 󵄩 󵄩∗ = |𝜆| ×̈󵄩𝑓󵄩 . for all 𝑥, 𝑦 ∈Ω and for all scalars 𝜆. ∗ 𝐶 (Ω) 𝑑∗ 𝐶 (Ω) Hence, the absolute homogeneity axiom (N2) also The -norm on ∗ defines a metric on ∗ given holds. by (iii) It is obtained by the similar way used in the proof of ∗ 󵄩 ̈󵄩∗ 𝑑 (𝑓,𝑔) = 󵄩𝑓−𝑔󵄩 ,𝑓,𝑔∈𝐶∗ (Ω) , (40) Lemma 9 that 󵄩 󵄩∗ ̈ ̈ 󵄩𝑓+𝑔̈󵄩 = max|𝑓 (𝑧) +𝑔̈(𝑧) | andiscalledtheinduced∗-metric by the ∗-norm. 󵄩 󵄩 𝑧∈Ω The definition of space of continuous functions makes ̈ ̈ ̈ ̈ ≤̈max|𝑓 (𝑧) |+̈max |𝑔 (𝑧) | (44) it possible to give a much more intuitive meaning to the 𝑧∈Ω 𝑧∈Ω classical notion of uniform convergence. Convergence in the 󵄩 󵄩∗ 󵄩 󵄩∗ space of continuous functions space turns into the uniform = 󵄩𝑓󵄩 +̈󵄩𝑔󵄩 . convergence. One of the most important results of the conceptofthespaceofcontinuousfunctionsisthefamous This means that the triangle inequality axiom (N3) is satisfied. ∗ Stone-Weierstrass approximation theorem which is a very Since (i)–(iii) are fulfilled, ‖⋅‖ , defined by (41), is a norm powerful tool for proof of general results on continuous for the space 𝐶∗(Ω). functions. Using this theorem, we can prove some results ∗ fits for functions of special type and later extend them Definition 12. Let 𝐴 be any set and let 𝐸⊂C be a complex to all continuous functions by a density argument. In this normed space. A mapping 𝑓 from 𝐴 into 𝐸 is bounded if 𝑓(𝐴) ∗ paper, we show, with the rules of non-Newtonian calculus, is bounded in 𝐸,orequivalentlyifmax𝑧∈𝐴‖𝑓(𝑧)‖ is finite. its advantages to Stone-Weierstrass theorem in the space of The set of all bounded mappings from 𝐴 into 𝐸 is denoted ∗ 𝐶∗(Ω),ornot.Theanswerofthisquestionisaffirmative by 𝐵 (𝐴; 𝐸). 8 Abstract and Applied Analysis

−1 −1 Corollary 13. Thesetofallboundedmappingsfrom𝐴 into 𝐸 𝛽 [𝑓(𝑧)] = lim𝑛→∞𝛽 [𝑓𝑛(𝑧)] for each 𝑧∈Ω.Since,given ∗ is denoted by 𝐵 (𝐴; 𝐸) is a complex vector space, since any 𝜀>0, 󵄩 󵄩∗ 󵄩 󵄩∗ 󵄩 󵄩∗ ∗ 󵄨 󵄨 󵄩𝑓+𝑔̈󵄩 ≤̈󵄩𝑓󵄩 +̈󵄩𝑔󵄩 ;𝑓,𝑔∈𝐵(𝐴; 𝐸) . 󵄨 −1 −1 󵄨 −1 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 (45) 󵄨𝛽 [𝑓𝑛 (𝑧)]−𝛽 [𝑓𝑚 (𝑧)]󵄨 <𝛽 (𝜀) (49)

Moreover, on this space, by letting 𝑛→∞, we obtain independent of 𝑧 that 󵄩 󵄩∗ ∗ 󵄨 󵄨 max 󵄩𝑓 (𝑧)󵄩 ,𝑓∈𝐵(𝐴; 𝐸) , (46) 󵄨 −1 −1 󵄨 −1 𝑧∈𝐴 󵄨𝛽 [𝑓 (𝑧)]−𝛽 [𝑓𝑚 (𝑧)]󵄨 <𝛽 (𝜀) , (50)

−1 −1 is a norm, as can be easily verified. for sufficiently large 𝑚.Hence,𝛽 [𝑓𝑛(𝑧)] → 𝛽 [𝑓(𝑧)] Ω Theorem 14. 𝐵∗(𝐴; 𝐸) 𝐸 uniformly in . is a Banach space if is a Banach Finally, since the limit of a uniformly convergent se- space. quence of continuous functions is continuous, then 𝑓∈ ∞ ∗ 𝐶∗(Ω) and 𝑓𝑛 →𝑓as 𝑛→∞.Thiscompletestheproof. Proof. Let (𝑓𝑛)𝑛=0 be a Cauchy sequence in 𝐵 (𝐴; 𝐸).Then, ̈̈ ∗ ̈ for any 𝜀 > 0,thereisan𝑛0 ∈ N such that ‖𝑓𝑛−𝑓̈𝑚‖ <𝜀 for all 𝑚, 𝑛0 ≥𝑛 .Fromsupnorm,itfollowsforany𝑧∈𝐴 ∗ ̈ that we have ‖𝑓𝑛(𝑧) −𝑓̈𝑚(𝑧)‖ <𝜀 for 𝑛, 𝑚0 ≥𝑛 .Hence,the 4. Applications ∞ sequence {𝑓𝑛(𝑧)}𝑛=0 converges to an element 𝑓(𝑧) ∈𝐸,since ∗ ̈ In this section, we study some properties of multiplicative 𝐸 iscomplete.Furthermore,wehave‖𝑓𝑛(𝑧) −̈ 𝑓(𝑧)‖ <𝜀 for calculus which is a kind of (NC). This calculus may be created any 𝑧∈𝐴and 𝑛≥𝑛0.Bythe∗-triangle inequality given by ∗ ̈ ∗ by taking 𝛼=𝐼, the identity function, and 𝛽=exp, Lemma 4,wefirstdeducethat‖𝑓(𝑧)‖ <‖𝑓𝑛(𝑧)‖ +𝜀̈ for all ∗ ̈ the exponential function. Multiplicative calculus, in short 𝑧∈𝐴;hence𝑓 is bounded. Moreover, we have ‖𝑓𝑛−𝑓‖̈ <𝜀for ∞ (MC), has many applications in some branch of mathematics all 𝑛≥𝑛0 and this means that the sequence (𝑓𝑛)𝑛=0 converges ∗ to 𝑓 in the space 𝐵 (𝐴; 𝐸). such as financial mathematics and elasticity. In the present paper, we investigate the complex multiplicative functions It is known from Mathematical Analysis that when we in a complex domain to make rational approximation for ∞ applied to Non-Newtonian calculus if (𝑓𝑛)𝑛=0 is a sequence of analytic functions. Later, as an application, we mention the functions from 𝐴 into a metric space 𝐸,wesaythesequence inner product property of (MC). (𝑓 )∞ ∗ 𝑓 𝐴 𝐸 𝑛 𝑛=0 is -convergent to a function from into if, for Let 𝑓 be a single-valued function defined on a set Ω which 𝑧∈𝐴 {𝑓 (𝑧)}∞ ∗ 𝐸 𝑓(𝑧) each ,thesequence 𝑛 𝑛=0 -converges in to ; is dense itself; that is, every point of Ω is a limit point of (𝑓 )∞ ∗ 𝐴 𝑓 we call that 𝑛 𝑛=0 -converges uniformly on to if the Ω.Then,𝑓 is said to be locally analytic on Ω if, given any followingequalityholds: 𝑧0 ∈Ω, there is a neighborhood ℵ(𝑧0) and a power series ∞ 𝑛 ∞ 𝑛 ∑ 𝑎𝑛(𝑧 −0 𝑧 ) such that 𝑓(𝑧0)=∑ 𝑎𝑛(𝑧 −0 𝑧 ) for all |𝑓̈ (𝑧) −𝑓̈ (𝑧) |̈= 0.̈ 𝑛=0 𝑛=0 𝑛→∞lim sup 𝑛 𝑧∈ℵ(𝑧)∩Ω 𝑧∈𝐴 (47) 0 . The concept of a locally analytic function on the domain Ω reduces to a single-valued analytic function. It is obvious that ∗-uniform convergence implies ∗-simple In (NC), particularly in (MC), when we consider single- convergence; however, the converse is not true. If 𝐸 is a valued analytic functions on a domain, we consider the 𝑛 ∗ ∞ (𝑧−𝑧0) ∗(𝑛) 1/𝑛! normed space, then -convergence of a sequence of functions product, 𝑓(𝑧0)=∏ 𝑎 ,where𝑎𝑛 =(𝑓 (𝑧0)) is 𝐵∗(𝐴; 𝐸) ∗ 𝑛=0 𝑛 in , therefore, corresponds to -uniform conver- the 𝑛th coefficient of the Taylor product of the function 𝑓 at gence of the sequence in 𝐴. ∗(𝑛) 𝑧=𝑧0 such that 𝑓 is the 𝑛th order ∗-derivative of the Finally, we give the theorem on the completeness of the 𝑓 𝐶 (Ω) ∗ function . space ∗ of -continuous functions. This is an application to show rational approximation ∗ also applicable for ∗-calculus (Multiplicative calculus). One Theorem 15. 𝐶∗(Ω) is a Banach space with the norm ‖⋅‖ question arises that which way of approximation is better, defined by (41). the classical one or the new one? Uzer made some numerical (𝑓 )∞ 𝐶 (Ω) solutions for the Bessel differential equations in3 [ ]and Proof. Let 𝑛 𝑛=0 beanyCauchysequencein ∗ .Foreach 𝐽 (𝑥+𝑖𝑦) 𝑧∈Ω,wehave he considered the function 5 with the modulus ‖𝐽5(𝑥+𝑖𝑦)‖. It is suggested that if the solution function varies ̈ ̈̈̈ ∗̈ |𝑓𝑛−𝑓̈𝑚 | ≤ |𝑓𝑛 (𝑧) −𝑓̈𝑚 (𝑧) | exponentially along a specific contour, then the method in 󵄨 󵄨 the (MC) sense shows a better performance. Otherwise, that 󵄨 −1 −1 󵄨 ⇐⇒ 󵄨𝛽 [𝑓𝑛 (𝑧)]−𝛽 [𝑓𝑚 (𝑧)]󵄨 (48) is if the solution function is oscillatory or linearly varying, 󵄩 󵄩 themethodinthe(CC)sensewillbebetter.Thesolution 󵄩 −1 −1 󵄩 𝐽 (𝑧) ≤ 󵄩𝛽 [𝑓𝑛 (𝑧)]−𝛽 [𝑓𝑚 (𝑧)]󵄩 , 5 in the given example exhibits exponential variations everywhere on the complex plane except near the real axis. and so {𝑓𝑛(𝑧)} is a Cauchy sequence of 𝑁-real numbers, and The other elements of the family of Bessel functions also −1 {𝛽 [𝑓𝑛(𝑧)} is a Cauchy sequence of real numbers as well; exhibit exponential variations. Indeed, there are many other hence they are ∗-convergent and convergent, respectively. functions exhibiting exponential variations on the complex ∗ −𝑧 −1 Let 𝑓:Ω → C be defined by 𝑓(𝑧) = lim𝑛→∞𝑓𝑛(𝑧) or plane such as the famous sigmoid function 𝜎(𝑧) = (1 +𝑒 ) Abstract and Applied Analysis 9 which plays an important role in decision making in Neutral If 𝑧 = (𝑥,, 𝑦) then, using again the equalities (24)and Networks. [10, p. 88], we conclude that ∗ As a second application of (MC), we mention the -inner 𝑧 ×̇𝑧=(̄ 𝑥,̇𝑦)̈×(̇𝑥,̇−𝑦)̈ product property. A ∗-inner product space 𝑋 is a vector space ⟨𝑧, 𝑤⟩ ∗ ‖⋅‖∗ with an inner product ∗ defined on it. A -norm =(𝛽(𝑥2 +𝑦2),0)̈ ∗ ∗ ̇ is defined by ‖𝑧‖ = √⟨𝑧, ∗𝑧⟩ and if ⟨𝑧, 𝑤⟩∗ = 0 holds, 𝑧 𝑤 ∗ ∗ (53) then and are called -orthogonal vectors. A -Hilbert =𝛽(𝑥2 +𝑦2) space 𝐻 is a complete ∗-inner product space. The spaces to be considered are defined as follows. 2̈ = |𝑧̈|̈. ∗ ∗ Definition 16. Let 𝑋 be a vector space over the field C or R . 2 2 Thus, in (MC), we have 𝑧⊙𝑧=̄ exp(𝑥 +𝑦 ). A ∗-inner product on 𝑋 is a mapping from 𝑋×𝑋into the ∗ ∗ ∗ 𝑋 scalar field 𝐾=C (or R )of𝑋;thatis,witheverypairof In a (real or complex) -inner product space ,two 𝑦, 𝑧 ∈𝑋 𝑧⊥𝑤 vectors 𝑧 and 𝑤,thereisascalar⟨𝑧, 𝑤⟩∗ called the ∗-inner vectors are called orthogonal and we write ⟨𝑧, 𝑤⟩ = 0̇ 𝐴⊆𝑋 𝐴⊥ product of 𝑧 and 𝑤, such that for all vectors 𝑦, 𝑧, 𝑤 and any provided ∗ .Forasubset ,theset is scalar 𝛼, the following axioms hold: defined by ⊥ ̇ 𝐴 ={𝑢∈𝑋|⟨𝑢,𝑧⟩∗ = 0∀𝑧∈𝐴}. (54) (IP1) ⟨𝑦 +𝑧,𝑤⟩̇ ∗ =⟨𝑦,𝑤⟩∗+̇ ⟨𝑧, 𝑤⟩∗, ̇ ̇ (IP2) ⟨𝛼 ×𝑧,𝑤⟩∗ =𝛼× ⟨𝑧, 𝑤⟩∗, Corollary 19. A multiplicative inner product space satisfies the parallelogram equality. Let 𝑧=(𝑥,̇𝑦)̇and 𝑤=(𝑢,̇V̇) such that (IP3) ⟨𝑧, 𝑤⟩∗ = ⟨𝑤, 𝑧⟩∗, 𝛼−1(𝑥)̇ = 𝑥, 𝛼−1(𝑦)̇ = 𝑦, 𝛼−1(𝑢)̇ = 𝑢,−1 𝛼 (V̇)=V ̇̇ ̇ ̇ .Consider (IP4) ⟨𝑧, ∗𝑧⟩ ≥ 0 and ⟨𝑧, 𝑧⟩∗ = 0⇔𝑧=0. ̈ 2̈̈ ̈ 2̈̈ Then, we say that 𝑋 is an inner product space provided (IP1)– ‖𝑧 +𝑤̈ ‖ + ‖𝑧 −𝑤̈ ‖ (IP4) hold. Here, ⟨𝑤, 𝑧⟩∗ denotes the complex conjugate of ̈ ̈ ̈ ̇ ̈ 2̈ ̈̈ ̇ ̈ 2̈ ⟨𝑤, 𝑧⟩∗. The conjugate of a ∗-complex number 𝑧=(𝑥,̇𝑦)̈ = ‖(𝑥̇+ 𝑢,̇𝑦̈+ V̈) ‖ + ‖(𝑥̇− 𝑢,̇𝑦̈− V̈) ‖ is 𝑧=(̄ 𝑥,̇ 𝑦) − ̈. Note that (IP2) and (IP3) imply that 2̈ 2̈ 2̈ 2̈ ⟨𝑧, 𝛼 ×𝑤⟩̇ ∗ = 𝛼 ×̇ ⟨𝑧, 𝑤⟩∗. =(𝜄[𝑥̇+̇𝑢])̇ ,(𝑦̈+̈V̈) +(𝜄[̈ 𝑥̇−̇𝑢])̇ ,(𝑦̈−̈V̈)

In (MC), the ∗-inner product properties turn into =𝛽{(𝑥+𝑢)2 +(𝑦+V)2} ⟨𝑦 +𝑧,𝑤⟩̇ =⟨𝑦,𝑤⟩ ⋅ ⟨𝑧, 𝑤⟩ (IP(MC)1) ∗ ∗ ∗, +𝛽̈ ( { 𝑥−𝑢)2 +(𝑦−V)2} ⟨𝑧,𝑤⟩∗ (IP(MC)2) ⟨𝛼 ×𝑧,𝑤⟩̇ ∗ =𝛼 , =𝛽{𝛽−1 [𝛽 {(𝑥+𝑢)2 +(𝑦+V)2}] (IP(MC)3) ⟨𝑧, 𝑤⟩∗ = ⟨𝑤, 𝑧⟩∗, ⟨𝑧, 𝑧⟩ ≥1 ⟨𝑧, 𝑧⟩ =1⇔𝑧=1 (IP(MC)4) ∗ and ∗ ; +𝛽−1 [𝛽 {(𝑥−𝑢)2 +(𝑦−V)2}]} (55) then we say that 𝑋 is multiplicative inner product space. =𝛽{(𝑥+𝑢)2 +(𝑦+V)2 + (𝑥−𝑢)2 +(𝑦−V)2} Corollary 17. It can easily be seen from the equality (24) and from [10]that =𝛽[2(𝑥2 +𝑢2 +𝑦2 + V2)] ∗ ∗ 𝑧 +𝑧̇ =(𝑎 +𝑎̇ ,𝑏 +𝑏̈ )=(𝑎 +𝑎,𝑏 ⋅𝑏), −1 2 2 −1 2 2 1 2 1 2 1 2 1 2 1 2 =𝛽{2[𝛽 (𝛽 (𝑥 +𝑦 )) + 𝛽 (𝛽 (𝑢 + V ))]} ∗ ̇∗ 𝑧1 ×𝑧2 (51) 2̈ 2̈ =𝛽{𝛽−1 [𝛽 (2)]𝛽−1 (‖𝑧̈‖̈)+𝛽−1 (‖𝑤̈‖̈)} =(𝑎1 ⋅𝑎2 − ln 𝑏1 ⋅ ln 𝑏2, exp {𝑎1 ⋅ ln 𝑏2 +𝑎2 ⋅ ln 𝑏1}) , 2̈ 2̈ =𝛽[𝛽−1 (2)̈ ⋅ 𝛽−1 (‖𝑧̈‖̈+̈‖𝑤̈‖̈)] where 𝑧1 =(𝑎1,𝑏1), 𝑧2 =(𝑎2,𝑏2) and 𝛼=𝐼, 𝛽=exp.For 2 example, if 𝑧1 =(1,𝑒)and 𝑧2 =(1,𝑒), then one has ̈ ̈ 2̈̈ ̈ 2̈̈ ∗ ̇∗ = 2×[̈‖𝑧‖ +̈‖𝑤‖ ]. 𝑧1 ×𝑧2 2 2 7 𝛼=𝐼 𝛽= =(3⋅1−ln 𝑒⋅ln 𝑒 , exp {3 ⋅ ln 𝑒 +1⋅ln 𝑒}) = (1, 𝑒 ). Of course, if we take and exp,wecaneasilyobtain these results for (MC).

(52) ∗ Definition 20. Let (𝐸, ‖ ⋅ ‖ ) be a ∗-normed space. If the cor- ∗ ∗ Remark 18. Since the product of (0,̇1)̈with itself equals responding metric 𝑑 is complete, we say that (𝐸, ‖ ⋅ ‖ ) is a ̈ ∗ ̇ (−1,̇ 0), we may define 𝑖 to be (0, −1)̈ .Ofcourse,the Banach space. If (𝐸, ⟨⋅, ⋅⟩∗) is an ∗-inner product space whose ̇ ̈ product of (0, −1)̈ with itself also equals (−1,̇ 0). Therefore, in corresponding metric is complete, we say that (𝐸, ⟨⋅, ⋅⟩∗) is a ∗ −1 (MC),𝑖 turns out (0, 𝑒 ). ∗-Hilbert space. 10 Abstract and Applied Analysis

Theorem 21 (Cauchy-Schwartz inequality). For all 𝑧= which gives by applying 𝛽 that ∗ (𝑥,̇𝑦),̈ 𝑤𝑢, =( ̇V̈)∈C ,thefollowinginequalityholds: ̈ 2 2√ 2 2 ⟨𝑧, 𝑤⟩∗ ≤𝛽(√𝑥 +𝑦 𝑢 + V )

∗ ∗ |⟨𝑧,̈ 𝑤⟩ |≤̈̈‖𝑧‖ ×̈‖𝑤‖ , (56) ∗ =𝛽{𝛽−1 [𝛽 (√𝑥2 +𝑦2)] ⋅ 𝛽−1 [𝛽√ ( 𝑢2 + V2)]}

−1 −1 −1 −1 where 𝛼 (𝑥)̇ = 𝑥, 𝛼 (𝑦)̈ = 𝑦, 𝛼 (𝑢)̇ = 𝑢,and𝛼 (V̈)=V. =𝛽(√𝑥2 +𝑦2) ×𝛽̈ (√𝑢2 + V2)

𝑧=(𝑥,̇𝑦),̈ 𝑤𝑢, =( ̇V̈)∈C∗ Proof. Let .Then, 󵄩 󵄩∗ ∗ = 󵄩(𝑥,̇𝑦)̈󵄩 ×̈‖(𝑢,̇V̈)‖

= ‖𝑧‖∗×‖𝑤‖̈ ∗, ⟨𝑧, 𝑤⟩∗ =⟨(𝑥,̇𝑦)̈ , (𝑢,̇V̈) ⟩∗ (59) =𝜄(𝑥̇×̇𝑢)̇+(̈𝑦̈×̈V̈) which completes the proof. =𝛽{𝛼−1 (𝑥̇×̇𝑢)}̇ Theorem 21 gives the following.

Corollary 22. The space 𝐶∗(Ω) is an ∗-inner product space +𝛽̈ −1{𝛽 (𝑦)̈ ⋅−1 𝛽 (V̈)} but is not a Hilbert space with the integral metric defined by ∗ =𝛽{𝛼−1 (𝛼 {𝛼−1 (𝑥̇) ⋅𝛼−1 (𝑢̇)})} ⟨𝑓,𝑔⟩∗ = ∫ 𝑓 (𝑧) 𝑔(𝑧)𝑑𝑧, (60) 𝑧∈Ω +𝛽̈ −1{𝛽 (𝑦)̈ ⋅−1 𝛽 (V̈)} where 𝑓, 𝑔 ∈𝐶 ∗(Ω). −1 −1 =𝛽{𝛼 (𝑥̇) ⋅𝛼 (𝑢̇)} The proof is easily obtained by the appropriate verifi- cations. Indeed, if we take 𝛼=𝐼=𝛽,weobtainthe +𝛽̈ −1{𝛽 (𝑦)̈ ⋅−1 𝛽 (V̈)} classical calculus (CC) and, in (CC), the results are the same for Corollary 22. It is an expected situation, because (CC) =𝛽{𝛼−1 (𝛼 (𝑥)) ⋅𝛼−1 (𝛼 (𝑢))} is a kind of (NC) and we cannot generalize the assertion of Corollary 22 differently. Let us consider the space 𝐶[−1, 1] with the inner product +𝛽̈ −1{𝛽 (𝛽 (𝑦)) ⋅𝛽−1 (𝛽 (V))} 1 ⟨𝑓, 𝑔⟩ = ∫ 𝑓 (𝑧) 𝑔 (𝑧)𝑑𝑧, (61) =𝛽{𝑥⋅𝑢} +̈ 𝛽 {𝑦 ⋅ V} 0

1 −1 −1 ‖𝑓‖ = √∫ |𝑓(𝑧)|2𝑑𝑧 =𝛽{𝛽 (𝛽 {𝑥⋅𝑢})+𝛽 (𝛽 {𝑦 ⋅ V})} whichgivestheassociatednorm 0 .The inner product space is not complete; the space of Riemannian = 𝛽 {𝑥 ⋅ 𝑢 +𝑦⋅ V}=𝜁. integrable functions on the interval [−1; 1] that are square- integrable, that is, (57) 1 󵄩 󵄩2 󵄨 󵄨2 󵄩𝑓󵄩 =∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 < ∞, (62) −1 𝛽−1 If we apply the function of to (57), then we have is not complete. As a final application, we give an inclusion relation 𝐶 (Ω) 𝐶󸀠 (Ω) −1 between the spaces ∗ and ∗ , the space of first-order 𝛽 (𝜁) =𝑥⋅𝑢+𝑦⋅V ∗-differentiable functions in Ω. 󵄨 󵄨 Theorem 23. 𝐶󸀠 (Ω) ⊂ 𝐶 (Ω) ≤ |𝑥⋅𝑢| + 󵄨𝑦⋅V󵄨 ∗ ∗ and the inclusion is strict. 𝑓 ∗ ≤ √𝑥2𝑢2 +𝑦2V2 Proof. In [10], if is a -continuous function in a given point 𝑎, then from the definition we have ∗-lim𝑥→𝑎𝑓(𝑥) = ∞ (58) 𝑏=𝑓(𝑎)if and only if {𝑓(𝑎𝑛)}𝑛=0 is 𝛽-convergent to 𝑓(𝑎) √ 2 2 2 2 2 2 2 2 ∞ ≤ 𝑥 𝑢 +𝑦 V +𝑥 V +𝑦 𝑢 whenever any sequence (𝑎𝑛)𝑛=0 of arguments is 𝛼-convergent to 𝑎. = √(𝑥2 +𝑦2)(𝑢2 + V2) Now, suppose that 𝑓 is a ∗-differentiable function in a given point 𝑎∈Ω.Then,thefollowinglimit = √𝑥2 +𝑦2√𝑢2 + V2, ∗ {[𝑓 (𝑥) −𝑓̈ (𝑎)] /[𝜄̈ (𝑥) −𝜄̈(𝑎)]} -lim𝑥→𝑎 (63) Abstract and Applied Analysis 11

∗ exists and is equal to the unique number [𝐷 𝑓](𝑎). There- The ∗-continuity of 𝑓 given by (69)isobtainedfrom ∞ 𝑥 fore for every infinite sequence (𝑎𝑛)𝑛=0 of arguments of uniform convergence of the function 𝑒 . Besides, as we distinct from 𝑎,is𝛼-convergent to 𝑎 which implies that already know from [10], and so forth, multiplicative differen- ̈ ̈ ̈ ∗ [𝑓(𝑎𝑛)−𝑓(𝑎)]/[𝜄(𝑎𝑛)−𝜄(𝑎)] is 𝛽-convergent to [𝐷 𝑓](𝑎) as tiation has a relationship between the classical differentiation 𝑛→∞.Itcanbewrittenas such as ∗ ̈ [𝐷 𝑓] (𝑎) =∗ {[𝑓 (𝑎𝑛) −𝑓̈ (𝑎)] /[𝜄(𝑎𝑛) −𝜄̈(𝑎)]} -lim𝑛→∞ ∗ 󸀠 𝑓󸀠(𝑥)/𝑓(𝑥) 𝑓 (𝑥) = exp [ln 𝑓 (𝑥)] =𝑒 . (70) 𝛽−1 [𝑓 (𝑎 )] − 𝛽−1 [𝑓 (𝑎)] =∗ =𝛽{ 𝑛 }. -lim𝑛→∞ −1 −1 ∗ 𝛽 [𝜄(𝑎 𝑛)] − 𝛽 [𝜄 (𝑎)] Therefore, this formula does not allow -differentiability to (64) the function (69), too. −1 If we apply 𝛽 to (64) and consider the iota function 𝜄=𝛽∘ −1 5. Conclusion 𝛼 ,weobtainthat 𝛽−1 ([𝐷∗𝑓] (𝑎)) One of the purposes of this work is to extend the classical calculus to the non-Newtonian real calculus for dealing with −1 −1 𝛽 [𝑓 𝑛(𝑎 )]−𝛽 [𝑓 (𝑎)] complex valued functions. Some of the analogies between = lim 𝑛→∞ 𝛽−1 [𝜄(𝑎 )] − 𝛽−1 [𝜄 (𝑎)] (CC) and the (NC) are demonstrated by theoretical examples. 𝑛 Wederiveclassicalcontinuousfunctionspaceinthesense −1 −1 −1 ∗ of non-Newtonian calculus and try to understand their 󳨐⇒ ( 𝛽 [𝜄(𝑎 𝑛)]−𝛽 [𝜄 (𝑎)])𝛽 {[𝐷 𝑓] (𝑎)} (65) structure of being non-Newtonian vector space. Generally, −1 −1 we work on the vector spaces which concern physics and = lim {𝛽 [𝑓 𝑛(𝑎 )] − 𝛽 [𝑓 (𝑎)]} 𝑛→∞ computing. There are lots of techniques that have been −1 −1 −1 ∗ developed in the sense of (CC). If (NC) is employed together 󳨐⇒ lim [𝛼 (𝑎𝑛)−𝛼 (𝑎)]𝛽 [𝐷 𝑓 (𝑎)] 𝑛→∞ with (CC) in the formulations, then many of the complicated phenomena in physics or engineering may be analyzed more = {𝛽−1 [𝑓 (𝑎 )] − 𝛽−1 [𝑓 (𝑎)]} . 𝑛→∞lim 𝑛 easily. Even some biological and finance problems can be −1 solved by exponential calculus, which is just a sort of non- Since (𝑎𝑛) is 𝛼-convergent to 𝑎,thedifferenceof𝛼 (𝑎𝑛)− −1 Newtonian calculus. 𝛼 (𝑎) converges to 0. Therefore, we conclude that Quiterecently,TaloandBas,ar have studied the certain 0= {𝛽−1 [𝑓 (𝑎 )] −𝛽−1 [𝑓 (𝑎)]} . sets of sequences of fuzzy numbers and introduced the classi- 𝑛→∞lim 𝑛 (66) cal sets ℓ∞(𝐹), 𝑐(𝐹), 𝑐0(𝐹),andℓ𝑝(𝐹) consisting of bounded, 𝑝 Now, we have by applying 𝛽 to (66)that convergent, null, and absolutely -summable sequences of fuzzy numbers in [12]. Next, they have defined the 𝛼-, 𝛽- 𝛽 (0) = 0=∗̈ 𝛽{𝛽−1 [𝑓 (𝑎 )]−𝛽−1 [𝑓 (𝑎)]} 𝛾 -lim𝑛→∞ 𝑛 ,and -duals of a set of sequences of fuzzy numbers and (67) gave the duals of the classical sets of sequences of fuzzy =∗ [𝑓 (𝑎 ) −𝑓̈ (𝑎)], numbers together with the characterization of the classes of -lim𝑛→∞ 𝑛 infinite matrices of fuzzy numbers transforming one of the which means that ∗-lim𝑛→∞𝑓(𝑎𝑛)=𝑓(𝑎)and this step classical set into another one. Following Bashirov et al. [2] concludes the proof. and Uzer [3], we have given the corresponding results for Following Wen [11], we give a counterexample such non-Newtonian calculus to the results obtained for fuzzy that there is a nowhere differentiable continuous function valued sequences in Talo and Bas¸ar [12], as a beginning. As constructed by infinite products. Suppose 0<𝑎𝑛 <1and 𝑝𝑛 a natural continuation of this paper, we should record that it 𝑛 ∑∞ 𝑎 is an even integer for each ,and 𝑛=1 𝑛 is convergent and is meaningful to define the 𝛼-, 𝛽-, and 𝛾-duals of a space of 𝑏 =∏𝑛 𝑝 2𝑛/(𝑎 𝑝 )→0 𝑛→∞ set 𝑛 𝑘=1 𝑘.If 𝑛 𝑛 as ,then sequences of non-Newtonian real elements and to determine ∗ ∗ ∗ ∗ ∞ the duals of classical spaces ℓ∞, 𝑐 , 𝑐0 ,andℓ𝑝̈together with 𝑓 (𝑥) = ∏ (1+𝑎𝑛 sin 𝑏𝑛𝜋𝑥) (68) the characterization of matrix transformations between the 𝑛=1 classical sequence spaces over the non-Newtonian complex C∗ is a continuous nowhere differentiable function. field . Further, one can obtain the similar results by using another type of calculus instead of non-Newtonian Now, let us consider that the Non-Newtonian ∗-calculus calculus. is multiplicative calculus, which means that the generator 𝑥 Non-Newtonian calculus is a new area in mathematics functions 𝛼 and 𝛽 are equal to 𝐼(𝑥) =𝑥 and exp(𝑥) = 𝑒 , and has very pristine subjects to discuss. We just begin with respectively. Then, the function 𝑓 defined by Wen [11]asin the space of continuous and bounded functions which would (68)is step us to investigate more complicated theoretical structures ∞∗ ⋅⋅ and properties of (NC). We are trying to develop something valuable about non-Newtonian Functional Analysis, but only 𝑓 (𝑥) = ∑ (1 + 𝑎𝑛 sin 𝑏𝑛𝜋𝑥) . (69) 𝑛=1 the mathematical authorities can decide that. 12 Abstract and Applied Analysis

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The authors have benefited a lot from the referee’s report. So, they would like to express their gratitude for his/her constructive suggestions which improved the presentation and readability.

References

[1] D. Stanley, “A multiplicative calculus,” Primus,vol.9,no.4,pp. 310–326, 1999. [2]A.E.Bashirov,E.M.Kurpınar,andA.Ozyapıcı,¨ “Multiplicative calculus and its applications,” JournalofMathematicalAnalysis and Applications,vol.337,no.1,pp.36–48,2008. [3] A. Uzer, “Multiplicative type complex calculus as an alternative to the classical calculus,” Computers & Mathematics with Appli- cations, vol. 60, no. 10, pp. 2725–2737, 2010. [4] A. Bashirov and M. Riza, “On complex multiplicative differenti- ation,” TWMS Journal of Applied and Engineering Mathematics, vol. 1, no. 1, pp. 75–85, 2011. [5]A.E.Bashirov,E.Mısırlı,Y.Tandogdu,˘ and A. Ozyapıcı,¨ “On modeling with multiplicative differential equations,” Applied Mathematics,vol.26,no.4,pp.425–438,2011. [6]A.F.C¸akmakandF.Bas¸ar, “On the classical sequence spaces and non-Newtonian calculus,” Journal of Inequalities and Appli- cations, vol. 2012, Article ID 932734, 12 pages, 2012. [7]S.TekinandF.Bas¸ar, “Certain sequence spaces over the non- Newtonian complex field,” Abstract and Applied Analysis,vol. 2013, Article ID 739319, 11 pages, 2013. [8] Z. C¸ akır, “Spaces of continuous and bounded functions over the field of geometric complex numbers,” Journal of Inequalities and Applications,vol.2013,article363,2013. [9] A. Uzer, “Exact solution of conducting half plane problems as a rapidly convergent series and an application of the multi- plicative calculus,” Turkish Journal of Electrical Engineering & Computer Sciences.Inpress. [10] M. Grossman and R. Katz, Non-Newtonian Calculus,LeePress, Pigeon Cove, Mass, USA, 1972. [11] L. Wen, “A nowhere differentiable continuous function con- structed by infinite products,” The American Mathematical Monthly,vol.109,no.4,pp.378–380,2002. [12] O.¨ Talo and F.Bas¸ar, “Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transforma- tions,” Computers & Mathematics with Applications,vol.58,no. 4, pp. 717–733, 2009. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 524761, 6 pages http://dx.doi.org/10.1155/2014/524761

Research Article A 𝑘-Dimensional System of Fractional Neutral Functional Differential Equations with Bounded Delay

Dumitru Baleanu,1,2,3 Sayyedeh Zahra Nazemi,4 and Shahram Rezapour4

1 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O.Box80204,Jeddah21589,SaudiArabia 2 DepartmentofMathematics,CankayaUniversity,OgretmenlerCaddesi14,Balgat,06530Ankara,Turkey 3 Institute of Space Sciences, Magurele, 76900 Bucharest,Romania 4 Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran

Correspondence should be addressed to Dumitru Baleanu; [email protected]

Received 6 October 2013; Accepted 15 March 2014; Published 10 April 2014

Academic Editor: M. Mursaleen

Copyright © 2014 Dumitru Baleanu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In 2010, Agarwal et al. studied the existence of a one-dimensional fractional neutral functional differential equation. In this paper, we study an initial value problem for a class of k-dimensional systems of fractional neutral functional differential equations by using Krasnoselskii’s fixed point theorem. In fact, our main result generalizes their main result in a sense.

1. Introduction (𝑖 = 1,2,...,𝑘) satisfying some assumptions that will be 𝑛 specified later, xt =(𝑥1 ,𝑥2 ,...,𝑥𝑘 ),and𝜙𝑖 ∈𝐶([−𝑟,0],R ) As you know, many researchers are interested in developing 𝑡 𝑡 𝑡 𝑛 for 𝑖 = 1,2,...,𝑘.If𝑥∈𝐶([𝑡0 −𝑟,𝑡0 +𝑎],R ),thenfor the theoretical analysis and numerical methods of fractional each 𝑡∈[𝑡0,𝑡0 +𝑎]define 𝑥𝑡 by 𝑥𝑡(𝜃) = 𝑥(𝑡 +𝜃) for all equations, because different applications of this area have 𝜃∈[−𝑟,0]. One-dimensional version of the problem has been founded in various fields of sciences and engineering been studied by Agarwal et al. (see [4]). We show that the (see, e.g., [1–37]). In this paper, we investigate the initial 𝑘 problem (1) is equivalent to an integral equation and by using value problem of a -dimensional system of fractional neutral Krasnoselskii’s fixed point theorem, we conclude that the functional differential equations with bounded delay: equivalent operator has (at least) a fixed point. This implies 𝑐 𝛼 that the problem (1) has at least one solution. One can find 𝐷 1 (𝑥 (𝑡) −𝑔 (𝑡, x )) = 𝑓 (𝑡, x ), 1 1 t 1 t thefollowinglemmain[38]. 𝑐 𝛼 𝐷 2 (𝑥 (𝑡) −𝑔 (𝑡, x )) = 𝑓 (𝑡, x ), 2 2 t 2 t Lemma 1 (Krasnoselskii’s fixed point theorem). Let 𝑋 be a 𝐸 𝑋 . Banach space and a closed convex subset of .Supposethat . (1) 𝑆 and 𝑈 are two maps of 𝐸 into 𝑋 such that 𝑆𝑥+𝑈𝑦∈ 𝐸for all 𝑥, 𝑦.If ∈𝐸 𝑆 is a contraction and 𝑈 is completely continuous, 𝑐 𝛼𝑘 𝐷 (𝑥𝑘 (𝑡) −𝑔𝑘 (𝑡, xt)) = 𝑓𝑘 (𝑡, xt), then the equation 𝑆𝑥 + 𝑈𝑥 =𝑥 has a solution on 𝐸.

𝑛 𝑥1 =𝜙1,𝑥2 =𝜙2, ..., 𝑥𝑘 =𝜙𝑘, 𝐼 R 𝑋=𝐶(𝐼,R ) 𝑡0 𝑡0 𝑡0 Let be an interval in and with the norm ‖𝑥‖ = sup𝑡∈𝐼|𝑥(𝑡)|,where|⋅|denotes a suitable complete 𝑛 𝑘 where 𝑡0 ≥0, 𝑎>0,and𝑟>0are constants, 𝑡∈ R (𝑋 = 𝑐 norm on . Consider the product Banach space (𝑡0,∞), 0<𝛼𝑖 <1,for𝑖 = 1,2,...,𝑘, 𝐷 is the standard 𝑋×𝑋×⋅⋅⋅×𝑋⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟,‖⋅‖ ) ‖(𝑥 ,𝑥 ,...,𝑥 )‖ = 𝑛 ∗ with the norm 1 2 𝑘 ∗ Caputo’s fractional derivative, 𝑓𝑖,𝑔𝑖 :[𝑡0,∞)×𝐶([−𝑟,0],R )× 𝑘 𝑛 𝑛 𝑛 𝐶([−𝑟, 0], R )×⋅⋅⋅×𝐶([−𝑟,0],R )→R are given functions max{‖𝑥1‖, ‖𝑥2‖,...,‖𝑥𝑘‖}. The fractional integral of order 𝑞 2 Abstract and Applied Analysis

𝑞 with the lower limit 𝑡0 for a function 𝑓 is defined by 𝐼 𝑓(𝑡) = Proof. It is easy to see that 𝑓𝑖(𝑡, xt) is Lebesgue measurable 𝑡 1−𝑞 𝐼 𝑖= (1/Γ(𝑞)) ∫ (𝑓(𝑠)/(𝑡 −𝑠) )𝑑𝑠 for 𝑡>𝑡0 and 𝑞>0,provided on 0 by using conditions (H1)and(H2)forall 𝑡0 𝛼 −1 [𝑡 ,∞) Γ 1,2,...,𝑘. Also, a direct calculation shows that (𝑡 − 𝑠) 𝑖 ∈ the right-hand side is pointwise defined on 0 .Here, is 1/(1−𝛼𝑖1) 𝑞 𝐿 ([𝑡0,𝑡])for 𝑡∈𝐼0. By using Holder’s inequality and the gamma function. Also, Caputo’s derivative of order with 𝛼 −1 (𝑡 − 𝑠) 𝑖 𝑓(𝑠, x ) the lower limit 𝑡0 for a function 𝑓:[𝑡0,∞) → R is defined condition (H3), we get that 𝑖 s is Lebesgue 𝑠∈[𝑡,𝑡] 𝑡∈𝐼 𝑖= by integrable with respect to 0 for all 0, 1,2,...,𝑘,and(𝑥1,𝑥2,...,𝑥𝑘)∈𝐴(𝛿,𝛾),and 𝑡 (𝑛) 𝑐 𝑞 1 𝑓 (𝑠) 𝑛−𝑞 (𝑛) 𝐷 𝑓 (𝑡) = ∫ 𝑑𝑠 =𝐼 𝑓 (𝑡) (2) 𝑡 Γ(𝑛−𝑞) 𝑞+1−𝑛 󵄨 𝛼 −1 󵄨 𝑡0 (𝑡−𝑠) 󵄨 𝑖 󵄨 ∫ 󵄨(𝑡−𝑠) 𝑓𝑖 (𝑠, xs)󵄨 𝑑𝑠 𝑡0 for 𝑡>𝑡0 and 𝑛−1<𝑞<𝑛([34]). (6) 󵄩 󵄩 󵄩 𝛼𝑖−1󵄩 󵄩 󵄩 ≤ 󵄩(𝑡−𝑠) 󵄩 󵄩𝑚 󵄩 1/𝛼 . 󵄩 󵄩𝐿1/(1−𝛼𝑖1)([𝑡 ,𝑡])󵄩 𝑖󵄩𝐿 𝑖1 (𝐼 ) 2. Main Results 0 0 It is easy to see that if 𝑥=(𝑥1,𝑥2,...,𝑥𝑘) is a solution of Consider the problem (1). Let 𝛿 and 𝛾 be positive constants, the problem (1), then 𝑥 is a solution of (∗).Now,suppose 𝐼0 =[𝑡0,𝑡0 +𝛿],and that 𝑥=(𝑥1,𝑥2,...,𝑥𝑘) is a solution of the equation (∗) and 𝑐 𝛼𝑖 𝑡∈(𝑡0,𝑡0 +𝛿].Then𝑥𝑖 =𝜙𝑖 and 𝐷 (𝑥𝑖(𝑡) −𝑖 𝑔 (𝑡, xt)) = 𝐴(𝛿,𝛾)={(𝑥1,𝑥2,...,𝑥𝑘):𝑥𝑖 =𝜙𝑖, 𝑡0 𝑡0 𝑓𝑖(𝑡, xt) for all 𝑡∈(𝑡0,𝑡0 +𝛿]and 𝑖 = 1,2,...,𝑘.Thus, 󵄨 󵄨 𝑥=(𝑥1,𝑥2,...,𝑥𝑘) is a solution of the problem (1). This sup 󵄨𝑥𝑖 (𝑡) −𝜙𝑖 (0)󵄨 ≤𝛾, ∀𝑖=1,2,...,𝑘}, 𝑡0≤𝑡≤𝑡0+𝛿 completes the proof. (3) Theorem 3. Suppose that there exist 𝛿 ∈ (0, 𝑎) and 𝛾 ∈ (0, ∞) 𝑥 ∈ 𝐶([𝑡 −𝑟,𝑡 +𝛿],R𝑛) where 𝑖 0 0 . For obtaining our results, such that (𝐻1)–(𝐻6) hold. Then the problem (1) has at least one we need the following conditions: solution on [𝑡0,𝑡0 +𝜂]for some positive number 𝜂. (H1) 𝑓𝑖(𝑡,1 𝜑 ,𝜑2,...,𝜑𝑘) is measurable with respect to 𝑡 on Proof. Since condition (H4)holds,theequation(∗) is equiv- 𝐼0 for all 𝑖=1,2,...,𝑘, alent to the equation (H2) 𝑓𝑖(𝑡,1 𝜑 ,𝜑2,...,𝜑𝑘) is continuous with respect to 𝜑𝑗 on 𝐶([−𝑟, 0], R𝑛) 𝑖,𝑗=1,2,...,𝑘 for all , 𝑥𝑖 (𝑡) =𝜙𝑖 (0) −𝑔𝑖1 (𝑡0,𝜙1,𝜙2,...,𝜙𝑘) (H3) there exist 𝛼𝑖1 ∈(0,𝛼𝑖) and a real-valued function 1/𝛼𝑖1 −𝑔𝑖2 (𝑡0,𝜙1,𝜙2,...,𝜙𝑘)+𝑔𝑖1 (𝑡, xt)+𝑔𝑖2 (𝑡, xt) 𝑚𝑖(𝑡) ∈ 𝐿 (𝐼0) such that (7) 󵄨 󵄨 𝑡 󵄨𝑓 (𝑡, x )󵄨 ≤𝑚 𝑡 1 󵄨 𝑖 t 󵄨 𝑖 ( ) (4) 𝛼𝑖−1 + ∫ (𝑡−𝑠) 𝑓𝑖 (𝑠, xs)𝑑𝑠 Γ(𝛼𝑖) 𝑡0 for all (𝑥1,𝑥2,...,𝑥𝑘)∈𝐴(𝛿,𝛾), 𝑡∈𝐼0,and𝑖= 1,2,...,𝑘, and 𝑥𝑖 =𝜙𝑖 for all 𝑡∈𝐼0 and 𝑖 = 1,2,...,𝑘.Let 𝑡0 (H4) 𝑔𝑖(𝑡, xt)=𝑔𝑖1(𝑡, xt)+𝑔𝑖2(𝑡, xt) for all (𝑥1,𝑥2,...,𝑥𝑘)∈ ̃ ̃ ̃ ̃ ̃ (𝜙1, 𝜙2,...,𝜙𝑘)∈𝐴(𝛿,𝛾)be defined by 𝜙𝑖 =𝜙𝑖 and 𝜙𝑖(𝑡0 + 𝐴(𝛿,, 𝛾) 𝑡0 𝑡) = 𝜙 (0) 𝑡∈[0,𝛿] 𝑖 = 1,2,...,𝑘 𝑥= 𝑔 𝑖 for all and .If (H5) 𝑖1 is continuous and (𝑥 ,𝑥 ,...,𝑥 ) 𝑥 (𝑡 +𝑡)= 󵄨 󵄨 󵄩 󵄩 1 2 𝑘 is a solution of problem (1)and 𝑖 0 󵄨𝑔 (𝑡, x ) −𝑔 (𝑡, y )󵄨 ≤𝑙󵄩𝑥−𝑦󵄩 ̃ 󵄨 𝑖1 t 𝑖1 t 󵄨 𝑖󵄩 󵄩∗ (5) 𝜙𝑖(𝑡0 +𝑡)+𝑦𝑖(𝑡) for 𝑡∈[−𝑟,𝛿]and 𝑖 = 1,2,...,𝑘,then ̃ 𝑥=(𝑥,𝑥 ,...,𝑥 ), 𝑦 = (𝑦 ,𝑦 ,...,𝑦 )∈ 𝑥𝑖 = 𝜙𝑖 +𝑦𝑖 for 𝑡∈[0,𝛿]and 𝑖=1,2,...,𝑘.Thus, for all 1 2 𝑘 1 2 𝑘 𝑡0+𝑡 𝑡0+𝑡 𝑡 𝐴(𝛿,,and 𝛾) 𝑡∈𝐼0,where𝑙𝑖 ∈ (0, 1) is a constant, for all 𝑖=1,2,...,𝑘, 𝑦𝑖 (𝑡) =−𝑔𝑖1 (𝑡0,𝜙1,𝜙2,...,𝜙𝑘)−𝑔𝑖2 (𝑡0,𝜙1,𝜙2,...,𝜙𝑘) 𝑔 {𝑡 ⊢ (H6) 𝑖2 is completely continuous and the family ̃ ̃ ̃ +𝑔𝑖1 (𝑡0 +𝑡,𝑦1 + 𝜙1 ,𝑦2 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 ) 𝑔𝑖2(𝑡, xt):(𝑥1,𝑥2,...,𝑥𝑘)∈Λ}is equicontinu- 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑛 𝑛 𝑛 ous on 𝐶(𝐼⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟0, R )×𝐶(𝐼0, R )×⋅⋅⋅×𝐶(𝐼0, R ) for all 𝑘 +𝑔 (𝑡 +𝑡,𝑦 + 𝜙̃ ,𝑦 + 𝜙̃ ,...,𝑦 + 𝜙̃ ) 𝑖2 0 1𝑡 1𝑡 +𝑡 2𝑡 2𝑡 +𝑡 𝑘𝑡 𝑘𝑡 +𝑡 bounded set Λ in 𝐴(𝛿, 𝜆) and 𝑖=1,2,...,𝑘. 0 0 0 𝑡 1 𝛼 −1 + ∫ (𝑡−𝑠) 𝑖 𝑓 (𝑡 +𝑠,𝑦 + 𝜙̃ ,𝑦 Lemma 2. Supposethatthereexist𝛿 ∈ (0, 𝑎) and 𝛾 ∈ (0, ∞) 𝑖 0 1𝑠 1𝑡 +𝑠 2𝑠 Γ(𝛼𝑖) 0 0 such that (𝐻1)–(𝐻3) hold. Then the problem (1) for 𝑡∈(𝑡0,𝑡0 + 𝛿] is equivalent to the equation ̃ ̃ +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )𝑑𝑠 𝑡0+𝑠 𝑠 𝑡0+𝑠 𝑥𝑖 (𝑡) =𝜙𝑖 (0) −𝑔𝑖 (𝑡0,𝜙1,𝜙2,...,𝜙𝑘)+𝑔𝑖 (𝑡, xt) (∗∗) 1 𝑡 (∗) 𝛼𝑖−1 𝑡 ∈ [0, 𝛿] 𝑖 = 1,2,...,𝑘 𝑔 𝑔 + ∫ (𝑡−𝑠) 𝑓𝑖 (𝑠, xs)𝑑𝑠 for and .Since 𝑖1, 𝑖2 are continuous Γ(𝛼𝑖) 𝑡0 𝑥 𝑡 𝑖 = 1,2,...,𝑘 and 𝑖𝑡 is continuous in for all , there exists 󸀠 ̃ ̃ with conditions 𝑥𝑖 =𝜙𝑖 for 𝑖=1,2,...,𝑘and t ∈𝐼0. 𝛿 >0such that |𝑔𝑖1(𝑡0 +𝑡,𝑦1 + 𝜙1 ,𝑦2 + 𝜙2 ,...,𝑦𝑘 + 𝑡0 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 Abstract and Applied Analysis 3

̃ 𝜙𝑘 )−𝑔𝑖1(𝑡0,𝜙1,𝜙2,...,𝜙𝑘)| < 𝛾/3 and |𝑔𝑖2(𝑡0 +𝑡,𝑦1 + 𝑆+𝑈on 𝐸(𝜂,. 𝛾) Hence, it is sufficient that we show that 𝑆+𝑈 𝑡0+𝑡 𝑡 ̃ ̃ ̃ has a fixed point in 𝐸(𝜂,. 𝛾) We prove it in three steps. 𝜙1 ,𝑦2 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )−𝑔𝑖2(𝑡0,𝜙1,𝜙2,...,𝜙𝑘)| < 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝛾/3 0<𝑡<𝛿󸀠 𝑖 = 1,2,...,𝑘 𝜂= for and .Put Step I. 𝑆𝑧 + 𝑈𝑦 ∈ 𝐸(𝜂, 𝛾) for all 𝑧=(𝑧1,𝑧2,...,𝑧𝑘), 𝑦 = 1/(1+𝛽 )(1−𝛼 ) 󸀠 1−𝛼𝑖1 𝑖 𝑖1 (𝑦 ,𝑦 ,...,𝑦 ) ∈ 𝐸(𝜂, 𝛾) min1≤𝑖≤𝑘{𝛿, 𝛿 ,(𝛾Γ(𝛼𝑖)(1 + 𝛽𝑖) /3𝑀𝑖) },where 1 2 𝑘 . 𝑧, 𝑦 ∈ 𝐸(𝜂, 𝛾) 𝑆 𝑧+𝑈𝑦∈ 𝛽𝑖 =(𝛼𝑖 − 1)/(1 − 𝛼𝑖1) ∈ (−1, 0) and 𝑀𝑖 =‖𝑚𝑖‖𝐿1/𝛼𝑖1 (𝐼 ) for Let be given. Then, 𝑖 𝑖 0 𝐶([−𝑟, 𝜂], R𝑛) 𝑖 = 1,2,...,𝑘 all 𝑖=1,2,...,𝑘.Define for all .Itiseasytocheckthat (𝑆𝑧 + 𝑈𝑦)(𝑡) =0 for all 𝑡∈[−𝑟,0]. Also, we have 𝑛 𝐸(𝜂,𝛾)={(𝑦1,𝑦2,...,𝑦𝑘):𝑦𝑖 ∈ 𝐶 ([−𝑟, 𝜂], R ), 𝑦𝑖 (𝑠) =0, 󵄨 󵄨 󵄨𝑆𝑖𝑧 (𝑡) +𝑈𝑖𝑦 (𝑡)󵄨 󵄩 󵄩 󵄩𝑦𝑖󵄩 ≤𝛾for 𝑠∈[−𝑟,] 0 , 𝑖=1,2,...,𝑘}. 󵄨 ≤ 󵄨 −𝑔 (𝑡 ,𝜙 ,𝜙 ,...,𝜙 ) (8) 󵄨 𝑖1 0 1 2 𝑘 󵄨 𝐸(𝜂, 𝛾) ̃ ̃ ̃ 󵄨 In fact, is a closed, bounded, and convex subset of +𝑔𝑖1 (𝑡0 +𝑡,𝑧1 + 𝜙1 ,𝑧2 + 𝜙2 ,...,𝑧𝑘 + 𝜙𝑘 )󵄨 𝑛 𝑛 𝑛 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 󵄨 𝐶([−𝑟, 𝜂], R )×𝐶([−𝑟,𝜂],R )×⋅⋅⋅×𝐶([−𝑟,𝜂],R ).Define 𝑆 𝑈 𝐸(𝜂, 𝛾) 󵄨 the operators and on by + 󵄨 −𝑔 (𝑡 ,𝜙 ,𝜙 ,...,𝜙 ) 󵄨 𝑖2 0 1 2 𝑘 𝑆 (𝑦 ,𝑦 ,...,𝑦 ) (𝑡) 󵄨 1 1 2 𝑘 ̃ ̃ ̃ 󵄨 +𝑔𝑖2 (𝑡0 +𝑡,𝑦1 + 𝜙1 ,𝑦2 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )󵄨 𝑆 (𝑦 ,𝑦 ,...,𝑦 ) (𝑡) 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 󵄨 𝑆(𝑦 ,𝑦 ,...,𝑦 ) 𝑡 =( 2 1 2 𝑘 ), 1 2 𝑘 ( ) . . 1 𝑡 󵄨 . 󵄨 𝛼𝑖−1 ̃ + ∫ 󵄨(𝑡−𝑠) 𝑓𝑖 (𝑡0 +𝑠,𝑦1 + 𝜙1 ,𝑦2 𝑆 (𝑦 ,𝑦 ,...,𝑦 ) (𝑡) 󵄨 𝑠 𝑡0+𝑠 𝑠 𝑘 1 2 𝑘 Γ(𝛼𝑖) 0 (9) 󵄨 𝑈1 (𝑦1,𝑦2,...,𝑦𝑘) (𝑡) ̃ ̃ 󵄨 +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )󵄨 𝑑𝑠 𝑈 (𝑦 ,𝑦 ,...,𝑦 ) (𝑡) 𝑡0+𝑠 𝑠 𝑡0+𝑠 󵄨 𝑈(𝑦 ,𝑦 ,...,𝑦 ) (𝑡) =( 2 1 2 𝑘 ), 1 2 𝑘 . 𝑡 1−𝛼𝑖1 . 2𝛾 1 (𝛼 −1)/(1−𝛼 ) . ≤ + (∫ (𝑡−𝑠) 𝑖 𝑖1 𝑑𝑠) 𝑈𝑘 (𝑦1,𝑦2,...,𝑦𝑘) (𝑡) 3 Γ(𝛼𝑖) 0 𝛼 𝑡0+𝑡 𝑖1 where 1/𝛼𝑖1 ×(∫ (𝑚𝑖 (𝑠)) 𝑑𝑠) 𝑡0 𝑆𝑖 (𝑦1,𝑦2,...,𝑦𝑘) (𝑡) (1+𝛽 )(1−𝛼 ) 2𝛾 𝑀 𝜂 𝑖 𝑖1 ≤ + 𝑖 ≤𝛾 1−𝛼 0𝑡∈[−𝑟,] 0 , 3 Γ(𝛼)(1+𝛽) 𝑖1 { 𝑖 𝑖 { {−𝑔𝑖1 (𝑡0,𝜙1,𝜙2,...,𝜙𝑘) (11) = { ̃ { +𝑔𝑖1 (𝑡0 +𝑡,𝑦1 + 𝜙1 ,𝑦2 𝑡∈[0,𝜂] 𝑖 = 1,2,...,𝑘 ‖𝑆 𝑧+𝑈𝑦‖ = { 𝑡 𝑡0+𝑡 𝑡 for all and .Thus, 𝑖 𝑖 { ̃ ̃ |(𝑆 𝑧)(𝑡) − (𝑈 𝑦)(𝑡)| ≤𝛾 𝑖=1,2,...,𝑘 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )𝑡∈[0,𝜂], sup𝑡∈[0,𝜂] 𝑖 𝑖 for all .Hence, { 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑆𝑧 + 𝑈𝑦 ∈ 𝐸(𝜂, 𝛾) for all 𝑧, 𝑦 ∈ 𝐸(𝜂,. 𝛾) 𝑈𝑖 (𝑦1,𝑦2,...,𝑦𝑘) (𝑡) Step 𝐼𝐼. 𝑆 is a contraction on 𝐸(𝜂,. 𝛾) 󸀠 󸀠 󸀠 󸀠 󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠󸀠 Let 𝑦 =(𝑦,𝑦 ,...,𝑦 ), 𝑦 =(𝑦 ,𝑦 ,...,𝑦 ) ∈ 𝐸(𝜂,. 𝛾) 0𝑡∈[−𝑟,] 0 , 1 2 𝑘 1 2 𝑘 { Then, {−𝑔 (𝑡 ,𝜙 ,𝜙 ,...,𝜙 ) { 𝑖2 0 1 2 𝑘 󸀠 ̃ 󸀠 ̃ 󸀠 ̃ { (𝑦1 + 𝜙1 ,𝑦2 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 ), { +𝑔 (𝑡 +𝑡,𝑦 + 𝜙̃ ,𝑦 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 { 𝑖2 0 1𝑡 1𝑡 +𝑡 2𝑡 { 0 (12) { ̃ ̃ 󸀠󸀠 󸀠󸀠 󸀠󸀠 { +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 ) (𝑦 + 𝜙̃ ,𝑦 + 𝜙̃ ,...,𝑦 + 𝜙̃ )∈𝐴(𝛿,𝛾) 𝑡0+𝑡 𝑡 𝑡0+𝑡 1 1 2 2 𝑘 𝑘 = { 𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 { 1 𝛼 −1 { + ∫ (𝑡−𝑠) 𝑖 { Γ(𝛼) and so { 𝑖 0 󵄨 󵄨 { ̃ 󵄨𝑆 𝑦󸀠 (𝑡) −𝑆𝑦󸀠󸀠 (𝑡)󵄨 { ×𝑓𝑖 (𝑡0 +𝑠,𝑦1 + 𝜙1 ,𝑦2 󵄨 𝑖 𝑖 󵄨 { 𝑠 𝑡0+𝑠 𝑠 { ̃ ̃ 󵄨 { + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )𝑑𝑠 𝑡∈[0,𝜂], 󵄨 󸀠 ̃ 󸀠 ̃ 󸀠 ̃ 𝑡0+𝑠 𝑠 𝑡0+𝑠 = 󵄨𝑔𝑖1 (𝑡0 +𝑡,𝑦1 + 𝜙1 ,𝑦2 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 ) 󵄨 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 𝑡 𝑡0+𝑡 (10) 󸀠󸀠 ̃ 󸀠󸀠 −𝑔𝑖1 (𝑡0 +𝑡,𝑦1 + 𝜙1 ,𝑦2 for 𝑖=1,2,...,𝑘. It is easy to check that the operator equation 𝑡 𝑡0+𝑡 𝑡 𝑦=𝑆𝑦+𝑈𝑦has a solution 𝑦=(𝑦1,𝑦2,...,𝑦𝑘) if and only 󵄨 ̃ 󸀠󸀠 ̃ 󵄨 𝑦 (∗∗) 𝑖 = 1,2,...,𝑘 +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )󵄨 if 𝑖 is a solution for for all .Inthiscase, 𝑡0+𝑡 𝑡 𝑡0+𝑡 󵄨 ̃ 𝑥𝑖(𝑡0 +𝑡) =𝑖 𝑦 (𝑡)+𝜙𝑖(𝑡0 +𝑡)will be a solution of the problem (1) 󵄩 󵄩 ≤𝑙󵄩𝑦󸀠 −𝑦󸀠󸀠󵄩 on [0, 𝜂]. Thus, the existence of a solution of the problem (1) 𝑖󵄩 󵄩∗ is equivalent to the existence of a fixed point for the operator (13) 4 Abstract and Applied Analysis

󸀠 󸀠󸀠 for all 𝑖 = 1,2,...,𝑘. This implies that ‖𝑆𝑦 −𝑆𝑦 ‖∗ ≤ 𝑦∈𝐸(𝜂,𝛾)}is equicontinuous. Let 0≤𝑡1 <𝑡2 ≤𝜂and 󸀠 󸀠󸀠 𝑦∈𝐸(𝜂,𝛾) 𝑙‖𝑦 −𝑦 ‖∗,where𝑙=max{𝑙1,𝑙2,...,𝑙𝑘}.Since0<𝑙<1, be given. Then, we have 𝑆 is a contraction on 𝐸(𝜂,. 𝛾) 󵄨 󵄨 󵄨𝑈𝑖2𝑦(𝑡2)−𝑈𝑖2𝑦(𝑡1)󵄨 𝐼𝐼𝐼 𝑈 󵄨 𝑡 Step . is a completely continuous operator. 󵄨 1 1 𝛼 −1 𝛼 −1 󵄨 𝑖 𝑖 Suppose that = 󵄨 ∫ [(𝑡2 −𝑠) −(𝑡1 −𝑠) ] 󵄨Γ(𝛼𝑖) 0 ̃ 𝑈 (𝑦 ,𝑦 ,...,𝑦 ) (𝑡) ×𝑓𝑖 (𝑡0 +𝑠,𝑦1 + 𝜙1 ,𝑦2 𝑖1 1 2 𝑘 𝑠 𝑡0+𝑠 𝑠 ̃ ̃ +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )𝑑𝑠 {0𝑡∈[−𝑟.0] , 𝑡0+𝑠 𝑠 𝑡0+𝑠 { {−𝑔 (𝑡 ,𝜙 ,𝜙 ,...,𝜙 ) 𝑡 𝑖2 0 1 2 𝑘 1 2 𝛼 −1 = + ∫ (𝑡 −𝑠) 𝑖 𝑓 (𝑡 +𝑠,𝑦 + 𝜙̃ ,𝑦 { +𝑔 (𝑡 +𝑡,𝑦 + 𝜙̃ ,𝑦 2 𝑖 0 1𝑠 1𝑡 +𝑠 2𝑠 { 𝑖2 0 1 1 2 Γ(𝛼) 𝑡 0 { 𝑡 𝑡0+𝑡 𝑡 𝑖 1 + 𝜙̃ ,...,𝑦 + 𝜙̃ ) 𝑡∈[0,𝜂], 󵄨 { 2𝑡 +𝑡 𝑘𝑡 𝑘𝑡 +𝑡 󵄨 0 0 ̃ ̃ 󵄨 +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )𝑑𝑠󵄨 𝑡0+𝑠 𝑠 𝑡0+𝑠 󵄨 𝑈𝑖2 (𝑦1,𝑦2,...,𝑦𝑘) (𝑡) 󵄨 𝑡 1 1 𝛼 −1 𝛼 −1 ≤ ∫ [(𝑡 −𝑠) 𝑖 −(𝑡 −𝑠) 𝑖 ] 0𝑡∈[−𝑟.0] , 1 2 { Γ(𝛼𝑖) 0 { 𝑡 { 1 𝛼 −1 { ∫ (𝑡−𝑠) 𝑖 󵄨 {Γ(𝛼) × 󵄨𝑓 (𝑡 +𝑠,𝑦 + 𝜙̃ ,𝑦 = 𝑖 0 󵄨 𝑖 0 1𝑠 1𝑡 +𝑠 2𝑠 { 󵄨 0 { ×𝑓 (𝑡 +𝑠,𝑦 + 𝜙̃ ,𝑦 { 𝑖 0 1𝑠 1𝑡 +𝑠 2𝑠 󵄨 { 0 ̃ ̃ 󵄨 ̃ ̃ + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )󵄨 𝑑𝑠 +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )𝑑𝑠 𝑡∈[0,𝜂], 𝑡0+𝑠 𝑠 𝑡0+𝑠 󵄨 { 𝑡0+𝑠 𝑠 𝑡0+𝑠 𝑡 (14) 1 2 󵄨 𝛼𝑖−1 󵄨 ̃ + ∫ (𝑡2 −𝑠) 󵄨𝑓𝑖 (𝑡0 +𝑠,𝑦1 + 𝜙1 ,𝑦2 󵄨 𝑠 𝑡0+𝑠 𝑠 Γ(𝛼𝑖) 𝑡1 𝑖=1,2,...,𝑘 󵄨 for .Itisclearthat ̃ ̃ 󵄨 + 𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )󵄨 𝑑𝑠 𝑡0+𝑠 𝑠 𝑡0+𝑠 󵄨

𝑡 1−𝛼𝑖1 𝑈 +𝑈 𝑀 1 1/(1−𝛼 ) 11 12 𝑖 𝛼𝑖−1 𝛼𝑖−1 𝑖1 𝑈 +𝑈 ≤ (∫ [(𝑡1 −𝑠) −(𝑡2 −𝑠) ] 𝑑𝑠) 21 22 Γ(𝛼) 𝑈=( ). 𝑖 0 . (15) . 𝑡 1−𝛼𝑖1 𝑀 2 1/(1−𝛼 ) 𝑈 +𝑈 𝑖 𝛼𝑖−1 𝑖1 𝑘1 𝑘2 + (∫ [(𝑡2 −𝑠) ] 𝑑𝑠) Γ(𝛼𝑖) 𝑡1

𝑡 1−𝛼𝑖1 𝑔 𝑖 = 1,2,...,𝑘 𝑀 1 Since 𝑖2 is completely continuous for all , 𝑖 𝛽𝑖 𝛽𝑖 ≤ (∫ [(𝑡1 −𝑠) −(𝑡2 −𝑠) ]𝑑𝑠) 𝑈𝑖1 {𝑈𝑖1(𝑦):𝑦∈𝐸(𝜂,𝛾)} is continuous and also is uniformly Γ(𝛼𝑖) 0 bounded. By using condition (H6), it is easy to check that 𝑡 1−𝛼𝑖1 {𝑈𝑖1(𝑦) : 𝑦 ∈ 𝐸(𝜂, 𝛾)} is equicontinuous. On the other hand, 𝑀 2 𝑖 𝛽𝑖 + (∫ (𝑡2 −𝑠) 𝑑𝑠) Γ(𝛼𝑖) 𝑡 󵄨 󵄨 1 󵄨𝑈 𝑦 (𝑡)󵄨 󵄨 𝑖2 󵄨 𝑀 1+𝛽 1+𝛽 1+𝛽 1−𝛼𝑖1 ≤ 𝑖 (𝑡 𝑖 −𝑡 𝑖 +(𝑡 −𝑡 ) 𝑖 ) 1−𝛼 1 2 2 1 1 𝑡 󵄨 Γ(𝛼)(1+𝛽) 𝑖1 𝛼𝑖−1 󵄨 ̃ 𝑖 𝑖 ≤ ∫ (𝑡−𝑠) 󵄨𝑓𝑖 (𝑡0 +𝑠,𝑦1 + 𝜙1 ,𝑦2 Γ(𝛼) 0 󵄨 𝑠 𝑡0+𝑠 𝑠 𝑖 𝑀 (1+𝛽 )(1−𝛼 ) + 𝑖 (𝑡 −𝑡 ) 𝑖 𝑖1 󵄨 1−𝛼 2 1 ̃ ̃ 󵄨 Γ(𝛼)(1+𝛽) 𝑖1 +𝜙2 ,...,𝑦𝑘 + 𝜙𝑘 )󵄨 𝑑𝑠 𝑖 𝑖 𝑡0+𝑠 𝑠 𝑡0+𝑠 󵄨 (16) 2𝑀 𝑖 (1+𝛽𝑖)(1−𝛼𝑖1) 𝑡 1−𝛼𝑖1 1 ≤ (𝑡2 −𝑡1) (𝛼𝑖−1)/(1−𝛼𝑖1) 1−𝛼𝑖1 ≤ (∫ (𝑡−𝑠) 𝑑𝑠) Γ(𝛼𝑖)(1+𝛽𝑖) Γ(𝛼) 0 𝑖 (17)

𝑡 +𝑡 𝛼𝑖1 (1+𝛽 )(1−𝛼 ) 0 1/𝛼 𝑀 𝜂 𝑖 𝑖1 𝑖 = 1,2,...,𝑘 {𝑈 𝑦:𝑦∈𝐸(𝜂,𝛾)} ×(∫ (𝑚 (𝑠)) 𝑖1 𝑑𝑠) ≤ 𝑖 for all .Thus, 𝑖2 is 𝑖 1−𝛼 𝑡 𝑖1 𝑈 0 Γ(𝛼𝑖)(1+𝛽𝑖) equicontinuous. Moreover, it is clear that 𝑖2 is continuous for all 𝑖 = 1,2,...,𝑘. This implies that 𝑈 is a completely continuous operator. Now, by using Krasnoselskii’s fixed for all 𝑡∈[0,𝜂]and 𝑖 = 1,2,...,𝑘. This implies that {𝑈𝑖2𝑦: point theorem we get that 𝑆+𝑈has a fixed point on 𝐸(𝜂, 𝛾) 𝑦∈𝐸(𝜂,𝛾)}isuniformlybounded.Now,weprovethat{𝑈𝑖2𝑦: and so the problem (1)hasasolution𝑥=(𝑥1,...,𝑥𝑘),where Abstract and Applied Analysis 5

𝑥𝑖(𝑡) =𝑖 𝜙 (0) + 𝑦𝑖(𝑡 −0 𝑡 ) for all 𝑡∈[𝑡0,𝑡0 +𝜂]and 𝑖= [5] B. Ahmad, “Existence of solutions for fractional differential 1,2,...,𝑘. equations of order q ∈ (2,3] with anti-periodic boundary conditions,” Journal of Applied Mathematics and Computing,vol. If we put 𝑔𝑖1 =0for all 𝑖=1,2,...,𝑘,thenweobtainnext 34, no. 1-2, pp. 385–391, 2010. result. [6]B.Ahmad,“Newresultsforboundaryvalueproblemsof nonlinear fractional differential equations with non-separated Corollary 4. 𝛿 ∈ (0, 𝑎) 𝛾∈ Suppose that there exist and boundary conditions,” Acta Mathematica Vietnamica,vol.36, (0, ∞) such that conditions (𝐻1)–(𝐻3) hold, 𝑔𝑖 is continuous no.3,pp.659–668,2011. 𝑖 = 1,2,...,𝑘 |𝑔 (𝑡, x )−𝑔(𝑡, y )| ≤ 𝑙 ‖𝑥 − 𝑦‖ for all ,and 𝑖 t 𝑖 t 𝑖 ∗ for [7]B.AhmadandJ.J.Nieto,“Existenceresultsforacoupled all 𝑥=(𝑥1,𝑥2,...,𝑥𝑘), 𝑦 = 1(𝑦 ,𝑦2,...,𝑦𝑘)∈𝐴(𝛿,𝛾),and system of nonlinear fractional differential equations with three- 𝑡∈𝐼0,where𝑙𝑖 ∈ (0, 1) is a constant for all 𝑖=1,2,...,𝑘.Then point boundary conditions,” Computers & Mathematics with the problem (1) has at least one solution on [𝑡0,𝑡0 +𝜂]for some Applications,vol.58,no.9,pp.1838–1843,2009. positive number 𝜂. [8] C.-Z. Bai and J.-X. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential If we put 𝑔𝑖2 =0for all 𝑖=1,2,...,𝑘,thenweobtainnext equations,” Applied Mathematics and Computation,vol.150,no. result. 3, pp. 611–621, 2004. [9] Z. Bai and H. Lu,¨ “Positive solutions for boundary value Corollary 5. Suppose that there exist 𝛿 ∈ (0, 𝑎) and 𝛾∈ problem of nonlinear fractional differential equation,” Journal (0, ∞) such that conditions (𝐻1)–(𝐻3) hold, 𝑔𝑖 is completely of Mathematical Analysis and Applications,vol.311,no.2,pp. continuous for all 𝑖 = 1,2,...,𝑘,andthefamily{𝑡 ⊢𝑖 𝑔 (𝑡, xt): 495–505, 2005. 𝑛 (𝑥1,𝑥2,...,𝑥𝑘)∈Λ}is equicontinuous on 𝐶(𝐼0, R )× 𝑛 𝑛 [10] Z. Bai and W. Sun, “Existence and multiplicity of positive 𝐶(𝐼0, R ) × ⋅⋅⋅ × 𝐶(𝐼0, R ) for all bounded set Λ in 𝐴(𝛿,. 𝜆) solutions for singular fractional boundary value problems,” Then the problem (1) has at least one solution on [𝑡0,𝑡0 +𝜂]for Computers & Mathematics with Applications,vol.63,no.9,pp. some positive number 𝜂. 1369–1381, 2012. [11] D. Baleanu, H. Mohammadi, and Sh. Rezapour, “Positive 3. Conclusions solutions of an initial value problem for nonlinear fractional differential equations,” Abstract and Applied Analysis,vol.2012, In this work, we study an initial value problem for a class ArticleID837437,7pages,2012. of 𝑘-dimensional systems of fractional neutral functional [12] D. Baleanu, R. P. Agarwal, H. Mohammadi, and Sh. Rezapour, differential equations by using Krasnoselskii’s fixed point “Some existence results for a nonlinear fractional differential theorem. Our result generalizes some old related results in a equation on partially ordered Banach spaces,” Boundary Value Problems,vol.2013,article112,2013. sense. [13] D. Baleanu, Sh. Rezapour, and H. Mohammadi, “Some existence results on nonlinear fractional differential equations,” Philo- Conflict of Interests sophical Transactions of the Royal Society of London A,vol.371, no.1990,ArticleID20120144,2013. The authors declare that there is no conflict of interests [14] D. Baleanu, H. Mohammadi, and Sh. Rezapour, “On a nonlinear regarding the publication of this paper. fractional differential equation on partially ordered metric spaces,” Advances in Difference Equations, vol. 2013, article 83, Acknowledgment 2013. [15] D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “The existence of The research of the second and third authors was supported positive solutions for a new coupled system of multiterm sin- by Azarbaidjan Shahid Madani University. gular fractional integrodifferential boundary value problems,” Abstract and Applied Analysis,vol.2013,ArticleID368659,15 References pages, 2013. [16] D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “Existence and [1] R. P.Agarwal and B. Ahmad, “Existence theory for anti-periodic uniqueness of solutions for multi-term nonlinear fractional boundary value problems of fractional differential equations integro-differential equations,” Advances in Differential Equa- and inclusions,” Computers & Mathematics with Applications, tions,vol.2013,article368,2013. vol.62,no.3,pp.1200–1214,2011. [17] D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “Attractivity for [2]R.P.Agarwal,M.Belmekki,andM.Benchohra,“Asurvey a k-dimensional system of fractional functional differential on semilinear differential equations and inclusions involving equations and global attractivity for a k-dimensional system of Riemann-Liouville fractional derivative,” Advances in Difference nonlinear fractional differential equations,” Journal of Inequali- Equations,vol.2009,ArticleID981728,47pages,2009. ties and Applications,vol.4014,p.31,2014. [3]R.P.Agarwal,M.Benchohra,andS.Hamani,“Asurveyon [18] M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value existence results for boundary value problems of nonlinear frac- problems for differential equations with fractional order and tional differential equations and inclusions,” Acta Applicandae nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Mathematicae,vol.109,no.3,pp.973–1033,2010. Applications,vol.71,no.7-8,pp.2391–2396,2009. [4]R.P.Agarwal,Y.Zhou,andY.He,“Existenceoffractional [19] R. C. Cascaval, E. C. Eckstein, C. L. Frota, and J. A. Goldstein, neutral functional differential equations,” Computers & Math- “Fractional telegraph equations,” Journal of Mathematical Anal- ematics with Applications, vol. 59, no. 3, pp. 1095–1100, 2010. ysis and Applications,vol.276,no.1,pp.145–159,2002. 6 Abstract and Applied Analysis

[20] C. Cuevas and J. Cesar´ de Souza, “Existence of 𝑆-asymptotically [37] Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for 𝜔-periodic solutions for fractional order functional integro- fractional neutral differential equations with infinite delay,” differential equations with infinite delay,” Nonlinear Analysis: Nonlinear Analysis: Theory, Methods & Applications,vol.71,no. Theory, Methods & Applications,vol.72,no.3-4,pp.1683–1689, 7-8, pp. 3249–3256, 2009. 2010. [38] D. R. Smart, Fixed Point Theorems, Cambridge University Press, [21] C. Cuevas and J. C. de Souza, “𝑆-asymptotically 𝜔-periodic solu- New York, NY, USA, 1974. tions of semilinear fractional integro-differential equations,” Applied Mathematics Letters of Rapid Publication,vol.22,no.6, pp.865–870,2009. [22]C.Cuevas,M.Rabelo,andH.Soto,“Pseudo-almostautomor- phic solutions to a class of semilinear fractional differential equations,” Communications on Applied Nonlinear Analysis,vol. 17,no.1,pp.33–48,2010. [23] M. A. Darwish and S. K. Ntouyas, “On initial and boundary value problems for fractional order mixed type functional differential inclusions,” Computers & Mathematics with Appli- cations,vol.59,no.3,pp.1253–1265,2010. [24]J.P.C.dosSantosandC.Cuevas,“Asymptoticallyalmost automorphic solutions of abstract fractional integro-differential neutral equations,” Applied Mathematics Letters of Rapid Publi- cation,vol.23,no.9,pp.960–965,2010. [25] S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004. [26] R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics,vol.378,pp.223– 276,Springer,NewYork,NY,USA,1997. [27] H. Hilfer, Applications of Fractional Calculus in Physics,World Scientific Publishing, Singapore, 2000. [28] V. Lakshmikantham, “Theory of fractional functional differen- tial equations,” Nonlinear Analysis: Theory, Methods & Applica- tions,vol.69,no.10,pp.3337–3343,2008. [29] V. Lakshmikantham and J. V. Devi, “Theory of fractional differential equations in a Banach space,” European Journal of Pure and Applied Mathematics,vol.1,no.1,pp.38–45,2008. [30] J. A. Tenreiro MacHado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathemat- ical Problems in Engineering,vol.2010,ArticleID639801,34 pages, 2010. [31] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,JohnWiley& Sons,NewYork,NY,USA,1993. [32] H. Mohammadi and Sh. Rezapour, “Existence results for non- linear fractional differential equations on ordered gauge spaces,” JournalofAdvancedMathematicalStudies,vol.6,no.2,pp.154– 158, 2013. [33] S. K. Ntouyas and M. Obaid, “A coupled system of fractional differential equations with nonlocal integral boundary condi- tions,” Advances in Difference Equations,vol.2012,article130, 2012. [34] I. Podlubny, Fractional Differential Equations, vol. 198, Aca- demic Press, New York, NY, USA, 1999. [35] X. Su, “Boundary value problem for a coupled system of non- linear fractional differential equations,” Applied Mathematics Letters of Rapid Publication,vol.22,no.1,pp.64–69,2009. [36] Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for 𝑝-type fractional neutral differential equations,” Nonlinear Analysis: Theory, Methods & Applications,vol.71,no.7-8,pp.2724–2733, 2009. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 798080, 17 pages http://dx.doi.org/10.1155/2014/798080

Research Article Integral Equation-Wavelet Collocation Method for Geometric Transformation and Application to Image Processing

Lina Yang,1,2 Yuan Yan Tang,1,3 Xiang Chu Feng,4 and Lu Sun5

1 Department of Computer and Information Science, Faculty of Science and Technology, University of Macau, Macau 2 Department of Mathematics and Computer Science, Guangxi Normal University of Nationalities, Chongzuo 532200, China 3 College of Computer Science, Chongqing University, Chongging 40030, China 4 Department of Mathematics, Xidian University, Xi’an 710126, China 5 Sichuan Sunray Machinery Co., Ltd., Deyang 618000, China

Correspondence should be addressed to Yuan Yan Tang; [email protected]

Received 28 August 2013; Accepted 13 February 2014; Published 1 April 2014

Academic Editor: M. Mursaleen

Copyright © 2014 Lina Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet- based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

1. Introduction for calculating new coordinate positions of these points under a certain model. The geometric transformation is defined by Geometric (or shape) distortion may be produced in the data acquisition phase in pattern recognition, computer vision, 𝑇:(𝑥1,𝑥2)󳨀→(𝑢, V) , (1) and robot vision systems, and it can be characterized by geometric transformation model [1, 2]. An obvious example such that ofsuchadistortioncanbefoundinaphotographtakenbya 𝑢=𝑢(𝑥1,𝑥2), V = V (𝑥1,𝑥2). (2) camera in a computer vision system. In the acquisition phase shown in Figure 1, the trade mark “Coke” is printed on a Through this transformation, an image in Cartesian coordi- Coca-Cola bottle, due to the cylindrical shape of the bottle, nates 𝑥1𝑂𝑥2 is transformed into a new image in Cartesian and the square shape of the trade mark has been changed. coordinates 𝑢𝑂V as shown in Figure 2.Thepropertiesofthe This kind of distortion can be characterized by a geometric transformation 𝑇 are determined by functions 𝑢=𝑢(𝑥1,𝑥2) transformation model, specifically, the biquadratic geometric and V = V(𝑥1,𝑥2); that is different functions can produce transformation model in this example [2]. different kinds of geometric transformations. An image is displayed by a set of coordinate points; hence, There are many models of the geometric transformations, a geometric transformation can be viewed as the procedure which have been widely used in many disciplines [1–4]. In 2 Abstract and Applied Analysis

example, the geometric transformation in Figure 1 can be approximated by the biquadratic model, and its mathematical function can be written as 𝑢 [1−𝑥 𝑥 ]0 [ ]=[ 1 1 ] V 0[1−𝑥1 𝑥1]

𝑋𝑃 𝑋𝑃 (3) [ 1 4 ] [ 𝑋 𝑋 ] 1−𝑥 [ 𝑃2 𝑃3 ] 2 × [ ] [ ], 𝑌𝑃 𝑌𝑃 𝑥2 [ 1 4 ] [ 𝑌𝑃 𝑌𝑃 ] 750 mL 2 3 𝑋 𝑌 (𝑖= 1,2,3,4) where 𝑃𝑖 and 𝑃𝑖 denote the new coordinates of the vertices in the quadrangle, which is a distorted image of the trade mark “Coke” in Figure 1. Oncethedistortedimageisapproximatedbyacertain geometric transformation model, its inverse transformation can be applied to remove the distortion for the geometric restoration. Look at Figure 3, the original trade mark “Coke” is shown in Figure 3(a), and its distorted images are displayed Figure 1: Example of the biquadratic geometric transformation. in Figure 3(b). When the inverse biquadratic transformation is applied to these images, the normalized images can be produced and illustrated in Figure 3(c), which are much more our previous work, they are categorized into two main classes easy to be recognized by a recognition system. [1, 2]. If the function is fixed, we call it a fixed transformation Consequently, finding a mathematical form to approxi- model; otherwise we call it an elastic transformation model. mate the distortion of a distorted image plays a key role for The former consists of linear and nonlinear models including the restoration. bilinear, quadratic, biquadratic, cubic, and bicubic models, Unfortunately, the harmonic geometric transformation while the latter comprises Coons model and harmonic model. model, which converts an image into arbitrary shape, does These transformation models can be summarized as follows not have any fixed mathematical forms. An example can [2]: be found in Figure 4, where the image of Canadian flag is distorted as shown in Figure 4(b). Note that the shape of (A) fixed models: the flag is changed, which is so complex that it cannot be described by any fixed model; that is, it cannot be represented (1) linear models by any fixed functions in mathematics. In fact, this model (a) translation, is characterized by other kinds of mathematical formula, (b) rotation, that is, partial differential equation. Unlike solving a fixed (c) scaling, mathematical formula, solving a partial differential equation (2) nonlinear models: is difficult. The harmonic transformation model is the most impor- (a) bilinear model, tant and most complicated one in the geometric transforma- (b) quadratic model, tion models. Actually, all of the other models can be in it. (c) biquadratic model, The harmonic model is represented by the partial differential (d) cubic model, equation (PDE) with boundary conditions. (e) bicubic model, Let Ω be the region of the elastic plane, where the image is located, and let Γ be its boundary. Suppose that the functions (B) elastic models of the transformation of boundary Γ are 𝑢=𝑓(𝑥1,𝑥2) and V =𝑔(𝑥1,𝑥2). The harmonic transformation (1) coons model, (2) harmonic model. 𝑇:(𝑥1,𝑥2) 󳨀→ (𝑢, V) (4)

Generallyspeaking,bythefixedmodels,astandardimage satisfies the partial differential equation: can be transformed into some special shapes. An example Δ𝑢 (𝑥 ,𝑥 )=0, (𝑥,𝑥 )∈Ω, of the fixed transformation can be found in Figure 1.More 1 2 1 2 precisely, a fixed transformation has a certain mathematical 𝑢|Γ =𝑓(𝑥1,𝑥2), (𝑥1,𝑥2)∈Γ, form to approximate it, which can be described by a fixed (5) class of mathematical functions. In our previous work [1, 2], ΔV (𝑥1,𝑥2)=0, (𝑥1,𝑥2)∈Ω some significant forms (models) and algorithms have been developed to handle these geometric transformations. For V|Γ =𝑔(𝑥1,𝑥2), (𝑥1,𝑥2)∈Γ, Abstract and Applied Analysis 3

x2

T

OOx1 u

Figure 2: An image in Cartesian coordinates 𝑥1𝑂𝑥2 is transformed into a new image in Cartesian coordinates 𝑢𝑂V by geometric transformation 𝑇.

the boundary measure formula, which can change the forms of integral equation and integral representation from boundary to the plane, is presented. Based on the new forms, the wavelet collocation method is used to solve the equation, which is the main task in Section 3.3.Acoupleofalgorithms of IEWC are provided in Section 4. The experiments are illustrated in Section 5. Finally, the conclusions are given in Section 6. (a) 2. Review of the Existing Methods

(b) (c) Twosuccessfulapproacheshavebeenusedtosolvethiskind of partial differential equation, namely, finite element method Figure 3: Restoration of image from the biquadratic distortion by inverse transformation. [5] and finite difference method [6]. In our previous work [2], the finite element method has been employed solving the equation to handle the harmonic transformation, which gives where the following algorithm. 𝜕2 𝜕2 Δ: + Algorithm 1 (finite element method). One has the following. 2 2 (6) 𝜕𝑥1 𝜕𝑥2 Step 1 (discrete region Ω). Divide region Ω by many small is Laplace’s operator; thus the above partial differential triangular elements 𝑒𝑖 such that Ω≈∪𝑖𝑒𝑖. For example, region equation is called Laplace’s equation or harmonic equation. Ω in Figure 5 is divided into twenty two triangular elements Accordingly, the task of the restoration is solving the based on twelve dots, which produce a lattice Π. above harmonic equation. We return to the previous example as shown in Figure 4,andthedistortedimageinFigure 4(b) Step 2 (discrete solution 𝑢). Use piecewise linear function 𝑢ℎ can be approximated by harmonic transformation. As the to approximate the original solution 𝑢.Because𝑢ℎ is linear in corresponding harmonic equation is solved and its inverse each triangular element, therefore 𝑢ℎ is fully determined by transformation is utilized, the restored image can be obtained the values at lattice Π. To determine these values, replacing 𝑢 in Figure 4(c). Therefore, solving the harmonic equation (5) with 𝑢ℎ in the variational form of the equation produces a set plays a key role in the geometric restoration. of linear equations This paper proposes a novel approach based on wavelet analysis to handle the harmonic transformation. 𝐾𝑢 = 𝑓 , The paper is organized as follows. The existing methods ℎ ℎ (7) are reviewed in Section 2.Thecoreoftheproposedwavelet- based approach is presented in Section 3. As the first step where the elements in vector 𝑢ℎ are the values of 𝑢ℎ at lattice of this method, in Section 3.1, the conversion of the partial Π, 𝐾 is a known matrix, and 𝑓 is a right-hand side vector. differential equation into boundary integral equation and ℎ representation is discussed. Two integral methods can be Step 3 (consider the boundary conditions).Findtheboundary applied; here the indirect method is chosen and the boundary dots on lattice Π, and thereafter, the values of 𝑢ℎ on these dots integral equation of the first kind is produced. In Section 3.2, are assigned to satisfy boundary condition 𝑓(𝑥1,𝑥2),which 4 Abstract and Applied Analysis

−1 T T−1

(a) (b) (c)

Figure 4: Harmonic distortion and restoration.

6 5 6 5 7 7 9 9 10 10 T

12 4 11 12 11 8 4 1 Ω 1 8

3 Ω󳰀 2

2 3

Figure 5: Finite element method. canbedonebymodifyingthematrix𝐾 and the right-hand transformed, while the values of the remainder two side 𝑓ℎ in the equations 𝐾𝑢ℎ = 𝑓ℎ. points (11 and 12) are not required to be transformed. Step 4 (solve equations). Solve the linear algebraic equations and finally obtain 𝑢|Π ≈𝑢ℎ|Π = 𝑢ℎ,where𝑢|Π indicates the On the other hand, the values of some pixels on the value of 𝑢 at lattice Π. pattern but not at the lattice are still unknown after In our example, shown in Figure 5,twelvevaluesatlattice solving 𝐾𝑢ℎ = 𝑓ℎ. For example, in Figure 5,most Π are to be determined, seven dots are on the boundary, and ofthepixelsofletter“A,”whicharerequiredtobe the correct values of 𝑢ℎ at these seven boundary dots (dots transformed, are not at the lattice. Thus, the values of 1to7)aregivenduetotheboundarycondition𝑓(𝑥1,𝑥2). these pixels are unknown in solution of 𝑢|Π ≈𝑢ℎ|Π = The values at remainder five inner dots (dots 8 to 12) will be 𝑢ℎ. Consequently, the interpolation will be used to obtained by solving the linear equations. approximate these values. In this way, the extra cost The finite difference method is another common wayto and error will be brought in. solve partial differential equation numerically. It can change the partial differential equation into a set of corresponding algebraic linear equations. The details can be found in [6]. (ii) More attention must be paid to deal with the Both the finite element method and finite difference boundary conditions. Specifically, for a given lattice, method have some defects when they will be used in our we should know which dots are on the boundary cases. For example, two issues of the weakness will occur and assign them the values satisfying the boundary when the finite element method is applied to the harmonic conditions, as mentioned in Step 3 of Algorithm 1. transformation. Therefore, the program code of the lattice and the linear equations is required to be modified, when the (i) It depends on the lattice Π. In finite element method, domain or lattice is changed. the values of all points at lattice Π are solved, even though some points do not lie on the image. For example, in Figure 5, the values of all points at lattice, Thus, more efficient method must be developed to handle say points 8–12, are solved. However, only three points the geometric transformation. In this paper, a novel approach (8–10)arethepixelsofthepattern,Englishletter called Integral Equation-Wavelet Collocation (IEWC) is pre- “A.” The values of these points are required to be sented. Abstract and Applied Analysis 5

Phase 1 Phase 2 Phase 3 Plane integral Partial Boundary integral equation and differential equation and Solution equation boundary integral plane integral representation representation

Figure 6: Diagram of the new approach.

3. Integral Equation-Wavelet Collocation modified at all. In mathematics, this process can be presented (IEWC) Approach in the following:

This is the core section of the paper; a novel approach 󵄨 󵄨 BIE : ∫ 𝜔(𝑦)log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) ,∀(𝑥1,𝑥2)∈Γ basedontheintegralequationandwavelets,calledIntegral Γ Equation-Wavelet Collocation (IEWC), is presented in this section. BIR :𝑢(𝑥) = ∫ 𝜔(𝑦)⋅⋅⋅𝑑𝑠𝑦,∀(𝑥1,𝑥2)∈𝜔 The diagram of the new method is shown in Figure 6.The Γ basic idea of the method is briefly described in Figure 6. ⇓ Boundary Measure Formula (BMF) ⇓ Phase 1.First,thepartial differential equation (PDE) (Laplace’s (9) 󵄨 󵄨 equation) is changed into the form of integral equation PIE : ∫ 𝜔(𝑦)‖𝜕Ω‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) , 𝑅2 and integral representation on boundary Γ, which are called boundary integral equation (BIE) and boundary integral rep- ∀(𝑥1,𝑥2)∈Γ resentation (BIR), respectively.Therearetwowaystodoso, namely, direct method and indirect method [7, 8]. In this PIR :𝑢(𝑥) = ∫ 𝜔(𝑦)‖𝜕Ω‖ ⋅⋅⋅𝑑𝑠𝑦,∀(𝑥1,𝑥2)∈𝜔, paper, the indirect method is utilized. Mathematically, the 𝑅2 process for solving 𝑢 canbewrittenasfollows: where ‖𝜕Ω‖ stands for the boundary measure, which will be discussed in Section 3.2. Δ𝑢 (𝑥 ,𝑥 )=0, ∀(𝑥,𝑥 )∈Ω :{ 1 2 1 2 PDE Phase 3.Then,wavelet collocation techniqueisusedtosolve 𝑢|Γ =𝑓(𝑥1,𝑥2), ∀(𝑥1,𝑥2)∈Γ, the plane integral equation. In the integral equation, the ⇓ Boundary Integral (Indirect) Method ⇓ integrand has a discontinuity across boundary Γ.Hence,there (8) is a kind of singularity in it. Fortunately, wavelets have a 󵄨 󵄨 good property to approximate this kind of singularity. More BIE : ∫ 𝜔(𝑦)log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) ,∀(𝑥1,𝑥2)∈Γ Γ specifically, suppose that we have a function 𝑓,whichhasa discontinuity across curve Γ; otherwise it is smooth. When a BIR :𝑢(𝑥) = ∫ 𝜔(𝑦)⋅⋅⋅𝑑𝑠𝑦,∀(𝑥1,𝑥2)∈𝜔, standard Fourier representation is applied to approximate 𝑓 Γ ̃𝐹 with the best 𝑚 nonzero Fourier terms, 𝑓𝑚,wehave

󵄩 󵄩2 where 𝜔(𝑦) isaunknownfunction,whichcanbesolvedin 󵄩𝑓−𝑓̃𝐹󵄩 ≍𝑚−1/2,𝑚󳨀→∞. 󵄩 𝑚󵄩2 (10) BIE and will be used in BIE. 𝑑𝑠𝑦 indicates the curvilinear integrate with respect to variable 𝑦=(𝑦1,𝑦2).Thefirst 𝑓 shortcoming, which arises in the finite element method, can When a wavelet representation is used to approximate with ̃W be overcome by this way. the best 𝑚 nonzero wavelet terms, 𝑓𝑚 ,wecanobtain Similarly, V canalsobeobtainedinthesameway,which 󵄩 𝑊󵄩2 −1 will be omitted to save the space. In the remainder of this 󵄩𝑓−𝑓̃ 󵄩 ≍𝑚 ,𝑚󳨀→∞. (11) paper, we only discuss 𝑢. 󵄩 𝑚 󵄩2

Phase 2. In order to solve the boundary integral equation Note that if we compare the right-hand sides of the above −1/2 −1 (BIE) efficiently and to get rid of the second defect in the two equations, that is, 𝑚 and 𝑚 , it is clear that the rate finite element method, the boundary measure formula (BMF) of the approximation is very slow when Fourier approach is used. It changes the boundary integral equation and bound- is utilized, and the rate of wavelet approximation is better ary integral representation into an integral equation and an than that of Fourier approximation. Moreover, until now, the 2 integral representation on the whole plane 𝑅 rather than the wavelet approach is the best result for a fixed nonadaptive special boundary Γ.Theyarecalledplane integral equation method [9]. (PIE) and plane integral representation (PIR), respectively. In The last step is the choice of the points to be transformed this way, when Γ is changed, the program code will not to be inthedomainandcalculatethenewcoordinatesofthemby 6 Abstract and Applied Analysis integral representation. The mathematical representation of Boundary Measure Formula (BMF) these operations can be illustrated as follows: ⇓⇓ 󵄨 󵄨 󵄨 󵄨 PIE : ∫ 𝜔(𝑦)‖𝜕Ω‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) , PIE : ∫ 𝜔(𝑦)‖𝜕Ω‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) , 𝑅2 𝑅2

∀(𝑥1,𝑥2)∈Γ ∀(𝑥1,𝑥2)∈Γ

PIR :𝑢(𝑥) = ∫ 𝜔(𝑦)‖𝜕Ω‖ ⋅⋅⋅𝑑𝑠𝑦,∀(𝑥1,𝑥2)∈𝜔 PIR :𝑢(𝑥) = ∫ 𝜔(𝑦)‖𝜕Ω‖ ⋅⋅⋅𝑑𝑠𝑦,∀(𝑥1,𝑥2)∈𝜔 𝑅2 𝑅2 ⇓ Wavelet Collocation (WC) Technique ⇓ ⇓⇓ 𝑗 󵄨 󵄨 ∑ℎ ∫ 𝜙𝑗 (𝑦 )𝜙𝑗 (𝑦 ) 󵄨𝑥 −𝑦󵄨 𝑑𝑦 Wavelet Collocation (WC) Method (𝑝,𝑞) 𝑝 1 𝑞 2 log 󵄨 𝑘 󵄨 𝑝,𝑞 𝑅2 (12) 𝑦 ⇓⇓ −𝑓(𝑥𝑘)=0, ∀(𝑥1,𝑥2)∈Γ 𝑗 𝑗 𝑗 󵄨 󵄨 𝑗 𝑗 𝑗 𝑗 ∑ℎ ∫ 𝜙 (𝑦 )𝜙 (𝑦 ) 󵄨𝑥 −𝑦󵄨 𝑑𝑦 − 𝑓 (𝑥 )=0, 𝑝,𝑞 𝑝 1 𝑞 2 log 󵄨 𝑘 󵄨 𝑘 𝑢 (𝑥) = ∑ℎ ∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2)⋅⋅⋅𝑑𝑦, ( ) 𝑅2 (𝑝,𝑞) 2 𝑝,𝑞 𝑦 𝑝,𝑞 𝑅𝑦 ∀(𝑥1,𝑥2)∈Γ ∀(𝑥1,𝑥2)∈𝜔 𝑗 𝑗 𝑗 𝑗 ⇓⇓ 𝑢 (𝑥) = ∑ℎ ∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2)⋅⋅⋅𝑑𝑦, (𝑝,𝑞) 2 𝑝,𝑞 𝑅𝑦 Solutions. ∀(𝑥1,𝑥2)∈𝜔 The principal advantages of our method are as follows. ⇓⇓

(i) The algorithm is divided into two parts, integral Solutions . equation and integral representation. After solving (13) the plane integral equation, we can choose the pixels to be transformed in the domain arbitrarily and use 3.1. Boundary Integral Method. According to the boundary plane integral representation to evaluate their new integralmethod,thefirstphaseoftheIEWCconvertsthe coordinates. Therefore, only the pixels on the pattern partial differential equation (PDE)5 ( )whichisrecalled aretransformedtothenewcoordinatespace.In below, Figure 2, for example, we will choose all pixels on characters “Coke,” and find the new coordinates of Δ𝑢=0, ∀𝑥=(𝑥1,𝑥2)∈Ω, them by the integral representation. (14) 𝑢|Γ =𝑓(𝑥1,𝑥2), ∀𝑥=(𝑥1,𝑥2)∈Γ, (ii) The program code is independent of the domain considered; that is, the program code will require no into a boundary integral equation (BIE) and a boundary change for the different kind of boundaries. It benefits integral representation (BIR). from the boundary measure formula. In fact, we do There are two kinds of boundary integral equations, notneedthefunctionofboundaryΓ at all. What we namely, (1) integral equation of the first kind and (2) integral really need are the original coordinates of the pixels at equation of the second kind. the boundary and the coordinates of the new ones. If the equation takes the form of

Finally, the summary of this novel approach can be illustrated ∫ 𝐾 (𝑥,) 𝑦 𝑢 (𝑦) 𝑑𝑠𝑦 =𝑓(𝑥) ,∀𝑥=(𝑥1,𝑥2) ∈Γ, (15) in the following mathematical expression: Γ it is called integral equation of the first kind,where𝐾(𝑥, 𝑦) and Δ𝑢 1(𝑥 ,𝑥2)=0, ∀(𝑥1,𝑥2)∈Ω :{ 𝑓(𝑥) are known functions and 𝑑𝑠𝑦 indicates the integrate with PDE 𝑢| =𝑓(𝑥,𝑥 ), ∀(𝑥 ,𝑥 )∈Γ, Γ 1 2 1 2 variable 𝑦=(𝑦1,𝑦2). Otherwise, the equation with the form ⇓⇓ 𝜆𝑢 (𝑥) − ∫ 𝐾 (𝑥,) 𝑦 𝑢 (𝑦) 𝑑𝑠𝑦 =𝑓(𝑥) ,∀𝑥=(𝑥1,𝑥2) ∈Γ Boundary Integral (Indirect) Method Γ (16) ⇓⇓ is called an integral equation of the second kind,where𝜆 is a 󵄨 󵄨 BIE : ∫ 𝜔(𝑦)log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) ,∀(𝑥1,𝑥2)∈Γ known constant. Γ Most of the researchers work on the integral equations BIR :𝑢(𝑥) = ∫ 𝜔(𝑦)⋅⋅⋅𝑑𝑠𝑦,∀(𝑥1,𝑥2)∈𝜔 of the second kind in both of the theories and applications, Γ because some integral equations of the first kind are quite ill- ⇓⇓ conditioned; that is, the speed of the convergent will be slow if Abstract and Applied Analysis 7 an iterative algorithm is used. However, the integral equations can be solved easily. Furthermore, to ensure the uniqueness of the first kind have been an increasingly popular approach of the solution, we tacitly assume that the interior of domain to solve various boundary value problems [7]. In this paper, Ω has the property of the boundary integral equations of the first kind are applied diameter (Ω) <1. (22) to facilitate the use of the boundary measure formula, as well as the further application of the wavelet collocation method. As to the existence of the solution, [10]hasprovedthat(20) Furthermore,therearetwowaystoconvertthepartial is equivalent to differential equation (PDE), into the boundary integral equa- 󵄨 󵄨 tion of the first kind, namely, (1) direct method and (2) indirect ∫ 𝜔(𝑦)log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) +𝑐, ∀𝑥∈Γ, Γ method [7, 8]. They will briefly be presented in the following. (23)

∫ 𝜔(𝑦)𝑑𝑠𝑦 =𝐴, Γ (1) Direct Method.Inthismethod,basedonGreen’s formula, and for arbitrary function 𝑓 and constant 𝐴,(23) has unique solutions 𝜔 and 𝑐,whichensuresthat(20)isofviability(i.e., ∫ ∇⋅F𝑑Ω = ∫ F ⋅ n𝑑𝑆, (17) Ω Γ unique solvable). The above discussion gives us a hint that we can first where ∇=𝜕/𝜕𝑥1 +𝜕/𝜕𝑥2, F =(𝑓1(𝑥),2 𝑓 (𝑥)), 𝑥=(𝑥1,𝑥2), obtain the data 𝜔 from (20)andthenuse(21)tocalculate𝑢(𝑥) and n is the unit outer normal vector; the partial differential at any pixel 𝑥=(𝑥1,𝑥2) in the domain Ω.Notethatthepixel equation (5) can be changed to a system with a boundary 𝑥∈Ωis chosen arbitrarily in the integral representation, integral equation and a boundary integral representation. which makes us free from the lattice built in either the finite Thereafter, the following main steps are done. element method or finite difference one. If the function of the boundary can be determined, that is, 𝜔 Γ Step 1. Solve the integral equation to find on such Γ is parameterized, (20) can be changed into one-dimensional that integral equation on a closed interval. Thereafter, it can be 󵄨 󵄨 solved by the classical methods or periodic wavelet method ∫ 𝜔 (𝑦) log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 Γ [11–16]. In this way, the program code is dependent on the representation of curve Γ.Unfortunately,inmostofthecases, 𝜕 󵄨 󵄨 (18) the function of the boundary is unknown as we mentioned in =−𝜋𝑓(𝑥) + ∫ 𝑓(𝑦) log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦, Γ 𝜕𝑛 Section 1. In order to develop a new method, which can avoid knowing the function of boundary Γ,theboundary measure ∀𝑥=(𝑥,𝑥 )∈Γ. 1 2 formula is utilized in our study to change the forms of (20) and (21)intoothersuitableforms. Step 2. ∀𝑥 ∈ Ω,calculate𝑢(𝑥) by formula 𝑢 (𝑥) 3.2. Boundary Measure Formula. Based on the boundary measure formula, the second phase of the IEWC converts the boundary integral equation (BIE) and boundary integral 1 𝜕 󵄨 󵄨 󵄨 󵄨 = ∫ (𝑓 (𝑦) log 󵄨𝑥−𝑦󵄨 −𝜔(𝑦)log 󵄨𝑥−𝑦󵄨)𝑑𝑠𝑦. representation (BIR) into the plane integral equation (PIE) 2𝜋 Γ 𝜕𝑛 and plane integral representation (PIR). (19) 2 Assume that Ω is a bounded domain in 𝑅 ,whosebound- ary Γ can be presented by Lipschitz function 𝐹(𝑥1,𝑥2)=𝑐, (2) Indirect Method. This method is based on the potential and 𝜒Ω denotes its characteristic function, which has value 1 theory, and the procedure of solving (5)ispresentedinthe if the point (𝑥1,𝑥2) is in domain Ω;otherwiseis0.Theunit following. normal vector along Γ can be written as Step 1. Solve the integral equation to find 𝜔 on Γ such ∇𝐹 n = , that |∇𝐹| (24) 󵄨 󵄨 where ∇𝐹 is the gradient of 𝐹 and |∇𝐹| stands for its ∫ 𝜔 (𝑦) log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 =𝑓(𝑥) ,∀𝑥=(𝑥1,𝑥2) ∈Γ. (20) Γ norm. They can generalize the vector-valued measure and the Radon measure, respectively, if 𝐹 is only of bounded variation Step 2.Calculate𝑢(𝑥) by formula over domain Ω [17]. Hence, the boundary measure formula can be described below. 󵄨 󵄨 𝑢 (𝑥) = ∫ 𝜔 (𝑦) log (󵄨𝑥−𝑦󵄨) 𝑑𝑠𝑦,∀𝑥=(𝑥1,𝑥2) ∈Ω. Γ Theorem 2. For any integrable function 𝑓 defined on Γ,after 2 (21) extending 𝑓 from Γ to 𝑅 ,wehave

Itisobviousthateitherthepairof(18), (19)or(20), (21) ∫ 𝑓𝑑𝑠 = ∫ 𝑓 ‖𝜕Ω‖ 𝑑𝑥, (25) 2 is equivalent to (5). The latter is utilized in this paper, because Γ 𝑅 theright-handsidesoftheseequationsareverysimpleand where ‖𝜕Ω‖ = −∇𝜒Ω ⋅ n is called boundary measure. 8 Abstract and Applied Analysis

Proof. In fact, we can derive after phase 2, the plane integral equation (PIE) and plane integral representation (PIR) have been obtained; thereafter, ∫ 𝑓𝑑𝑠 = ∫ 𝑓n ⋅ n𝑑𝑠 inphase3,thewaveletcollocationtechniqueisemployedto Γ Γ arrive the solution. Let 𝜙 be the Daubechies scale function and = ∫ ∇𝑓 ⋅ n𝑑𝑥 + ∫ 𝑓 div n𝑑𝑥 Ω Ω 𝑗 𝑗/2 𝑗 𝜙𝑝 (𝑡) =2 𝜙(2 𝑡−𝑝). (30)

= ∫ ∇𝑓 ⋅ 𝜒 Ωn𝑑𝑥 + ∫ 𝑓𝜒 Ω div n𝑑𝑥 𝑅2 𝑅2 Basedontheboundarymeasureformula(25), the task now (26) is to solve plane integral equation: = ∫ ∇𝑓 ⋅ 𝜒 Ωn𝑑𝑥 − ∫ ∇(𝑓𝜒Ω)⋅n𝑑𝑥 𝑅2 𝑅2 󵄨 󵄨 ∫ 𝜔 ‖𝜕Ω‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑦 =𝑓 (𝑥) ,∀𝑥∈Γ. 2 (31) 𝑅𝑦 =−∫ 𝑓∇𝜒Ω ⋅ n𝑑𝑥 𝑅2 Recall that the support of the gradient of the characteristic 2 = ∫ 𝑓 ‖𝜕Ω‖ 𝑑𝑥. function ∇𝜒Ω is a compact domain in 𝑅 ;infact,itisatubular 𝑅2 neighborhood of Γ. Therefore, we need only finite number of scale functions to represent 𝜔‖𝜕Ω‖ = Ω−𝜔∇𝜒 ⋅ n.LetΛ be an This establishes (25). index set, and let |Λ| be the cardinal of Λ.Thekeypointhere Reference [18] has proved that the gradient of the char- is solving the product 𝜔‖𝜕Ω‖ instead of solving the unknown 𝜔 𝜔‖𝜕Ω‖ acteristic functions ∇𝜒Ω and n can be approximated by function .Infact,whenweknow ,thenfromintegral Daubechies scale or wavelet function in the Sobolev space representation (29), we can obtain 𝑢(𝑥) for any 𝑥∈Ω.That −1 1 𝐻 (Ω) or space 𝐻 (Ω), respectively. That ensures that iswhyweuseintegralequationofthefirstkindinthispaper. As 𝜔‖𝜕Ω‖ can be approximated by 𝑓 ‖𝜕Ω‖ =−𝑓∇𝜒Ω ⋅ n (27) (𝜔 ‖𝜕Ω‖)𝑗 = ∑ ℎ𝑗 𝜙𝑗 (𝑦 )𝜙𝑗 (𝑦 ), (𝑝,𝑞) 𝑝 1 𝑞 2 (32) can be approximated by Daubechies scale or wavelet function (𝑝,𝑞)∈Λ if 𝑓 is integrable. We will use this formula in our new ‖𝜕Ω‖ = −∇𝜒 ⋅ n 𝑗 approach.Itshouldbenotedthat Ω substituting 𝜔‖𝜕Ω‖ in (28)with(𝜔‖𝜕Ω‖) produces the error Γ has singularities along boundary , therefore the wavelet representation 𝑒(𝑥): representation is more effective to handle this problem than the Fourier representation due to its sparse representation 𝑗 󵄨 󵄨 𝑒 (𝑥) = ∫ (𝜔 ‖𝜕Ω‖) log 󵄨𝑥−𝑦󵄨 𝑑𝑦 −𝑓 (𝑥) ,∀𝑥∈Γ. of singularities. For example, to represent an edge, a type of 2 𝑅𝑦 √1/𝑁 𝑁2 singularity, with error ,roughlyspeaking,requires (33) sinusoids in Fourier form but needs only about 𝑁 wavelet items in wavelet representation. As we discussed in Section 1, Choose |Λ| collocation points {𝑥𝑘}⊂Γ, and let 𝑒(𝑥𝑘)=0,1≤ the wavelets outperform the classical method. That is the 𝑘≤|Λ|,andthenwehave main reason we use wavelet here. Using the boundary measure formula, the boundary inte- 𝑗 󵄨 󵄨 𝑒(𝑥𝑘)=∫ (𝜔 ‖𝜕Ω‖) log 󵄨𝑥𝑘 −𝑦󵄨 𝑑𝑦 − 𝑓 (𝑥𝑘)=0, gral equation (20) and the boundary integral representation 𝑅2 𝑦 (34) (21) become the following plane integral equation and plane integral representation: 1≤𝑘≤|Λ| .

󵄨 󵄨 That is, ∫ 𝜔 ‖𝜕Ω‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑦 =𝑓 (𝑥) ,∀𝑥∈Γ, 2 (28) 𝑅𝑦 𝑗 󵄨 󵄨 ∫ (𝜔 ‖𝜕Ω‖) log 󵄨𝑥𝑘 −𝑦󵄨 𝑑𝑦 = 𝑓 (𝑥𝑘), 1≤𝑘≤|Λ| . 󵄨 󵄨 2 󵄨 󵄨 𝑅𝑦 𝑢 (𝑥) = ∫ 𝜔 ‖𝜕Ω‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑦, ∀𝑥 ∈Ω. 2 (29) 𝑅𝑦 (35)

Theorem 3. Plane integral representation (29) is the solution That is, of partial differential equation (5),where𝜔 is the solution of plane integral equation (28). 𝑗 𝑗 𝑗 󵄨 󵄨 ∑ ℎ ∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2) log 󵄨𝑥𝑘 −𝑦󵄨 𝑑𝑦 = 𝑓 (𝑥𝑘). (𝑝,𝑞) 2 (𝑝,𝑞)∈Λ 𝑅𝑦 Proof. The proof of this theorem is omitted in this paper to (36) avoidthecomplicatedmathematicsandtosavespace. Equation (36) is a set of linear algebraic equations, which can 3.3. Wavelet Collocation Technique. Wavelet analysis has berewritteninformof been widely applied in image processing [19–22]. In this paper, the wavelet theory is used in IEWC method, in which, 𝐾ℎ=𝑓, (37) Abstract and Applied Analysis 9

𝑗 from which, {ℎ(𝑝,𝑞)} is obtained, where matrix 𝐾 with entries 4. Algorithms of the IEWC Approach of |Λ| × |Λ| is In this section, the wavelet-based algorithms of the IEWC 𝑗 𝑗 󵄨 󵄨 approach are presented in both of the general version and 𝐾𝑘,(𝑝,𝑞) = ∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2) log 󵄨𝑥𝑘 −𝑦󵄨 𝑑𝑦, 𝑅2 detailed version. To facilitate the implementation of the 𝑦 (38) algorithm, the computation of matrix, which will be useful

1≤(𝑝,𝑞), 𝑘≤|Λ| ,𝑥𝑘 ∈Γ, in performance of the algorithm, is also discussed in this section. 𝑓 and the right hand side vector is Algorithm 6 (boundary measure and wavelet (general)). One has the following. 𝑓𝑘 =𝑓(𝑥𝑘) , 1≤𝑘≤|Λ| . (39) {𝑥 }|Λ| Γ Remark 4. Note that matrix 𝐾 does not depend on Γ,except Step 1. Choose the collocation points 𝑘 𝑘=1 on ,and |Λ| 𝐾 𝑓 points {𝑥𝑘}𝑘=1 that should be chosen on Γ.Meantime,the evaluate matrixes and in (37)withtheformula(38). right-hand side in (37) does not need the representation of |Λ| Step 2.Solve(37) with least square method to obtain coeffi- function 𝑓(𝑥),exceptvalues{𝑓(𝑥𝑘)}𝑘=1 ofthediscretepixels 𝑗 |Λ| cients {ℎ(𝑝,𝑞)}(𝑝,𝑞)∈Λ. {𝑥𝑘}𝑘=1 on the boundary Γ. These points are called tiepoints. It indicates that, in the implementation, the program code is Step 3.Chooseapoint𝑥 in the domain needed to be independent of the boundary. transformed to new coordinate, and calculate the coefficients {𝑄(𝑝,𝑞)(𝑥)}(𝑝,𝑞)∈Λ with the formula (41). Remark 5. The condition number of matrix 𝐾 may be large, 𝑗 because it is from the integral equation of the first kind; Step 4. Obtain the approximation 𝑢 (𝑥) using (41). thus, usually the Tikhonov’s regularize method is used to solve it [23]. In our new approach, linear algebraic equations (37) are generated by the wavelet collocation technique. 4.1. Computation of the Matrix. IntheStep1andStep3 Therefore, the diagonal preconditioning method [14]canbe above, the main cost is the computation of employed to reduce the condition number of matrix 𝐾.It leads us to use a very easy way to solve (37) instead of using ∫ 𝑓(𝑥,𝑦)𝜙𝑗 (𝑦 )𝜙𝑗 (𝑦 )𝑑𝑦. 𝑝 1 𝑞 2 (43) Tikhonov’s regularize method. This approach was first used 𝑅2 in the Galerkin method, and thereafter, in 1995, Schneider 𝑦 proved that it also can be applied to the collocation method [24]. In this subsection we will give a simple method to calculate it approximately. Let us first introduce some notations. Now we can use integral representation (29)toevaluate theapproximatevalueof𝑢(𝑥) at any point 𝑥 in Ω;thatis, Definition 7. If a quadrature formula 󵄨 󵄨 󵄨 󵄨 𝑛 𝑢 (𝑥) = ∫ 𝜔 log 󵄨𝑥−𝑦󵄨 𝑑𝑠𝑦 Γ ∫ 𝜙 (𝑡) 𝑓 (𝑡) 𝑑𝑡 = ∑𝐴𝑘𝑓(𝑡𝑘) (44) 𝑘=0 󵄨 󵄨 = ∫ 𝜔 ‖𝜕𝜔‖ log 󵄨𝑥−𝑦󵄨 𝑑𝑦 2 𝑅𝑦 (40) holds for any polynomial of degree less than or equal to 2𝑛 + 𝑗 𝑗 𝑗 󵄨 󵄨 1,thenitiscalledGauss-typequadraturewithscalefunction ≈ ∑ ℎ(𝑝,𝑞) ∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2) log 󵄨𝑥−𝑦󵄨 𝑑𝑦 𝑅2 (𝑝,𝑞)∈Λ 𝑦 𝜙(𝑡) as its weight function, the dots {𝑡𝑘},andthecoefficients {𝐴𝑘} are called generalized Gauss-type quadrature dots and 𝑗 =𝑢 (𝑥) ,∀𝑥∈Ω. generalized Gauss-type quadrature weights.

We use notation 𝑄(𝑝,𝑞)(𝑥) to represent the integral expression Similar to the classical Gauss-type quadrature formula, in (40); that is we recall that [0,2𝑁−1]is the support of the scale function 𝜙; therefore, we have the following. 𝑗 𝑗 󵄨 󵄨 𝑄(𝑝,𝑞) (𝑥) = ∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2) log 󵄨𝑥−𝑦󵄨 𝑑𝑦, ∀𝑥 ∈Ω, 2 Proposition 8. In formula (44), the necessary and sufficient 𝑅𝑦 condition for {𝑡𝑘} being generalized Gauss-type quadrature dots (41) 𝑛 is 𝑤𝑛(𝑡) = Π𝑘=0(𝑡 −𝑘 𝑡 ) being orthogonal to any polynomial 𝑃 (𝑡)( ≤𝑛) 𝜙(𝑡) and at last, we have 𝑛 degree with as the weight function; that is,

𝑗 𝑗 2𝑁−1 𝑢 (𝑥) = ∑ ℎ(𝑝,𝑞)𝑄(𝑝,𝑞),∀𝑥∈Ω. (42) ∫ 𝜙 (𝑡) 𝑃𝑛 (𝑡) 𝑤𝑛 (𝑡) 𝑑𝑡 = 0. (45) (𝑝,𝑞)∈Λ 0 10 Abstract and Applied Analysis

(a) (b)

(c) (d)

Figure 7: Experiment 1: nonlinear harmonic and its inverse: (a) original image, (b) distorted image, (c) restored image by IEWC, and (d) restored image by bilinear method.

A significant task is to determine {𝑡𝑘} and {𝐴𝑘} in formula 𝑖 (44). Let 𝑓(𝑡) =𝑡 , 0≤𝑖≤2𝑛+1,weobtainanonlinearsystem [25]: 𝑛 𝑖 𝑖 ∑𝐴𝑘𝑡𝑘 = ∫ 𝜙 (𝑡) 𝑡 𝑑𝑡, 𝑖=0,...,2𝑛+1. (47) 𝑘=0

First, we compute the right-hand side of (47)recursively. Suppose that 2𝑛+1 ≤ 𝑁−1,fromthetheoryofwavelet, we can write 𝑡𝑖 +∞ Figure 8: A harmonic distorted image is approximated by piecewise = ∑ 𝑃 (𝑘) 𝜙 (𝑡−𝑘) , 𝑖=0,...,2𝑛+1, 𝑖! 𝑖 (48) bilinear model. 𝑘=−∞

from which we know +∞ 𝑚 Proposition 9. 𝑓(𝑡)2𝑛+2 ∈𝐶 [0, 2𝑁 − 1] 𝑡 In formula (44),if , 𝑐𝑚 = ∫ 𝜙 (𝑡) 𝑑𝑡𝑚 =𝑃 (0) , (49) the error of (44) is −∞ 𝑚! and it can be computed recursively using [26] 2𝑁−1 𝑛 𝑅=∫ 𝜙 (𝑡) 𝑓 (𝑡) 𝑑𝑡 − ∑𝐴𝑘𝑓(𝑡𝑘) 𝑐0 =1 0 𝑘=0 (46) 1 𝑖 (50) 𝑓(2𝑛+2) (𝜉) 𝜙 (𝜉) 2𝑁−1 𝑐 = ∑𝑀 𝑐 2 𝑖 𝑖 𝑟 𝑖−𝑟 = ∫ 𝑤𝑛 (𝑡) 𝑑𝑡, 2 −1𝑟=1 (2𝑛 + 2)! 0 √ 𝑟 with 𝑀𝑟 =(1/ 2)Σ𝑚ℎ𝑚(𝑚 /𝑟!), where reals ℎ𝑘 are coeffi- √ where 𝜉∈[0,2𝑁−1]. cients in 𝜙(𝑡) = 2Σℎ𝑘𝜙(2𝑡−𝑘). So far, the right-hand side in Abstract and Applied Analysis 11

×104 ×104 6 6

5 4

4

2 3

0 2 0100 200 300 0100 200 300 (a) Orig 𝑥-axis (b) Chang 𝑥-axis ×104 ×104 6 6

4 4

2 2

0 0 0100 200 300 0100 200 300 (c) Pde 𝑥-axis (d) Bi-128 𝑥-axis

Figure 9: Comparison in Experiment 1:projectionofimagestothe𝑥-axis: (a) original image, (b) distorted image, (c) restored image by IEWC, and (d) restored image by the piecewise bilinear method.

4 Table 1 For 𝑓∈𝐶, by use of spline function, we can prove that its error, 𝜀,is 𝑁𝑥1 =𝑐 8.17401𝐸 − 001 3 2𝑁 − 1 𝜀=𝑂(ℎ4) ,ℎ= . 4 1.00539 2𝑗 (53) 51.19390 61.38216Based on these results, we know that

1 𝑝+𝑐 𝑞+𝑐 (47) is computed. Then, we solve the nonlinear system (47), 𝑓( , ) 𝑗 𝑗 𝑗 (54) so that 2𝑛 + 2 coefficients {𝑡𝑘} and {𝐴𝑘} are obtained. For 2 2 2 example, when 𝑛=1, 𝑁 = 3, 4, 5, 6,wehave canbeusedtoapproximate 𝐴0 ≈0, 𝐴1 ≈1 (51) and obtain Table 1. 𝑗 𝑗 ∫ 𝑓(𝑥,𝑦)𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2)𝑑𝑦, Therefore, we have a very simple formula 2 (55) 𝑅𝑦 2𝑁−1 ∫ 𝜙 (𝑡) 𝑓 (𝑡) 𝑑𝑡 ≈𝑓 (𝑐) . (52) 0 and we will use it in our experiments. 12 Abstract and Applied Analysis

×104 ×104 6 6

5 4

4

2 3

0 2 0100 200 300 0100 200 300 (a) Orig 𝑦-axis (b) Chang 𝑦-axis

×104 ×104 6 6

4 4

2 2

0 0 0100 200 300 0100 200 300 (c) Pde 𝑦-axis (d) Bi-128 𝑦-axis

Figure 10: Comparison in Experiment 1:projectionofimagestothe𝑦-axis: (a) original image, (b) distorted image by harmonic transformation, (c) restored image by IEWC, and (d) restored image by the piecewise bilinear method.

Algorithm 10 (boundary measure and wavelet (detail)). Computematrixelements 𝑗 𝑗 󵄨 󵄨 𝐾𝑘,(𝑝,𝑞) (𝑥𝑘)=∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2) log 󵄨𝑥𝑘 −𝑦󵄨 𝑑𝑦 Input {𝑥𝑘}⊂Γand {𝑓(𝑥𝑘)}. 2 𝑅𝑦 1 𝑝+𝑐 2 𝑞+𝑐 2 ≈ [(𝑥 − ) +(𝑥 − ) ]. log 𝑘 𝑗 𝑘 𝑗 Step 1 2 1 2 2 2 (56) 𝑗 Choose a resolution level and fix a rectan- Obtain right-hand side {𝑓(𝑥𝑘)} from 𝐷 {𝑥 } gular domain covering 𝑘 . {𝑓(𝑥𝑘)}. Calculate the index set Λ={(𝑝,𝑞)}by End. 𝜙𝑗 ∩𝐷={0}̸ (𝑝,𝑞) . Step 3 |Λ| {𝑥 }⊂ Choose any collocation points 𝑘 Solve the linear systems 𝐾ℎ=𝑓 and obtain {𝑥𝑘}. 𝑗 ℎ={ℎ(𝑝,𝑞)}. Step 2 Step 4 𝑀 Choose the points {𝑧𝑘}𝑘=1 in the domain to For 𝑘=1to |Λ| Do be transformed to new coordinate space. Abstract and Applied Analysis 13

(a) (b)

(c) (d)

Figure 11: Experiment 2: nonlinear harmonic and its inverse for a circle area.

Step 5 Experiment 1 (nonlinear-harmonic and its inverse). The har- monic transformation 𝑇 : (𝑥, 𝑦) →(𝑢, V) satisfies (5), which 𝑘=1 𝑀 For to Do: is recalled as follows: Evaluate Δ𝑢 1(𝑥 ,𝑥2)=0, (𝑥1,𝑥2)∈Ω, 𝑗 𝑗 󵄨 󵄨 𝑄(𝑝,𝑞) (𝑧𝑘)=∫ 𝜙𝑝 (𝑦1)𝜙𝑞 (𝑦2) log 󵄨𝑧𝑘 −𝑦󵄨 𝑑𝑦 𝑢| =𝑓(𝑥,𝑥 ), (𝑥 ,𝑥 )∈Γ, 2 Γ 1 2 1 2 𝑅𝑦 ΔV (𝑥1,𝑥2)=0, (𝑥1,𝑥2)∈Ω, 1 𝑝+𝑐 2 𝑞+𝑐 2 ≈ log [(𝑧𝑘 − ) +(𝑧𝑘 − ) ]. V|Γ =𝑔(𝑥1,𝑥2), (𝑥1,𝑥2)∈Γ. 2 1 2𝑗 2 2𝑗 (57) (59) In this experiment, the equations of the boundary conditions Calculate the new coordinate 𝑢|Γ =𝑓(𝑥1,𝑥2), V|Γ =𝑔(𝑥1,𝑥2) and the region Ω are specified by the following forms: 𝑗 𝑗 𝑢 (𝑧𝑘)= ∑ ℎ 𝑄(𝑝,𝑞) (𝑧𝑘). (𝑝,𝑞) (58) 2 (𝑝,𝑞)∈Λ 𝑓(𝑥1,𝑥2) = 1.2𝑥1,𝑔(𝑥1,𝑥2)=𝑥2 +(𝑥1 − 0.5) , (60) Ω:0≤𝑥,𝑥≤ 0.5. End. 1 2 (61) Geometric transform is viewed to map the location of input 𝑗 𝑀 Output {𝑢 (𝑧𝑘)}𝑘=1. image to a location in the output image; it defines how the pixelvaluesintheoutputimagearetobearranged.The distorted image coordinates can be defined by the equations 5. Experiments 𝑢=𝑢(𝑥1,𝑥2), V = V (𝑥1,𝑥2). (62) Experiments have been conducted to evaluate the perfor- mances of the new approach. The primary idea is to find a mathematical model for the A couple of numerical experiments using the wavelet- geometric distortion process, specifically, the equations 𝑢= based IEWC approach is presented in this section. 𝑢(𝑥1,𝑥2), V = V(𝑥1,𝑥2) and then apply the inverse process 14 Abstract and Applied Analysis

4 ×104 ×10 6 5.5

5 5

4 4.5

3 4

2 3.5

1 3 0100 200 300 0100 200 300 (a) Orig 𝑥-axis (b) Chang 𝑥-axis ×104 ×104 6 6

5 5

4 4

3 3

2 2

1 1 0 100 200 300 0100 200 300 (c) Pde 𝑥-axis (d) Bi-128 𝑥-axis

Figure 12: Comparison in Experiment 2:projectionofimagestothe𝑥-axis: (a) original image, (b) distorted image, (c) restored image by IEWC, and (d) restored image by the piecewise bilinear method.

to find the restored image. To determine the necessary equa- In the traditional approach, a harmonic distorted image is tions, we need to identify a set of points in the original image approximated by piecewise bilinear model [27], as shown in that matches another set of points in the distorted image. Figure 8. In the bilinear model, four points of each subregion These sets of points are called tiepoints. In this experiment, areusedtogeneratetheequation: the relationship between these tiepoints in two images is determined by the boundary condition, which is described 𝑢=𝑘1𝑥1 +𝑘2𝑥2 +𝑘3𝑥1𝑥2 +𝑘4, in (60), and the proposed IEWC method can be used. (63) V =𝑘5𝑥1 +𝑘6𝑥2 +𝑘7𝑥1𝑥2 +𝑘8, We use Figure 7 as an example, where Figure 7(a) is the original image. As the IEWC approach is applied to solve the where 𝑘𝑖, 𝑖 = 1,2,...,8, are constants to be determined harmonic transformation with the boundary condition (60), by solving the eight simultaneous equations. Because we the original image is distorted and displayed in Figure 7(b). have defined four tiepoints, thus we have eight equations, To achieve the restoration, then the CSIM method is used to where 𝑥1, 𝑥2, 𝑢,andV are known. Now we can solve perform the inverse transformation, and the restored image the eight equations for the eight unknowns 𝑘𝑖,sothat is illustrated in Figure 7(c). The details of the CSIM method the necessary equations for the coordinate mapping can can be found in our previous work [1]. be obtained. The mapping equations 𝑢=𝑢(𝑥1,𝑥2), V = To quantify and clarify the advantages of IEWC over the V(𝑥1,𝑥2) then are applied to all (𝑥1,𝑥2) pairs needed. In traditional approach, a comparison is given in Figure 8. practice, the values of 𝑥1, 𝑥2, 𝑢,andV are not likely to Abstract and Applied Analysis 15

4 ×10 ×104 6 6

4 4

2 2

0 0 0100 200 300 0100 200 300 (a) Orig 𝑦-axis (b) Chang 𝑦-axis ×104 ×104 6 6

4 4

2 2

0 0 0100 200 300 0100 200 300 (c) Pde 𝑦-axis (d) Bi-128 𝑦-axis

Figure 13: Comparison in Experiment 2:projectionofimagestothe𝑦-axis: (a) original image, (b) distorted image, (c) restored image by IEWC, and (d) restored image by the piecewise bilinear method.

be integers. The simplest solution is the nearest neighbor 𝑓(𝑥1,𝑥2)=𝑥1 +𝑥2, method, where the pixel is assigned to the value of the 𝑔(𝑥 ,𝑥 )=𝑥2 −𝑥2. closestpixelintheimage.Therestoredimagewhichis 1 2 1 2 constructed by the classical piecewise bilinear method is pre- (64) sented in Figure 7(d),inwhich,128tiepointsareusedonthe Ω boundary. This method does not necessarily provide optimal In this experiment, the domain of the partial differential equation (PDE) is a circle area. Because the area is not a results. quadrilateral (four-sided polygon), it is difficult to use the Tocomparetheabovetworestoredimages,weproject piecewise bilinear method. It is shown that the program code both of them, as well as the original image and the distorted 𝑥 𝑦 of the IEWC approach is independent of the boundary form. one, to the -axis and -axis. The results are shown in Figures The results are shown in Figure 11.Tocomparetheresults,the 9 and 10, respectively. From these projections, it is clear that projections of them to the axis are shown in Figures 12 and 13. the results of IEWC approach are better than that of the traditional one. 6. Conclusions Experiment 2 (nonlinear-harmonic and its inverse for a circle Usuallythepiecewisebilinearmodelcanbeusedtoall area). Consider geometric degradation image regardless the different char- 2 2 2 𝑇2 :(𝑥1,𝑥2)󳨀→(𝑢, V) ,Ω:𝑥1 +𝑥2 ≤ 0.4 , acteristicsofimages.Wehavefoundanewmodel(partial 16 Abstract and Applied Analysis differential equation PDE) for the transformation in our pre- References vious work [1, 2] and aim to construct an efficient algorithm (IEWC) to solve the PDE in this paper. [1]Z.C.Li,T.D.Bui,Y.Y.Tang,andC.Y.Suen,Computer In this paper, we have presented a wavelet-based approach Transformation of Digital images and Patterns,vol.17,World Scientific, Singapore, 1989. for the harmonic transformation. Unlike the traditional methods, in the IEWC approach, the pixels needed to be [2]Y.Y.TangandC.Y.Suen,“Imagetransformationapproach to nonlinear shape restoration,” IEEE Transactions on Systems, transformed to new coordinates can be chosen arbitrarily. Man and Cybernetics,vol.23,no.1,pp.155–171,1993. Meanwhile, the program code of our method is independent [3]H.D.Cheng,Y.Y.Tang,andC.Y.Suen,“Parallelimagetrans- of the boundary, and we need only a set of the original formation and its VLSI implementation,” Pattern Recognition, coordinates of the pixels on the boundary of the image as vol. 23, no. 10, pp. 1113–1129, 1990. well as their new coordinates in the transformed image. [4] G. Wolberg, Digital Image Warping, IEEE Computer Society To make the algorithm more efficiency, Daubechies wavelet Press, Los Alamitos, CA, USA, 1990. (scale) functions and a Gauss-type quadrature formula have [5] O. C. Zienkiewics, The Finite Element Method, McGraw-Hill, been used. Different examples have been tested with the London, UK, 3rd edition, 1977. anticipated results. [6] A. R. Mitchell, The Finite Difference Method in Partial Differen- Some further work is still under study, which is presented tial Equations, Addision-Wesley, New York, NY, USA, 1980. below. The tiepoints are unknown and should be guessed [7]K.E.Atkinson,The Numerical Solution of Integral Equations or calculated by some other methods. In this paper, a GA of the Second Kind,vol.4,CambridgeUniversityPress,Cam- approach has been applied to extract the outer contours and bridge, UK, 1997. find the tiepoints. In our further work, other GA, wavelet- [8] G. C. Hsiao and W. L. Wendland, “A finite element method for based method, local search, immune approach, and so forth some integral equations of the first kind,” Journal of Mathemat- will be used to find their usage in this direction to construct ical Analysis and Applications,vol.58,no.3,pp.449–481,1977. a good interpolation method. [9] D. L. Donoho, “Orthonormal ridgelets and linear singularities,” In addition, as discussed above, there are singularities SIAMJournalonMathematicalAnalysis,vol.31,no.5,pp.1062– along the boundary, which can be treated efficiently by 1099, 2000. wavelet. More recently, Donoho [9] has constructed a tool [10]G.HsiaoandR.C.MacCamy,“Solutionofboundaryvalue called curvelet to handle this kind of singularity, which is built problems by integral equations of the first kind,” SIAM Review, from Meyer wavelet basis. In our further work, the curvelet vol.15,no.4,pp.77–93,1973. will be utilized. In this way, the boundary measure will be [11] A. Antoniadis, X. Brossat, J. Cugliari, and J.-M. Poggi, “Clus- approximated by the curvelet with the same accuracy as we tering functional data using wavelets,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 11, no. use wavelet. Meantime, compared with wavelet or sinusoid 1, Article ID 1350003, 30 pages, 2013. basis, fewer terms will be computed when the curvelet will be [12] G. Bhatnagar and Q. M. J. Wu, “An image fusion framework used. It will make our algorithm more efficient. based on human visual system in framelet domain,” Inter- Besides the further improvement of the algorithm, the national Journal of Wavelets, Multiresolution and Information proposed method will be combined with other techniques Processing,vol.10,no.1,ArticleID1250002,30pages,2012. to solve more sophisticated problems. For example, when we [13] G. Chen, S.-E. Qian, J.-P. Ardouin, and W. Xie, “Super- restore the distorted image, some blurred pictures may occur. resolution of hyperspectral imagery using complex Ridgelet To obtain a clean image, which will be easier to be recognized transform,” International Journal of Wavelets, Multiresolution by a pattern recognition system, the fusion technique will be and Information Processing,vol.10,no.3,ArticleID1250025, applied. 22 pages, 2012. [14] W.Dahmen, S. Prossdorf, and R. Schneider, Multiscale Methods for Pseudodifferential Equations. Equations, Recent Advances in Conflict of Interests Wavelet Analysis, 1993, edited by, L. L. Schumaker and G. Webb. [15] R. L. Wagner and W.C. Chew, “Study of wavelets for the solution The authors declare that there is no conflict of interests of electromagnetic integral equations,” IEEE Transactions on regarding the publication of this paper. Antennas and Propagation,vol.43,no.8,pp.802–810,1995. [16] Z. Xiang and Y. Lu, “An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations,” IEEE Transactions on Antennas and Propagation, Acknowledgments vol.45,no.8,pp.1205–1213,1997. [17] R. O. Wells, Jr. and X. Zhou, “Wavelet solutions for the Dirichlet This work was financially supported by the Multi-Year problem,” Numerische Mathematik,vol.70,no.3,pp.379–396, Research of University of Macau under Grants no. 1995. MYRG205(Y1-L4)-FST11-TYY and no. MYRG187(Y1-L3)- [18] H. L. Resnikoff and R. O. Wells Jr., Wavelet Analysis-The Scalable FST11-TYY and by the National Natural Science Foundation Structure of Information, Springer, New York, NY, USA, 1998. of China under Grant no. 61273244. This research project was [19] S. Kumar, S. Kumar, B. Raman, and N. Sukavanam, “Image dis- also supported by the Science and Technology Development parity estimation based on fractional dual-tree complex wavelet Fund (FDCT) of Macau, under Contract nos. 100-2012-A3, transform: a multi-scale approach,” International Journal of 026-2013-A, and also by Guangxi Science and Technology Wavelets, Multiresolution and Information Processing, vol. 11, no. Fund of China, under Contract no. 201203YB179. 1, Article ID 1350004, 21 pages, 2013. Abstract and Applied Analysis 17

[20] S. P. Maity and M. K. Kundu, “Performance improvement in spread spectrum image watermarking using wavelets,” Inter- national Journal of Wavelets, Multiresolution and Information Processing, vol. 9, no. 1, pp. 1–33, 2011. [21] S. P. Maity, A. Phadikar, and M. K. Kundu, “Image error concealment based on QIM data hiding in dual-tree complex wavelets,” International Journal of Wavelets, Multiresolution and Information Processing,vol.10,no.2,ArticleID1250016,30 pages, 2012. [22] M. Zaied, S. Said, O. Jemai, and C. Ben Amar, “A novel approach for face recognition based on fast learning algorithm and wavelet network theory,” International Journal of Wavelets, Multiresolution and Information Processing,vol.9,no.6,pp. 923–945, 2011. [23] L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985. [24] W. Dahmen, “Wavelet and multiscale methods for operator equations,” ACTA Numerica,vol.6,pp.55–228,1997. [25] Y. H. Zhou and J. Z. Wang, “A generalized Gaussian integral method for the calculation of scaling function transforms of wavelets and its applications,” Acta Mathematica Scientia A: Shuxue Wuli Xuebao,vol.19,no.3,pp.293–300,1999. [26] A. Cohen, I. Daubechies, and P. Vial, “Wavelets on the inter- val and fast wavelet transforms,” Applied and Computational Harmonic Analysis: Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications,vol.1,no.1, pp. 54–81, 1993. [27] S. E. Umbaugh, Computer Vision and Image Processing, Prentice-Hall, New York, NY, USA, 1998. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 858704, 9 pages http://dx.doi.org/10.1155/2014/858704

Research Article On the Paranormed Nörlund Sequence Space of Nonabsolute Type

Medine YeGilkayagil1 and Feyzi BaGar2

1 Department of Mathematics, Us¸ak University, 1 Eylul¨ Campus, 64200 Us¸ak, Turkey 2 Department of Mathematics, Fatih University, Hadımkoy¨ Campus, Buy¨ ukc¨ ¸ekmece, 34500 Istanbul,˙ Turkey

Correspondence should be addressed to Feyzi Bas¸ar; [email protected]

Received 3 December 2013; Accepted 2 February 2014; Published 26 March 2014

Academic Editor: M. Mursaleen

Copyright © 2014 M. Yes¸ilkayagil and F. Bas¸ar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

∞ 𝑝𝑘 Maddox defined the space ℓ(𝑝) of the sequences 𝑥=(𝑥𝑘) such that ∑𝑘=0 |𝑥𝑘| <∞, in Maddox, 1967. In the present paper, the 𝑡 𝑡 Norlund¨ sequence space 𝑁 (𝑝) of nonabsolute type is introduced and proved that the spaces 𝑁 (𝑝) and ℓ(𝑝) are linearly isomorphic. 𝑡 𝑡 Besides this, the alpha-, beta-, and gamma-duals of the space 𝑁 (𝑝) arecomputedandthebasisofthespace𝑁 (𝑝) is constructed. 𝑡 𝑡 𝑡 The classes (𝑁 (𝑝) : 𝜇) and (𝜇 : 𝑁 (𝑝)) of infinite matrices are characterized. Finally, some geometric properties of the space 𝑁 (𝑝) are investigated.

1. Introduction

󵄨 󵄨𝑝𝑘 We denote the space of all sequences of complex entries by ℓ∞ (𝑝) = {𝑥𝑘 =(𝑥 )∈𝜔:sup󵄨𝑥𝑘󵄨 <∞} 𝜔. Any vector subspace of 𝜔 is called a sequence space.We 𝑘∈N write ℓ∞, 𝑐,and𝑐0 for the spaces of all bounded, convergent, (1) and null sequences, respectively. Also by 𝑏𝑠, 𝑐𝑠, ℓ1,andℓ𝑝,we denote the spaces of all bounded, convergent, absolutely and whicharethecompletespacesparanormedby 𝑝-absolutely convergent series, respectively. A linear topological space 𝑋 over the real field R is said 1/𝑀 󵄨 󵄨𝑝 to be a paranormed space if there is a subadditive function 𝑔 (𝑥) =(∑󵄨𝑥 󵄨 𝑘 ) , 𝑔:𝑋 → R 𝑔(𝜃) =0 𝑔(𝑥) = 𝑔(−𝑥) 1 󵄨 𝑘󵄨 such that , and 𝑘 scalar multiplication is continuous; that is, |𝛼𝑛 −𝛼|→ 0 (2) 𝑔(𝑥 −𝑥) → 0 𝑔(𝛼 𝑥 −𝛼𝑥) →0 󵄨 󵄨𝑝𝑘/𝑀 and 𝑛 imply 𝑛 𝑛 for all 𝑔2 (𝑥) = sup󵄨𝑥𝑘󵄨 iff inf𝑝𝑘 >0, 𝛼’s in R and all 𝑥’s in 𝑋,where𝜃 is the zero vector in the 𝑘∈N 𝑘∈N linear space 𝑋. Assume here and after that (𝑝𝑘) is a bounded sequence of strictly positive real numbers with sup 𝑝𝑘 =𝐻 respectively, where N = {0, 1, 2, . . .}. For simplicity in and 𝑀=max{1, 𝐻}.Then,thelinearspacesℓ(𝑝) and ℓ∞(𝑝) notation, here and in what follows, the summation without −1 were defined by Maddox in [1](seealso[2, 3]) as follows: limits runs from 0 to ∞. We assume throughout that 𝑝𝑘 + 󸀠 −1 (𝑝𝑘) =1,provided1

with 0<𝑝𝑘 ≤𝐻<∞, 𝑆 (𝜆,𝜇) = {𝑧=(𝑧𝑘) ∈𝜔:𝑥𝑧=(𝑥𝑘𝑧𝑘) ∈𝜇∀𝑥∈𝜆} . (3) 2 Abstract and Applied Analysis

𝑡 𝑡 𝑡 𝑡 With the notation of (3), the alpha-, beta-, and gamma-duals The inverse matrix 𝑈 =(𝑢𝑛𝑘) of the matrix 𝑁 =(𝑎𝑛𝑘) is 𝛼 of a sequence space 𝜆, which are, respectively, denoted by 𝜆 , given by Mears in [5] as follows: 𝛽 𝛾 𝜆 ,and𝜆 , are defined by 𝑛−𝑘 𝛼 𝛽 𝛾 𝑡 (−1) 𝐷 𝑇 , 0≤𝑘≤𝑛, 𝜆 =𝑆(𝜆,ℓ), 𝜆 =𝑆(𝜆,𝑐𝑠) ,𝜆=𝑆(𝜆,𝑏𝑠) . (4) 𝑢 ={ 𝑛−𝑘 𝑘 1 𝑛𝑘 0, 𝑘>𝑛 (10) If a sequence space 𝜆 paranormed by 𝑔 contains a (𝑏 ) 𝑥∈𝜆 sequence 𝑛 with the property that, for every ,there for all 𝑘, 𝑛 ∈ N. Also, one can derive by straightforward (𝛼 ) is a unique sequence of scalars 𝑛 such that calculation for all 𝑘∈{1,2,3,...}that 𝑛 𝑔 (𝑥−∑𝛼 𝑏 ) =0, 𝑘−1 𝑛→∞lim 𝑘 𝑘 (5) 𝑗−1 𝑘−1 𝑘=0 𝐷𝑘 = ∑(−1) 𝑡𝑗𝐷𝑘−𝑗 + (−1) 𝑡𝑘. (11) 𝑗=1 then (𝑏𝑛) is called a Schauder basis (or briefly basis)for𝜆. The series ∑ 𝛼𝑘𝑏𝑘 which has the sum 𝑥 is then called the 𝑘 The rest of this paper is organized as follows. expansion of 𝑥 with respect to (𝑏𝑛) and written as 𝑥=∑ 𝛼𝑘𝑏𝑘. 𝑘 In Section 2,thecompleteparanormedNorlund¨ Let 𝜆, 𝜇 be any two sequence spaces, and let 𝐴=(𝑎𝑛𝑘) be 𝑡 𝑡 sequence space 𝑁 (𝑝) isintroducedandprovedthat𝑁 (𝑝) an infinite matrix of complex numbers 𝑎𝑛𝑘,where𝑘, 𝑛 ∈ N. is linearly isomorphic to the space ℓ(𝑝) and the basis for the Then, we say that 𝐴 defines a matrix transformation from 𝜆 𝑡 space 𝑁 (𝑝) is determined. Section 3 is devoted to the alpha-, into 𝜇 andwedenoteitbywriting𝐴:𝜆→𝜇,ifforevery 𝑡 𝑥=(𝑥)∈𝜆 𝐴𝑥 = {(𝐴𝑥) } 𝐴 beta-, and gamma-duals of the space 𝑁 (𝑝).InSection 4,the sequence 𝑘 ,thesequence 𝑛 ,the - (𝑁𝑡(𝑝) : 𝜇) (𝜇 : 𝑁𝑡(𝑝)) transform of 𝑥,isin𝜇,where classes and of infinite matrices are characterized, where 𝜇 denotes any given sequence space. In 𝑡 (𝐴𝑥) = ∑𝑎 𝑥 𝑛∈N. Section 5, the rotundity of the space 𝑁 (𝑝) is characterized 𝑛 𝑛𝑘 𝑘 for each (6) 𝑘 andsomeresultsrelatedtothisconceptaregiven.Inthe (𝜆: 𝜇) 𝐴 final section of the paper, the significance of the space is By , we denote the class of all matrices such that mentioned and further suggestions are recorded. 𝐴:𝜆→.Thus, 𝜇 𝐴 ∈ (𝜆: 𝜇) if and only if the series on the 𝑛∈N 𝑥∈𝜆 right side of (6) converges for each and every ,and 𝑡 we have 𝐴𝑥 ∈𝜇 for all 𝑥∈𝜆.Also,wewrite𝐴𝑛 =(𝑎𝑛𝑘)𝑘∈N 2. The Nörlund Sequence Space 𝑁 (𝑝) of for the sequence in the 𝑛th row of 𝐴. Nonabsolute Type Now, following Peyerimhoff [4,pp.17–19]andMears[5], 𝑁𝑡(𝑝) we give short knowledge on the Norlund¨ means. Let (𝑡𝑘) be a In this section, we define theorlund N¨ sequence space 𝑁𝑡(𝑝) ℓ(𝑝) sequence of nonnegative real numbers with 𝑡0 >0and write and prove that is linearly isomorphic to the space , 𝑛 0<𝑝 ≤𝐻<∞ 𝑘∈N 𝑇𝑛 =∑𝑘=0 𝑡𝑘 for all 𝑛∈N.Then,theNorlund¨ means with where 𝑘 for all .Finally,wegivethe 𝑡 𝑁𝑡(𝑝) respect to the sequence 𝑡=(𝑡𝑘) is defined by the matrix 𝑁 = basis for the space . 𝑡 𝜆 𝜆 (𝑎𝑛𝑘) which is given by Let be any sequence space. Then, the matrix domain 𝐴 of an infinite matrix 𝐴 in 𝜆 is defined by 𝑡 { 𝑛−𝑘 , 0≤𝑘≤𝑛, 𝑎𝑡 = 𝑇 𝑛𝑘 { 𝑛 (7) 𝜆𝐴 ={𝑥=(𝑥𝑘)∈𝜔:𝐴𝑥∈𝜆}. (12) {0, 𝑘 > 𝑛 𝑡 for all 𝑘, 𝑛 ∈ N. It is known that the Norlund¨ matrix 𝑁 is a In [6], Choudhary and Mishra defined the sequence space ℓ(𝑝) 𝐵 Toeplitz matrix if and only if 𝑡𝑛/𝑇𝑛 →0,as𝑛→∞,andis which consists of all sequences such that -transforms ℓ(𝑝) 𝐵=(𝑏 ) reducedinthecase𝑡 = 𝑒 = (1,1,1,...)to the matrix 𝐶1 of of them are in ,where 𝑛𝑘 is defined by 𝑟−1 arithmetic means. Additionally, for 𝑡𝑛 =𝐴𝑛 for all 𝑛∈N, 𝑁𝑡 𝐶 1, 0≤𝑘≤𝑛 the method is reduced to the Cesaro` method 𝑟 of order 𝑏 ={ 𝑟>−1,where 𝑛𝑘 0, 𝑘 > 𝑛. (13) (𝑟+1)(𝑟+2) ⋅⋅⋅(𝑟+𝑛) 𝑟 { , 𝑛=1,2,3,..., 𝐴𝑛 = { 𝑛! (8) Bas¸ar and Altay [7] examined the space 𝑏𝑠(𝑝) which was {1, 𝑛 = 0. formerly defined by Bas¸ar [8] as the set of all series whose sequences of partial sums are in the space ℓ∞(𝑝).Withthe Let 𝑡0 =𝐷0 =1and define 𝐷𝑛 for 𝑛∈{1,2,3,...}by notation of (12), the spaces ℓ(𝑝) and 𝑏𝑠(𝑝) can be redefined 󵄨 𝑡 100⋅⋅⋅0󵄨 󵄨 1 󵄨 by 󵄨 𝑡 𝑡 10⋅⋅⋅0󵄨 󵄨 2 1 󵄨 󵄨 𝑡 𝑡 𝑡 1⋅⋅⋅0󵄨 󵄨 3 2 1 󵄨 ℓ(𝑝)=[ℓ(𝑝)] ,𝑏𝑠(𝑝)=[ℓ(𝑝)] . (14) 𝐷 = 󵄨 󵄨 . 𝐵 ∞ 𝐵 𝑛 󵄨 . . . . . 󵄨 (9) 󵄨 . . . . d . 󵄨 󵄨 󵄨 𝑞 󵄨𝑡𝑛−1 𝑡𝑛−2 𝑡𝑛−3 𝑡𝑛−4 ⋅⋅⋅ 1󵄨 In [9], Bas¸ar and Altay defined the sequence space 𝑟 (𝑝) 󵄨 󵄨 𝑞 󵄨 𝑡𝑛 𝑡𝑛−1 𝑡𝑛−2 𝑡𝑛−3 ⋅⋅⋅ 𝑡1󵄨 which consists of all sequences such that 𝑅 -transforms of Abstract and Applied Analysis 3

𝑞 𝑞 them are in ℓ(𝑝),where𝑅 =(𝑟𝑛𝑘) is the matrix of Riesz mean; Let us take any 𝑦∈ℓ(𝑝)and define the sequence 𝑥=(𝑥𝑘) that is, by

𝑞 𝑞 𝑘 𝑟 (𝑝) = {ℓ (𝑝)} ,𝑟=(ℓ ) . 𝑘−𝑖 𝑅𝑞 𝑝 𝑝 𝑞 (15) 𝑅 𝑥𝑘 = ∑(−1) 𝐷𝑘−𝑖𝑇𝑖𝑦𝑖 ∀𝑘 ∈ N. (20) 𝑖=0 In [10], Wang defined the sequence space 𝑋𝑎(𝑝) consisting 𝑡 Therefore, we see from (19)that of all sequences whose 𝑁 -transforms are in ℓ𝑝 which is a 󵄨 󵄨𝑝 1/𝑀 Banach space with the norm 󵄨 𝑘 󵄨 𝑘 󵄨 1 󵄨 𝑔 (𝑥) =(∑󵄨 ∑𝑡 𝑥 󵄨 ) 󵄨𝑇 𝑘−𝑗 𝑗󵄨 󵄨 󵄨𝑝 1/𝑝 𝑘 󵄨 𝑘 𝑗=0 󵄨 󵄨 1 𝑘 󵄨 󵄨 󵄨 𝑥 = (∑󵄨 ∑𝑡 𝑥 󵄨 ) 1≤𝑝<∞. ‖ ‖𝑝 󵄨 𝑘−𝑗 𝑗󵄨 with (16) 1/𝑀 󵄨𝑇 󵄨 󵄨 󵄨𝑝𝑘 𝑘 󵄨 𝑘 𝑗=0 󵄨 󵄨 𝑘 𝑗 󵄨 󵄨 󵄨 󵄨 1 𝑗−𝑖 󵄨 =(∑ 󵄨 ∑𝑡 ∑(−1) 𝐷 𝑇 𝑦 󵄨 ) (21) 󵄨𝑇 𝑘−𝑗 𝑗−𝑖 𝑖 𝑖󵄨 𝑡 𝑘 󵄨 𝑘 𝑗=0 𝑖=0 󵄨 Now, we introduce the Norlund¨ sequence space 𝑁 (𝑝) defined by 1/𝑀 󵄨 󵄨𝑝𝑘 =(∑󵄨𝑦𝑘󵄨 ) =𝑔1 (𝑦) < ∞. 󵄨 󵄨𝑝 { 󵄨 𝑘 󵄨 𝑘 } 𝑘 𝑡 󵄨 1 󵄨 𝑁 (𝑝) := 𝑥=(𝑥)∈𝜔:∑󵄨 ∑𝑡 𝑥 󵄨 <∞ { 𝑘 󵄨𝑇 𝑘−𝑗 𝑗󵄨 } 𝑥∈𝑁𝑡(𝑝) 𝑇 { 𝑘 󵄨 𝑘 𝑗=0 󵄨 } (17) This means that .Consequently, is surjective and is paranorm preserving. Hence, 𝑇 is linear bijection and this 𝑁𝑡(𝑝) ℓ(𝑝) with 0<𝑝𝑘 ≤𝐻<∞. says us that the spaces and are linearly isomorphic. Therefore, the proof is completed. 𝑡 It is natural that the space 𝑁 (𝑝) canalsobedefinedwiththe 𝑁𝑡(𝑝) 𝑡 We determine the basis for the paranormed space . notation of (12)that𝑁 (𝑝) = {ℓ(𝑝)}𝑁𝑡 . 𝑦=(𝑦) 𝑁𝑡 (𝑘) (𝑘) Define the sequence 𝑘 by the -transform of a Theorem 4. Define the sequence 𝑏 (𝑡) = 𝑛{𝑏 (𝑡)}𝑛∈N of the 𝑥=(𝑥) 𝑡 sequence 𝑘 ;thatis, elements of the space 𝑁 (𝑝) for every fixed 𝑘∈N by

𝑘 𝑛−𝑘 1 (𝑘) (−1) 𝐷𝑛−𝑘𝑇𝑘, 0≤𝑘≤𝑛, 𝑦 = ∑𝑡 𝑥 ∀𝑘 ∈ N. 𝑏 (𝑡) ={ 𝑘 𝑘−𝑗 𝑗 (18) 𝑛 0, 𝑘>𝑛. (22) 𝑇𝑘 𝑗=0 (𝑘) 𝑡 Then, the sequence {𝑏 (𝑡)}𝑘∈N is a basis for the space 𝑁 (𝑝) Theorem 1. 𝑁𝑡(𝑝) 𝑡 is a complete linear metric space para- and any 𝑥∈𝑁(𝑝) has a unique representation of the form normed by 𝑔 defined by 𝑥=∑𝜆 (𝑡) 𝑏(𝑘) (𝑡) , 𝑘 (23) 󵄨 󵄨𝑝 1/𝑀 󵄨 𝑘 󵄨 𝑘 𝑘 󵄨 1 󵄨 𝑔 (𝑥) = (∑󵄨 ∑𝑡𝑘−𝑗𝑥𝑗󵄨 ) 𝑤𝑖𝑡ℎ𝑘 0<𝑝 ≤𝐻<∞. 𝑡 󵄨𝑇 󵄨 where 𝜆𝑘(𝑡) = (𝑁 𝑥)𝑘 for all 𝑘∈N and 0<𝑝𝑘 ≤𝐻<∞. 𝑘 󵄨 𝑘 𝑗=0 󵄨 (19) (𝑘) 𝑡 Proof. It is clear that {𝑏 (𝑡)} ⊂ 𝑁 (𝑝),since

𝑡 (𝑘) (𝑘) Proof. Since this can be shown by a routine verification, we 𝑁 𝑏 (𝑡) =𝑒 ∈ℓ(𝑝) ∀𝑘∈N, (24) omit the detail. (𝑘) where 𝑒 isthesequencewhoseonlynonzerotermisa1 in Remark 2. One can easily see that the absolute property does 𝑡 the 𝑘th place for each 𝑘∈N and 0<𝑝𝑘 ≤𝐻<∞. not hold on the space 𝑁 (𝑝);thatis,𝑔(𝑥) ≠ 𝑔(|𝑥|) for at least 𝑥∈𝑁𝑡(𝑝) 𝑚 𝑁𝑡(𝑝) 𝑁𝑡(𝑝) Let be given. For every nonnegative integer , one sequence in the space , and this says that is we put a sequence space of nonabsolute type, where |𝑥| = (|𝑥𝑘|). 𝑚 𝑡 [𝑚] (𝑘) Theorem 3. The Norlund¨ sequence space 𝑁 (𝑝) of nonabsolute 𝑥 = ∑𝜆𝑘 (𝑡) 𝑏 (𝑡) . (25) 𝑘=0 type is linearly isomorphic to the space ℓ(𝑝),where0<𝑝𝑘 ≤ 𝐻<∞for all 𝑘∈N. 𝑡 Then, we obtain by applying 𝑁 to (25)with(24)that Proof. To prove the theorem, we should show the existence 𝑚 𝑚 𝑁𝑡(𝑝) ℓ(𝑝) 𝑁𝑡𝑥[𝑚] = ∑𝜆 (𝑡) 𝑁𝑡𝑏(𝑘) (𝑡) = ∑(𝑁𝑡𝑥) 𝑒(𝑘), of a linear bijection between the spaces and for 𝑘 𝑘 0<𝑝𝑘 ≤𝐻<∞. Consider the transformation 𝑇 defined, 𝑘=0 𝑘=0 𝑡 with the notation of (18), from 𝑁 (𝑝) to ℓ(𝑝) by 𝑥 󳨃→𝑦= (26) 𝑡 0, 0 ≤ 𝑖 ≤ 𝑚, 𝑇𝑥 =𝑁 𝑥.Thelinearityof𝑇 is clear. Further, it is trivial that {𝑁𝑡 (𝑥−𝑥[𝑚])} ={ 𝑖 𝑡 𝑥=𝜃whenever 𝑇𝑥 =𝜃 and hence 𝑇 is injective. (𝑁 𝑥)𝑖,𝑖>𝑚, 4 Abstract and Applied Analysis

where 𝑖, 𝑚 ∈ N.Given𝜖>0, then there is an integer 𝑚0 such Lemma 6 (see [12], Theorem 1). The following statements that hold.

∞ 1/𝑀 1<𝑝 ≤𝐻<∞ 𝑘∈N 𝐴= 󵄨 󵄨𝑝 (i) Let 𝑘 for every .Then, [ ∑ 󵄨(𝑁𝑡𝑥) 󵄨 𝑘 ] <𝜖 (𝑎 ) ∈ (ℓ(𝑝) : ℓ ) 󵄨 𝑖󵄨 (27) 𝑛𝑘 ∞ if and only if there exists an integer 𝑖=𝑚+1 𝐵>1such that

(𝑚 + 1) ≥ 𝑚 󵄨 󵄨𝑝󸀠 for all 0.Hence, 󵄨 −1󵄨 𝑘 sup∑󵄨𝑎𝑛𝑘𝐵 󵄨 <∞. (32) 𝑛∈N 𝑘 ∞ 1/𝑀 󵄨 󵄨𝑝 𝑔[𝑁𝑡 (𝑥−𝑥[𝑚])] = [ ∑ 󵄨(𝑁𝑡𝑥) 󵄨 𝑘 ] 󵄨 𝑖󵄨 𝑖=𝑚+1 (ii) Let 0<𝑝𝑘 ≤1for every 𝑘∈N.Then,𝐴=(𝑎𝑛𝑘)∈ (28) ∞ 1/𝑀 (ℓ(𝑝) :∞ ℓ ) if and only if 󵄨 󵄨𝑝 ≤[∑ 󵄨(𝑁𝑡𝑥) 󵄨 𝑘 ] <𝜖 󵄨 𝑖󵄨 𝑖=𝑚 󵄨 󵄨𝑝𝑘 0 sup 󵄨𝑎𝑛𝑘󵄨 <∞. 𝑛,𝑘∈N (33) 𝑡 for all (𝑚 + 1) ≥0 𝑚 which proves that 𝑥∈𝑁(𝑝) is represented as in (23). Lemma 7 0<𝑝 ≤𝐻<∞ Letusshowtheuniquenessoftherepresentationfor𝑥∈ (see [12], Theorem 1). Let 𝑘 for 𝑡 𝑘∈N 𝐴=(𝑎 ) ∈ (ℓ(𝑝) : 𝑐) 𝑁 (𝑝) given by (23). Suppose, on the contrary, that there every .Then, 𝑛𝑘 ifandonlyif (32), (𝑘) (33) hold and there is 𝛽𝑘 ∈ C such that 𝑎𝑛𝑘 →𝛽𝑘 for each exists a representation 𝑥=∑𝑘 𝜇𝑘(𝑡)𝑏 (𝑡). Since the linear 𝑡 𝑘∈N. transformation 𝑇,from𝑁 (𝑝) to ℓ(𝑝),usedintheproofof Theorem 3 is continuous, we have at this stage that Theorem 8. Let 1<𝑝𝑘 ≤𝐻<∞for every 𝑘∈N.Definethe sets 𝐷1(𝑝), 𝐷2(𝑝),and𝐷3(𝑝) as follows: (𝑁𝑡𝑥) = ∑𝜇 (𝑡) {𝑁𝑡𝑏(𝑘) (𝑡)} = ∑𝜇 (𝑡) 𝑒(𝑘) =𝜇 (𝑡) 𝑛 𝑘 𝑛 𝑘 𝑛 𝑛 𝑘 𝑘 { (29) 𝐷1 (𝑝) : = {𝑎=(𝑎𝑘)∈𝜔

𝑡 { for all 𝑛∈N which contradicts the fact that (𝑁 𝑥)𝑛 =𝜆𝑛(𝑡) 𝑡 󵄨 󵄨𝑝󸀠 for all 𝑛∈N. Hence, the representation (23)of𝑥∈𝑁(𝑝) is 󵄨 󵄨 𝑘 } 󵄨 𝑛−𝑘 −1󵄨 : sup∑󵄨 ∑ (−1) 𝑎𝑛𝐷𝑛−𝑘𝑇𝑘𝐵 󵄨 <∞} , unique. This completes the proof. 󵄨 󵄨 𝑁∈F 𝑘 󵄨𝑛∈𝑁 󵄨 }

󵄨 󵄨𝑝󸀠 3. The Alpha-, Beta-, and Gamma-Duals of { 󵄨 𝑛 󵄨 𝑘 𝑡 󵄨 𝑖−𝑘 −1󵄨 the Space 𝑁 (𝑝) 𝐷2 (𝑝) : = {𝑎=(𝑎𝑘)∈𝜔:sup∑󵄨∑(−1) 𝑎𝑖𝐷𝑖−𝑘𝑇𝑘𝐵 󵄨 𝑛∈N 󵄨 󵄨 { 𝑘 󵄨𝑖=𝑘 󵄨 In this section, we determine the alpha-, beta-, and gamma- 𝑡 duals of the space 𝑁 (𝑝).Wewillquotesomelemmaswhich 󸀠 } −1 𝑝𝑘 are needed in proving our theorems. <∞,{(𝑎𝑛𝑇𝑛𝐵 ) }∈ℓ∞} , } Lemma 5 (see [11], Theorem 5.1.0). The following statements hold. 𝐷3 (𝑝) = 𝑐𝑠. (34) (i) Let 1<𝑝𝑘 ≤𝐻<∞for every 𝑘∈N.Then,𝐴= (𝑎 ) ∈ (ℓ(𝑝) : ℓ ) 𝑛𝑘 1 if and only if there exists an integer Then, the following statements hold: 𝐵>1such that {𝑁𝑡(𝑝)}𝛼 =𝐷(𝑝) 𝑝󸀠 (i) 1 ; 󵄨 󵄨 𝑘 󵄨 󵄨 ∑󵄨 ∑ 𝑎 𝐵−1󵄨 <∞. (30) 𝑡 𝛾 sup 󵄨 𝑛𝑘 󵄨 (ii) {𝑁 (𝑝)} =𝐷2(𝑝); 𝑁∈F 𝑘 󵄨𝑛∈𝑁 󵄨 𝑡 𝛽 (iii) {𝑁 (𝑝)} =𝐷2(𝑝) ∩3 𝐷 (𝑝).

(ii) Let 0<𝑝𝑘 ≤1for every 𝑘∈N.Then,𝐴=(𝑎𝑛𝑘)∈ 𝑎=(𝑎)∈𝜔 (ℓ(𝑝) : ℓ ) Proof. (i) Let us take 𝑘 .Weeasilyderivewith(20) 1 if and only if that

󵄨 󵄨𝑝𝑘 𝑛 󵄨 󵄨 󵄨 ∑ 𝑎 󵄨 <∞. 𝑎 𝑥 = ∑(−1)𝑛−𝑘𝑎 𝐷 𝑇 𝑦 =(𝐶𝑦) ∀𝑛 ∈ N, supsup󵄨 𝑛𝑘󵄨 (31) 𝑛 𝑛 𝑛 𝑛−𝑘 𝑘 𝑘 𝑛 (35) 𝑁∈F𝑘∈N󵄨𝑛∈𝑁 󵄨 𝑘=0 Abstract and Applied Analysis 5

where 𝐶=(𝑐𝑛𝑘) is defined by Proof. Thisiseasilyobtainedbyproceedingasintheproof of Theorem 8 by using Lemma 7 and the second parts of 𝑛−𝑘 (−1) 𝑎𝑛𝐷𝑛−𝑘𝑇𝑘, 0≤𝑘≤𝑛, Lemmas 5 and 6 instead of the first parts. So, we omit the 𝑐𝑛𝑘 ={ (36) 0, 𝑘>𝑛 detail.

𝑘, 𝑛 ∈ N for all . Thus, we observe by combining35 ( )withPart 4. Some Matrix Transformations Related to (i) of Lemma 5 that 𝑎𝑥 =𝑛 (𝑎 𝑥𝑛)∈ℓ1 whenever 𝑥=(𝑥𝑘)∈ 𝑡 𝑡 the Sequence Space 𝑁 (𝑝) 𝑁 (𝑝) if and only if 𝐶𝑦 ∈ℓ1 whenever 𝑦=(𝑦𝑘)∈ℓ(𝑝).This {𝑁𝑡(𝑝)}𝛼 =𝐷(𝑝) gives the result that 1 . In the present section, we characterize the matrix transfor- 𝑡 (ii) Consider the equality mations from the space 𝑁 (𝑝) into any given sequence space 𝑡 𝜇 andfromagivensequencespace𝜇 into the space 𝑁 (𝑝). 𝑛 𝑛−1 𝑛 𝜇 ≅𝜇 𝐴 𝜇 ∑𝑎 𝑥 = ∑ ∑(−1)𝑖−𝑘𝑎 𝐷 𝑇 𝑦 +𝑎 𝑇 𝑦 Since 𝐴 for any triangle and any sequence space ,itis 𝑘 𝑘 𝑖 𝑖−𝑘 𝑘 𝑘 𝑛 𝑛 𝑛 𝑥∈𝜇 𝑦=𝐴𝑥∈𝜇 𝑘=0 𝑘=0 𝑖=𝑘 (37) trivial that the equivalence “ 𝐴 if and only if ” holds. =(𝐸𝑦) ∀𝑛 ∈ N, 𝑛 Now, we can give the following theorem. 𝐸=(𝑒 ) where 𝑛𝑘 is defined by Theorem 10. Suppose that the elements of the infinite matrices 𝑛 𝐴=(𝑎𝑛𝑘) and 𝐹=(𝑓𝑛𝑘) are connected with the relation { 𝑖−𝑘 {∑(−1) 𝑎 𝐷 𝑇 , 0≤𝑘≤𝑛−1, { 𝑖 𝑖−𝑘 𝑘 ∞ 𝑖=𝑘 𝑓 := ∑(−1)𝑗−𝑘𝐷 𝑇 𝑎 𝑒𝑛𝑘 = { (38) 𝑛𝑘 𝑗−𝑘 𝑘 𝑛𝑗 (40) {𝑎 𝑇 , 𝑘=𝑛, 𝑗=𝑘 { 𝑛 𝑛 0, 𝑘>𝑛 { for all 𝑘, 𝑛 ∈ N and 𝜇 is any given sequence space. Then, 𝐴∈ 𝑡 𝑡 𝛽 for all 𝑘, 𝑛 ∈ N.Thus,wededucefromPart(i)ofLemma 6 (𝑁 (𝑝) : 𝜇) if and only if 𝐴𝑛 ∈{𝑁(𝑝)} for all 𝑛∈N and 𝑡 with (37)that𝑎𝑥 =𝑛 (𝑎 𝑥𝑛)∈𝑏𝑠whenever 𝑥=(𝑥𝑘)∈𝑁(𝑝) 𝐹 ∈ (ℓ(𝑝) :𝜇). if and only if 𝐸𝑦 ∈ℓ∞ whenever 𝑦=(𝑦𝑘)∈ℓ(𝑝). Therefore, 𝑡 𝛾 Proof. Let 𝜇 be any given sequence space. Suppose that (40) we obtain from Part (i) of Lemma 6 that {𝑁 (𝑝)} =𝐷2(𝑝). holds between the elements of the matrices 𝐴=(𝑎𝑛𝑘) and (iii) We see from Lemma 7 that 𝑎𝑥 =𝑛 (𝑎 𝑥𝑛)∈𝑐𝑠 𝑡 𝑡 𝐹=(𝑓𝑛𝑘),andtakeintoaccountthatthespaces𝑁 (𝑝) and whenever 𝑥=(𝑥𝑘)∈𝑁(𝑝) if and only if 𝐸𝑦 ∈𝑐 whenever ℓ(𝑝) are linearly isomorphic. 𝑦=(𝑦𝑘)∈ℓ(𝑝). Therefore, we derive from Lemma 7 that 𝑡 𝑡 𝛽 Let 𝐴∈(𝑁(𝑝) : 𝜇) andtakeany𝑦∈ℓ(𝑝).Then {𝑁 (𝑝)} =𝐷2(𝑝) ∩3 𝐷 (𝑝). ∞ ∞ ∞ 𝑡 Therefore, the proof is completed. (𝐹𝑁𝑡) = ∑𝑓 𝑎𝑡 = ∑ ∑(−1)𝑖−𝑗𝐷 𝑎 𝑇 𝑗−𝑘 =𝑎 . 𝑛𝑘 𝑛𝑗 𝑗𝑘 𝑖−𝑗 𝑛𝑖 𝑗 𝑇 𝑛𝑘 𝑗=𝑘 𝑗=𝑘 𝑖=𝑗 𝑗 Theorem 9. Let 0<𝑝𝑘 ≤1for every 𝑘∈N.Definethesets (41) 𝐷4(𝑝) and 𝐷5(𝑝) by 𝑡 𝑡 𝛽 That is, 𝐹𝑁 exists and 𝐴𝑛 ∈{𝑁(𝑝)} which yields that 𝐹𝑛 ∈ ℓ 𝑛∈N 𝐹𝑦 𝐷4 (𝑝) : = {𝑎 𝑘= (𝑎 )∈𝜔 1 for each .Hence, exists and thus ∞ 𝑖−𝑘 󵄨 󵄨𝑝 ∑𝑓𝑛𝑘𝑦𝑘 = ∑∑(−1) 𝐷𝑖−𝑘𝑎𝑛𝑖𝑇𝑘 󵄨 󵄨 𝑘 󵄨 𝑛−𝑘 󵄨 𝑘 𝑘 𝑖=𝑘 : supsup 󵄨 ∑ (−1) 𝑎𝑛𝐷𝑛−𝑘𝑇𝑘󵄨 <∞}, 𝑁∈F𝑘∈N 󵄨 󵄨 (42) 󵄨𝑛∈𝑁 󵄨 1 𝑘 ×( ∑𝑡 𝑥 )=∑𝑎 𝑥 𝑇 𝑘−𝑗 𝑗 𝑛𝑘 𝑘 󵄨 𝑛 󵄨𝑝𝑘 𝑘 𝑗=0 󵄨 󵄨 𝑘 𝐷 (𝑝) : = {𝑎 = (𝑎 )∈𝜔: 󵄨∑(−1)𝑖−𝑘𝑎 𝐷 𝑇 󵄨 5 𝑘 sup 󵄨 𝑖 𝑖−𝑘 𝑘󵄨 𝑛,𝑘∈N󵄨𝑖=𝑘 󵄨 for all 𝑛∈N.So,wehavethat𝐹𝑦 = 𝐴𝑥, which leads us to the consequence 𝐹 ∈ (ℓ(𝑝) :𝜇). 𝑡 𝛽 𝑝 Conversely, let 𝐴𝑛 ∈{𝑁(𝑝)} for each 𝑛∈N and <∞,{(𝑎𝑇 ) 𝑘 }∈ℓ }. 𝑡 𝑛 𝑛 ∞ 𝐹 ∈ (ℓ(𝑝) :𝜇),andtake𝑥=(𝑥𝑘)∈𝑁(𝑝).Then,𝐴𝑥 exists. Therefore, we obtain from the equality (39) 𝑘 𝑘−𝑖 Then, the following statements hold: ∑𝑎𝑛𝑘𝑥𝑘 =∑𝑎𝑛𝑘 [∑(−1) 𝐷𝑘−𝑖𝑇𝑖𝑦𝑖]=∑𝑓𝑛𝑘𝑦𝑘 ∀𝑛 ∈ N 𝑘 𝑘 𝑖=0 𝑘 {𝑁𝑡(𝑝)}𝛼 =𝐷(𝑝) (i) 4 ; (43) 𝑡 𝛾 (ii) {𝑁 (𝑝)} =𝐷5(𝑝); 𝑡 that 𝐴𝑥=𝐹𝑦and this shows that 𝐴∈(𝑁(𝑝) : 𝜇).This 𝑡 𝛽 (iii) {𝑁 (𝑝)} =𝐷3(𝑝) ∩5 𝐷 (𝑝). completes the proof. 6 Abstract and Applied Analysis

𝑡 By changing the roles of the spaces 𝑁 (𝑝) with 𝜇 in (iv) the modular 𝜎 is called convex if 𝜎(𝛼𝑥+𝛽𝑦) ≤ 𝛼𝜎(𝑥)+ Theorem 10,wehavethefollowing. 𝛽𝜎(𝑦) for all 𝑥, 𝑦 ∈𝑋 and 𝛼, 𝛽 >0 with 𝛼+𝛽=1. Amodular𝜎 on 𝑋 is called Theorem 11. Suppose that 𝜇 is any given sequence space and the elements of the infinite matrices 𝐴=(𝑎𝑛𝑘) and 𝐺=(𝑔𝑛𝑘) 𝑛 (a) right continuous if lim𝛼→1+ 𝜎(𝛼𝑥) = 𝜎(𝑥) for all are connected with the relation 𝑔𝑛𝑘 = ∑ (𝑡𝑛−𝑗/𝑇𝑛)𝑎𝑗𝑘 for all 𝑗=0 𝑥∈𝑋𝜎; 𝑘, 𝑛 ∈ N 𝐴∈(𝜇:𝑁𝑡(𝑝)) 𝐺 ∈ (𝜇 : ℓ(𝑝)) .Then, if and only if . (b) left continuous if lim𝛼→1− 𝜎(𝛼𝑥) = 𝜎(𝑥) for all 𝑥∈𝑋𝜎; Proof. Let 𝑥=(𝑥𝑘)∈𝜇and consider the following equality: (c)continuousifitisbothrightandleftcontinuous, 𝑛 𝑡 𝑚 𝑚 where ∑ 𝑛−𝑗 ∑𝑎 𝑥 = ∑𝑔 𝑥 ∀𝑛 ∈ N. 𝑇 𝑗𝑘 𝑘 𝑛𝑘 𝑘 (44) 𝑗=0 𝑛 𝑘=0 𝑘=0 𝑋𝜎 ={𝑥∈𝑋: lim 𝜎 (𝛼𝑥) =0}. (47) 𝛼→0+ 𝑡 Then, by letting 𝑚→∞in (44), we have {𝑁 (𝐴𝑥)}𝑛 =(𝐺𝑥)𝑛 𝑡 𝑡 𝜎 𝑁𝑡(𝑝) 𝜎 (𝑥) = for all 𝑛∈N.Since𝐴𝑥 ∈𝑁 (𝑝), 𝑁 (𝐴𝑥) = 𝐺𝑥 ∈ ℓ(𝑝).This We define 𝑝 on by 𝑝 𝑘 𝑝𝑘 completes the proof. ∑𝑘 |(1/𝑇𝑘) ∑𝑗=0 𝑡𝑘−𝑗𝑥𝑗| .If𝑝𝑘 ≥1for all 𝑘∈N1 = {1, 2, . . .}, 𝑝 𝑡 󳨃→|𝑡| 𝑘 𝑘∈N 𝜎 𝑡 by the convexity of the function for each , 𝑝 𝑁 (𝑝) 𝑡 𝑡 5. The Rotundity of the Space is a convex modular on 𝑁 (𝑝). We consider 𝑁 (𝑝) equipped with Luxemburg norm given by In functional analysis, the rotundity of Banach spaces is one of the most important geometric properties. For details, 𝑥 ‖𝑥‖ = {𝛼 > 0 : 𝜎 ( )≤1}. the reader may refer to [13–15]. In this section, we give the inf 𝑝 (48) 𝑡 𝛼 necessary and sufficient condition in order to the space 𝑁 (𝑝) 𝑡 be rotund and present some results related to this concept. 𝑁 (𝑝) is a Banach space with this norm. This can be shown bythesimilarwayusedintheproofofTheorem7in[16]. 𝑆(𝑋) Definition 12. Let be the unit sphere of a Banach space We establish some basic properties for the modular 𝜎𝑝. 𝑋.Then,apoint𝑥∈𝑆(𝑋)is called an extreme point if 2𝑥 = 𝑦+𝑧 𝑦=𝑧 𝑦, 𝑧 ∈ 𝑆(𝑋) 𝑋 𝑡 implies for every .ABanachspace Proposition 16. The modular 𝜎𝑝 on 𝑁 (𝑝) satisfies the follow- 𝑆(𝑋) is said to be rotund (strictly convex) if every point of is ing properties with 𝑝𝑘 ≥1for all 𝑘∈N. an extreme point. 𝑀 (i) If 0<𝛼≤1,then𝛼 𝜎𝑝(𝑥/𝛼)𝑝 ≤𝜎 (𝑥) and 𝜎𝑝(𝛼𝑥) ≤ 𝑋 Definition 13. ABanachspace is said to have Kadec- 𝛼𝜎𝑝(𝑥). Klee property (or property (𝐻)) if every weakly convergent 𝛼≥1 𝜎 (𝑥) ≤ 𝛼𝑀𝜎 (𝑥/𝛼) sequence on the unit sphere is convergent in norm. (ii) If ,then 𝑝 𝑝 . (iii) If 𝛼≥1,then𝛼𝜎𝑝(𝑥/𝛼)𝑝 ≤𝜎 (𝑥). Definition 14. ABanachspace𝑋 is said to have (iv) The modular 𝜎𝑝 is continuous. (i)theOpialpropertyifeverysequence(𝑥𝑛) weakly 𝑀 𝑝𝑘 convergent to 𝑥0 ∈𝑋satisfies Proof. (i) Let 0<𝛼≤1.Then𝛼 /𝛼 ≤1for all 𝑝𝑘 ≥1.So, 󵄩 󵄩 󵄩 󵄩 we have 󵄩𝑥 −𝑥 󵄩 < 󵄩𝑥 +𝑥󵄩 lim𝑛→∞ inf 󵄩 𝑛 0󵄩 lim𝑛→∞ inf 󵄩 𝑛 󵄩 (45) 󵄨 󵄨𝑝𝑘 𝑥 𝛼𝑀 󵄨 1 𝑘 󵄨 𝛼𝑀𝜎 ( )=∑ 󵄨 ∑𝑡 𝑥 󵄨 𝑝 𝑝 󵄨 𝑘−𝑗 𝑗󵄨 𝑥∈𝑋 𝑥 =𝑥̸ 𝛼 𝛼 𝑘 󵄨𝑇 󵄨 for every with 0; 𝑘 󵄨 𝑘 𝑗=0 󵄨 (ii) the uniform Opial property if for each 𝜖>0,there 󵄨 󵄨𝑝𝑘 𝑟>0 󵄨 1 𝑘 󵄨 exists an such that ≤ ∑󵄨 ∑𝑡 𝑥 󵄨 =𝜎 (𝑥) , 󵄩 󵄩 󵄨 𝑘−𝑗 𝑗󵄨 𝑝 󵄩 󵄩 󵄨𝑇𝑘 𝑗=0 󵄨 1+𝑟≤lim inf 󵄩𝑥𝑛 +𝑥󵄩 (46) 𝑘 󵄨 󵄨 𝑛→∞ (49) 󵄨 󵄨𝑝𝑘 󵄨 1 𝑘 󵄨 𝑥∈𝑋 ‖𝑥‖ ≥ 𝜖 (𝑥 ) 𝑝𝑘 󵄨 󵄨 for each with and each sequence 𝑛 𝜎𝑝 (𝛼𝑥) = ∑𝛼 󵄨 ∑𝑡𝑘−𝑗𝑥𝑗󵄨 𝑤 󵄨𝑇 󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 in 𝑋 such that 𝑥𝑛 󳨀→0and lim inf𝑛→∞‖𝑥𝑛‖≥1. 󵄨 󵄨 󵄨 󵄨𝑝 󵄨 𝑘 󵄨 𝑘 Definition 15. Let 𝑋 be a real vector space. A functional 𝜎: 󵄨 1 󵄨 ≤𝛼∑󵄨 ∑𝑡 𝑥 󵄨 =𝛼𝜎 (𝑥) . 𝑋→[0,∞)is called a modular if 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑝 𝑘 󵄨 𝑘 𝑗=0 󵄨 (i) 𝜎(𝑥) =0 if and only if 𝑥=𝜃; 𝑀 𝑝𝑘 (ii) Let 𝛼≥1.Then1≤𝛼 /𝛼 for all 𝑝𝑘 ≥1.So,wehave (ii) 𝜎(𝛼𝑥) = 𝜎(𝑥) for all scalars 𝛼 with |𝛼| =; 1 𝑀 (iii) 𝜎(𝛼𝑥 + 𝛽𝑦) ≤ 𝜎(𝑥) +𝜎(𝑦) for all 𝑥, 𝑦 ∈𝑋 and 𝛼, 𝛽 ≥0 𝛼 𝑀 𝑥 𝛼+𝛽=1 𝜎𝑝 (𝑥) ≤ 𝜎𝑝 (𝑥) =𝛼 𝜎𝑝 ( ). (50) with ; 𝛼𝑝𝑘 𝛼 Abstract and Applied Analysis 7

𝑝𝑘 𝑡 (iii) Let 𝛼≥1.Then𝛼/𝛼 ≤1for all 𝑝𝑘 ≥1. Therefore, Proof. Let 𝑥∈𝑁(𝑝). one can easily see that (i) Let 𝜖>0such that 0 < 𝜖 < 1−‖𝑥‖. By the definition of 󵄨 󵄨𝑝𝑘 ‖⋅‖ 𝛼>0 ‖𝑥‖ + 𝜖 >𝛼 𝑥 𝛼 󵄨 1 𝑘 󵄨 in (48), there exists an such that 𝛼𝜎 ( )=∑ 󵄨 ∑𝑡 𝑥 󵄨 𝜎 (𝑥/𝛼) ≤1 𝑝 𝑝 󵄨 𝑘−𝑗 𝑗󵄨 and 𝑝 .So,wehave 𝛼 𝛼 𝑘 󵄨𝑇 󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 󵄨 󵄨𝑝 (51) 𝑝 󵄨 𝑘 󵄨 𝑘 󵄨 󵄨𝑝 ‖𝑥‖ +𝜖 𝑘 󵄨 1 󵄨 󵄨 𝑘 󵄨 𝑘 󵄨 󵄨 󵄨 1 󵄨 𝜎𝑝 (𝑥) ≤ ∑( ) 󵄨 ∑𝑡𝑘−𝑗𝑥𝑗󵄨 ≤ ∑󵄨 ∑𝑡 𝑥 󵄨 =𝜎 (𝑥) . 𝛼 󵄨𝑇 󵄨 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑝 𝑘 󵄨 𝑘 𝑗=0 󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 (56) 𝑥 ≤ (‖𝑥‖ +𝜖) 𝜎 ( )≤‖𝑥‖ +𝜖. (iv) If 𝛼>1,thenwehave 𝑝 𝛼

󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 𝑘 󵄨 𝑘 󵄨 𝑘 󵄨 𝑘 󵄨 1 󵄨 𝑝 󵄨 1 󵄨 𝜖 𝜎 (𝑥) ≤ ‖𝑥‖ ∑𝛼󵄨 ∑𝑡 𝑥 󵄨 ≤ ∑𝛼 𝑘 󵄨 ∑𝑡 𝑥 󵄨 Since is arbitrary, we have 𝑝 from (56). 󵄨𝑇 𝑘−𝑗 𝑗󵄨 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 (ii) If we choose 𝜖>0such that 0 < 𝜖 < 1 − 1/‖𝑥‖,then 1 < (1 − 𝜖)‖𝑥‖ < ‖𝑥‖ ‖⋅‖ 󵄨 󵄨𝑝 (52) . By the definition of in (48) 󵄨 𝑘 󵄨 𝑘 𝑀󵄨 1 󵄨 and Part (iii) of Proposition 16,wehave ≤ ∑𝛼 󵄨 ∑𝑡 𝑥 󵄨 ; 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 𝑥 1 1<𝜎 [ ]≤ 𝜎 (𝑥) . 𝑝 (1−𝜖) ‖𝑥‖ (1−𝜖) ‖𝑥‖ 𝑝 (57) that is,

𝑀 𝛼𝜎𝑝 (𝑥) ≤𝜎𝑝 (𝛼𝑥) ≤𝛼 𝜎𝑝 (𝑥) . (53) So, (1 − 𝜖)‖𝑥‖ < ‖𝑥‖ for all 𝜖 ∈ (0, 1 − (1/‖𝑥‖)).This implies that ‖𝑥‖ <𝑝 𝜎 (𝑥). + By passing to limit as 𝛼→1 in (53), we have 𝜎𝑝(𝛼𝑥) → (iii) Since 𝜎𝑝 is continuous, by Theorem 1.4 of15 [ ]we 𝜎 (𝑥) 𝜎 𝑝 .Hence, 𝑝 is right continuous. directly have (iii). If 0<𝛼<1,wehave (iv) This follows from Parts (i) and (iii). 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 𝑘 󵄨 𝑘 󵄨 𝑘 󵄨 𝑘 𝑀󵄨 1 󵄨 𝑝 󵄨 1 󵄨 (v) This follows from Parts (ii) and (iii). ∑𝛼 󵄨 ∑𝑡 𝑥 󵄨 ≤ ∑𝛼 𝑘 󵄨 ∑𝑡 𝑥 󵄨 󵄨𝑇 𝑘−𝑗 𝑗󵄨 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 (vi) This follows from Part (ii) and Part (i) of 󵄨 󵄨𝑝 (54) Proposition 16. 󵄨 𝑘 󵄨 𝑘 󵄨 1 󵄨 ≤ ∑𝛼󵄨 ∑𝑡 𝑥 󵄨 ; (vii) This follows from Part (i) and Part (ii) of 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 Proposition 16. that is, 𝑡 Theorem 18. The space 𝑁 (𝑝) is rotund if and only if 𝑝𝑘 >1 𝑀 𝛼 𝜎𝑝 (𝑥) ≤𝜎𝑝 (𝛼𝑥) ≤𝛼𝜎𝑝 (𝑥) . (55) for all 𝑘∈N. 𝑡 − Proof. Let 𝑁 (𝑝) be rotund and choose 𝑘∈N such that 𝑝𝑘 =1 By letting 𝛼→1in (55), we have 𝜎𝑝(𝛼𝑥)𝑝 →𝜎 (𝑥). for all 𝑘<3. Consider the following sequences given by Hence, 𝜎𝑝 is left continuous. Since 𝜎𝑝 is both right and left continuous, it is continuous. 𝑥=(1,−𝐷1,𝐷2,−𝐷3,𝐷4,...), (58) Now, we give some relationships between the modular 𝜎𝑝 𝑦=(0,𝑇,−𝑇 𝐷 ,𝑇 𝐷 ,−𝑇 𝐷 ,...). 𝑡 1 1 1 1 2 1 3 and the Luxemburg norm on 𝑁 (𝑝). 𝑥 =𝑦̸ 𝑡 Then, obviously and Proposition 17. For any 𝑥∈𝑁(𝑝), the following statements 𝑥+𝑦 hold. 𝜎 (𝑥) =𝜎 (𝑦) = 𝜎 ( )=1. 𝑝 𝑝 𝑝 2 (59) (i) If ‖𝑥‖ <,then 1 𝜎𝑝(𝑥) ≤ ‖𝑥‖. 𝑥, 𝑦, (𝑥 + 𝑦)/2 ∈𝑆[𝑁𝑡(𝑝)] (ii) If ‖𝑥‖ >,then 1 𝜎𝑝(𝑥) ≥ ‖𝑥‖. By Part (iii) of Proposition 17, which leads us to the contradiction that the sequence space (iii) ‖𝑥‖ = 1 if and only if 𝜎𝑝(𝑥) =. 1 𝑡 𝑁 (𝑝) is not rotund. Hence, 𝑝𝑘 >1for all 𝑘∈N. 𝑡 𝑡 (iv) ‖𝑥‖ < 1 if and only if 𝜎𝑝(𝑥) <. 1 Conversely, let 𝑥∈𝑆[𝑁(𝑝)] and V,𝑧∈𝑆[𝑁(𝑝)] with 𝑥= (V + 𝑧)/2. By convexity of 𝜎𝑝 and Part (iii) of Proposition 17, (v) ‖𝑥‖ > 1 if and only if 𝜎𝑝(𝑥) >. 1 we have 𝑀 (vi) If 0<𝛼<1and ‖𝑥‖ > 𝛼,then𝜎𝑝(𝑥) > 𝛼 . 𝜎𝑝 (V) +𝜎𝑝 (𝑧) 𝑀 1=𝜎 (𝑥) ≤ =1, (60) (vii) If 𝛼≥1and ‖𝑥‖ < 𝛼,then𝜎𝑝(𝑥) < 𝛼 . 𝑝 2 8 Abstract and Applied Analysis

(𝑛) 𝑡 which gives that we have 𝜎𝑝(𝑥 )→1as 𝑛→∞.Also,𝑥∈𝑆[𝑁(𝑝)] implies ‖𝑥‖ =. 1 By Part (iii) of Proposition 17,weobtain𝜎𝑝(𝑥) =. 1 𝜎 (V) +𝜎 (𝑧) 𝑝 𝑝 𝜎 (𝑥(𝑛))→𝜎(𝑥) 𝑛→∞ 𝜎𝑝 (𝑥) = . (61) Therefore, we have 𝑝 𝑝 as . 2 (𝑛) 𝑤 𝑡 Since 𝑥 󳨀→𝑥and 𝑞𝑘 :𝑁(𝑝) → R (or C) defined by 𝑥=(V + 𝑧)/2 (𝑛) Also, since and from (61), we obtain that 𝑞𝑘(𝑥) =𝑘 𝑥 is continuous, 𝑥𝑘 →𝑥𝑘 as 𝑛→∞. Therefore, (𝑛) 󵄨 󵄨𝑝 𝑥 →𝑥 𝑛→∞ 󵄨 𝑘 (V +𝑧)󵄨 𝑘 as .Thiscompletestheproof. 󵄨 1 𝑗 𝑗 󵄨 ∑󵄨 ∑𝑡 󵄨 󵄨𝑇 𝑘−𝑗 2 󵄨 Theorem 22. 1<𝑝<∞ 𝑋 𝑘 󵄨 𝑘 𝑗=0 󵄨 For any ,thespace 𝑎(𝑝) has the uniform Opial property. 󵄨 󵄨𝑝 󵄨 󵄨𝑝 (62) 󵄨 𝑘 󵄨 𝑘 󵄨 𝑘 󵄨 𝑘 1 󵄨 1 󵄨 󵄨 1 󵄨 = (∑󵄨 ∑𝑡 V 󵄨 + ∑󵄨 ∑𝑡 𝑧 󵄨 ). Proof. Since the proof can be given by the similar way used 2 󵄨𝑇 𝑘−𝑗 𝑗󵄨 󵄨𝑇 𝑘−𝑗 𝑗󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 𝑘 󵄨 𝑘 𝑗=0 󵄨 in proving Theorem 13 of Nergiz and Bas¸ar [16], we omit the detail. This implies that 󵄨 󵄨𝑝 󵄨 󵄨𝑝 𝑝 󵄨 󵄨 𝑘 󵄨 󵄨 𝑘 6. Conclusion 󵄨V +𝑧 󵄨 𝑘 󵄨V 󵄨 + 󵄨𝑧 󵄨 󵄨 𝑗 𝑗 󵄨 󵄨 𝑗󵄨 󵄨 𝑗󵄨 󵄨 󵄨 = (63) 󵄨 2 󵄨 2 Wang introduced the sequence space 𝑋𝑎(𝑝),in[10]. Although the domain of several triangle matrices in the classical 𝑝𝑘 for all 𝑘∈N. Since the function 𝑡→|𝑡| is strictly convex sequence spaces ℓ𝑝, 𝑐0, 𝑐,andℓ∞ andintheMaddoxspaces for all 𝑘∈N,itfollowsby(63)thatV𝑘 =𝑧𝑘 for all 𝑘∈N. ℓ(𝑝), 𝑐0(𝑝), 𝑐(𝑝),andℓ∞(𝑝) was investigated by researchers, 𝑡 Hence, V =𝑧.Thatis,𝑁 (𝑝) is rotund. the domain of Norlund¨ mean neither in a normed sequence space nor in a paranormed sequence space was not studied Theorem 19. (𝑥 ) 𝑁𝑡(𝑝) Let 𝑛 be a sequence in .Then,the and is still as an open problem. So, we have worked on following statements hold: thedomainofNorlund¨ mean in the Maddox space ℓ(𝑝). Additionally, we emphasize on some geometric properties (i) lim𝑛→∞‖𝑥𝑛‖=1implies lim𝑛→∞𝜎𝑝(𝑥𝑛)=1; 𝑡 𝑡 of the new space 𝑁 (𝑝). It is obvious that the matrix 𝑁 is 𝜎 (𝑥 )=0 ‖𝑥 ‖=0 𝑟 𝑟 (ii) lim𝑛→∞ 𝑝 𝑛 implies lim𝑛→∞ 𝑛 . not comparable with the matrices 𝐸 , 𝐴 ,or𝐵(𝑟,.So,the 𝑠) present results are new. Proof. The proof is similar to that of Theorem 10 in[16]. It is clear that by depending on the choice of the sequence space 𝜇, the characterization of several classes of matrix Theorem 20. 𝑥∈𝑁𝑡(𝑝) (𝑥(𝑛))⊂𝑁𝑡(𝑝) 𝑡 Let and .If transformations from the space 𝑁 (𝑝) andintothespace 𝜎 (𝑥(𝑛))→𝜎(𝑥) 𝑛→∞ 𝑥(𝑛) →𝑥 𝑛→∞ 𝑡 𝑝 𝑝 as and 𝑘 𝑘 as for 𝑁 (𝑝) canbeobtainedfromTheorems10 and 11,respectively. (𝑛) all 𝑘∈N,then𝑥 →𝑥as 𝑛→∞. As a natural continuation of this paper, we will study the domain of the Norlund¨ mean in Maddox’s spaces ℓ∞(𝑝), 𝑡 (𝑛) 𝑡 Proof. Let 𝜖>0be given. Since 𝑥∈𝑁(𝑝) and (𝑥 )⊂𝑁(𝑝), 𝑐(𝑝),and𝑐0(𝑝). (𝑛) 𝑡 (𝑛) 𝑝𝑘 𝜎𝑝(𝑥 −𝑥)=∑𝑘 |{𝑁 (𝑥 −𝑥)}𝑘| <∞. So, there exists an 𝑘 ∈ N 0 such that Conflict of Interests ∞ 𝑝 󵄨 𝑡 (𝑛) 󵄨 𝑘 𝜖 The authors declare that there is no conflict of interests ∑ 󵄨{𝑁 (𝑥 −𝑥)} 󵄨 < . 󵄨 𝑘󵄨 2 (64) regarding the publication of this paper. 𝑘=𝑘0+1

(𝑛) Also, since 𝑥𝑘 →𝑥𝑘 as 𝑛→∞,wehave References

𝑘 [1] I. J. Maddox, “Spaces of strongly summable sequences,” Quar- 0 𝑝 󵄨 𝑡 (𝑛) 󵄨 𝑘 𝜖 ∑󵄨{𝑁 (𝑥 −𝑥)} 󵄨 < . (65) terly Journal of Mathematics,vol.18,no.1,pp.345–355,1967. 󵄨 𝑘󵄨 2 𝑘=1 [2] H. Nakano, “Modulared sequence spaces,” Proceedings of the Japan Academy,vol.27,no.2,pp.508–512,1951. (𝑛) Therefore, we obtain from (64)and(65)that𝜎𝑝(𝑥 −𝑥)<. 𝜖 [3] S. Simons, “The sequence spaces ℓ(𝑝𝜐) and 𝑚(𝑝𝜐),” Proceedings (𝑛) of the London Mathematical Society,vol.15,no.3,pp.422–436, This means that 𝜎𝑝(𝑥 −𝑥)→0as 𝑛→∞.Thisresult (𝑛) 1965. implies ‖𝑥 −𝑥‖ → 0as 𝑛→∞from Part (ii) of 𝑥 →𝑥 𝑛→∞ [4] A. Peyerimhoff, Lectures on Summability,LectureNotesin Theorem 19.Hence, 𝑛 as . Mathematics, Springer, New York, NY, USA, 1969. 𝑡 [5]F.M.Mears,“TheinverseNorlund¨ mean,” Annals of Mathemat- Theorem 21. The sequence space 𝑁 (𝑝) has the Kadec-Klee ics,vol.44,no.3,pp.401–409,1943. property. [6]B.ChoudharyandS.K.Mishra,“OnKothe-Toeplitz¨ duals 𝑥∈𝑆[𝑁𝑡(𝑝)] (𝑥(𝑛))⊂𝑁𝑡(𝑝) of certain sequence spaces and thair matrix transformations,” Proof. Let and such that Indian Journal of Pure and Applied Mathematics,vol.24,no.59, (𝑛) (𝑛) 𝑤 ‖𝑥 ‖→1and 𝑥 󳨀→𝑥aregiven.ByPart(i)ofTheorem 19, pp.291–301,1993. Abstract and Applied Analysis 9

[7] F. Bas¸ar and B. Altay, “Matrix mappings on the space bs(p) and its 𝛼-, 𝛽-and𝛾-duals,” The Aligarh Bulletin of Mathematics,vol. 21,no.1,pp.79–91,2002. [8] F. Bas¸ar, “Infinite matrices and almost boundedness,” Bollettino della Unione Matematica Italiana A,vol.6,no.7,pp.395–402, 1992. [9] B. Altay and F. Bas¸ar, “On the paranormed Riesz sequence space of non-absolute type,” Southeast Asian Bulletin of Mathematics, vol. 26, pp. 701–715, 2002. [10] C. S. Wang, “On Norlund¨ sequence space,” Tamkang Journal of Mathematics,vol.9,pp.269–274,1978. [11] K. G. Grosse-Erdmann, “Matrix transformations between the sequence spaces of Maddox,” Journal of Mathematical Analysis and Applications,vol.180,no.1,pp.223–238,1993. [12] C. G. Lascarides and I. J. Maddox, “Matrix transformations between some classes of sequences,” Proceedings of the Cam- bridge Philosophical Society, vol. 68, pp. 99–104, 1970. [13] S. Chen, “Geometry of Orlicz spaces,” Dissertationes Mathemat- icae,vol.356,pp.1–224,1996. [14] J. Diestel, Geomety of Banach Spaces-Selected Topics,Springer, Berlin, Germany, 1984. [15] L. Maligranda, Orlicz Spaces and Interpolation,Instituteof Mathematics Polish Academy of Sciences, Poznan, Poland, 1985. [16] H. Nergiz and F. Bas¸ar, “Some geometric properties of the domain of the double sequential band matrix 𝐵(𝑟,̃ 𝑠)̃ in the sequence space ℓ(𝑝),” Abstract and Applied Analysis,vol.2013, ArticleID421031,7pages,2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 248725, 13 pages http://dx.doi.org/10.1155/2014/248725

Research Article Sobolev-Type Spaces on the Dual of the Chébli-Trimèche Hypergroup and Applications

Mourad Jelassi1 and Hatem Mejjaoli2

1 Department of Mathematics, ISSAT Mateur, Carthage University, Bizerte, 7030 Mateur, Tunisia 2 Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah Al Munawarah, Saudi Arabia

Correspondence should be addressed to Mourad Jelassi; [email protected]

Received 4 December 2013; Accepted 30 December 2013; Published 4 March 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 M. Jelassi and H. Mejjaoli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑠,𝑝 We define and study Sobolev-type spaces 𝑊𝐴 (R+) associated with singular second-order differential operator on (0, ∞).Some properties are given; in particular we establish a compactness-type imbedding result which allows a Reillich-type theorem. Next, we introduce a generalized Weierstrass transform and, using the theory of reproducing kernels, some applications are given.

1. Introduction (ii) 𝐴 is increasing on R+ and lim𝑥→∞𝐴(𝑥). =∞ 󸀠 (iii) 𝐴 /𝐴 is decreasing on (0, ∞),and TheSobolevspaceshaveservedasaveryusefultoolinthe (𝐴󸀠(𝑥)/𝐴(𝑥)) =2𝜌 theory of partial differential equations, mostly those related to lim𝑥→∞ . continuum mechanics or physics. Their uses and the study of (iv) There exists a constant 𝜎>0,suchthatforall𝑥∈ their properties were facilitated by the theory of distributions [𝑥0, ∞), 𝑥0 >0,wehave 𝑠,𝑝 and Fourier analysis. The Sobolev space 𝑊 (R+) is defined −𝜎𝑥 by the use of the classical Fourier transform as the set of 𝐴󸀠 (𝑥) {2𝜌 + 𝑒 𝐹 (𝑥) , if 𝜌>0 𝑢 = {2𝛼 + 1 −𝜎𝑥 (3) all tempered distribution such that its classical Fourier 𝐴 (𝑥) +𝑒 𝐹 (𝑥) , if 𝜌=0, transform 𝑢̂ satisfying { 𝑥

󵄨 󵄨2 𝑠/2 𝑝 𝐹 𝐶∞ (0, ∞) (1+󵄨𝜉󵄨 ) 𝑢∈𝐿̂ (R+) . (1) where is on , bounded together with its deriva- tives. 2𝛼+1 Generalization of the Sobolev space has been studied by For 𝐴(𝑥) =𝑥 , 𝛼>−1/2,and𝜌=0,weregainthe replacing the classical Fourier transform by a generalized one. Bessel operator In this paper we consider the differential operator on 𝑑2𝑓 2𝛼 + 1 𝑑𝑓 (0, ∞), 𝑙 𝑓= +( ) . (4) 𝛼 𝑑𝑥2 𝑥 𝑑𝑥 2 󸀠 𝑑 𝐴 (𝑥) 𝑑 2 Δ = + +𝜌 ,𝜌>0, (2) 2𝛼+1 2𝛽+1 𝐴 𝑑𝑥2 𝐴 (𝑥) 𝑑𝑥 For 𝐴(𝑥) = sinh (𝑥)cosh (𝑥), 𝛼≥𝛽≥−1/2, 𝛼 =−̸ 1/2,and𝜌=𝛼+𝛽+1, we regain the Jacobi operator where 𝐴 is the Chebli-Trimeche function (cf. [1, Section 3.5]) [0, ∞) 𝑑2𝑓 defined on and satisfies the following conditions. 𝑙 𝑓= +[(2𝛼 + 1) 𝑥+(2𝛽+1) 𝑥] 𝛼,𝛽 𝑑𝑥2 coth tanh (i) There exists a positive even infinitely differentiable (5) 𝑑𝑓 function 𝐵 on R,with𝐵(𝑥), ≥1 𝑥∈R+,suchthat 2 2𝛼+1 × +𝜌 . 𝐴(𝑥) =𝑥 𝐵(𝑥), 𝛼>−1/2. 𝑥 2 Abstract and Applied Analysis

The purpose of this paper is to introduce and study new where 𝑤(𝑥, 𝑦, ⋅) is a measurable positive function on 𝑠,𝑝 [0, ∞) [|𝑥 − 𝑦|, 𝑥 +𝑦] Sobolev-type spaces 𝑊𝐴 (R+),associatedwiththesingular ,withsupportin . Δ operator 𝐴 that generalizes the corresponding classical (iii) spaces.TheBesselcasewastreatedbyAssalandNessibi[2], 󵄨 󵄨 while Ben Salem and Dachraoui [3] studied the generalized ∀𝜆 ≥ 0, 𝑥 ∈ R, 󵄨𝜑𝜆 (𝑥)󵄨 ≤1. (9) Soblev spaces in the Jacobi setting theory. The paper is organized as follows. In Section 2 we recall (iv) For 𝜌>0,wehave the main results about the harmonic analysis associated with the operator Δ 𝐴.InSection 3 Sobolev-type spaces on the ∀𝑥≥0, ∀𝜆∈R, dual of the Chebli-Trim´ eche` hypergroup are studied. Some (10) 󵄨 󵄨 properties including completeness and Sobolev embedding 󵄨𝜑𝜆 (𝑥)󵄨 ≤𝜑0 (𝑥) ≤𝑚(1+𝑥) exp (−𝜌𝑥) , theorems are established. Next, we prove a Reillich-type theorem. Finally, in Section 4, as applications, we give prac- where 𝑚 is a positive constant. ticalrealinversionformulasusingthetheoryofreproducing (v) For 𝜌=0,wehave kernels for the generalized Weierstrass transform.

∀𝑥≥0,0 𝜑 (𝑥) =1. (11) 2. Preliminaries (vi) We have the following integral representation of In this section, we collect some harmonic analysis results Mehler type, related to the operator Δ 𝐴. For details, we refer the reader to [1, 4–8]. 𝑥 ∀𝑥>0, ∀𝜆∈C,𝜑𝜆 (𝑥) = ∫ 𝑘 (𝑥,) 𝑡 cos (𝜆𝑡) 𝑑𝑡, (12) 0 2.1. Eigenfunctions of the Operator Δ 𝐴. Inthefollowing,we ∞ denote by where 𝑘(𝑥, ⋅) is an even positive 𝐶 function on ∞ (−𝑥, 𝑥) [−𝑥, 𝑥] E∗(R) the space of even 𝐶 -functions on R, with support in .

S∗(R) the subspace of E∗(R), consisting of functions 𝑓 2.2. Generalized Fourier Transform. For a Borel positive rapidly decreasing together with their derivatives, 𝑝 2 measure 𝜇 on R,and1≤𝑝≤∞,wewrite𝐿𝜇(R+) for the S∗(R)=𝜑0S∗(R),where𝜑0 is the eigenfunction of Lebesgue space equipped with the norm ‖⋅‖𝐿𝑝 (R ) defined by the operator Δ 𝐴 associated with the value 𝜆=0, 𝜇 + 󸀠 S (R) the dual topological space of S∗(R), 1/𝑝 ∗ 󵄩 󵄩 󵄨 󵄨𝑝 2 󸀠 2 󵄩𝑓󵄩𝐿𝑝 (R ) =(∫ 󵄨𝑓 (𝑥)󵄨 𝑑𝜇 (𝑥)) , if 𝑝<∞, (13) (S∗) (R) the dual topological space of S∗(R), 𝜇 + R 󸀠 E (R+) the dual topological space of E∗(R), ∗ ‖𝑓‖ ∞ = |𝑓(𝑥)| 𝜇(𝑥) = 𝑤(𝑥)𝑑𝑥 and 𝐿𝜇 (R+) ess sup𝑥∈R+ .When , H∗(C) the space of even entire functions on C which with 𝑤 a nonnegative function on R+,wereplacethe𝜇 in the are of exponential type and slowly increasing, norms by 𝑤. H (C) H (C) 1 ∗,𝑎 the subspace of ∗ satisfying For 𝑓∈𝐿𝐴(R+), the generalized Fourier transform is ∃𝑚 ∈ N. defined by

2 −𝑚 󵄨 󵄨 (6) 𝑃 (𝑓) = (1 + 𝜆 ) 󵄨𝑓 (𝜆)󵄨 (−𝑎 | 𝜆|) <+∞. F (𝑓) (𝜆) = ∫ 𝑓 (𝑥) 𝜑𝜆 (𝑥) 𝐴 (𝑥) 𝑑𝑥, ∀𝜆 ∈ R. (14) 𝑚 sup 󵄨 󵄨 exp Im R 𝜆∈C + The inverse generalized Fourier transform of a suitable We have H∗(C)=⋃𝑎≥0 H∗,𝑎(C). function 𝑔 on R+ is given by For every 𝜆∈C, let us denote by 𝜑𝜆 the unique solution of the eigenvalue problem: J𝑔 (𝑥) = F−1𝑔 (𝑥) = ∫ (𝜆) 𝜑 (𝑥) 𝑑𝛾 (𝜆) , 2 𝜆 (15) Δ 𝐴𝑓 (𝑥) =−𝜆𝑓 (𝑥) , R+ (7) 𝑓 (0) =1, 𝑓󸀠 (0) =0. where 𝑑𝛾(𝜆) is the spectral measure given by 𝑑𝜆 Remark 1. This function satisfies the following properties. 𝑑𝛾 (𝜆) = . 󵄨 󵄨2 (16) 󵄨𝑐 (𝜆)󵄨 (i) ∀𝑥⩾0,thefunction𝜆 󳨃→𝜑𝜆(𝑥) is analytic on C. 󵄨 𝐴 󵄨 (ii) Product formula: Remark 2. The function 𝜆 󳨃→𝑐𝐴(𝜆) satisfies the following ∀𝑥, 𝑦 ⩾ 0; properties. ∞ (8) 𝜆∈R 𝑐 (−𝜆) = 𝑐 (𝜆) 𝜑 (𝑥) 𝜑 (𝑦) = ∫ 𝜑 (𝑧) 𝑤(𝑥,𝑦,𝑧)𝐴(𝑧) 𝑑𝑧, (i) For ,wehave 𝐴 𝐴 . 𝜆 𝜆 𝜆 −2 0 (ii) The function |𝑐𝐴(𝜆)| is continuous on [0, ∞[. Abstract and Applied Analysis 3

(iii) There exist positive constants 𝑘1, 𝑘2,and𝑘3,suchthat Proposition 8 (see [10]). For a suitable function 𝑓 on R+,we have If 𝜌⩾0:∀𝜆∈C,Im𝜆⩽0, |𝜆| >3 𝑘 ; 𝜏 𝑓(𝑦) =𝜏 𝑓(𝑥) 2𝛼+1 󵄨 󵄨−2 2𝛼+1 (i) 𝑥 𝑦 , 𝑘1|𝜆| ⩽ 󵄨𝑐𝐴 (𝜆)󵄨 ⩽𝑘2|𝜆| . (17) (ii) 𝜏0𝑓(𝑦) = 𝑓(𝑦), If 𝜌=0, 𝛼>0:∀𝜆∈C, |𝜆| ⩽3 𝑘 ; (iii) 𝜏𝑥𝜏𝑦 =𝜏𝑦𝜏𝑥, 2𝛼+1 󵄨 󵄨−2 2𝛼+1 𝑘1|𝜆| ⩽ 󵄨𝑐𝐴 (𝜆)󵄨 ⩽𝑘2|𝜆| . (18) (iv) 𝜏𝑥𝜑𝜆(𝑦) =𝜆 𝜑 (𝑥)𝜑𝜆(𝑦), F(𝜏 𝑓)(𝜆) =𝜑 (𝑥)F(𝑓)(𝜆) If 𝜌>0:∀𝜆∈C, |𝜆| ⩽3 𝑘 ; (v) 𝑥 𝜆 , Δ (𝜏 )𝑓 = 𝜏 (Δ 𝑓) 2 󵄨 󵄨−2 2 (vi) 𝐴 𝑥 𝑥 𝐴 . 𝑘1|𝜆| ⩽ 󵄨𝑐𝐴 (𝜆)󵄨 ⩽𝑘2|𝜆| . (19) Proposition 3 F Definition 9 (see [10]). For suitable functions 𝑓 and 𝑔,we (see [7, 9]). (i) The generalized transform 𝑓∗ 𝑔 and its inverse J are topological isomorphisms between the define the convolution product 𝐴 by 2 generalized Schwartz space S∗(R) and the Schwartz space 𝑓∗ 𝑔 (𝑥) = ∫ 𝜏 𝑓 (𝑦) 𝑔 (𝑦) 𝐴 (𝑦) 𝑑𝑦, S(R∗). 𝐴 𝑥 (25) R (ii) The transform F is a topological isomorphism from + E󸀠 (R ) H (C) 𝑇∈E󸀠 (R ) ∗ + onto ∗ .Moreover,forall ∗ + ,wehave Remark 10. It is clear that this convolution product is both (𝑇) ⊆ [−𝑎, 𝑎] F(𝑇) ∈ H (C) supp if and only if ∗,𝑎 . commutative and associative:

Next, we give some properties of this transform. (i) 𝑓∗𝐴𝑔=𝑔∗𝐴𝑓. 1 (𝑓∗ 𝑔)∗ ℎ=𝑓∗ (𝑔∗ ℎ) (i) For 𝑓 in 𝐿𝐴(R+) we have (ii) 𝐴 𝐴 𝐴 𝐴 . 󵄩 󵄩 󵄩 󵄩 󵄩F (𝑓)󵄩 ≤ 󵄩𝑓󵄩 . 󵄩 󵄩𝐿∞(R ) 󵄩 󵄩𝐿1 (R ) (20) Proposition 11 (see [10]). (i) Assume that 1≤𝑝, 𝑞, 𝑟 ≤∞ 𝛾 + 𝐴 + 𝑝 satisfy (1/𝑝) + (1/𝑞) − 1.Then,forevery =1/𝑟 𝑓∈𝐿𝐴(R+) 2 𝑔∈𝐿𝑞 (R ) 𝑓∗ 𝑔∈𝐿𝑟 (R ) (ii) For 𝑓 in S∗(R) we have and 𝐴 + ,wehave 𝐴 𝐴 + ,and F (Δ 𝑓) (𝑦)2 =−𝑦 F (𝑓) (𝑦) , ∀𝑦 ∈ R . 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝐴 + (21) 󵄩𝑓∗𝐴𝑔󵄩 𝑟 ≤𝐶󵄩𝑓󵄩 𝑝 󵄩𝑔󵄩 𝑞 . 𝐿𝐴(R+) 𝐿𝐴(R+) 𝐿𝐴(R+) (26) Proposition 4 (see [7, 9]). Plancherel formula for F.Forall 𝜌>0 1≤𝑝<𝑞≤2 2 (ii) If and ,then 𝑓 in S∗(R),wehave 𝑝 𝑞 𝑞 𝐿𝐴 (R+)∗𝐴𝐿𝐴 (R+)󳨅→𝐿𝐴 (R+). (27) 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨𝑓 (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥 = ∫ 󵄨F (𝑓) (𝜉)󵄨 𝑑𝛾 (𝜉) . (22) R R + + (iii) If 𝜌>0and 2<𝑝,𝑞<∞such that 𝑞/2≤𝑝<𝑞,then F (ii) Plancherel Theorem. The transform extends uniquely to 𝑝 𝑞󸀠 𝑞 2 2 𝐿 (R+)∗𝐴𝐿 (R+)󳨅→𝐿 (R+), (28) an isomorphism from 𝐿𝐴(R+) onto 𝐿𝛾(R+). 𝐴 𝐴 𝐴 󸀠 2 𝑝 where 𝑞 is the conjugate exponent of 𝑞. Remark 5. We have S∗(R)⊂𝐿𝐴(R+) for all 2≤𝑝≤∞,but 2 𝑝 (iv) If 𝜌>0and 1<𝑝<2such that 𝑝 < 𝑞 ≤ 𝑝/(2, −𝑝) 𝑆 (R) ⊆𝐿̸ (R+) for all 0<𝑝<2. ∗ 𝐴 then Proposition 6. Let 1≤𝑝≤2. The Fourier transform F (resp. 𝑝 𝑝 𝑞 𝑝 𝐿𝐴 (R+)∗𝐴𝐿𝐴 (R+)󳨅→𝐿𝐴 (R+). (29) J) can be extended as a continuous mapping from 𝐿𝐴(R+) 󸀠 󸀠 𝐿𝑝 (R ) 𝐿𝑝(R ) 𝐿𝑝 (R ) 2 𝑝 onto 𝛾 + (resp. from 𝛾 + onto 𝐴 + )andwehave Proposition 12. For 𝑓∈𝐿𝐴(R+) and 𝑔∈𝐿𝐴(R+),with1≤ 𝑝<2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 we have 󵄩F𝑓󵄩 𝑝󸀠 ⩽ 󵄩𝑓󵄩 𝑝 ; 󵄩J𝑔󵄩 𝑝󸀠 ⩽ 󵄩𝑔󵄩 𝑝 󵄩 󵄩𝐿 (R ) 󵄩 󵄩𝐿 (R+) 󵄩 󵄩𝐿 (R ) 󵄩 󵄩𝐿𝛾(R+) 𝛾 + 𝐴 𝐴 + F (𝑓∗ 𝑔) = F (𝑓) (𝜆) F (𝑔) (𝜆) . (23) 𝐴 (30) 󸀠 2 2 with (1/𝑝 ) + (1/𝑝) = 1. Proposition 13. Let 𝑓, 𝑔 ∈𝐿 𝐴(R+).Then𝑓∗𝐴𝑔∈𝐿𝐴(R+) if 2 and only if F(𝑓)F(𝑔) belongs to 𝐿𝐴(R+), and in this case we 2.3. Generalized Convolution have F (𝑓∗ 𝑔) = F (𝑓) F (𝑔) . Definition 7 (see [10]). The translation operator associated 𝐴 (31) 1 with the operator Δ 𝐴 is defined on 𝐿 (R+),by 𝐴 Definition 14. The generalized Fourier transform of a distri- ∞ 2 󸀠 bution 𝜏 in (S∗) (R) is defined by ∀𝑥, 𝑦 ⩾ 0;𝑥 𝜏 𝑓(𝑦)=∫ 𝑓 (𝑧) 𝑤(𝑥,𝑦,𝑧)𝐴(𝑧) 𝑑𝑧, (24) 0 ⟨F (𝜏) , 𝜙⟩ = ⟨𝜏, F−1 (𝜙)⟩ , ∀𝜙 ∈ S (R) . where 𝑤 isthefunctiondefinedintherelation(9). ∗ (32) 4 Abstract and Applied Analysis

Proposition 15. The generalized Fourier transform F is a Proof. (i) is clear. 2 󸀠 󸀠 𝑠 = (1−𝑡)𝑠 +𝑡𝑠 𝑡∈]0,1[ topological isomorphism from (S∗) (R) onto S∗(R). (ii) We consider 1 2,with .Moreover it is easy to see 󸀠 𝜏 (S2 ) (R ) Δ 𝜏 1−𝑡 𝑡 Let be in ∗ + . We define the distribution 𝐴 ,by 𝑠,𝑝 ‖𝑢‖𝑊 (R ) ≤ ‖𝑢‖ 𝑠1,𝑝 ‖𝑢‖ 𝑠2,𝑝 . 𝐴 + 𝑊𝐴 (R+) 𝑊𝐴 (R+) (42) ⟨Δ 𝜏, 𝜓⟩ = ⟨𝜏,Δ 𝜓⟩ , ∀𝜓 ∈ S2 (R ). 𝐴 𝐴 ∗ + (33) Thus, 1−𝑡 𝑡 This distribution satisfy the following property: −𝑡/(1−𝑡) ‖𝑢‖𝑊𝑠,𝑝(R ) ≤(𝜀 ‖𝑢‖𝑊𝑠1,𝑝(R )) (𝜀‖𝑢‖𝑊𝑠2,𝑝(R )) 2 𝐴 + 𝐴 + 𝐴 + F (Δ 𝐴𝜏) = −𝑦 F (𝜏) . (34) (43) −𝑡/(1−𝑡) ≤𝜀 ‖𝑢‖ 𝑠1,𝑝 +𝜀‖𝑢‖ 𝑠2,𝑝 . 𝑊𝐴 (R+) 𝑊𝐴 (R+) 3. Sobolev-Type Spaces on the Dual of −𝑡/(1−𝑡) Hence, the proof is completed for 𝐶𝜀 =𝜀 . the Chébli-Trimèche Hypergroup Proposition 19. (i) Let 𝑝∈[1,∞]if 𝜌=0and 𝑝 in [1, 2],if Definition 16. Let 𝑠∈R and 𝑝∈[1,∞]. We define the 𝑠,𝑝 𝑠,𝑝 𝜌>0.Thespace𝑊𝐴 (R+) provided with the norm ‖⋅‖𝑊𝑠,𝑝(R ) Sobolev-type spaces 𝑊𝐴 (R+) as the set of tempered dis- 𝐴 + 󸀠 is a Banach space. tributions 𝑢∈𝑆 (R) such that ∗ (ii) Let 𝑠, 𝑡 ∈ R and 𝑝∈[1,∞]. Then, the operator (𝐼 + 𝑡 𝑠 𝑝 󸀠 2 D𝐴) defined, on S∗(R),by (1 + 𝑥 ) J (𝑢) ∈𝐿𝐴 (R+). (35) 𝑡 2 𝑡 𝑠,𝑝 (𝐼 + D𝐴) 𝑢=F ((1 + 𝑥 ) J (𝑢)) (44) We provide the space 𝑊𝐴 (R+) with the norm: 𝑠,𝑝 𝑠−𝑡,𝑝 󵄩 2 𝑠 󵄩 𝑊 (R ) 𝑊 (R ) 𝑢 𝑠,𝑝 = 󵄩(1 + 𝑥 ) J 𝑢 󵄩 . is an isometric isomorphism from 𝐴 + onto 𝐴 + . ‖ ‖𝑊 (R ) 󵄩 ( )󵄩 𝑝 𝐴 + 󵄩 󵄩𝐿 (R ) (36) 𝑠,𝑝 𝐴 + Moreover, for all 𝑢∈𝑊𝐴 (R+), 𝑝∈[1,2],andforall𝑡≤𝑠, In the sequel, we will give some properties of the space the function 𝑊𝑠,𝑝(R ) 𝑡 𝐴 + . 𝑥󳨀→(𝐼+D𝐴) 𝑢 (𝑥) (45)

󸀠 Proposition 17. Let 𝑠∈R.ThespaceS∗(R) is dense in 𝑝 󸀠 𝑠,𝑝 belongs to the space 𝐿𝛾 (R+),with(1/𝑝) + (1/𝑝 )=1. 𝑊𝐴 (R+),for (𝑓 ) 𝑊𝑠,𝑝(R ) [1, ∞) ,𝑖𝑓𝜌=0 Proof. (i) Let 𝑚 𝑚∈N be a Cauchy sequence in 𝐴 + . 𝑝∈{ 2 𝑠 𝑝 (37) Then ((1 + 𝑥 ) J(𝑓𝑚))𝑚∈N is a Cauchy sequence in 𝐿𝐴(R+). [2, ∞) ,𝑖𝑓𝜌>0. 𝑝 But 𝐿𝐴(R+) is complete, so, there exists a function 𝑔 such that S (R) 2 𝑠 𝑝 Proof. Firstly, we want to prove that the space ∗ is a (1 + 𝑥 ) 𝑔∈𝐿𝐴(R+) and 𝑊𝑠,𝑝(R ) 𝑓∈S (R) subset of 𝐴 + .Indeed,let ∗ .ByProposition 󵄩 𝑠 𝑠 󵄩 2 𝑠 2 󵄩(1 + 𝑥2) J (𝑓 )−(1+𝑥2) 𝑔󵄩 =0. 3(i) the function (1+𝑥 ) J(𝑓) ∈ S∗(R).Thus,usingRemark 𝑚→∞lim 󵄩 𝑚 󵄩 𝑝 (46) 𝐿𝐴(R+) 5, we deduce the claim. Now, we prove the density. Let 𝑓∈ 𝑊𝑠,𝑝(R ) 𝐷 (R ) 𝐿𝑝 (R ) But since 𝑝∈[1,∞]when 𝜌=0and 𝑝∈[1,2]when 𝜌>0, 𝐴 + . Then, from the density of ∗ + in 𝐴 + ,we 2 󸀠 󸀠 (𝑔 ) 𝐷 (R ) then 𝑔∈(𝑆∗) (R) and consequently 𝑓=F(𝑔) ∈∗ 𝑆 (R).This deduce the existence of a sequence 𝑛 𝑛 in ∗ + such that 𝑠,𝑝 󵄩 󵄩 implies that 𝑓∈𝑊𝐴 (R+) and from relation (46), we get 󵄩 2 𝑠 󵄩 󵄩(1 + 𝑥 ) J (𝑓) −𝑛 𝑔 󵄩 𝑝 =0. 󵄩 󵄩 𝑛→∞lim 󵄩 󵄩𝐿 (R ) (38) 󵄩 󵄩 𝐴 + lim 󵄩𝑓𝑚 −𝑓󵄩 𝑠,𝑝 =0. 𝑚→∞ 𝑊𝐴 (R+) (47) 𝑛∈ On the other hand, according to Proposition 3(i), for all This achieves the proof of (i). 2 −𝑠 𝑠,𝑝 N,thefunction𝑓𝑛 = F((1+𝑥 ) 𝑔𝑛) is in S∗(R) and we have (ii) Let 𝑢∈𝑊𝐴 (R+).ByremarkingthatJ is an 󵄩 𝑠 󵄩 󸀠 2 󸀠 󵄩 󵄩 󵄩 2 󵄩 isomorphism from S∗(R) onto (𝑆∗(R)) and using the fact 󵄩𝑓−𝑓󵄩 𝑠,𝑝 = 󵄩(1 + 𝑥 ) J (𝑓) − 𝑔 󵄩 . 󵄩 𝑛󵄩 󵄩 𝑛󵄩 𝑝 𝑊𝐴 (R+) 󵄩 󵄩 (39) 𝐿𝐴(R+) that 2 𝑠−𝑡 𝑡 2 𝑠 Therefore, the result follows by combining (38)and(39). (1 + 𝑥 ) J ((𝐼 + D𝐴) 𝑢) = (1 + 𝑥 ) J (𝑢) , Proposition 18. 1≤𝑝<∞ 𝑠 𝑠 R 𝑡 −𝑡 −𝑡 𝑡 (i) Let and let 1 and 2 in such (𝐼+D𝐴) ∘(𝐼+D𝐴) (𝑢)=(𝐼+D𝐴) ∘(𝐼+D𝐴) (𝑢)=𝑢, 𝑠 ≥𝑠 that 2 1 then (48) 𝑠 ,𝑝 𝑠 ,𝑝 2 1 𝑠,𝑝 𝑊𝐴 (R+)󳨅→𝑊𝐴 (R+). (40) wededucethefirstpartof(ii).Now,let𝑢∈𝑊𝐴 (R+), 𝑝∈ [1, 2].Then,forall𝑡≤𝑠,thefunction (ii) Let 1≤𝑝<∞,andlet𝑠1, 𝑠 and 𝑠2 be three real 𝑡 numbers: 𝑠1 <𝑠<𝑠2.Then,forall𝜀>0, there exists a 2 𝑠,𝑝 𝑥󳨀→(1+𝑥) J (𝑢) (49) nonnegative constant 𝐶𝜀 such that for all 𝑢 in 𝑊𝐴 (R+) 𝑝 belongs to the space 𝐿𝐴(R+). Therefore, we obtain the second ‖𝑢‖𝑊𝑠,𝑝(R ) ≤𝐶𝜀‖𝑢‖𝑊𝑠1,𝑝(R ) +𝜀‖𝑢‖𝑊𝑠2,𝑝(R ). (41) 𝐴 + 𝐴 + 𝐴 + part of (ii) by using inequality (23). Abstract and Applied Analysis 5

In the following, we prove a Hardy-Littlewood-Paley type (2) Let 𝜌>0, 𝛼>−1/2,and𝛿>0. According to 𝑝 inequality for the transform J. Proposition 3(ii), for all 𝑓∈𝐿𝛾(R+) ⋂ H∗,𝛿(C),itfollows that supp(J(𝑓)) ⊆ [−𝛿, 𝛿]. Proposition 20. (1) Let 𝑝∈[1,2].Then,for𝜌=0and 𝛼>0, 𝑝 there exists a positive constant 𝐶1 such that for all 𝑓∈𝐿𝛾(R+) (i) If 𝑝∈[1,2)and 𝑠 > 2(𝛼 + 1)(1 − ,onecan(2/𝑝)) ∞ easily see by using Holder inequality, Lemma 21(i), 󵄨 󵄨𝑝 󵄩 󵄩𝑝 𝑝 ∫ 𝑥2(𝛼+1)(𝑝−2)󵄨J (𝑓) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥󵄩 ≤𝐶 𝑓󵄩 . 𝑓∈𝐿 (R )∩H (C) 󵄨 󵄨 󵄩 󵄩𝐿𝑝 R (50) and inequality (23)thatfor 𝛾 + ∗,𝛿 0 𝛾( +) (2) (i) For 𝜌>0, 1≤𝑝<2, 𝛼>−1/2,and𝛿>0,wehavefor ∞ 1/𝑝 𝑠𝑝󵄨 󵄨𝑝 all 𝑠 > 2(𝛼 + 1)(1 − (2/𝑝)) (∫ 𝑥 󵄨J (𝑓) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥) 0 ∞ 𝑠𝑝󵄨 󵄨𝑝 󵄩 󵄩𝑝 󵄩 󵄩 ∫ 𝑥 󵄨J (𝑓) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥≤𝐶(𝛿,𝑠,𝑝)󵄩𝑓󵄩 𝑝 , 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩𝐿 R ≤𝐶(𝛿,𝑠,𝑝)󵄩1 J (𝑓)󵄩 𝑝󸀠 (57) 0 𝛾( +) 󵄩 [0,𝛿] 󵄩𝐿 (R ) (51) 𝐴 + 𝑝 󵄩 󵄩 𝑓∈𝐿𝛾 (R+)∩H∗,𝛿 (C) . ≤𝐶(𝛿,𝑠,𝑝)󵄩𝑓󵄩 𝑝 . 󵄩 󵄩𝐿𝛾(R+) (ii) For 𝜌>0, 𝑝=2, 𝛼>−1/2,and𝛿>0,wehave(51) 𝑠≥0 for all . (ii) If 𝑝=2, then by virtue of Plancherel Theorem for the transform J,wededucethatforall𝑠≥0, We start with the following lemma deduced from the 𝐴 ∞ hypothesis of the function . 2𝑠󵄨 󵄨2 2𝑠󵄩 󵄩2 ∫ 𝑥 󵄨J (𝑓) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥≤𝛿 󵄩𝑓󵄩2,𝛾, 0 Lemma 21. (i) For any real 𝑎>0, there exist positive constants (58) 𝐶 (𝑎) 𝐶 (𝑎) 𝑥∈[0,𝑎] 2 1 and 2 such that for all , 𝑓∈𝐿𝛾 (R+)∩H∗,𝛿 (C) . 2𝛼+1 2𝛼+1 𝐶1 (𝑎) 𝑥 ≤𝐴(𝑥) ≤𝐶2 (𝑎) 𝑥 . (52) (ii) For 𝜌>0, This completes the proof of the proposition. 2𝜌𝑥 𝐴 (𝑥) ∼𝑒 , (𝑥󳨀→∞) . (53) Proposition 22. (1) Let 𝑝 ∈ (1, 2].Thenfor𝜌=0, 𝛼>0,and 𝑠 ≤ (𝛼 + 1)(1 − (2/𝑝)) (iii) For 𝜌=0, , 𝐴 (𝑥) ∼𝑥2𝛼+1, (𝑥󳨀→∞) . 𝑝 𝑠,𝑝 (54) 𝐿𝛾 (R+) 󳨅→𝐴 𝑊 (R+) . (59) Proof of Proposition 20. (1) Let 𝜌=0and 𝛼>0.Clearly,the 𝑝 operator 𝐾 defined on 𝐿𝛾(R+), 1≤𝑝≤2,by and we have 2(𝛼+1) 󵄩 󵄩 󵄩 󵄩 𝑝 𝐾 (𝑓) (𝑥) =𝑥 J (𝑓) (𝑥) 󵄩𝑓󵄩 𝑠,𝑝 ≤𝐶󵄩𝑓󵄩 𝑝 ,𝑓∈𝐿𝛾 (R+). (55) 𝑊𝐴 (R+) 𝐿𝛾(R+) (60) is of strong type (2, 2) between the spaces (R+, 𝑑𝛾(𝜆)) 4(𝛼+1) (2) (i) Let 𝑝∈[1,2).Thenfor𝜌>0, 𝛼>−1/2,and𝛿>0, and (R+, 𝐴(𝑥)𝑑𝑥/𝑥 ). Therefore, according to the we have for all 𝑠 > (𝛼 + 1)(1 − (2/𝑝)) Marcinkiewicz theorem (cf. [11]),toobtaintheresult,it 𝐾 (1, 1) suffices to show that is of weak type between the 𝐿𝑝 (R ) ∩ H (C) 󳨅→ 𝑊 𝑠,𝑝 (R ) , spaces under consideration. Indeed, using assertions (i) and 𝛾 + ∗,𝛿 𝐴 + (61) (iii) of Lemma 21 and inequality (23), we obtain for all 𝜆>0 󵄩 󵄩 󵄩 󵄩 𝑝 1 󵄩𝑓󵄩 𝑠,𝑝 ≤𝐶󵄩𝑓󵄩 𝑝 ,𝑓∈𝐿(R+)∩H∗,𝛿 (C) . 󵄩 󵄩𝑊 (R+) 󵄩 󵄩𝐿𝛾(R+) 𝛾 and 𝑓∈𝐿𝛾(R+) 𝐴 𝐴 (𝑥) 𝑑𝑥 𝑝=2 𝜌>0𝛼>−1/2 𝛿>0 ∫ (ii) For .Thenfor , ,and ,we 4(𝛼+1) 𝑠≥0 {𝑥∈R+,𝐾(𝑓)(𝑥)>𝜆} 𝑥 have for all ,

𝑥2𝛼+1𝑑𝑥 𝑊𝑠,2 (R )⊂𝐿2 (R )∩H (C) , ≤𝐶∫ 𝐴 + 𝛾 + ∗,𝛿 2(𝛼+1) 𝑥4(𝛼+1) {𝑥∈R+,𝑥 ‖J(𝑓)‖𝐿∞(R )>𝜆} (62) 𝐴 + 󵄩 󵄩 󵄩 󵄩 2 󵄩𝑓󵄩𝑊𝑠,2(R ) ≤𝐶󵄩𝑓󵄩2,𝛾,𝑓∈𝐿𝛾 (R+)∩H∗,𝛿 (C) . 𝑑𝑥 𝐴 + ≤𝐶∫ 2(𝛼+1) 2𝛼+3 (56) {𝑥∈R ,𝑥 ‖𝑓‖ 1 >𝜆} 𝑥 + 𝐿𝛾(R+) Proof. (1) The result follows from Proposition 20(1) and the fact that, for all 𝑠 ≤ (𝛼 + 1)(1 − (2/𝑝)), 𝑑𝑥 =𝐶∫ 𝑠 1/2(𝛼+1) 2𝛼+3 2 2(𝛼+1)(1−(2/𝑝)) (𝜆/‖𝑓‖ 1 ) 𝑥 (1 + 𝑥 ) ≤𝑥 ,𝑥∈(0, ∞) . 𝐿𝛾(R+) (63) 󵄩 󵄩 󵄩𝑓󵄩 󵄩 󵄩𝐿1 (R ) (2) (i) If 𝑝∈[1,2)and using the fact that, for all 𝑠∈R, ≤𝐶 𝛾 + , 𝜆 𝑠 (1 + 𝑥2) ≤𝐶(𝑥2𝑠 +1), 𝑥∈(0, ∞) , and the desired result follows. (64) 6 Abstract and Applied Analysis it follows, from Holder inequality, that for all 𝛿>0and 𝑠> Proposition 24. For 1≤𝑝<∞and 𝑠∈R,thespace (𝛼 + 1)(1 − (2/𝑝)) 𝑠,𝑝 𝑊𝐴 (R+) is separable. 󵄩 󵄩 󵄩𝑓󵄩 𝑠,𝑝 ≤𝐶(𝛿,𝑠,𝑝) 𝑝 󵄩 󵄩𝑊 (R ) 𝑝∈[1,∞) 𝐿 (R ) 𝐴 + Proof. Let .Itiswellknownthat 𝐴 + is ∞ 1/𝑝 separable. More precisely, the set 2𝑠𝑝󵄨 󵄨𝑝 ×[(∫ 𝑥 󵄨J (𝑓) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥) 0 (65) 𝐸={𝐴−1/𝑝 (𝑥) ∑ 𝛼 1 (𝑥) ;𝛼,𝑎,𝑏 ∈ Q} 𝑖 ]𝑎𝑖,𝑏𝑖[ 𝑖 𝑖 𝑖 (69) 󵄩 󵄩 finite +󵄩1 J (𝑓)󵄩 󸀠 ]. 𝑝 𝑠,𝑝 󵄩 [0,𝛿] 󵄩𝐿𝑝 (R ) 𝐿 (R ) 𝑢∈𝑊 (R ) 𝐴 + is countable and dense in 𝐴 + .Thus,forall 𝐴 + , 𝑠∈R, there exists a sequence (𝑢𝑛)𝑛∈N in 𝐸 such that Thus, we deduce the result using Proposition 20(2) and 󵄩 𝑠 󵄩 󵄩𝑢 −(1+𝑥2) J (𝑢)󵄩 =0. inequality (23). 𝑛→∞lim 󵄩 𝑛 󵄩 𝑝 (70) 𝐿𝐴(R+) 𝑝=2 (ii) By virtue of (64), we obtain the result, for , 𝑛∈N 𝑢 ∈(𝐿1 ∩𝐿𝑝 )(R ) from Plancherel Theorem and Proposition 20(2). On the other hand, for all , 𝑛 𝐴 𝐴 + ,and 2 󸀠 so 𝑢𝑛 ∈(𝑆∗) (R+). Therefore, for all 𝑛∈N, there exists V𝑛 ∈ 𝑠,𝑝 2 𝑠 𝑊 (R+) such that 𝑢𝑛 =(1+𝑥) J(V𝑛).Hence,from(70), Proposition 23. 𝑠 𝐴 Let be a non negative real number. Then, we we obtain have 󵄩 󵄩 lim 󵄩V𝑛 −𝑢󵄩 𝑠,𝑝 =0. 𝑠,1 2 2𝑚 𝑛→∞ 𝑊𝐴 (R+) (71) (i) For 𝜌≥0, 𝑊𝐴 (R+)∩𝐿𝐴(R+)⊂𝐶∗ (R+), 𝑚∈N; 𝑚≤𝑠 2 −𝑠 . This implies that 𝐹={F((1 + 𝑥 ) 𝑓) : 𝑓 ∈ 𝐸} is countable 𝑠,2 2𝑚 𝑠,𝑝 (ii) If 𝜌=0, 𝑊𝐴 (R+)⊂𝐶∗ (R+), 𝑚∈N; 𝑚 + (1/2)(𝛼 + and dense in 𝑊𝐴 (R+) and the proposition is proved. 1) < 𝑠, 𝑘 3.1. Reillich-Type Theorem. In this subsection, using Hahn where 𝐶∗(R+) is the space of even functions with 𝑘 Banach’s and Riesz’s theorems [12, 13], we describe the class 𝐶 on R. 𝑠,𝑝 ∗ 𝑠,𝑝 dual space (𝑊𝐴 (R+)) of 𝑊𝐴 (R+).Weprovealsothata 𝑠,1 2 compact imbedding theorem and a Reillich-type theorem are Proof. (i) Let 𝑢 be in 𝑊𝐴 (R+)∩𝐿𝐴(R+) with 𝑠∈R+.Itis 1 2 established. We need firstly the following lemmas. clear that J(𝑢) belongs to 𝐿𝐴(R+)∩𝐿𝐴(R+). Thus, from (14)andProposition 4(ii), we have Lemma 25. Let 𝑠∈R and 𝑥, 𝑡. ≥0 For all positive continuous function 𝑓,wehave 𝑢 (𝜆) = ∫ J (𝑢)(𝑥) 𝜑𝜆 (𝑥) 𝐴 (𝑥) 𝑑𝑥, a.e.𝜆∈R+. (66) 󵄨 󵄨 R 2 𝑠 󵄨 󵄨 |𝑠| 2 𝑠 󵄨 2 |𝑠| 󵄨 + (1 + 𝑥 ) 󵄨𝜏 𝑓 (𝑡)󵄨 ≤2 (1 + 𝑡 ) 󵄨𝜏 [(1 + 𝑦 ) 𝑓] (𝑡)󵄨 . 󵄨 𝑥 󵄨 󵄨 𝑥 󵄨 We identify 𝑢 with the second member, then we deduce that 𝑠,1 (72) 𝑢 belongs to 𝐶∗(R) and the injection of 𝑊𝐴 (R+) into 𝐶∗(R) is continuous. Proof. The result follows by using the following classical 𝑠,1 Now, let 𝑢 be in 𝑊𝐴 (R+) with 𝑠∈R+ such that 𝑠>𝑚 Peetre’s inequality: with 𝑚∈N \{0}.From(12), for all 𝑥, 𝜆 ∈ R+,and𝑛∈N such 𝑠 𝑠 |𝑠| (1 + 𝑥2) ≤2|𝑠|(1 + 𝑡2) (1 + |𝑥−𝑡|2) , that 𝑛≤𝑝,wehave (73) 󵄨 𝑛 󵄨 𝑛 𝑊(𝑥, 𝑡, ⋅) 󵄨𝐷𝜆𝜑𝜆 (𝑥)󵄨 ≤𝑥 . (67) and the fact that the kernel is positive with support in [|𝑥−𝑡|,𝑥+𝑡]. Using the same method as for 𝑚=0and the derivation Lemma 26. 𝑢∈S󸀠 (R) 𝜙∈S (R) theorem under the integral sign, we deduce that For all ∗ and ∗ ,wehave

J (Φ𝑢) = J (𝑢) ∗𝐴J (Φ) , ∀𝑥 ∈ R+, (74) 2 󸀠 2 𝑛 𝑛 (68) where, for 𝑆 in (𝑆∗) (R) and 𝜓 in 𝑆∗(R),thefunction𝑆∗𝜓is 𝐷𝜆𝑢 (𝜆) = ∫ J (𝑢)(𝑥) 𝐷𝜆𝜑𝜆 (𝑥) 𝐴 (𝑥) 𝑑𝑥. 𝑆 R the generalized convolution product of the distribution and + the function 𝜓 defined by 𝑛 Then, for all 𝑛∈N such that 𝑛 ≤ 2𝑚,𝜆 𝐷 𝑢 belongs to 2𝑚 𝑠,1 𝑆∗𝐴𝜓 (𝑥) = ⟨𝑆,𝑥 𝜏 (𝜓)⟩ . (75) 𝐶∗(R). Thus, 𝑢 is in 𝐶∗ (R) and the injection of 𝑊𝐴 (R+) 2𝑚 2 󸀠 2 into 𝐶∗ (R) is continuous. Proof. For all 𝑆∈(𝑆∗) (R) and 𝜓∈𝑆∗(R),thefunction𝑆∗𝜓 𝜌=0 𝑢 𝑊𝑠,2(R ) 2 󸀠 (ii) If and in 𝐴 + ; then using assertions (i) belongs to (𝑆∗) (R) (cf. [4]) and we have and (iii) of Lemma 21 and Holder inequality, we deduce that 2 for all 𝑚∈N;𝑚+(1/2)(𝛼+1)<𝑠,thefunction𝑢 belongs to ⟨𝑆∗𝐴𝜓, 𝜑⟩ = ⟨𝑆, 𝜓∗𝐴𝜙⟩ , 𝜙∗ ∈𝑆 (R) , 𝑚,1 2 𝑊 (R+)∩𝐿 (R+). (76) 𝐴 𝐴 F (𝑆∗ 𝜓) = F (𝑆) F (𝜓) . Therefore, (ii) follows from (i). 𝐴 Abstract and Applied Analysis 7

|𝑠|,𝑝 Therefore, the result follows by using the fact that F is an (iii) Let 𝑝∈(max(2, 𝑞/2), .Then𝑞) Φ∈𝑊𝐴 (R+). 2 2 󸀠 isomorphism from 𝑆∗(R) (resp. (𝑆∗) (R))ontoS∗(R) (resp. Therefore, from (80)andusingProposition 11(iii), we 󸀠 S ∗(R)). deduce the result. (iv) Using (23) and Holder’s inequality, we obtain, for all Theorem 27. Let 𝑠∈R, Φ∈S∗(R),and1≤𝑞≤𝑟≤∞. 𝑠,𝑞 Φ∈S∗(R) and 𝑢∈𝑊𝐴 (R+), (i) If 𝜌=0, then the mapping 𝑠,𝑞 𝑠,𝑟 ‖Φ𝑢‖𝑊0,𝑟(R ) 𝑊𝐴 (R+)󳨀→𝑊𝐴 (R+) 𝐴 + (77) 𝑢 󳨃󳨀→ Φ 𝑢 ≤ ‖Φ𝑢‖ 𝑟󸀠 𝐿𝛾 (R+) is continuous. {‖𝑢‖ 𝑞󸀠 ‖Φ‖ 𝑞𝑟/(𝑟−𝑞) , for 𝑟 =𝑞̸ 𝐿 (R ) 𝐿𝛾 (R+) (ii) If 𝜌>0.Wehavethesameresultasin(i)if1/2 ≤ ≤𝐶{ 𝛾 + (84) ‖𝑢‖𝐿2 (R )‖Φ‖𝐿∞(R ), for 𝑟=𝑞=2 (1/𝑞) − (1/𝑟) ≤1. { 𝛾 + 𝛾 + (iii) If 𝜌>0and 2<𝑞<∞, then the mapping ‖𝑢‖𝑊𝑠,𝑞(R )‖Φ‖ 𝑞𝑟/(𝑟−𝑞) , for 𝑟 =𝑞̸ 𝐴 + 𝐿𝛾 (R+) 𝑠,𝑞󸀠 𝑠,𝑞 ≤𝐶{ 𝑊𝐴 (R+)󳨀→𝑊𝐴 (R+) ‖𝑢‖𝑊𝑠,2(R )‖Φ‖𝐿∞(R ), for 𝑟=𝑞=2, (78) 𝐴 + 𝛾 + 𝑢 󳨃󳨀→ Φ 𝑢 󸀠 󸀠 where 𝑞 and 𝑟 are,respectively, the conjugates of 𝑞 and is continuous. 𝑟. Therefore, we deduce the result by remarking that the 𝜌>0 𝑠≥0 𝑞≤2≤𝑟 𝑝 (iv) Let , ,and .Then,themapping embedding S∗(R)󳨅→𝐿𝛾(R+) is continuous. This achieves 𝑠,𝑞 0,𝑟 the proof of theorem. 𝑊𝐴 (R+)󳨀→𝑊𝐴 (R+) (79) 𝑢 󳨃󳨀→ Φ 𝑢 𝑠,𝑝 ∗ Now, we shall characterize the dual space (𝑊𝐴 (R+)) of 𝑊𝑠,𝑝(R ) is continuous. 𝐴 + . Theorem 28. Let 𝑝∈[1,∞)when 𝜌=0and 𝑝∈[2,∞) Proof. (1) (i) According to Lemmas 25 and 26,wehave,for 𝑠,𝑝 ∗ 𝑠,𝑝 𝑠,𝑝 𝜌>0 (𝑊 (R )) 𝑊 (R ) Φ∈S (R) 𝑢∈𝑊 (R ) when . The dual space 𝐴 + of 𝐴 + can be ∗ and 𝐴 + , 󸀠 𝑊−𝑠,𝑝 (R ) 𝑞󸀠 𝑝 󵄨 𝑠 󵄨 identified with 𝐴 + ,where is the conjugate of . 󵄨(1 + 𝑥2) J (Φ𝑢)(𝑥)󵄨 󵄨 󵄨 Proof. Let 𝑝∈[1,+∞)when 𝜌=0and 𝑝∈[2,∞)when 󵄨 󸀠 󵄨 2 𝑠 𝜌>0 𝑢∈𝑊−𝑠,𝑝 (R ) 𝜑∈𝑆 (R) = 󵄨 ⟨(1 + 𝑡 ) J (𝑢) , .Then,for 𝐴 + ,wehave,forall ∗ , 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2 −𝑠 2 𝑠 󵄨 2 −𝑠 2 𝑠 󵄨 󵄨⟨𝑢, 𝜑⟩󵄨 = 󵄨⟨(1 + 𝑥 ) J (𝑢) ,(1+𝑥 ) J (𝜑)⟩ 󵄨 (1 + 𝑡 ) (1 + 𝑥 ) 𝜏𝑥 (J (Φ))⟩ 󵄨 󵄨 󵄨 󵄨 𝐿2 (R )󵄨 𝐿2 (R )󵄨 󵄨 𝐴 + 󵄨 𝐴 + 󵄨 (85) 󵄩 󵄩 𝑠 |𝑠| ≤ ‖𝑢‖ −𝑠,𝑝󸀠 󵄩𝜑󵄩 𝑠,𝑝 . |𝑠| 2 2 󵄩 󵄩𝑊 (R+) ≤2 [(1 + 𝑡 ) |J (𝑢)|]∗[(1+𝑡 ) |J (Φ)|] (𝑥) . 𝑊𝐴 (R+) 𝐴 (80) 𝑠,𝑝 This proves, from the density of 𝑆∗(R) in 𝑊𝐴 (R+),that𝑢 𝑠 On the other hand, from the hypothesis on 𝑞 and 𝑟,there admits a unique continuous extension to 𝑊𝑝. 𝑠,𝑝 ∗ exists Conversely, suppose that 𝑢∈(𝑊𝐴 (R+)) .Thenthe 1 1 1 𝑝∈[1, ∞] : + −1= . mapping 𝑝 𝑞 𝑟 (81) V :𝐿𝑝 (R )󳨀→C Thus, using Proposition 11(i), we obtain the result. 𝐴 + |𝑠|,𝑝 −1 (86) (ii) As Φ∈S∗(R),itiseasytoseethatΦ∈𝑊𝐴 (R+) 𝜓 󳨃󳨀→ ⟨𝑢, U (𝜓)⟩ for 2≤𝑝≤∞. Moreover, we proceed as above and using Proposition 11(i), we obtain is continuous, where U is the isometric isomorphism from 𝑠,𝑝 𝑝 |𝑠| (𝑊 (R ), ‖ ⋅ ‖ 𝑠,𝑝 ) (𝐿 (R ), ‖ ⋅ ‖ 𝑝 ) 𝑠,𝑟 𝑠,𝑞 𝐴 + 𝑊 (R ) into 𝐴 + 𝐿 (R ) , defined by ‖Φ𝑢‖𝑊 (R ) ≤2 ‖𝑢‖𝑊 (R )‖Φ‖ |𝑠|,𝑝 , (82) 𝐴 + 𝐴 + 𝐴 + 𝐴 + 𝑊𝐴 (R+) 2 𝑠 for U (𝜑) = (1 + 𝑥 ) J (𝜑) . (87) 1 1 1 𝑝∈[2, ∞] : + −1= . (83) 𝑝 𝑞 𝑟 with inverse

Hence, using the fact that the imbedding S∗(R)󳨅→ −1 2 −𝑠 |𝑠|,𝑝 U (𝜓) = F ((1 + 𝑥 ) 𝜓) . (88) 𝑊𝐴 (R+) is continuous, the desired result follows. 8 Abstract and Applied Analysis

󸀠 Thus, and using the fact that 𝑢∈𝑆∗(R), there exists 𝐶>0 Now, according to Theorem 27,forall𝑘∈N, Φ⋅(𝑢𝑚 −𝑢) ∈ 𝑠,∞ 𝑘 such that 𝑊𝐴 (R+) and using (82)weget

∀𝜓 ∈∗ 𝐷 (R) , ∀𝑘 ∈ N,∀𝑥∈R,

󵄨 󵄨 󵄨 2 𝑠 󵄨 󵄨 󵄨 󵄨 2 −𝑠 󵄨 󵄩 󵄩 (89) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨(1 + 𝑥 ) J (Φ ⋅ (𝑢𝑚 −𝑢))(𝑥)󵄨 󵄨V (𝜓)󵄨 = 󵄨⟨(1 + 𝑥 ) J (𝑢) ,𝜓⟩ 󵄨 ≤𝐶󵄩𝜓󵄩 . 󵄨 𝑘 󵄨 (93) 󵄨 󵄨 󵄨 2 󵄨 󵄩 󵄩𝑝,] 󵄨 𝐿𝐴(R+)󵄨 |𝑠| 𝑠,𝑞 ≤2 (1 + ‖𝑢‖ )⋅‖Φ‖ |𝑠|,𝑞󸀠 . 𝑊𝐴 (R+) 𝑝 𝑊𝐴 (R+) Hence, from the density of 𝐷∗(R) in 𝐿𝐴(R+) and using 󸀠 −𝑠,𝑝 Hence, by Lebesgue’s theorem, we deduce that for all 𝑅>0, Riez’s theorem, we deduce that 𝑢 belongs to 𝑊𝐴 (R+).This completes the proof of Theorem 28. 𝑅 𝑟𝑡󵄨 󵄨𝑟 ∫ (1 + 𝑥2) 󵄨J (Φ ⋅ (𝑢 −𝑢))(𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥=0, lim 󵄨 𝑚𝑘 󵄨 Proposition 29. Let 𝑞∈(1,∞)when 𝜌=0and 𝑞=2when 𝑘→∞ 0 𝜌>0 Φ S (R) 𝑟∈[𝑞,∞) 𝑠, 𝑡 ∈ R .Let be in ∗ and .Thenforall 𝑟∈[𝑞,∞). such that 𝑡<𝑠if 𝜌=0and 𝑡<0≤𝑠if 𝜌>0,themapping

𝑠,𝑞 𝑡,𝑟 (94) 𝑊𝐴 (R+)󳨀→𝑊𝐴 (R+) (90) On the other hand, for 𝜌=0with 𝑡<𝑠, 𝑞∈(1,∞),and 𝑢 󳨃→Φ𝑢 𝑟∈[𝑞,∞),itfollowsfromTheorem 27(i) that for 𝑝∈[1,∞] and satisfying (1/𝑝) + (1/𝑞) − 1 =1/𝑟,wehave is compact. ∞ 2 𝑟𝑡󵄨 󵄨𝑟 𝑠,𝑞 ∫ (1 + 𝑥 ) 󵄨J (Φ ⋅ (𝑢 −𝑢))(𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥 󵄨 𝑚𝑘 󵄨 Proof. Let (𝑢𝑚)𝑚 be a sequence in 𝑊𝐴 (R+) such that 𝑅 ‖𝑢𝑚‖𝑊𝑠,𝑞(R ) ≤1,forall𝑚∈N.Then,fromTheorem 28, 󵄩 󵄩𝑟 𝐴 + 󵄩Φ⋅(𝑢 −𝑢)󵄩 (𝑢 ) 󵄩 𝑚𝑘 󵄩 𝑠,𝑟 we deduce that 𝑚 𝑚 can be regarded as a sequence in 𝑊𝐴 (R+) 󸀠 ≤ −𝑠,𝑞 ∗ 󸀠 𝑟(𝑠−𝑡) (𝑊𝐴 (R+)) , with (1/𝑞) + (1/𝑞 )=1, and using Holder (1 + 𝑅2) (95) inequality, we obtain for all 𝑚∈N 𝑟 𝑟 (1 + ‖𝑢‖𝑊𝑠,𝑞(R )) ‖Φ‖ |𝑠|,𝑝 𝐴 + 𝑊 (R ) 󵄩 󵄩 𝐴 + 󵄩𝑢 󵄩 󸀠 ∗ ≤𝐶 . 󵄩 𝑚󵄩(𝑊−𝑠,𝑞 (R )) 2 𝑟(𝑠−𝑡) 𝐴 + (1 + 𝑅 ) 󵄨 󵄨 𝑠 = 󵄨 ⟨(1 + 𝑥2) J (𝑢 ), 𝜌>0 𝑡<0≤𝑠 𝑞=2 𝑟∈[2,∞) sup 󵄨 𝑚 And for with , ,and ,weobtain −𝑠,𝑞󸀠 󵄨 𝜓∈𝑊𝐴 (R+) from Theorem 27(iv) ‖𝜓‖ ≤1 −𝑠,𝑞󸀠 ∞ 𝑊 (R+) (91) 𝐴 2 𝑟𝑡󵄨 󵄨𝑟 ∫ (1 + 𝑥 ) 󵄨J (Φ ⋅ (𝑢𝑚 −𝑢))(𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥 󵄨 󵄨 𝑘 󵄨 −𝑠 󵄨 𝑅 (1+𝑥2) J (𝜓)⟩ 󵄨 2 󵄨 󵄩 󵄩𝑟 𝐿 (R+)󵄨 󵄩 󵄩 𝐴 󵄨 󵄩Φ⋅(𝑢 −𝑢)󵄩 0,𝑟 󵄩 𝑚𝑘 󵄩𝑊 (R ) 󵄩 󵄩 𝐴 + 󵄩 󵄩 ≤ −𝑟𝑡 ≤ 󵄩𝑢𝑚󵄩 𝑠,𝑞 ≤1. 2 𝑊𝐴 (R+) (1 + 𝑅 )

𝑟 𝑟 (96) 󸀠 (1 + 𝑢 𝑠,2 ) Φ −𝑠,𝑞 { ‖ ‖𝑊 (R ) ‖ ‖ 2𝑟/(𝑟−2) 𝑊 (R ) { 𝐴 + 𝐿𝛾 (R+) Therefore, by virtue of Propositions 24 and 19(i), 𝐴 + {𝐶 , 𝑟 =2̸ { −𝑟𝑡 for is a separable Banach space. Which implies, from Alaoglu { (1 + 𝑅2) ≤ (𝑢 ) { 2 2 theorem (cf. [14]), that there exists a subsequence 𝑚𝑘 𝑘 { { (1 + ‖𝑢‖ 𝑠,2 ) ‖Φ‖ ∞ 󸀠 { 𝑊 (R+) 𝐿 (R ) −𝑠,𝑞 ∗ { 𝐴 𝛾 + weakly converging in (𝑊 (R+)) . We denote by 𝑢 its weak {𝐶 , 𝑟=2. 𝐴 2 −2𝑡 for limit. Using Lemma 26,forall𝑘∈N,wehave { (1 + 𝑅 ) Which implies that these last integrals, (95)and(96), tend to J [Φ (𝑢 −𝑢)](𝑥) =[J (𝑢 −𝑢)∗ J (Φ)] (𝑥) 𝑚𝑘 𝑚𝑘 𝐴 zero when 𝑅 tends to ∞ uniformly with respect to 𝑘∈N (Φ ⋅ (𝑢 −𝑢)) and so, by virtue of (94), we conclude that 𝑚𝑘 𝑘 is =⟨J (𝑢 −𝑢),𝜏 (J (Φ))⟩ 𝑡,𝑟 𝑚𝑘 𝑥 (92) 𝑊 (R ) ‖Φ.(𝑢 − strongly converging in 𝐴 + ; that is, lim𝑘→∞ 𝑚𝑘 𝑢)‖𝑊𝑡,𝑟(R ) =0. This achieves the proof of Proposition 29. =⟨𝑢 −𝑢,𝜑 (𝑥) Φ⟩ . 𝐴 + 𝑚𝑘 (⋅) Notation. Let 𝐾 be a compact contained in R. We denote by 𝑠,𝑞 𝑠,𝑞 −𝑠,𝑞󸀠 ∗ 𝑊 (R ) 𝑊 (R ) (𝑢 −𝑢) (𝑊 (R )) 𝐴,𝐾 + the subspace of 𝐴 + defined by But 𝑚𝑘 𝑘 is weakly converging to zero in 𝐴 + , then it follows that for all 𝑥∈Rlim𝑘→∞J(Φ ⋅ (𝑢𝑚 − 𝑢)(𝑥) = 𝑠,𝑞 𝑠,𝑞 𝑘 𝑊 (R ) = {𝑢∈𝑊 (R ) ; 𝑢⊂𝐾} . 0. 𝐴,𝐾 + 𝐴 + supp (97) Abstract and Applied Analysis 9

As a consequence of Proposition 29,with𝜑∈S∗(R) Hence, by tending 𝑛 to infinity and from (101), it follows that supportedonaboundedset𝑉 containing the compact 𝐾 and 1≤0which is impossible. Thus, the required inequality is satisfying 𝜑=1on 𝐾,wededucethemainresultofthis satisfied. Combining the left hand side inequality of(98)and section. (99), we obtain Theorem 30 𝑞 ∈ (1, ∞) 1 ̇ (Reillich-type theorem). Let when ‖𝑢‖𝑊𝑠,𝑞(R ) ≤ 𝐽𝑠 (𝑢) ≤𝐶‖𝑢‖𝑊𝑠,𝑞(R ). (104) 𝜌=0and 𝑞=2when 𝜌>0.Let𝐾 be a compact contained 𝐶 𝐴 + 𝐴 + in R and let 𝑟∈[𝑞,∞).Thenforall𝑠, 𝑡 ∈ R such that 𝑡<𝑠 This completes the proof. if 𝜌=0and 𝑡<0≤𝑠if 𝜌>0, the canonical imbedding 𝑠,𝑞 𝑡,𝑟 𝑊𝐴,𝐾(R+)󳨅→𝑊𝐴,𝐾(R+) is compact. 4. Applications Using Reillich-type theorem, we prove the following inequalities. 4.1. Weierstrass Transform on the Dual of the Chebli-Trim´ eche` Hypergroup. Thissubsectionisdevotedtodefineandestab- T𝑠 Corollary 31. Let 𝐾 be a compact contained in R and 𝑠≥0. lish some properties for the Weierstrass transform 𝐴,onthe Then, for 𝑞∈(1,∞)when 𝜌=0and 𝑞=2when 𝜌>0,there dual of the Chebli-Trim´ eche` hypergroup, which we need later. 𝑠,𝑞 exists 𝐶>0such that, for all 𝑢∈𝑊𝐴,𝐾(R+),wehave Definition 32. Let 𝑠≥0. We define the generalized Weier- 1 ̇ 𝑠 𝑆󸀠 (R) ‖𝑢‖𝑊𝑠,𝑞(R ) ≤ 𝐽𝑠 (𝑢) ≤𝐶‖𝑢‖𝑊𝑠,𝑞(R ), (98) strass transform of order on ∗ as follows: 𝐶 𝐴 + 𝐴 + 2 2 ∞ 1/𝑞 T𝑠 (𝑢) = F (𝑒−𝑠(𝜆 +𝜌 )F−1 (𝑢)). 𝐽̇(𝑢) = (∫ 𝑥𝑞𝑠|J(𝑢)(𝑥)|𝑞𝐴(𝑥)𝑑𝑥) 𝐴 (105) where 𝑠 0 . 𝑠,𝑞 𝑞 For 1≤𝑝<∞, we denote by Proof. It is clear that, for all 𝑢∈𝑊𝐴,𝐾(R+), J(𝑢) ∈ 𝐿𝐴(R+) and we have 𝑠,𝑝 󸀠 𝑠 𝑝 W𝐴 (R+):={𝜙∈S∗ (R) : T𝐴 (𝜙) ∈ 𝐿𝛾 (R+)} . (106) 𝐽̇(𝑢) ≤ ‖𝑢‖ 𝑠,𝑞 . 𝑠 𝑊𝐴 (R+) (99) W𝑠,𝑝(R ) Now, let us prove the left hand side inequality. Suppose that The norm in 𝐴 + is given by 𝑠,𝑞 for all positive integer 𝑘, there exists 𝑢𝑘 ∈𝑊𝐴,𝐾(R+) such that 󵄩 󵄩 󵄩 𝑠 󵄩 󵄩𝜙󵄩 𝑠,𝑝 = 󵄩T𝐴 (𝜙)󵄩 𝑝 . W𝐴 (R+) 𝐿𝛾(R+) (107) 1 󵄩 󵄩 ̇ 󵄩𝑢𝑘󵄩𝑊𝑠,𝑞(R ) > 𝐽𝑠 (𝑢𝑘) . (100) 𝑘 𝐴 + Remark 33. Let 𝑠>0.Forall𝜆∈R,wehave

Without loss of generality, one can suppose that ‖𝑢𝑘‖𝑊𝑠,𝑞(R ) = 2 2 1 𝐴 + 𝑒−𝑠(𝜆 +𝜌 ) = F (𝑤 (𝑠,) ⋅ ) (𝜆) , 1.Thenwehave (𝜆2 +𝜌2)𝑛 𝑛 (108) ̇ lim 𝐽𝑠 (𝑢𝑘)=0 𝑘→∞ (101) where 𝑤𝑛(𝑠, ⋅), 𝑛∈N, are the heat functions on the Chebli-´ and, by using Reillich-type theorem, we deduce that there Trimeche` hypergroup (R+,∗𝐴). In particular, 𝑤𝑛(𝑠, ⋅) =𝑠 𝐸 is (𝑢 ) (𝑢 ) R ,∗ exists a subsequence 𝑘𝑛 𝑛 of 𝑘 𝑘 strongly converging in theGaussiankernelon( + 𝐴), see [15]. 𝑡,𝑞 𝑊 (R+), 𝑡<0.Let𝑢 be its strong limit. Therefore, by (101) In the case of the Bessel-Kingman hypergroup (when the 𝐴 𝐴 𝐴(𝑥)2𝛼+1 =𝑥 𝜌=0 and the fact that for 𝑅>0 function is of the form and ), the 𝑅 1/𝑞 Weierstrass transform associated with the Hankel transform (∫ 𝑥2𝑞𝑠|J (𝑢)(𝑥)|𝑞𝐴 (𝑥) 𝑑𝑥) is studied in [16]. For the classical Weierstrass transform, one 0 (102) can see [17–19]. 2 𝑠−𝑡󵄩 󵄩 ̇ 𝑠 ≤(1+𝑅) 󵄩𝑢−𝑢 󵄩 𝑡,𝑞 + 𝐽 (𝑢 ), T 󵄩 𝑘𝑛 󵄩𝑊 (R ) 𝑠 𝑘𝑛 Inthefollowing,weshowsomepropertiesfor 𝐴 and 𝐴 + W𝑠,𝑝(R ) ̇ 𝐴 + . it follows that 𝐽𝑠(𝑢) =. 0 This implies that J(𝑢) = 0 and so 𝑢=0 𝐶 󸀠 . On the other hand, there exists a positive constant Proposition 34. (i) Let 𝑠, 𝑡.Forall ≥0 𝑓∈S∗(R), such that 󵄩 󵄩 𝑠 𝑡 𝑠+𝑡 0 1=󵄩𝑢 󵄩 T𝐴 (T𝐴 (𝑓)) = T𝐴 (𝑓) , T𝐴 (𝑓) = 𝑓. (109) 󵄩 𝑘𝑛 󵄩 𝑠 𝑊𝐴,𝐾(R+) 2 1/𝑞 𝑓∈𝐿 (R ) 1 󵄨 󵄨𝑞 (ii) For all 𝛾 + ,wehave 󵄨 2 𝑡 󵄨 ≤𝐶{(∫ 󵄨(1 + 𝑥 ) J (𝑢𝑘 ) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥) 󵄨 𝑛 󵄨 0 T𝑠 (𝑓) = 𝑓, 𝐿2 (R ). lim+ 𝐴 in 𝛾 + (110) (103) 𝑠→0 1/𝑞 ∞ 󵄨 󵄨𝑞 +(∫ 󵄨𝑥2𝑠J (𝑢 ) (𝑥)󵄨 𝐴 (𝑥) 𝑑𝑥) }. 1≤𝑝<∞ 𝑠≥𝑡≥0 𝜙∈W𝑠,𝑝(R ) 󵄨 𝑘𝑛 󵄨 (iii) Let and .Then,forall 𝐴 + , 1 we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ̇ 󵄩 𝑡 󵄩 󵄩 󵄩 ≤𝐶{󵄩𝑢 󵄩 𝑡,𝑞 + 𝐽 (𝑢 )} . 󵄩T (𝜙)󵄩 𝑠−𝑡,𝑝 = 󵄩𝜙󵄩 𝑠,𝑝 . 󵄩 𝑘𝑛 󵄩𝑊 (R ) 𝑠 𝑘𝑛 󵄩 𝐴 󵄩 󵄩 󵄩W (R ) (111) 𝐴 + W𝐴 (R+) 𝐴 + 10 Abstract and Applied Analysis

𝑡,2 𝑠,2 (iv) Let 𝑠≥𝑡≥0.Then,W𝐴 (R+)⊂W𝐴 (R+).Moreover,for (i) for all 𝑦>0,thefunction𝑥 󳨃→𝐾𝑠(𝑥, 𝑦) belongs 𝑡,2 H𝑠 (R ) all 𝑢∈W𝐴 (R+),wehave to 𝐴 + . 𝑠 (ii) The reproducing property: for all 𝑓∈H𝐴(R+) and 𝑦> ‖𝑢‖ 𝑠,2 ≤ ‖𝑢‖ 𝑡,2 . 0 W𝐴 (R+) W𝐴 (R+) (112) , 𝑓 (𝑦) = ⟨𝑓, 𝐾 (𝑥,)⟩ 𝑦 . 󸀠 𝑠 H𝑠 (R ) (118) Proof. (i) For all 𝑓∈S∗(R),wehave 𝐴 +

2 2 2 2 Proof. (i) It is clear from Lemma 21 and relations (9)and(10) T𝑠 (T𝑡 (𝑓)) = F (𝑒−𝑠(𝜆 +𝜌 )𝑒−𝑡(𝜆 +𝜌 )F−1 (𝑓)) 𝐴 𝐴 that, for all 𝑦>0,thefunction

2 2 = F (𝑒−(𝑡+𝑠)(𝜆 +𝜌 )F−1 (𝑓)) (113) 𝜑𝑦 (𝜉) Θ𝑦 :𝜉󳨃󳨀→ (119) (1 + 𝜉2)2𝑠 = T𝑠+𝑡 (𝑓) 𝐴 1 2 belongs to 𝐿𝐴(R+)∩𝐿𝐴(R+) when 𝑠>3/4when 𝜌>0and 0 −1 𝑠 > (𝛼 + 1)/2 𝜌=0 𝐾 (⋅, 𝑦) and T𝐴(𝑓) = F(F (𝑓)) = 𝑓. when .Thus,thefunction 𝑠 is well (ii) Clear. defined and we can write 𝑠,𝑝 (iii) Let 𝜙∈W (R+).By(i)andDefinition 32,weobtain 𝐴 𝐾𝑠 (𝑥, 𝑦) = F (Θ𝑦) (𝑥) ,∀𝑥∈R+. (120) 󵄩 󵄩 󵄩T𝑡 (𝜙)󵄩 󵄩 𝐴 󵄩W𝑠−𝑡,𝑝(R ) Moreover, from Proposition 4,wecanseethatthefunction 𝐴 + 2 𝐾𝑠(⋅, 𝑦) belongs to 𝐿𝛾(R+),andwehave 󵄩 𝑠−𝑡 𝑡 󵄩 󵄩 𝑠 󵄩 = 󵄩T (T (𝜙))󵄩 𝑝 = 󵄩T (𝜙)󵄩 𝑝 󵄩 𝐴 𝐴 󵄩𝐿 (R ) 󵄩 𝐴 󵄩𝐿𝛾(R ) (114) 𝛾 + + 𝜑𝑦 (𝜉) J (𝐾𝑠 (⋅, 𝑦)) (𝜉) = ,∀𝜉∈R+. 󵄩 󵄩 2 2𝑠 (121) = 󵄩𝜙󵄩 𝑠,𝑝 . (1 + 𝜉 ) W𝐴 (R+) Therefore, according to Lemma 21 and using relations (9)and (iv) We deduce the result by using Proposition 4(ii). (10), we deduce that Now, under a sufficient condition on 𝑝 and 𝑞,weshall 󵄩 󵄩2 󵄩𝐾𝑠 (⋅, 𝑦)󵄩 𝑠 <∞. (122) 𝑠 𝑡,𝑝 H𝐴(R+) prove that T𝐴 is a bounded operator from 𝑊𝐴 (R+) into 𝑠,𝑞 W (R ) This proves that for all 𝑦>0,thefunction𝐾𝑠(⋅, 𝑦) belongs to 𝐴 + . 𝑠 H𝐴(R+). 𝑠 Proposition 35. Let 𝑠>0and 𝑡∈R.Then,forall𝑞≥2and (ii) Let 𝑓 be in H𝐴(R+) and 𝑦>0.Thenby(121), we get 𝑝 ≥ 𝑞/(𝑞 −1), there exists a positive constant 𝐶𝑠,𝑡(𝑝, 𝑞) such 𝑡,𝑝 𝑓∈𝑊 (R ) ⟨𝑓, 𝐾𝑠 (⋅, 𝑦)⟩ 𝑠 = ∫ J (𝑓) (𝜉) 𝜑𝑦 (𝜉) 𝐴 (𝜉) 𝑑𝜉, that for all 𝐴 + ,wehave H𝐴(R+) (123) R+ 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩 𝑠,𝑞 ≤𝐶𝑠,𝑡 (𝑝, 𝑞) 󵄩𝑓󵄩 𝑡,𝑝 . W𝐴 (R+) 𝑊𝐴 (R+) (115) and from inversion formula, we obtain the reproducing property Proof. According to Lemma 21 and Definition 32,weobtain 𝑓 (𝑦) = ⟨𝑓, 𝐾 (𝑥,)⟩ 𝑦 . 𝑠 H𝑠 (R ) (124) the result by using Proposition 6 and applying Holder 𝐴 + inequality. This completes the proof of the theorem.

𝑠 4.2. Kernel Reproducing. Let 𝑠∈R.ThespaceH𝐴(R+)= Definition 37. For all positive real numbers 𝑟, 𝑠,and𝑡, 𝑠,2 𝑟,𝑠,𝑡 𝑊 (R ) we define the Hilbert space H𝐴 (R+) as the subspace of 𝐴 + provided with the inner product, 𝑠 H𝐴(R+) with the inner product: 2𝑠 ⟨𝑓, 𝑔⟩ = ∫ (1+𝑥2) J (𝑓) (𝑥) J (𝑔) (𝑥) 𝐴 (𝑥) 𝑑𝑥, 𝑡 𝑡 H𝑠 (R ) ⟨𝑓,𝑔⟩ 𝑟,𝑠,𝑡 = 𝑟⟨𝑓,ℎ⟩ +⟨T 𝑓, T 𝑔⟩ , 𝐴 + H (R ) H𝑠 (R ) 𝐴 𝐴 2 R+ 𝐴 + 𝐴 + 𝐿𝛾(R+) (125) (116) 𝑠 𝑓, 𝑔 ∈ H𝐴 (R+). 2 and the norm ‖𝑓‖H𝑠 (R ) =⟨𝑓,𝑓⟩H𝑠 (R ),isaHilbertspace. 𝐴 + 𝐴 + The norm associated to the inner product is defined by Proposition 36. 𝑠>3/4 𝜌>0 𝑠 > (𝛼 + 1)/2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 For when and 󵄩𝑓󵄩 𝑟,𝑠,𝑡 := 𝑟󵄩𝑓󵄩 𝑠 + 󵄩𝑓󵄩 𝑡,2 . (126) 𝑠 H𝐴 (R+) H𝐴(R+) W𝐴 (R+) when 𝜌=0,theHilbertspaceH𝐴(R+) admits the following reproducing kernel: Proposition 38. Let 𝑠>3/4when 𝜌>0and 𝑠 > (𝛼 + 1)/2 𝑟,𝑠,𝑡 when 𝜌=0.Forall𝑟, 𝑡,theHilbertspace >0 H𝐴 (R+) admits 𝜑𝑥 (𝜉) 𝜑𝑦 (𝜉) 𝐴 (𝜉) 𝑑𝜉 the following reproducing kernel: 𝐾𝑠 (𝑥, 𝑦) = ∫ ; 2 2𝑠 (117) R+ (1 + 𝜉 ) 𝜑 (𝜉) 𝜑 (𝜉) K𝑡 (𝑥, 𝑦) = ∫ 𝑥 𝑦 𝐴 (𝜉) 𝑑𝜉. 𝑟,𝑠 2𝑠 2 2 (127) R 𝑟(1 + 𝜉2) +𝑒−2𝑡(𝜉 +𝜌 ) that is, + Abstract and Applied Analysis 11

Proof. As in Proposition 36,wecandeducethatforall𝑦>0, (i) 𝐻𝐾 is a Hilbert space with the reproducing kernel 𝑡 2 𝐾 (𝑝,𝑟 𝑞) 𝐸 there exists a function 𝑥 󳨃→ K𝑟,𝑠(𝑥, 𝑦) in 𝐿𝛾(R+) such that 𝑟 on and satisfying the equation we have 𝐾(⋅,𝑞)=(𝑟𝐼+𝐿∗𝐿) 𝐾 (⋅, 𝑞) , 𝜑 𝑟 (135) J (K𝑡 (⋅, 𝑦)) = 𝑦 , 𝑟,𝑠 2 2𝑠 −2𝑡(𝜉2+𝜌2) ∗ 𝑟(1 + 𝜉 ) +𝑒 where 𝐿 is the adjoint operator of 𝐿:𝐻𝐾 →𝐻. (128) 󵄩 󵄩2 (ii) For any 𝑟>0and for any ℎ in 𝐻,theinfinitum 󵄩K𝑡 (⋅, 𝑦)󵄩 <∞. 󵄩 𝑟,𝑠 󵄩H𝑟,𝑠 R 𝐴 ( +) 󵄩 󵄩2 󵄩 󵄩2 {𝑟󵄩𝑓󵄩 + 󵄩𝐿𝑓 −ℎ󵄩 } inf 󵄩 󵄩𝐻𝐾 󵄩 󵄩𝐻 (136) 𝑡 𝑓∈𝐻𝐾 Thisprovesthatforall𝑦>0the function K𝑟,𝑠(⋅, 𝑦) belongs H𝑟,𝑠,𝑡(R ) to 𝐴 + . 𝑓∗ 𝐻 𝑓 H𝑟,𝑠,𝑡(R ) 𝑦>0 is attained by a unique function 𝑟,ℎ in 𝐾 and this On the other hand, for in 𝐴 + and ,wehave extremal function is given by ⟨𝑓, K𝑡 (⋅, 𝑦)⟩ =𝑟𝐼 +𝐼, ∗ 𝑟,𝑠 𝑟,𝑠,𝑡 1 2 (129) 𝑓 (𝑝) = ⟨ℎ,𝑟 𝐿𝐾 (⋅, 𝑝)⟩ . H𝐴 (R+) 𝑟,ℎ 𝐻 (137) where We can now state the main result of this paragraph. 𝐼 =⟨𝑓,K𝑡 (⋅, 𝑦)⟩ , 1 𝑟,𝑠 H𝑠 (R ) 𝐴 + Theorem 40. Let 𝑠>3/4when 𝜌>0and 𝑠 > (𝛼 + 1)/2 when (130) 𝜌=0. 𝐼 =⟨T𝑡 𝑓, T𝑡 (K𝑡 (⋅, 𝑦))⟩ . 2 𝐴 𝐴 𝑟,𝑠 2 𝐿𝛾(R+) 2 (i) For any 𝑔∈𝐿𝛾(R+) and for any 𝑟>0,thebest ∗ But from (116)and(128), we have approximate function 𝑓𝑟,𝑔 in the sense 2𝑠 2 󵄩 󵄩2 󵄩 𝑡 󵄩2 (1 + 𝜉 ) J (𝑓) (𝜉) 𝜑𝑦 (𝜉) {𝑟󵄩𝑓󵄩 + 󵄩𝑔−T 𝑓󵄩 } inf 󵄩 󵄩 𝑠 󵄩 𝐴 󵄩 2 𝐼 =∫ 𝐴 (𝜉) 𝑑𝜉. 𝑠 H𝐴(R+) 󵄩 󵄩𝐿 (R ) 1 2𝑠 2 2 (131) 𝑓∈H (R ) 𝛾 + R 𝑟(1 + 𝜉2) +𝑒−2𝑡(𝜉 +𝜌 ) 𝐴 + + (138) 󵄩 󵄩2 󵄩 󵄩2 =𝑟󵄩𝑓∗ 󵄩 + 󵄩𝑔−T𝑡 𝑓∗ 󵄩 and from (128),itfollows,byusingParsevalformulaforthe 󵄩 𝑟,𝑔󵄩 𝑠 󵄩 𝐴 𝑟,𝑔󵄩 2 H𝐴(R+) 𝐿𝛾(R+) transform F,that ∗ −𝑡(𝑥2+𝜌2) exists uniquely and 𝑓𝑟,𝑔 is represented by 𝐼2 = ∫ F (𝑒 J (𝑓)) (𝜆) F R+ 𝑓∗ (𝑦) = ∫ 𝑔 (𝑥) 𝑄 (𝑥,) 𝑦 𝑑𝛾 (𝑥) , −𝑡(𝑥2+𝜌2) 𝑡 𝑟,𝑔 𝑟 (139) ×(𝑒 J (K (⋅, 𝑦))) (𝜆) 𝑑𝛾 (𝜆) R+ 𝑟,𝑠 (132) −2𝑡(𝜉2+𝜌2) where 𝑒 J (𝑓) (𝜉) 𝜑𝑦 (𝜉) = ∫ 𝐴 (𝜉) 𝑑𝜉. 2 2 2𝑠 2 2 −2𝑡(𝜉 +𝜌 ) R 𝑟(1 + 𝜉2) +𝑒−2𝑡(𝜉 +𝜌 ) 𝑒 𝜑 (𝜉) 𝜑 (𝜉) + 𝑄 (𝑥, 𝑦) = ∫ 𝑥 𝑦 𝐴 (𝜉) 𝑑𝜉. 𝑟 2𝑠 2 2 (140) R 𝑟(1 + 𝜉2) +𝑒−2𝑡(𝜉 +𝜌 ) Thus, by virtue of (129), and combining (131)and(132), we + deduce that 𝑠 𝑡 (ii) Let 𝑓∈H (R+).Then,Ifwetake𝑔=T 𝑓,wehave ⟨𝑓, K𝑡 (⋅, 𝑦)⟩ =𝑓(𝑦), . . 𝐴 𝐴 𝑟,𝑠 H𝑟,𝑠,𝑡(R ) a e (133) 𝐴 + ∗ + 𝑓𝑟,𝑔 󳨀→ 𝑓 𝑎 𝑠 𝑟󳨀→0 ,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦. (141)

4.3. Extremal Function for Generalized Weierstrass Transform. 𝛿>0 𝑔 𝑔𝛿 ‖𝑔 −𝛿 𝑔 ‖ 2 ≤𝛿 (iii) Let and let and satisfy 𝐿𝛾(R+) . In this subsection, we show the existence and unicity of Then the extremal function related to the generalized Weierstrass 𝑡 T 󵄩 ∗ ∗ 󵄩 𝛿 transform 𝐴. We start with the following fundamental 󵄩𝑓 −𝑓 󵄩 ≤ . 󵄩 𝑟,𝑔 𝑟,𝑔𝛿 󵄩H𝑠 (R ) (142) theorem (cf. [20]). 𝐴 + √𝑟

Theorem 39. Let 𝐻𝐾 be a Hilbert space admitting the repro- Proof. (i) By Proposition 38 and Theorem 39(ii), the infini- 𝐾(𝑝, 𝑞) 𝐸 𝐻 ∗ ducing kernel on a set and let be a Hilbert space. tum given by (138) is attained by a unique function 𝑓𝑟,𝑔,and Let 𝐿:𝐻𝐾 →𝐻be a bounded linear operator on 𝐻𝐾 into 𝐻. ∗ the extremal function 𝑓𝑟,𝑔 is represented by For 𝑟>0,weintroducetheinnerproductin𝐻𝐾 and we call it 𝐻 ∗ 𝑡 𝑡 𝐾𝑟 as 𝑓 (𝑦) = ⟨𝑔, T (K (⋅, 𝑦))⟩ ,𝑦>0, 𝑟,𝑔 𝐴 𝑔,𝑟 2 (143) 𝐿𝛾(R+) ⟨𝑓1,𝑓2⟩𝐻 = 𝑟⟨𝑓1,𝑓2⟩𝐻 +⟨𝐿𝑓1,𝐿𝑓2⟩𝐻. 𝐾𝑟 𝐾 (134) 𝑡 where K𝑔,𝑟 is the kernel given by Proposition 38.Hence,by Then, (128), we obtain the expression (140)of𝑄𝑟(𝑥, 𝑦). 12 Abstract and Applied Analysis

(ii) From Proposition 35,thefunction𝑔 belongs to Conflict of Interests 2 𝐿𝛾(R+). According to Lemma 21 and relations (9)and(10), it The authors declare that there is no conflict of interests can be observed, using Cauchy-Schwartz inequality, that for regarding the publication of this paper. all 𝑦>0the function

𝜉 󳨃󳨀→ J (𝑓) (𝜉) 𝜑𝑦 (𝜉) (144) References

1 [1] W. R. Bloom and H. Heyer, Harmonic Analysis of Probability belongs to 𝐿𝐴(R+), which implies, from inversion formula for J Measures on Hypergroups,vol.20ofde Gruyter Studies in the transform ,that Mathematics,WalterdeGruyter,Berlin,Germany,1995. [2] M. Assal and M. M. Nessibi, “Bessel-Sobolev type spaces,” 𝑓 (𝑦) = ∫ J (𝑓) (𝜉) 𝜑𝑦 (𝜉) 𝐴 (𝜉) 𝑑𝜉. (145) Mathematica Balkanica,vol.18,no.3-4,pp.227–234,2004. R + [3] N. Ben Salem and A. Dachraoui, “Sobolev type spaces associ- ated with the Jacobi operator,” Integral Transforms and Special Therefore, by (132), it follows that Functions,vol.7,no.1,1998. 2𝑠 [4] W. R. Bloom and Z. Xu, “Fourier multipliers for local Hardy −𝑟(1 + 𝜉2) J (𝑓) (𝜉) 𝜑 (𝜉) (𝑓∗ −𝑓)(𝑦) =∫ 𝑦 𝐴 (𝜉) 𝑑𝜉. spaces on Chebli-Trim´ eche` hypergroups,” Canadian Journal of 𝑟,𝑔 2 2𝑠 −2𝑡(𝜉2+𝜌2) Mathematics, vol. 50, no. 5, pp. 897–928, 1998. R+ 𝑟(1 + 𝜉 ) +𝑒 [5] M. Dziri, M. Jelassi, and L. T. Rachdi, “Spaces of 𝐷𝐿 𝑝 type (146) and a convolution product associated with a singular second order differential operator,” Journal of Concrete and Applicable Hence, by dominated convergence theorem we deduce the Mathematics,vol.10,no.3-4,pp.207–232,2012. result. [6] K. Trimeche,` “Inversion of the Lions transmutation operators (iii) It is clear, from (140), that using generalized wavelets,” Applied and Computational Har-

2 2 monic Analysis,vol.4,no.1,pp.97–112,1997. 𝑒−2𝑡(𝜉 +𝜌 )𝜑 𝜉 𝑥 ( ) [7] K. Trimeche,` “Transformation integrale´ de Weyl et theor´ eme` de 𝑄𝑟 (𝑥, 𝑦) = F ( )(𝑦), (147) 𝑟(1 + 𝜉2)2𝑠 +𝑒−2𝑡(𝜉2+𝜌2) Paley-Wiener associes´ aunop` erateur´ differentiel´ singulier sur (0, ∞),” Journal de Mathematiques´ Pures et Appliquees´ ,vol.60, then, using Parseval formula for the Fourier transform F,it no. 1, pp. 51–98, 1981. follows, by (139), that [8] Z. Xu, Harmonic Analysis on Chebli-Trim´ eche` Hypergroups [Ph.D. thesis], Murdoch University, Perth, Australia, 1994. −2𝑡(𝜉2+𝜌2) [9] H. Chebli, “Theor´ eme` de Paley-Wiener associe´ aunop` erateur´ ∗ 𝑒 J (𝑔) (𝜉) 𝜑𝑥 (𝜉) 𝑓 (𝑥) = ∫ 𝐴 (𝜉) 𝑑𝜉 (148) differentiel´ singulier sur(0, ∞),” Journal de Mathematiques´ 𝑟,𝑔 2 2𝑠 −2𝑡(𝜉2+𝜌2) R+ 𝑟(1 + 𝜉 ) +𝑒 Pures et Appliquees´ ,vol.58,no.1,pp.1–19,1979. [10] H. Chebli, “Operateurs´ de translation gen´ eralis´ ee´ et semi- and so groupes de convolution,” in Theorie´ du Potentiel et Analyse Harmonique,vol.404ofLecture Notes in Mathematics,pp.35– −2𝑡(𝜉2+𝜌2) ∗ 𝑒 J (𝑔) (𝜉) 59, Springer, Berlin, Germany, 1974. J (𝑓 ) (𝜉) = . (149) 𝑟,𝑔 𝑟(1 + 𝜉2)2𝑠 +𝑒−2𝑡(𝜉2+𝜌2) [11] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, UK, 1937. Hence, [12] G. B. Folland, Real Analysis Modern Techniques and Their Appli- cations, Pure and Applied Mathematics (New York), John Wiley 2 2 & Sons, A Wiley-Interscience Publication, New York, NY, USA, 𝑒−2𝑡(𝜉 +𝜌 )J (𝑔−𝑔 ) (𝜉) J (𝑓∗ −𝑓∗ ) (𝜉) = 𝛿 . 1984. 𝑟,𝑔 𝑟,𝑔 2𝑠 2 2 (150) 𝛿 𝑟(1 +2 𝜉 ) +𝑒−2𝑡(𝜉 +𝜌 ) [13] W. Rudin, Analyse Fonctionnelle, Ediscience International, Paris, France, 1995. 2 Using the inequality (𝑥 + 𝑦) ≥4𝑥𝑦,weobtain [14] H. Brezis, Analyse Fonctionnelle,Masson,Paris,France,3rd edition, 1983. 2𝑠󵄨 󵄨2 1 2 󵄨 ∗ ∗ 󵄨 󵄨 󵄨2 [15] L.BouattourandK.Trimeche,` “Beurling-Hormander’stheorem¨ (1 + 𝜉 ) 󵄨J (𝑓𝑟,𝑔 −𝑓𝑟,𝑔 ) (𝜉)󵄨 ≤ 󵄨J (𝑔 −𝛿 𝑔 ) (𝜉)󵄨 . 󵄨 𝛿 󵄨 4𝑟 for the Chebli-Trim´ eche` transform,” Global Journal of Pure and (151) Applied Mathematics,vol.1,no.3,pp.342–357,2005. [16] S. Omri and L. T. Rachdi, “Weierstrass transform associated Thus, and from Proposition 4(ii), we obtain with the Hankel operator,” Bulletin of Mathematical Analysis and Applications,vol.1,no.2,pp.1–16,2009. 󵄩 ∗ ∗ 󵄩2 1 󵄩 󵄩2 [17]T.Matsuura,S.Saitoh,andD.D.Trong,“Approximateandana- 󵄩𝑓 −𝑓 󵄩 ≤ 󵄩J (𝑔−𝑔 )󵄩 2 󵄩 𝑟,𝑔 𝑟,𝑔𝛿 󵄩 𝑠 󵄩 𝛿 󵄩𝐿 (R ) H𝐴(R+) 4𝑟 𝐴 + lytical inversion formulas in heat conduction on multidimen- (152) sional spaces,” Journal of Inverse and Ill-Posed Problems,vol.13, 1 󵄩 󵄩2 = 󵄩𝑔−𝑔 󵄩 , no.3–6,pp.479–493,2005. 󵄩 𝛿󵄩𝐿2 (R ) 4𝑟 𝛾 + [18] S. Saitoh, “Approximate real inversion formulas of the Gaussian convolution,” Applicable Analysis,vol.83,no.7,pp.727–733, which gives the desired result. 2004. Abstract and Applied Analysis 13

[19] S. Saitoh, “The Weierstrass transform and an isometry in the heat equation,” Applicable Analysis,vol.16,no.1,pp.1–6,1983. [20] S. Saitoh, TheoryofReproducingKernelsanditsApplications,vol. 189 of Pitman Research Notes in Mathematics Series,Longman Scientific & Technical, Harlow, UK, 1988. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 238036, 29 pages http://dx.doi.org/10.1155/2014/238036

Research Article On Positive Solutions of a Fourth Order Nonlinear Neutral Delay Difference Equation

Zeqing Liu,1 Xiaoping Zhang,1 Shin Min Kang,2 and Young Chel Kwun3

1 Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China 2 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 3 Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Correspondence should be addressed to Young Chel Kwun; [email protected]

Received 19 August 2013; Accepted 17 December 2013; Published 4 March 2014

Academic Editor: Adem Kılıc¸man

Copyright © 2014 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.

1. Introduction Krasnoselskii’s fixed point theorem, coincidence degree the- ory and critical point theory, the authors [3, 4, 9, 13–16, 18, 25, In the past few decades, the researchers [1–31]andothers 27, 28] and others proved the existence of nonoscillatory solu- studied oscillation, asymptotic behavior, and solvability for a tions, uncountably many bounded nonoscillatory solutions lot of second and third order nonlinear difference equations, and periodic solutions for the difference equations above, some of which are as follows: where Lipschitz conditions were used in [14, 16]. Recently, the Δ(𝑎𝑛Δ(𝑥𝑛 +𝑝𝑥𝑛−𝜏)) + 𝐹 (𝑛 +𝑛+1−𝜎 1,𝑥 ) = 0, 𝑛 ≥ 1, authors [32] used the Krasnoselskii’s fixed point theorem to obtain ℎ-asymptotic stability results about the zero solution Δ(𝑎𝑛Δ(𝑥𝑛 +𝑏𝑥𝑛−𝜏)) for a very general first order nonlinear neutral differential equation with functional delay. +𝑓(𝑛,𝑥 ,...,𝑥 )=𝑐,𝑛≥𝑛, 𝑛−𝑑1𝑛 𝑛−𝑑𝑘𝑛 𝑛 0 However, to our knowledge, no one studied the following 3 fourth order nonlinear neutral delay difference equation: Δ 𝑥𝑛 +𝑓(𝑛,𝑥𝑛,𝑥𝑛−𝑟)=0, 𝑛≥𝑛0, 3 3 Δ (𝑎𝑛Δ(𝑥𝑛 +𝛾𝑛𝑥𝑛−𝜏)) + Δ 𝑓(𝑛,𝑥𝑏 ,...,𝑥𝑏 ) Δ(𝑎 Δ2 (𝑥 +𝑃𝑥 )) 1𝑛 𝑘𝑛 𝑛 𝑛 𝑛 𝑛−𝜏 (1) 2 +Δ 𝑔(𝑛,𝑥𝑐 ,...,𝑥𝑐 )+Δℎ(𝑛,𝑥𝑑 ,...,𝑥𝑑 ) (2) +𝑓(𝑛,𝑥 ,...,𝑥 )=𝑔,𝑛≥𝑛, 1𝑛 𝑘𝑛 1𝑛 𝑘𝑛 𝑛−𝑑1𝑛 𝑛−𝑑𝑘𝑛 𝑛 0 +𝑝(𝑛,𝑥𝑜 ,...,𝑥𝑜 )=𝑟𝑛,𝑛≥𝑛0, Δ2 (𝑎 Δ(𝑥 +𝛾𝑥 )) + Δ2𝑓(𝑛,𝑥 ,...,𝑥 ) 1𝑛 𝑘𝑛 𝑛 𝑛 𝑛 𝑛−𝜏 𝑛−𝑑1𝑛 𝑛−𝑑𝑘𝑛 𝑘 where 𝜏, 𝑘 ∈ N,𝑛0 ∈ N0,𝑓,𝑔,ℎ,𝑝∈N 𝐶( 𝑛 × R , R), +Δ𝑔(𝑛,𝑥𝑛−𝑑 ,...,𝑥𝑛−𝑑 ) 0 1𝑛 𝑘𝑛 {𝑎𝑛}𝑛∈N ,and{𝛾𝑛}𝑛∈N are real sequences with 𝑎𝑛 =0̸ for 𝑛∈ 𝑛0 𝑛0 N {𝑏 ,𝑐 ,𝑑 ,𝑜 :𝑛∈N ,𝑙∈{1,2,...,𝑘}}⊂Z =ℎ(𝑛,𝑥 ,...,𝑥 ), 𝑛≥𝑛 . 𝑛0 and 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑛0 with 𝑛−𝑑1𝑛 𝑛−𝑑𝑘𝑛 0 𝑏 = 𝑐 = 𝑑 = 𝑜 =+∞, By employing a few famous tools in nonlinear analysis 𝑛→∞lim 𝑙𝑛 𝑛→∞lim 𝑙𝑛 𝑛→∞lim 𝑙𝑛 𝑛→∞lim 𝑙𝑛 including the nonlinear alternative of Leray-Schauder type, (3) Banach’s fixed point theorem, Schauder’s fixed point theorem, 𝑙∈{1,2,...,𝑘} . 2 Abstract and Applied Analysis

∞ The purpose of this paper is to fill this gap in the literature Definition 1 (see [5]). A subset 𝐷 of 𝑙𝛽 is said to be uniformly and to study solvability of (2). Under certain conditions, we Cauchy (or equi-Cauchy) if for every 𝜀>0there exists prove the existence of uncountably many bounded positive a positive integer 𝐾>𝛽such that solutions of (2) by means of the Krasnoselskii’s fixed point 󵄨 󵄨 theorem and Schauder’s fixed point theorem, respectively. 󵄨 󵄨 󵄨𝑥𝑖 −𝑥𝑗󵄨 <𝜀, (6) Nine examples are included. This paper is organized as follows. In Section 2 we ∞ whenever 𝑖, 𝑗 >𝐾 for any 𝑥={𝑥𝑛}𝑛∈Z ∈𝑙𝛽 . present some notations, definitions, and lemmas. In Section 3 𝛽 we establish nine sufficient conditions which guarantee the Lemma 2 (see [5]). Each bounded and uniformly Cauchy existence of uncountably many bounded positive solutions ∞ subset of 𝑙 is relatively compact. of (2) by using fixed point theorems and new techniques. In 𝛽 Section 4 we give nine examples to illustrate the effectiveness Lemma 3 (Krasnoselskii’s fixed point theorem). Let 𝑋 be a and applications of the results presented in Section 3. Banach space, let 𝐷 be a bounded closed convex subset of 𝑋, and let 𝑆, 𝐺 be mappings from 𝐷 into 𝑋 such that 𝑆𝑥+𝐺𝑦∈𝐷 𝑥, 𝑦 ∈𝐷 𝑆 𝐺 2. Preliminaries for every pair .If is a contraction and is completely continuous, then the equation Throughout this paper, we assume that R =(−∞,+∞), Z, N 𝑆𝑥 + 𝐺𝑥 =𝑥 and N0 stand for the sets of all integers, positive integers, and (7) nonnegative integers, respectively: has a solution in 𝐷.

N𝑛 ={𝑛:𝑛∈N with 𝑛≥𝑛0},𝑛0 ∈ N0, 0 Lemma 4 (Schauder’s fixed point theorem). Let 𝐷 be a nonempty closed convex subset of a Banach space 𝑋, 𝑇:𝐷 → 𝛼=inf {𝑏𝑙𝑛,𝑑𝑙𝑛,𝑐𝑙𝑛,𝑜𝑙𝑛 :1≤𝑙≤𝑘,𝑛∈N𝑛 }, 0 𝐷 continuous, and 𝑓(𝐷) relatively compact. Then 𝑓 has at least (4) one fixed point in 𝐷. 𝛽=min {𝑛0 −𝜏,𝛼},

Lemma 5. Let 𝜏∈N, 𝑛0 ∈ N0, {𝑎𝑛}𝑛∈N , {𝑏𝑛}𝑛∈N ,and Z𝛽 ={𝑛:𝑛∈Z with 𝑛≥𝛽}; 𝑛0 𝑛0 {𝑐𝑛}𝑛∈N be nonnegative sequences. If 𝑛0

Δ denotes the forward different operator defined by Δ𝑥𝑛 = 𝑖 𝑖−1 ∞ ∞ ∞ ∞ 𝑥𝑛+1 −𝑥𝑛 and Δ 𝑥𝑛 =Δ(Δ 𝑥𝑛) for 𝑖∈{2,3,4}.Let𝑙 denote 𝛽 ∑ ∑ ∑𝑗𝑐𝑗𝑎𝑠𝑏𝑡 <+∞, Z (8) the Banach space of all bounded sequences on 𝛽 with norm: 𝑗=𝑛0 𝑠=𝑗 𝑡=𝑠

󵄨 󵄨 ∞ then ‖𝑥‖ = sup 󵄨𝑥𝑛󵄨 for 𝑥={𝑥𝑛}𝑛∈Z ∈𝑙𝛽 , 𝑛∈Z 𝛽 𝛽 ∞ ∞ ∞ ∞ ∑ ∑ ∑ ∑𝑐𝑗𝑎𝑠𝑏𝑡 Ω1 (𝑁,) 𝑀 𝑖=1 𝑗=𝑛0+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 ∞ ∞ ∞ ∞ ={𝑥={𝑥𝑛} ∈𝑙𝛽 :𝑁≤𝑥𝑛 ≤𝑀,𝑛∈Z𝛽}; 𝑛∈Z𝛽 𝑗−𝑛0 = ∑ ∑ ∑ [ ]𝑐𝑗𝑎𝑠𝑏𝑡 (9) 𝑡=𝑠 𝜏 ∞ 𝑁 𝑀 𝑗=𝑛0+𝜏 𝑠=𝑗 Ω2𝑇 (𝑁,) 𝑀 ={𝑥={𝑥𝑛}𝑛∈Z ∈𝑙𝛽 : ≤𝑥𝑛 ≤ , 𝛽 𝛾 𝛾 𝑛 𝑛 1 ∞ ∞ ∞ ≤ ∑ ∑ ∑𝑗𝑐 𝑎 𝑏 , 𝑁 𝑀 𝑗 𝑠 𝑡 𝜏 𝑗=𝑛 +𝜏 𝑠=𝑗 𝑡=𝑠 𝑛≥𝑇; ≤𝑥𝑛 ≤ , 𝛽≤𝑛<𝑇}; 0 𝛾𝑇 𝛾𝑇 where [(𝑗−𝑛0)/𝜏] denotes the integer part of number (𝑗−𝑛0)/𝜏. ∞ −𝑁 −𝑀 Ω3𝑇 (𝑁,) 𝑀 = {𝑥={𝑥𝑛}𝑛∈Z ∈𝑙𝛽 : ≤𝑥𝑛 ≤ , 𝛽 𝛾 𝛾 𝑛 𝑛 Proof. Notice that −𝑁 −𝑀 𝑛≥𝑇; ≤𝑥 ≤ ,𝛽≤𝑛<𝑇}. ∞ ∞ ∞ ∞ 𝛾 𝑛 𝛾 𝑇 𝑇 ∑ ∑ ∑ ∑𝑐𝑗𝑎𝑠𝑏𝑡 (5) 𝑖=1 𝑗=𝑛0+𝑖𝜏 𝑠=𝑗 𝑡=𝑠

∞ ∞ ∞ ∞ ∞ ∞ Ω (𝑁, 𝑀) Ω (𝑁, 𝑀) Ω (𝑁, 𝑀) Obviously, 1 , 2𝑇 ,and 3𝑇 ,are = ∑ ∑ ∑𝑐𝑗𝑎𝑠𝑏𝑡 + ∑ ∑ ∑𝑐𝑗𝑎𝑠𝑏𝑡 𝑙∞ 𝑀>𝑁>0 closed bounded and convex subsets of 𝛽 for any . 𝑗=𝑛0+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗=𝑛0+2𝜏 𝑠=𝑗 𝑡=𝑠 {𝑥 } By a solution of (2), we mean a real sequence 𝑛 𝑛∈Z𝛽 ∞ ∞ ∞ 𝑇≥𝑛 +𝜏+|𝛽| with a positive integer 0 such that (2) is satisfied + ∑ ∑ ∑ 𝑐𝑗𝑎𝑠𝑏𝑡 +⋅⋅⋅ 𝑛≥𝑇 for all . 𝑗=𝑛0+3𝜏 𝑠=𝑗 𝑡=𝑠 Abstract and Applied Analysis 3

𝑛0+2𝜏−1 ∞ ∞ 𝑗−𝑛0 Proof. Set 𝐿 ∈ (𝑁 + 𝑐𝑀, 𝑀(1−𝑐)). It follows from (13)that = ∑ ∑ ∑ [ ]𝑐𝑗𝑎𝑠𝑏𝑡 𝜏 there exists 𝑇≥𝑛0 +𝑛1 + 𝜏 + |𝛽| sufficiently large such that 𝑗=𝑛0+𝜏 𝑠=𝑗 𝑡=𝑠 ∞ 𝑊 ∞ ∞ 1 𝑛 +3𝜏−1 𝑡 0 ∞ ∞ ∑ 󵄨 󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 𝑗−𝑛0 󵄨 󵄨 󵄨 󵄨 + ∑ ∑ ∑ [ ]𝑐𝑎 𝑏 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝜏 𝑗 𝑠 𝑡 𝑗=𝑛0+2𝜏 𝑠=𝑗 𝑡=𝑠 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) (14) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑛0+4𝜏−1 ∞ ∞ 󵄨𝑎 󵄨 𝑗−𝑛0 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 + ∑ ∑ ∑ [ ]𝑐𝑗𝑎𝑠𝑏𝑡 +⋅⋅⋅ 𝑗=𝑛 +3𝜏 𝑠=𝑗 𝑡=𝑠 𝜏 0 < min {𝑀 (1−𝑐) −𝐿,𝐿−𝑐𝑀−𝑁} .

∞ ∞ ∞ ∞ 𝑗−𝑛0 Define two mappings 𝑈𝐿 and 𝑆𝐿: Ω1(𝑁, 𝑀)𝛽 →𝑙 by = ∑ ∑ ∑ [ ]𝑐𝑗𝑎𝑠𝑏𝑡 𝑗=𝑛 +𝜏 𝑠=𝑗 𝑡=𝑠 𝜏 0 {1 𝐿−𝛾𝑛𝑥𝑛−𝜏,𝑛≥𝑇, ∞ ∞ ∞ (𝑈𝐿𝑥)𝑛 = {2 (15) 1 (𝑈 𝑥) , 𝛽≤𝑛<𝑇, ≤ ∑ ∑ ∑𝑗𝑐 𝑎 𝑏 . { 𝐿 𝑇 𝜏 𝑗 𝑠 𝑡 𝑗=𝑛0+𝜏 𝑠=𝑗 𝑡=𝑠 (𝑆𝐿𝑥) (10) 𝑛 ∞ {1 1 { 𝐿+∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) {2 𝑎 1𝑡 𝑘𝑡 { 𝑡=𝑛 𝑡 That is, (9)holds.Thiscompletestheproof. { ∞ ∞ { 1 { + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) { 𝑎 1𝑡 𝑘𝑡 { 𝑠=𝑛 𝑡=𝑠 𝑠 = −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 3. Existence of Uncountably Many Bounded { 𝑐1𝑡 𝑐𝑘𝑡 { ∞ ∞ ∞ Positive Solutions { 𝑡−𝑠+1 { − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡], { 𝑎 1𝑡 𝑘𝑡 { 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 { Now we study the existence of uncountably many bounded { 𝑛≥𝑇, positive solutions for (2) by using the Krasnoselskii’s fixed { (𝑆 𝑥) , 𝛽≤𝑛<𝑇 point theorem and Schauder’s fixed point theorem, respec- { 𝐿 𝑇 tively. (16)

Theorem 6. 𝑛 ∈ N 𝑀 𝑥={𝑥𝑛}𝑛∈Z ∈Ω1(𝑁, 𝑀) Assume that there exist constants 1 𝑛0 , , for each 𝛽 . 𝑁,and𝑐 with 𝑀>𝑁>0and 𝑐 ∈ [0, (𝑀 − 𝑁)/2𝑀) and Now we prove that nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N ,and 𝑛0 𝑛0 𝑛0 𝑈𝐿𝑥+𝑆𝐿𝑦∈Ω1 (𝑁,) 𝑀 ,∀𝑥,𝑦∈Ω1 (𝑁,) 𝑀 , {𝑅𝑛}𝑛∈N satisfying 𝑛0 󵄩 󵄩 󵄩 󵄩 󵄩𝑈𝐿𝑥−𝑈𝐿𝑦󵄩 ≤𝑐󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦∈Ω1 (𝑁,) 𝑀 , (17) 󵄨 󵄨 󵄩 󵄩 󵄨𝛾𝑛󵄨 ≤𝑐, ∀𝑛≥𝑛1, (11) 󵄩𝑆𝐿𝑥󵄩 <𝑀, ∀𝑥,𝑦∈Ω1 (𝑁,) 𝑀 . 󵄨 󵄨 󵄨 󵄨 󵄨𝑓(𝑛,𝑢 ,𝑢 ,...,𝑢 )󵄨 ≤𝑊, 󵄨𝑔(𝑛,𝑢 ,𝑢 ,...,𝑢 )󵄨 ≤𝑃, In view of (11), (12), and (14)–(16), we conclude that for any 󵄨 1 2 𝑘 󵄨 𝑛 󵄨 1 2 𝑘 󵄨 𝑛 𝑥={𝑥} 𝑦={𝑦} ∈Ω(𝑁, 𝑀) 𝑛≥𝑇 𝑛 𝑛∈Z𝛽 , 𝑛 𝑛∈Z𝛽 1 ,and , 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨ℎ(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑄𝑛, 󵄨𝑝(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑅𝑛, 󵄨 󵄨 󵄨(𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 −𝐿󵄨

𝑘 󵄨 ∞ ∀(𝑛,𝑢 ,𝑢 ,...,𝑢 )∈N × [𝑁,] 𝑀 , 󵄨 1 1 2 𝑘 𝑛0 = 󵄨−𝛾 𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) 󵄨 𝑛 𝑛−𝜏 𝑏1𝑡 𝑏𝑘𝑡 (12) 󵄨 𝑡=𝑛 𝑎𝑡 ∞ ∞ ∞ ∞ ∞ 1 𝑊𝑡 1 +∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) { ∑ , ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃], 𝑑1𝑡 𝑑𝑘𝑡 max 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 󵄨𝑎 󵄨 𝑠=𝑛 󵄨𝑎 󵄨 𝑡=𝑛0 󵄨 𝑡󵄨 0 𝑡=𝑠 󵄨 𝑠󵄨 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] (13) 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 } ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) <+∞. ∞ ∞ ∞ 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 } 𝑡−𝑠+1 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 󵄨𝑎 󵄨 − ∑ ∑ ∑ 0 󵄨 𝑗󵄨 } 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 󵄨 󵄨 󵄨 Then (2) possesses uncountably many bounded positive solu- ×[𝑝(𝑡,𝑦𝑜 ,...,𝑦𝑜 )−𝑟𝑡] 󵄨 1𝑡 k𝑡 󵄨 tions in Ω1(𝑁, 𝑀). 󵄨 4 Abstract and Applied Analysis

∞ ∞ ∞ ∞ ∞ ∞ 𝑊𝑡 1 𝑡−𝑠+1 󵄨 󵄨 ≤𝑐𝑀+∑ 󵄨 󵄨 + ∑∑󵄨 󵄨 + ∑∑∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨𝑎 󵄨 󵄨𝑎 󵄨 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑡=𝑇 󵄨 𝑡󵄨 𝑠=𝑇𝑡=𝑠 󵄨 𝑠󵄨 𝑗=𝑇𝑠=𝑗𝑡=𝑠 󵄨𝑎𝑗󵄨

×[(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 1 < 𝐿+min {𝑀 (1−𝑐) −𝐿,𝐿−𝑐𝑀−𝑁} ∞ ∞ ∞ 2 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) <𝑀, 󵄨 󵄨 󵄨 󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 (18) <𝑐𝑀+ {𝑀 (1−𝑐) −𝐿,𝐿−𝑐𝑀−𝑁} , min which yield that (17)hold. 𝑆 󵄩 󵄩 󵄨 󵄨 In order to prove that 𝐿 is completely continuous 󵄩𝑈𝐿𝑥−𝑈𝐿𝑦󵄩 = 󵄨(𝑈𝐿𝑥) −(𝑈𝐿𝑦) 󵄨 󵄩 󵄩 sup 󵄨 𝑛 𝑛󵄨 in Ω1(𝑁, 𝑀),wehavetoshowthat𝑆𝐿 is continuous in 𝑛∈Z𝛽 Ω1(𝑁, 𝑀) and 𝑆𝐿(Ω1(𝑁, 𝑀)) is relatively compact. Suppose 𝑚 that {𝑥 }𝑚∈N is an arbitrary sequence in Ω1(𝑁, 𝑀) and 𝑥∈ 󵄨 󵄨 𝑚 𝑚 𝑚 󵄨 󵄨 Ω1(𝑁, 𝑀) with lim𝑚→∞𝑥 =𝑥,where𝑥 ={𝑥𝑛 }𝑛∈Z for = max { sup 󵄨(𝑈𝐿𝑥)𝑛 −(𝑈𝐿𝑦)𝑛󵄨 , 𝛽 𝑇>𝑛≥𝛽 𝑚∈N 𝑥={𝑥} each and 𝑛 𝑛∈Z𝛽 . With the help of (12), (13), 𝑚 lim𝑚→∞𝑥 =𝑥,andthecontinuityof𝑓, 𝑔, ℎ,and𝑝,weknow 󵄨 󵄨 that for given 𝜀>0, there exist 𝑇1,𝑇2,𝑇3,and𝑇4 ∈ N with sup 󵄨(𝑈𝐿𝑥)𝑛 −(𝑈𝐿𝑦)𝑛󵄨} 𝑛≥𝑇 𝑇4 >𝑇3 >𝑇2 >𝑇1 >𝑇satisfying 󵄨 󵄨 󵄨 󵄨 ∞ ∞ ∞ = sup (󵄨𝛾𝑛󵄨 󵄨𝑥𝑛−𝜏 −𝑦𝑛−𝜏󵄨) 𝑊 1 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑛≥𝑇 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 󵄩 󵄩 1 1 ≤𝑐󵄩𝑥−𝑦󵄩 , 󵄩 󵄩 ∞ ∞ ∞ 󵄩 󵄩 󵄨 󵄨 𝑡−𝑠+1 𝜀 󵄩𝑆 𝑥󵄩 = 󵄨(𝑆 𝑥) 󵄨 + ∑ ∑ ∑ 󵄨 󵄨 𝑅𝑡 < ; 󵄩 𝐿 󵄩 sup 󵄨 𝐿 𝑛󵄨 󵄨 󵄨 18 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎 󵄨 𝑛∈Z𝛽 1 󵄨 𝑗󵄨

󵄨 󵄨 󵄨 󵄨 { ∞ } = { 󵄨(𝑆 𝑥) 󵄨 , 󵄨(𝑆 𝑥) 󵄨} 𝑡−𝑠+1 max sup 󵄨 𝐿 𝑛󵄨 sup 󵄨 𝐿 𝑛󵄨 ∑ 󵄨 󵄨 𝑅𝑡 :𝑇≤𝑗≤𝑇1,𝑗≤𝑠≤𝑇2 𝑇>𝑛≥𝛽 𝑛≥𝑇 max { 󵄨 󵄨 } 𝑡=𝑇 󵄨𝑎 󵄨 { 3 󵄨 𝑗󵄨 } 󵄨 󵄨1 ∞ 1 𝜀 = 󵄨 𝐿+∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) < ; sup 󵄨 𝑏1𝑡 b𝑘𝑡 𝑛≥𝑇 󵄨2 𝑡=𝑛 𝑎𝑡 18𝑇1𝑇2

∞ ∞ 1 { ∞ ∞ 𝑡−𝑠+1 } 𝜀 + ∑ ∑ [ (𝑡−𝑠+1) ∑ ∑ 𝑅 :𝑇≤𝑗≤𝑇 < ; 𝑎 max { 󵄨 󵄨 𝑡 1} 18𝑇 𝑠=𝑛 𝑡=𝑠 𝑠 𝑠=𝑇 𝑡=𝑠 󵄨𝑎 󵄨 1 { 2 󵄨 𝑗󵄨 } ∞ ×ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) { 1 } 1𝑡 𝑘𝑡 ∑ [(𝑡−𝑠+1) 𝑄 +𝑃]:𝑇≤𝑠≤𝑇 max { 󵄨 󵄨 𝑡 𝑡 1} 󵄨𝑎𝑠󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑡=𝑇2 󵄨 󵄨 𝑐1𝑡 𝑐𝑘𝑡 { } 𝜀 ∞ ∞ ∞ 𝑡−𝑠+1 < ; − ∑ ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 ) 𝑜1𝑡 𝑜𝑘𝑡 18𝑇1 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 󵄨 󵄨 𝑚 𝑚 󵄨 max {󵄨𝑞(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨 󵄨 1𝑡 𝑘𝑡 −𝑟 ] 󵄨 𝑡 󵄨 󵄨 󵄨 −𝑞 (𝑡, 𝑥 ,...,𝑥 )󵄨 :𝑞∈{𝑓,𝑔,ℎ,𝑝}} 𝑏1𝑡 𝑏𝑘𝑡 󵄨 1 ∞ 𝑊 ∞ ∞ 1 𝜀 ≤ ( 𝐿+∑ 𝑡 + ∑∑ [(𝑡−𝑠+1) 𝑄 < ,∀𝑚≥𝑇, 𝑇≤𝑡≤𝑇, sup 󵄨 󵄨 󵄨 󵄨 𝑡 4 3 𝑛≥𝑇 2 𝑡=𝑛 󵄨𝑎𝑡󵄨 𝑠=𝑛𝑡=𝑠 󵄨𝑎𝑠󵄨 18𝑇1𝑇2𝑇3 (𝐴+𝐵+𝐸) (19) +𝑃𝑡] where ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 +∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨)) 1 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝐴= { :𝑇≤𝑠≤𝑇}, 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 max 󵄨 󵄨 1 󵄨 󵄨 󵄨𝑎𝑠󵄨 ∞ ∞ ∞ 1 𝑊𝑡 1 𝑡−𝑠+1 ≤ 𝐿+∑ 󵄨 󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 𝐵= { :𝑇≤𝑠≤𝑇,𝑠≤𝑡≤𝑇}, 2 󵄨𝑎 󵄨 󵄨𝑎 󵄨 max 󵄨 󵄨 1 2 𝑡=𝑇 󵄨 𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨 𝑠󵄨 󵄨𝑎𝑠󵄨 Abstract and Applied Analysis 5

𝑡−𝑠+1 ∞ ∞ ∞ 𝑡−𝑠+1󵄨 + ∑ ∑ ∑ 󵄨𝑝(𝑡,𝑥𝑚 ,...,𝑥𝑚 ) 𝐸=max { 󵄨 󵄨 :𝑇≤𝑗≤𝑇1,𝑗≤𝑠≤𝑇2,𝑠≤𝑡≤𝑇3}. 󵄨 󵄨 󵄨 𝑜 𝑜 󵄨 󵄨 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 󵄨𝑎𝑗󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 (20) 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨

𝑇 By virtue of (16), (19), and (20), we get that 1 1 󵄨 󵄨 󵄨 𝑚 𝑚 󵄨 = ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )−𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )󵄨 󵄨 󵄨 󵄨 1t 𝑘𝑡 1𝑡 𝑘𝑡 󵄨 𝑡=𝑇 󵄨𝑎𝑡󵄨 󵄩 󵄩 󵄩𝑆 𝑥𝑚 −𝑆 𝑥󵄩 󵄩 𝐿 𝐿 󵄩 ∞ 1 󵄨 𝑚 𝑚 󵄨 𝑚 󵄨 + ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 ) = 󵄨(𝑆 𝑥 ) −(𝑆 𝑥) 󵄨 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 sup 󵄨 𝐿 𝑛 𝐿 𝑛󵄨 󵄨𝑎𝑡󵄨 𝑡=𝑇1+1 󵄨 󵄨 𝑛∈𝑍𝛽 󵄨 −𝑓 (𝑡,𝑏 𝑥 ,...,𝑥𝑏 )󵄨 󵄨 𝑚 󵄨 1𝑡 𝑘𝑡 󵄨 = max { sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨 , 𝑇>𝑛≥𝛽 𝑇1 𝑇2 1 󵄨 𝑚 𝑚 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 𝑚 󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨} 𝑛≥𝑇 󵄨 −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 󵄨 ∞ 1𝑡 𝑘𝑡 󵄨 󵄨 1 𝑚 𝑚 = 󵄨∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) 󵄨 sup 󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨 𝑚 𝑚 𝑛≥𝑇 󵄨 𝑎𝑡 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 󵄨𝑡=𝑛 󵄨 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ 󵄨 1 𝑚 𝑚 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑐1𝑡 𝑐𝑘𝑡 󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 𝑇1 ∞ 1 󵄨 𝑚 𝑚 𝑚 𝑚 + ∑ ∑ 󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑐 𝑐 󵄨𝑎𝑠󵄨 1𝑡 𝑘𝑡 𝑠=𝑇 𝑡=𝑇2+1 󵄨 󵄨 ∞ ∞ ∞ 󵄨 𝑡−𝑠+1 𝑚 𝑚 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 − ∑ ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟] 𝑑1𝑡 𝑑𝑘𝑡 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑎𝑗 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 󵄨 𝑚 𝑚 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 ∞ 1 󵄨 −(∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 1𝑡 𝑘𝑡 𝑐1𝑡 𝑐𝑘𝑡 󵄨 𝑡=𝑛 𝑎𝑡 ∞ ∞ 1 󵄨 𝑚 𝑚 ∞ ∞ 1 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨𝑎𝑠󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑠=𝑇1+1 𝑡=s 󵄨 󵄨 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 󵄨 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 −𝑔(𝑡,𝑥𝑐 ,...,𝑥𝑐 )] 1𝑡 𝑘𝑡 󵄨 󵄨 𝑚 𝑚 + 󵄨𝑔(𝑡,𝑥𝑐 ,...,𝑥𝑐 ) ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 1𝑡 𝑘𝑡 − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 ) 󵄨 𝑎 1𝑡 𝑘𝑡 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 𝑐1𝑡 𝑐𝑘𝑡 󵄨

𝑇 𝑇 𝑇 󵄨 1 2 3 𝑡−𝑠+1󵄨 󵄨 󵄨 𝑚 𝑚 −𝑟 ])󵄨 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 ) 𝑡 󵄨 󵄨 󵄨 󵄨 1𝑡 𝑘𝑡 󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 󵄨 ∞ 󵄨 1 󵄨 𝑚 𝑚 󵄨 −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 ≤ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 )−𝑓(𝑡,𝑥 ,...,𝑥 )󵄨 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑏1𝑡 𝑏𝑘𝑡 󵄨 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑇1 𝑇2 ∞ 𝑡−𝑠+1󵄨 𝑚 𝑚 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) ∞ ∞ 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 1 󵄨 𝑚 𝑚 󵄨𝑎 󵄨 󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑇3+1 󵄨 𝑗󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 𝑠=𝑇 𝑡=𝑠 󵄨 𝑠󵄨 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 󵄨 −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 1𝑡 𝑘𝑡 󵄨 𝑇1 ∞ ∞ 𝑡−𝑠+1󵄨 𝑚 𝑚 󵄨 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑚 𝑚 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 𝑡=𝑠 󵄨𝑎 󵄨 󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑗=𝑇 𝑠=𝑇2+1 󵄨 𝑗󵄨 󵄨 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 𝑐1𝑡 𝑐𝑘𝑡 󵄨 1𝑡 𝑘𝑡 󵄨 6 Abstract and Applied Analysis

∞ ∞ ∞ 𝑡−𝑠+1󵄨 𝑚 𝑚 Next we prove that 𝑆𝐿(Ω1(𝑁, 𝑀)) is uniformly Cauchy. + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝜀>0 𝑉>𝑇 𝑡=𝑠 󵄨𝑎 󵄨 It follows from (13)thatforgiven , there exists 𝑗=𝑇1+1 𝑠=𝑗 󵄨 𝑗󵄨 satisfying 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 ∞ ∞ ∞ 𝑇 𝑊𝑡 1 1 ∑ + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 𝑚 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 ≤ ∑𝐴 󵄨𝑓(𝑡,𝑥 ,...,𝑥 )−𝑓(𝑡,𝑥 ,...,𝑥 )󵄨 󵄨𝑎𝑡󵄨 𝑡=𝑠 󵄨𝑎𝑠󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑏1𝑡 𝑏𝑘𝑡 󵄨 𝑡=𝑉 󵄨 󵄨 𝑠=𝑉 󵄨 󵄨 𝑡=𝑇 (22) ∞ ∞ ∞ 𝑡−𝑠+1 𝜀 𝑇 𝑇 󵄨 󵄨 1 2 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨)< . 󵄨 𝑚 𝑚 󵄨 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 4 + ∑ ∑ [𝐵 󵄨ℎ(𝑡,𝑥 ,...,𝑥 )−ℎ(𝑡,𝑥 ,...,𝑥 )󵄨 𝑗=𝑉 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑑1𝑡 𝑑𝑘𝑡 󵄨 󵄨 󵄨 𝑠=𝑇 𝑡=𝑠 󵄨 󵄨 +𝐴 󵄨𝑔(𝑡,𝑥𝑚 ,...,𝑥𝑚 )−𝑔(𝑡,𝑥 ,...,𝑥 )󵄨] 󵄨 𝑐 𝑐 𝑐1𝑡 𝑐𝑘𝑡 󵄨 𝑥={𝑥} ∈ 󵄨 1𝑡 𝑘𝑡 󵄨 Using (16)and(22), we know that for any 𝑛 𝑛∈Z𝛽 Ω1(𝑁, 𝑀) and 𝑚, 𝑛 ≥𝑉, 𝑇1 𝑇2 𝑇3 󵄨 𝑚 𝑚 󵄨 + ∑ ∑ ∑𝐸 󵄨𝑝(𝑡,𝑥 ,...,𝑥 )−𝑝(𝑡,𝑥 ,...,𝑥 )󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨 󵄨 󵄨(𝑆𝐿𝑥)𝑚 −(𝑆𝐿𝑥)𝑛󵄨

∞ 𝑇1 ∞ 𝑊𝑡 1 󵄨 ∞ ∞ +2 ∑ 󵄨 󵄨 +2∑ ∑ 󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 1 1 󵄨𝑎 󵄨 󵄨𝑎 󵄨 𝑡 𝑡 = 󵄨∑ 𝑓(𝑡,𝑥 ,...,𝑥 )−∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) 𝑡=𝑇 +1 󵄨 𝑡󵄨 𝑠=𝑇 𝑡=𝑇 +1 󵄨 𝑠󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑏1𝑡 𝑏𝑘𝑡 1 2 󵄨𝑡=𝑚 𝑎𝑡 𝑡=𝑛 𝑎𝑡 ∞ ∞ 1 ∞ ∞ 1 +2 ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 󵄨 󵄨 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 𝑠=𝑇 +1 𝑡=𝑠 󵄨𝑎𝑠󵄨 1𝑡 𝑘𝑡 1 𝑠=𝑚 𝑡=𝑠 𝑎𝑠

𝑇 𝑇 1 2 ∞ 𝑡−𝑠+1 −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] +2∑ ∑ ∑ 󵄨 󵄨 𝑅 1𝑡 𝑘𝑡 󵄨𝑎 󵄨 𝑡 𝑗=𝑇 𝑠=𝑗 𝑡=𝑇3+1 󵄨 𝑗󵄨 ∞ ∞ 1 − ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑇 ∞ ∞ 𝑑1𝑡 𝑑𝑘𝑡 1 𝑡−𝑠+1 𝑠=𝑛 𝑎𝑠 +2∑ ∑ ∑ 𝑅 𝑡=𝑠 󵄨 󵄨 𝑡 𝑗=𝑇 𝑠=𝑇 +1 𝑡=𝑠 󵄨𝑎 󵄨 2 󵄨 𝑗󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ ∞ 𝑡−𝑠+1 +2 ∑ ∑ ∑ 𝑅 ∞ ∞ ∞ 󵄨 󵄨 𝑡 𝑡−𝑠+1 𝑡=𝑠 󵄨𝑎 󵄨 + ∑ ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟] 𝑗=𝑇1+1 𝑠=𝑗 󵄨 𝑗󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗

𝜀𝐴1 (𝑇 −𝑇+1) < ∞ ∞ ∞ 󵄨 18𝑇 𝑇 𝑇 𝐴+𝐵+𝐸 𝑡−𝑠+1 󵄨 1 2 3 ( ) − ∑ ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟]󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡 󵄨 𝑗=𝑚 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 󵄨 𝜀 (𝐵+𝐴) (𝑇 −𝑇+1)(𝑇 −𝑇+1) 󵄨 + 1 2 18𝑇 𝑇 𝑇 (𝐴+𝐵+𝐸) ∞ ∞ 1 2 3 𝑊𝑡 𝑊𝑡 ≤ ∑ 󵄨 󵄨 + ∑ 󵄨 󵄨 𝜀𝐸 (𝑇 −𝑇+1)(𝑇 −𝑇+1)(𝑇 −𝑇+1) 𝑡=𝑚 󵄨𝑎𝑡󵄨 𝑡=𝑛 󵄨𝑎𝑡󵄨 + 1 2 3 18𝑇 𝑇 𝑇 (𝐴+𝐵+𝐸) 1 2 3 ∞ ∞ 1 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 2𝜀 (𝑇 −𝑇+1) 󵄨𝑎 󵄨 𝑡 𝑡 + 1 𝑠=𝑚 𝑡=𝑠 󵄨 𝑠󵄨 18𝑇1 ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 2𝜀 1(𝑇 −𝑇+1)(𝑇2 −𝑇+1) 󵄨 󵄨 𝑡 𝑡 + 𝑠=𝑛 𝑡=𝑠 󵄨𝑎𝑠󵄨 18𝑇1𝑇2 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 2𝜀 (𝑇 −𝑇+1) 𝜀 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) + 1 + <𝜀, 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 18𝑇1 18 󵄨 󵄨

∀𝑚 ≥ 𝑇 , ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 4 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 (21) 𝑗=𝑚 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 which means that 𝑆𝐿 is continuous in Ω1(𝑁, 𝑀). (23) Abstract and Applied Analysis 7

∞ 𝑊 ∞ ∞ 1 ≤2∑ 𝑡 +2∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] which means that 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨𝑎𝑡󵄨 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝑡=𝑉 𝑠=𝑉 Δ3 (𝑎 Δ(𝑥 +𝛾𝑥 )) + Δ3𝑓(𝑛,𝑥 ,...,𝑥 ) 𝑛 𝑛 𝑛 𝑛−𝜏 𝑏1𝑛 𝑏𝑘𝑛 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 (24) 2 +2∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) +Δ 𝑔(𝑛,𝑥 ,...,𝑥 )+Δℎ(𝑛,𝑥 ,...,𝑥 ) 󵄨 󵄨 𝑐1𝑛 𝑐𝑘𝑛 𝑑1𝑛 𝑑𝑘𝑛 (27) 𝑗=𝑉 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 +𝑝(𝑛,𝑥 ,...,𝑥 )=𝑟,∀𝑛≥𝑇; <𝜀, 𝑜1𝑛 𝑜𝑘𝑛 𝑡 𝑆 (Ω (𝑁, 𝑀)) which yields that 𝐿 1 is uniformly Cauchy. It fol- that is, 𝑥={𝑥𝑛}𝑛∈Z is a bounded positive solution of (2)in 𝑆 (Ω (𝑁, 𝑀)) 𝛽 lows from Lemma 2 that 𝐿 1 is relatively compact. Ω1(𝑁, 𝑀). 𝑥= Consequently Lemma 3 guarantees that there exists Finally we prove that (2) has uncountably many bounded {𝑥 } ∈Ω(𝑁, 𝑀) 𝑈 𝑥+𝑆𝑥=𝑥 𝑛 𝑛∈Z𝛽 1 satisfying 𝐿 𝐿 ,which positive solutions in Ω1(𝑁, 𝑀).Let𝐿1,𝐿2 ∈ (𝑐𝑀 + 𝑁, 𝑀(1 − together with (15)and(16)givesthat 𝑐)) and 𝐿1 =𝐿̸ 2.Foreach𝑗∈{1,2},weconcludesimilarlythat ∞ there exist a positive integer 𝑇𝑗 ≥𝑛0 +𝑛1 +𝜏+|𝛽|and two 1 𝑈 𝑆 𝐿 𝑇 𝑥𝑛 =𝐿−𝛾𝑛𝑥𝑛−𝜏 + ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) mappings 𝐿𝑗 and 𝐿𝑗 satisfying (14)–(16), where and are 𝑎 1𝑡 𝑘𝑡 𝑡=𝑛 𝑡 𝐿 𝑇 𝑈 +𝑆 replaced by 𝑗 and 𝑗,respectively,and 𝐿𝑗 𝐿𝑗 has a fixed point 𝑧𝑗 ={𝑧𝑗𝑛}𝑛∈Z , which is a bounded positive solution (2) ∞ ∞ 1 𝛽 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) Ω (𝑁, 𝑀) 𝑑1𝑡 𝑑𝑘𝑡 in 1 ;thatis, 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 ∞ 1 (25) 𝑧 =𝐿 −𝛾𝑧 + ∑ 𝑓(𝑡,𝑧 ,...,𝑧 ) −𝑔(𝑡,𝑥𝑐 ,...,𝑥𝑐 )] 𝑗𝑛 𝑗 𝑛 𝑗𝑛−𝜏 𝑗𝑏1𝑡 𝑗𝑏𝑘𝑡 1𝑡 𝑘𝑡 𝑡=𝑛 𝑎𝑡 ∞ ∞ ∞ 𝑡−𝑠+1 ∞ ∞ 1 − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡], 1𝑡 𝑘𝑡 + ∑ ∑ [(𝑡−𝑠+1) ℎ(t,𝑧𝑗𝑑 ,...,𝑧𝑗𝑑 ) 𝑡=𝑠 𝑎𝑗 1𝑡 𝑘𝑡 𝑗=𝑛 𝑠=𝑗 𝑠=𝑛 𝑡=𝑠 𝑎𝑠

∀𝑛 ≥ 𝑇, −𝑔 (𝑡, 𝑧 ,...,𝑧 )] (28) 𝑗𝑐1𝑡 𝑗𝑐𝑘𝑡 which yields that ∞ ∞ ∞ 𝑡−𝑠+1 1 − ∑ ∑ ∑ [𝑝 (𝑡,𝑗𝑜 𝑧 ,...,𝑧𝑗𝑜 )−𝑟𝑡], Δ(𝑥 +𝛾𝑥 )+ 𝑓(𝑛,𝑥 ,...,𝑥 ) 𝑎 1𝑡 𝑘𝑡 𝑛 𝑛 𝑛−𝜏 𝑏1𝑛 𝑏𝑘𝑛 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 𝑎𝑛

∞ 1 ∀𝑛 ≥𝑗 𝑇 ,𝑗∈{1, 2} . =−∑ [(𝑡−𝑛+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑑1𝑡 𝑑𝑘𝑡 𝑡=𝑛 𝑎𝑛 Equation (13) ensures that there exists some 𝑇3 > max{𝑇1,𝑇2} satisfying −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡 ∞ 𝑊 ∞ ∞ 1 ∞ ∞ ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑡−𝑠+1 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 + ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟], 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡 3 3 𝑠=𝑛 𝑡=𝑠 𝑎𝑛 󵄨 󵄨 (29) ∞ ∞ ∞ 𝑡−𝑠+1 󵄨𝐿 −𝐿 󵄨 ∀𝑛 ≥ 𝑇, + ∑ ∑ ∑ 𝑅 < 󵄨 1 2󵄨. 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 4 𝑗=𝑇3 𝑠=𝑗 󵄨 𝑗󵄨 𝑎 Δ(𝑥 +𝛾𝑥 )+𝑓(𝑛,𝑥 ,...,𝑥 ) 𝑛 𝑛 𝑛 𝑛−𝜏 𝑏1𝑛 𝑏𝑘𝑛 Combining (11), (12), (28), and (29), we deduce that for any ∞ 𝑛≥𝑇3, =−∑ (𝑡−𝑛+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑑1𝑡 𝑑𝑘𝑡 𝑡=𝑛 󵄨 󵄨 󵄨𝑧1𝑛 −𝑧2𝑛󵄨 ∞ 󵄨 󵄨 + ∑𝑔(𝑡,𝑥𝑐 ,...,𝑥𝑐 ) 󵄨 1𝑡 𝑘𝑡 = 󵄨𝐿1 −𝐿2 −𝛾𝑛𝑧1𝑛−𝜏 +𝛾𝑛𝑧2𝑛−𝜏 𝑡=𝑛 󵄨 󵄨 ∞ ∞ ∞ + ∑ ∑ (𝑡−𝑠+1) [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟], 1 𝑜1𝑡 𝑜𝑘𝑡 𝑡 + ∑ 𝑓(𝑡,𝑧1𝑏 ,...,𝑧1𝑏 ) 𝑠=𝑛 1𝑡 𝑘𝑡 𝑡=𝑠 𝑡=𝑛 𝑎𝑡 ∀𝑛 ≥ 𝑇, ∞ 1 − ∑ 𝑓(𝑡,𝑧 ,...,𝑧 ) 2𝑏1𝑡 2𝑏𝑘𝑡 (26) 𝑡=𝑛 𝑎𝑡 8 Abstract and Applied Analysis

∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑧 ,...,𝑧 ) which implies that 1𝑑1𝑡 1𝑑𝑘𝑡 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 󵄨 󵄨 −𝑔 (𝑡,1𝑐 𝑧 ,...,𝑧1𝑐 )] 󵄩 󵄩 󵄨𝐿1 −𝐿2󵄨 1𝑡 𝑘𝑡 󵄩𝑧 −𝑧 󵄩 > >0. (31) 󵄩 1 2󵄩 2 (1+𝑐) ∞ ∞ 1 − ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑧 ,...,𝑧 ) 2𝑑1𝑡 2𝑑𝑘𝑡 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 That is, 𝑧1 =𝑧̸ 2.Thus(2) has uncountably many bounded −𝑔 (𝑡, 𝑧 ,...,𝑧 )] positive solutions in Ω1(𝑁, 𝑀).Thiscompletestheproof. 2𝑐1𝑡 2𝑐𝑘𝑡

∞ ∞ ∞ Theorem 7. Assume that there exist constants 𝑛1 ∈ N𝑛 , 𝑀, 𝑡−𝑠+1 𝑁 𝑐 𝑀>𝑁>0 𝑐 ∈ [0, (𝑀 − 0𝑁)/𝑀) − ∑ ∑ ∑ [𝑝 (𝑡,1𝑜 𝑧 ,...,𝑧1𝑜 )−𝑟𝑡] ,and with and and 𝑎 1𝑡 𝑘𝑡 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N ,and 𝑛0 𝑛0 𝑛0 󵄨 {𝑅𝑛}𝑛∈N satisfying (12), (13),and ∞ ∞ ∞ 󵄨 𝑛0 𝑡−𝑠+1 󵄨 +∑ ∑ ∑ [𝑝 (𝑡,2𝑜 𝑧 ,...,𝑧2𝑜 )−𝑟𝑡]󵄨 𝑎 1𝑡 𝑘𝑡 󵄨 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 󵄨 0≤𝛾𝑛 ≤𝑐, ∀𝑛≥𝑛1. (32) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≥ 󵄨𝐿1 −𝐿2󵄨 − 󵄨𝛾𝑛󵄨 󵄨𝑧1𝑛−𝜏 −𝑧2𝑛−𝜏󵄨

∞ 1 󵄨 󵄨 − ∑ 󵄨𝑓(𝑡,𝑧 ,...,𝑧 )−𝑓(𝑡,𝑧 ,...,𝑧 )󵄨 Then (2) possesses uncountably many bounded positive solu- 󵄨 1𝑏1𝑡 1𝑏𝑘𝑡 2𝑏1𝑡 2𝑏𝑘𝑡 󵄨 𝑡=𝑛 𝑎𝑡 tions in Ω1(𝑁, 𝑀).

∞ ∞ 1 󵄨 Proof. Let 𝐿 ∈ (𝑁 + 𝑐𝑀, 𝑀). It follows from (13)thatthere − ∑ ∑ [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑧1𝑑 ,...,𝑧1𝑑 ) 󵄨 1𝑡 𝑘𝑡 exists 𝑇≥𝑛0 +𝑛1 +𝜏+|𝛽|sufficiently large such that 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 󵄨 −ℎ (𝑡, 𝑧 ,...,𝑧 )󵄨 2𝑑1𝑡 2𝑑𝑘𝑡 󵄨 ∞ ∞ ∞ 𝑊𝑡 1 󵄨 ∑ 󵄨 󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] + 󵄨𝑔(𝑡,𝑧 ,...,𝑧 ) 󵄨𝑎 󵄨 󵄨𝑎 󵄨 𝑡 𝑡 󵄨 1𝑐1𝑡 1𝑐𝑘𝑡 𝑡=𝑇 󵄨 𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨 𝑠󵄨 󵄨 −𝑔 (𝑡,2𝑐 𝑧 ,...,𝑧2𝑐 )󵄨] ∞ ∞ ∞ 1𝑡 𝑘𝑡 󵄨 𝑡−𝑠+1 󵄨 󵄨 (33) + ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) ∞ ∞ ∞ 󵄨 󵄨 󵄨 󵄨 𝑡−𝑠+1󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 − ∑ ∑ ∑ 󵄨𝑝(𝑡,𝑧 ,...,𝑧 ) 󵄨 󵄨 󵄨 1𝑜1𝑡 1𝑜𝑘𝑡 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 < min {𝑀−𝐿, 𝐿−𝑐𝑀−𝑁} . 󵄨 −𝑝 (𝑡, 𝑧 ,...,𝑧 )󵄨 2𝑜1𝑡 2𝑜𝑘𝑡 󵄨 󵄨 󵄨 󵄩 󵄩 ≥ 󵄨𝐿 −𝐿 󵄨 −𝑐󵄩𝑧 −𝑧 󵄩 ∞ 󵄨 1 2󵄨 󵄩 1 2󵄩 Let the mappings 𝑈𝐿 and 𝑆𝐿 :Ω1(𝑁,𝑀)→𝑙𝛽 be defined ∞ ∞ ∞ by (15)and(16), respectively. By means of (12), (15), (16), (32), 𝑊𝑡 1 −2(∑ + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] and (33),weinferthatforany𝑥={𝑥𝑛}𝑛∈Z , 𝑦={𝑦𝑛}𝑛∈Z ∈ 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝛽 𝛽 𝑡=𝑛 󵄨𝑎𝑡󵄨 𝑠=𝑛 𝑡=𝑠 󵄨𝑎𝑠󵄨 Ω1(𝑁, 𝑀),and𝑛≥𝑇, ∞ ∞ ∞ 𝑡−𝑠+1 +∑ ∑ ∑ 𝑅 ) 󵄨 󵄨 𝑡 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 󵄨𝑎 󵄨 󵄨 𝑗󵄨 (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 󵄨 󵄨 󵄩 󵄩 ≥ 󵄨𝐿 −𝐿 󵄨 −𝑐󵄩𝑧 −𝑧 󵄩 󵄨 1 2󵄨 󵄩 1 2󵄩 ∞ 1 =𝐿−𝛾𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) ∞ ∞ ∞ 𝑛 𝑛−𝜏 𝑏1𝑡 𝑏𝑘𝑡 𝑊 1 𝑡=𝑛 𝑎𝑡 −2(∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨𝑎𝑡󵄨 󵄨𝑎𝑠󵄨 ∞ ∞ 𝑡=𝑇3 󵄨 󵄨 𝑠=𝑇3 𝑡=𝑠 󵄨 󵄨 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦𝑑 ,...,𝑦𝑑 ) 𝑎 1𝑡 𝑘𝑡 ∞ ∞ ∞ 𝑡−𝑠+1 𝑠=𝑛 𝑡=𝑠 𝑠 + ∑ ∑ ∑ 𝑅 ) 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 𝑗=𝑇3𝑠=𝑗 󵄨 𝑗󵄨 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 󵄨 󵄨𝐿 −𝐿 󵄨 󵄩 󵄩 > 󵄨 1 2󵄨 −𝑐󵄩𝑧 −𝑧 󵄩 , ∞ ∞ ∞ 2 󵄩 1 2󵄩 𝑡−𝑠+1 − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 )−𝑟𝑡] 𝑎 1𝑡 𝑘𝑡 (30) 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 Abstract and Applied Analysis 9

∞ 𝑊 ∞ ∞ 1 𝐿∈(𝑁,(1−𝑐)𝑀) ≤𝐿+∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] Proof. Let . It follows from (13)thatthere 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑇≥𝑛 +𝑛 +𝜏+|𝛽| 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 exists 0 1 sufficiently large such that

∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) 󵄨 󵄨 ∞ ∞ ∞ 𝑗=𝑇𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 𝑊 1 ∑ 𝑡 + ∑∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇𝑡=𝑠 󵄨𝑎𝑠󵄨 <𝐿+min {𝑀−𝐿,𝐿−𝑐𝑀−𝑁} ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 (36) ≤𝑀, + ∑ ∑∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗=𝑇 𝑠=𝑗𝑡=𝑠 󵄨𝑎𝑗󵄨

(𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 < min {(1−𝑐) 𝑀−𝐿,𝐿−𝑁} . ∞ 1 =𝐿−𝛾𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) 𝑛 𝑛−𝜏 𝑏1𝑡 𝑏𝑘𝑡 𝑡=𝑛 𝑎𝑡 𝑈 𝑆 :Ω(𝑁, 𝑀) →𝑙∞ ∞ ∞ 1 Let the mappings 𝐿 and 𝐿 1 𝛽 be defined by + ∑∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦𝑑 ,...,𝑦𝑑 ) 𝑎 1𝑡 𝑘𝑡 (15)and(16), respectively. Making use of (12), (15), (16), (35), 𝑠=𝑛𝑡=𝑠 𝑠 𝑥={𝑥} 𝑦={𝑦} ∈ and (36), we derive that for any 𝑛 𝑛∈Z𝛽 , 𝑛 𝑛∈Z𝛽 Ω1(𝑁, 𝑀) and 𝑛≥𝑇, −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑐1𝑡 𝑐𝑘𝑡

∞ ∞ ∞ 𝑡−𝑠+1 − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 )−𝑟𝑡] 1𝑡 𝑘𝑡 (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗

∞ ∞ 1 𝑊𝑡 =𝐿−𝛾𝑛𝑥𝑛−𝜏 + ∑ 𝑓(𝑡,𝑦𝑏 ,...,𝑦𝑏 ) ≥𝐿−𝑐𝑀−∑ 𝑎 1𝑡 𝑘𝑡 󵄨 󵄨 𝑡=𝑛 𝑡 𝑡=𝑇 󵄨𝑎𝑡󵄨 ∞ ∞ 1 ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) 𝑑1𝑡 𝑑𝑘𝑡 − ∑∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] =𝑛 𝑎 󵄨𝑎 󵄨 𝑡 𝑡 s 𝑡=𝑠 𝑠 s=𝑇𝑡=𝑠 󵄨 𝑠󵄨 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] ∞ ∞ ∞ 𝑐1𝑡 𝑐𝑘𝑡 𝑡−𝑠+1 󵄨 󵄨 − ∑∑∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) 󵄨𝑎 󵄨 𝑗=𝑇𝑠=𝑗𝑡=𝑠 󵄨 𝑗󵄨 ∞ ∞ ∞ 𝑡−𝑠+1 − ∑ ∑ ∑ [𝑝 (𝑡, 𝑦 ,...,𝑦 )−𝑟] 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 >𝐿−𝑐𝑀−min {𝑀−𝐿,𝐿−𝑐𝑀−𝑁} ∞ 𝑊𝑡 ≥𝑁; ≤𝐿+𝑐𝑀+∑ 󵄨 󵄨 𝑡=𝑇 󵄨𝑎𝑡󵄨 (34) ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 that is, 𝑈𝐿𝑥+𝑆𝐿𝑦∈Ω1(𝑁, 𝑀) for any 𝑥, 𝑦1 ∈Ω (𝑁, 𝑀).The 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof. ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 Theorem 8. 𝑛 ∈ N 𝑀 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 Assume that there exist constants 1 𝑛0 , , 󵄨 󵄨 𝑁,and𝑐 with 𝑀>𝑁>0and 𝑐∈[0,(𝑀−𝑁)/𝑀)and four nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N , <𝐿+𝑐𝑀+ {(1−𝑐) 𝑀−𝐿,𝐿−𝑁} 𝑛0 𝑛0 𝑛0 min and {𝑅𝑛}𝑛∈N satisfying (12), (13),and 𝑛0 ≤𝑀,

−𝑐 ≤𝑛 𝛾 ≤0, ∀𝑛≥𝑛1 (35) (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛

∞ 1 Then (2) possesses uncountably many bounded positive solu- =𝐿−𝛾𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) 𝑛 𝑛−𝜏 𝑏1𝑡 𝑏𝑘𝑡 tions in Ω1(𝑁, 𝑀). 𝑡=𝑛 𝑎𝑡 10 Abstract and Applied Analysis

∞ ∞ 1 𝑈 𝑆 :Ω (𝑁, 𝑀) →𝑙∞ + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) Define two mappings 𝐿 and 𝐿 2𝑇 𝛽 by 𝑑1𝑡 𝑑𝑘𝑡 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 1 { (𝐿 − 𝑥 ), 𝑛≥𝑇, −𝑔 (𝑡, 𝑦 ,...,𝑦 )] (𝑈 𝑥) = 𝛾 𝑛+𝜏 𝑐1𝑡 𝑐𝑘𝑡 𝐿 𝑛 { 𝑛+𝜏 (41) {(𝑈𝐿𝑥)𝑇,𝛽≤𝑛<𝑇 ∞ ∞ ∞ 𝑡−𝑠+1 − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 )−𝑟𝑡] ∞ 𝑎 1𝑡 𝑘𝑡 { 1 1 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑗 { { ∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) { 𝑏1𝑡 𝑏𝑘𝑡 {𝛾𝑛+𝜏 𝑡=𝑛+𝜏 𝑎𝑡 { ∞ ∞ ∞ ∞ ∞ { 1 𝑊𝑡 1 { + ∑ ∑ [ (𝑡−𝑠+1) ≥𝐿−∑ 󵄨 󵄨 − ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] { 𝑎 󵄨𝑎 󵄨 󵄨𝑎 󵄨 { 𝑠=𝑛+𝜏𝑡=𝑠 𝑠 𝑡=𝑇 󵄨 𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨 𝑠󵄨 { { ×ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) { 1𝑡 𝑘𝑡 ∞ ∞ ∞ { −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑡−𝑠+1 󵄨 󵄨 { 𝑐1𝑡 𝑐𝑘𝑡 − (𝑅 + 󵄨𝑟 󵄨) { ∞ ∞ ∞ ∑ ∑ ∑ 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 (𝑆 𝑥) = 𝑡−𝑠+1 󵄨𝑎 󵄨 𝐿 𝑛 { 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 { − ∑ ∑∑ { 𝑎𝑗 { 𝑗=𝑛+𝜏 𝑠=𝑗𝑡=𝑠 { >𝐿− {(1−𝑐) 𝑀−𝐿,𝐿−𝑁} { ×[𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 ) min { 1𝑡 𝑘𝑡 { { { −𝑟𝑡]}, ≥𝑁; { { { 𝑛≥𝑇, (37) { (𝑆 𝑥) , 𝛽≤𝑛<𝑇, { 𝐿 𝑇 𝑈 𝑥+𝑆 𝑦∈Ω(𝑁, 𝑀) 𝑥, 𝑦 ∈Ω (𝑁, 𝑀) that is, 𝐿 𝐿 1 for any 1 .The (42) rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof. 𝑥={𝑥} ∈Ω (𝑁, 𝑀) for each 𝑛 𝑛∈Z𝛽 2𝑇 .Bymeansof(13)and 𝑥={𝑥} 𝑦={𝑦} ∈ (39)–(42), we get that for any 𝑛 𝑛∈Z𝛽 , 𝑛 𝑛∈Z𝛽 Theorem 9. Assume that there exist constants 𝑛1 ∈ N𝑛 , 𝑀, 0 Ω (𝑁, 𝑀) 𝑛≥𝑇 𝑁,and𝑐 with 𝑀>𝑁>0and 𝑐 > 𝑀/(𝑀 −𝑁) and 2𝑇 ,and , nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N ,and 𝑛0 𝑛0 𝑛0 (𝑈 𝑥) +(𝑆 𝑦) {𝑅𝑛}𝑛∈N satisfying (13): 𝐿 𝑛 𝐿 𝑛 𝑛0 1 ∞ 1 𝛾𝑛 ≥𝑐, ∀𝑛≥𝑛1, (38) = (𝐿 − 𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) 𝑛+𝜏 𝑏1𝑡 𝑏𝑘𝑡 𝛾𝑛+𝜏 𝑡=𝑛+𝜏 𝑎𝑡 󵄨 󵄨 󵄨 󵄨 󵄨𝑓(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑊𝑛, 󵄨𝑔(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑃𝑛, ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) 󵄨 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨ℎ(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑄𝑛, 󵄨𝑝(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑅𝑛, 𝑠=𝑛+𝜏 𝑡=𝑠 𝑎𝑠

𝑀 𝑘 −𝑔 (𝑡,𝑐 𝑦 ,...,𝑦𝑐 )] ∀(𝑛,𝑢1,𝑢2,...,𝑢𝑘)∈N𝑛 ×[0, ] . 1𝑡 𝑘𝑡 0 𝑐 (39) ∞ ∞ ∞ 𝑡−𝑠+1 − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 ) 𝑎 1𝑡 𝑘𝑡 Then (2) possesses uncountably many bounded positive solu- 𝑗=𝑛+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 ∞ tions in 𝑙𝛽 . −𝑟 ]) Proof. Let 𝐿 ∈ (𝑁 + 𝑀/𝑐,.Itfollowsfrom( 𝑀) 13)thatthere 𝑡 exists an integer 𝑇≥𝑛0 +𝑛1 +𝜏+|𝛽|satisfying 1 𝑁 ∞ 𝑊 ∞ 𝑊 ∞ ∞ 1 ≤ (𝐿 − + ∑ 𝑡 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝛾 𝛾 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑛+𝜏 𝑛+𝜏 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 ∞ ∞ ∞ ∞ ∞ 1 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] + ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) (40) 󵄨𝑎 󵄨 𝑡 𝑡 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨

∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 𝑀 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨)) < min {𝑀 − 𝐿, 𝐿 −𝑁 }. 𝑎 𝑡 󵄨 𝑡󵄨 𝑐 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 Abstract and Applied Analysis 11

1 𝑀 𝑚 < (𝐿 + {𝑀−𝐿,𝐿−𝑁− }) that {𝑥 }𝑚∈N is an arbitrary sequence in Ω2𝑇(𝑁, 𝑀) and 𝑥∈ min 𝑚 𝑚 𝑚 𝛾𝑛+𝜏 𝑐 Ω (𝑁, 𝑀) 𝑥 =𝑥 𝑥 ={𝑥 } 2𝑇 with lim𝑚→∞ ,where 𝑛 𝑛∈Z𝛽 for 𝑚∈N 𝑥={𝑥} 𝑀 each and 𝑛 𝑛∈Z𝛽 . On account of (13), (39), ≤ , 𝑥𝑚 =𝑥 𝑓, 𝑔, ℎ 𝑝 𝛾𝑛+𝜏 lim𝑚→∞ ,andthecontinuityof ,and ,we obtain that for given 𝜀>0, there exist 𝑇1,𝑇2,𝑇3,and𝑇4 ∈ N 𝑇 >𝑇 >𝑇 >𝑇 >𝑇 (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 with 4 3 2 1 satisfying 1 ∞ 1 ∞ ∞ ∞ = (𝐿 −𝑛+𝜏 𝑥 + ∑ 𝑓(𝑡,𝑦𝑏 ,...,𝑦𝑏 ) 𝑊 1 𝛾 𝑎 1𝑡 𝑘𝑡 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨𝑎𝑡󵄨 󵄨𝑎𝑠󵄨 𝑡=𝑇1+𝜏 󵄨 󵄨 𝑠=𝑇1+𝜏 𝑡=𝑠 󵄨 󵄨 ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦𝑑 ,...,𝑦𝑑 ) ∞ ∞ ∞ 𝑎 1𝑡 𝑘𝑡 𝑡−𝑠+1 𝜀𝑐 𝑠=𝑛+𝜏 𝑡=𝑠 𝑠 + ∑ ∑ ∑ 𝑅 < ; 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 18 𝑗=𝑇1+𝜏 𝑠=𝑗 󵄨 𝑗󵄨 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑐1𝑡 𝑐𝑘𝑡 { ∞ 𝑡−𝑠+1 } ∞ ∞ ∞ ∑ 𝑅 :𝑇≤𝑗≤𝑇 +𝜏, 𝑗≤𝑠≤𝑇 +𝜏 𝑡−𝑠+1 max { 󵄨 󵄨 𝑡 1 2 } − ∑ ∑ ∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 )−𝑟𝑡]) 𝑡=𝑇 +𝜏 󵄨𝑎 󵄨 1𝑡 𝑘𝑡 { 3 󵄨 𝑗󵄨 } 𝑗=𝑛+𝜏 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 𝜀𝑐 < ; 1 𝑀 ∞ 𝑊 18𝑇 𝑇 ≥ (𝐿 − − ∑ 𝑡 1 2 𝛾 𝛾 󵄨 󵄨 𝑛+𝜏 𝑛+𝜏 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨 { ∞ ∞ 𝑡−𝑠+1 } ∑ ∑ 𝑅 :𝑇≤𝑗≤𝑇 +𝜏 ∞ ∞ max { 󵄨 󵄨 𝑡 1 } 1 𝑠=𝑇 +𝜏 𝑡=𝑠 󵄨𝑎 󵄨 − ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] { 2 󵄨 𝑗󵄨 } 󵄨𝑎 󵄨 𝑡 𝑡 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 𝜀𝑐 < ; 18𝑇1 ∞ ∞ ∞ 𝑡− +1 󵄨 󵄨 − ∑ ∑ ∑ s (𝑅 + 󵄨𝑟 󵄨)) 𝑡 󵄨 𝑡󵄨 ∞ 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 { 1 𝑗=𝑇+𝜏 ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] max { 󵄨 󵄨 𝑡 𝑡 𝑡=𝑇 +𝜏 󵄨𝑎𝑠󵄨 1 𝑀 𝑀 { 2 > (𝐿 − − {𝑀 − 𝐿, 𝐿 −𝑁 }) 𝛾 𝑐 min 𝑐 𝑛+𝜏 } 𝜀𝑐 :𝑇≤𝑠≤𝑇 +𝜏 < ; 𝑁 1 } 18𝑇 ≥ , } 1 𝛾𝑛+𝜏 󵄨 𝑚 𝑚 (43) {󵄨𝑞(𝑡,𝑥 ,...,𝑥 ) max 󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨 which yield that 𝑈𝐿𝑥+𝑆𝐿𝑦∈Ω2𝑇(𝑁, 𝑀) for any 𝑥, 𝑦 ∈ −𝑞 (𝑡, 𝑥 ,...,𝑥 )󵄨 :𝑞∈{𝑓,𝑔,ℎ,𝑝}} 𝑏1𝑡 𝑏𝑘𝑡 󵄨 Ω2𝑇(𝑁, 𝑀).Byvirtueof(38)and(41), we infer that 𝜀𝑐 󵄩 󵄩 󵄨 󵄨 < ,∀𝑚≥𝑇, 𝑇≤𝑡≤𝑇, 󵄩𝑈𝐿𝑥−𝑈𝐿𝑦󵄩 = sup 󵄨(𝑈𝐿𝑥) −(𝑈𝐿𝑦) 󵄨 4 3 𝑛 𝑛 18𝑇1𝑇2𝑇3 (𝐴+𝐵+𝐸) 𝑛∈Z𝛽 (45) 󵄨 󵄨 = max { sup 󵄨(𝑈𝐿𝑥)𝑛 −(𝑈𝐿𝑦)𝑛󵄨 , 𝑇>𝑛≥𝛽 where

󵄨 󵄨 󵄨(𝑈 𝑥) −(𝑈 𝑦) 󵄨} (44) 1 sup 󵄨 𝐿 𝑛 𝐿 𝑛󵄨 𝐴= { :𝑇+𝜏≤𝑠≤𝑇 +𝜏}, 𝑛≥𝑇 max 󵄨 󵄨 1 󵄨𝑎𝑠󵄨 1 󵄨 󵄨 = ( 󵄨𝑥 −𝑦 󵄨) 𝑡−𝑠+1 sup 󵄨 󵄨 󵄨 𝑛+𝜏 𝑛+𝜏󵄨 𝐵= { :𝑇+𝜏≤𝑠≤𝑇 +𝜏,𝑠≤𝑡≤𝑇 +𝜏}, 𝑛≥𝑇 󵄨𝛾𝑛+𝜏󵄨 max 󵄨 󵄨 1 2 󵄨𝑎𝑠󵄨 1 󵄩 󵄩 ≤ 󵄩𝑥−𝑦󵄩 ; 𝑐 󵄩 󵄩 𝑡−𝑠+1 𝐸=max { 󵄨 󵄨 :𝑇+𝜏≤𝑗≤𝑇1 +𝜏, 󵄨 󵄨 󵄨𝑎𝑗󵄨 that is, 𝑈𝐿 is a contraction in Ω2𝑇(𝑁, 𝑀) because 𝑐 > 𝑀/(𝑀− 𝑁) >. 1 𝑆 In order to prove that 𝐿 is completely continuous in 𝑗≤𝑠≤𝑇2 +𝜏,𝑠≤𝑡≤𝑇3 +𝜏}. Ω2𝑇(𝑁, 𝑀),wehavetoshowthat𝑆𝐿 is continuous in Ω2𝑇(𝑁, 𝑀) and 𝑆𝐿(Ω2𝑇(𝑁, 𝑀) is relatively compact. Suppose (46) 12 Abstract and Applied Analysis

󵄨 It follows from (42)–(46)that −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 ) 𝑜1𝑡 𝑜𝑘𝑡 󵄨 󵄩 𝑚 󵄩 󵄩𝑆𝐿𝑥 −𝑆𝐿𝑥󵄩 𝑇 +𝜏−1 󵄨 𝑚 󵄨 1 1 1 󵄨 󵄨 = sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨 󵄨 𝑚 𝑚 󵄨 ≤ ( ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 )−𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )󵄨 𝑛∈Z𝛽 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 1𝑡 𝑘𝑡 󵄨 𝑐 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨 󵄨 󵄨 ∞ = { 󵄨(𝑆 𝑥𝑚) −(𝑆 𝑥) 󵄨 , 1 󵄨 𝑚 𝑚 max sup 󵄨 𝐿 𝑛 𝐿 𝑛󵄨 + ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 ) 𝑇>𝑛≥𝛽 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨𝑎𝑡󵄨 𝑡=𝑇1+𝜏 󵄨 󵄨

󵄨 𝑚 󵄨 󵄨 󵄨(𝑆 𝑥 ) −(𝑆 𝑥) 󵄨} −𝑓 (𝑡, 𝑥 ,...,𝑥 )󵄨 sup 󵄨 𝐿 𝑛 𝐿 𝑛󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨 𝑛≥𝑇 𝑇 +𝜏−1 𝑇 +𝜏−1 󵄨 1 2 1 󵄨 󵄨 ∞ 󵄨 𝑚 𝑚 󵄨 1 1 𝑚 𝑚 + ∑ ∑ 󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨 󵄨 󵄨 󵄨 1𝑡 𝑘𝑡 = sup 󵄨 ( ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑡=𝑠 󵄨𝑎𝑠󵄨 󵄨𝛾 𝑎 1𝑡 𝑘𝑡 𝑠=𝑇+𝜏 𝑛≥𝑇 󵄨 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 󵄨 ∞ ∞ −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 1 𝑚 𝑚 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 𝑎 1𝑡 𝑘𝑡 󵄨 𝑚 𝑚 𝑠=𝑛+𝜏 𝑡=𝑠 𝑠 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 −𝑔 (𝑡,𝑚 𝑥 ,...,𝑥𝑚 )] 󵄨 𝑐1𝑡 𝑐𝑘𝑡 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 ∞ ∞ ∞ 𝑡−𝑠+1 𝑚 𝑚 𝑇1+𝜏−1 ∞ − ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡]) 1 󵄨 𝑚 𝑚 𝑎 1𝑡 𝑘𝑡 + ∑ ∑ 󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑗=𝑛+𝜏𝑠=𝑗t=𝑠 𝑗 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨𝑎𝑠󵄨 𝑠=𝑇+𝜏 𝑡=𝑇2+𝜏 󵄨 󵄨 ∞ 1 1 󵄨 − ( ∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑏1𝑡 𝑏𝑘𝑡 −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 𝛾𝑛+𝜏 𝑡=𝑛+𝜏 𝑎𝑡 1𝑡 𝑘𝑡 󵄨 󵄨 𝑚 𝑚 ∞ ∞ + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 1 󵄨 𝑐1𝑡 𝑐𝑘𝑡 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 𝑎 1𝑡 𝑘𝑡 󵄨 𝑠=𝑛+𝜏𝑡=𝑠 𝑠 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ 1 󵄨 𝑚 𝑚 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) ∞ ∞ ∞ 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑡−𝑠+1 𝑠=𝑇 +𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 − ∑ ∑∑ 1 𝑡=𝑠 𝑎𝑗 󵄨 𝑗=𝑛+𝜏𝑠=𝑗 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 󵄨 󵄨 󵄨 𝑚 𝑚 󵄨 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) ×[𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 )−𝑟𝑡])󵄨 󵄨 𝑐1𝑡 𝑐𝑘𝑡 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 1 ∞ 1 󵄨 ≤ ( ∑ 󵄨𝑓(𝑡,𝑥𝑚 ,...,𝑥𝑚 ) 󵄨 󵄨 󵄨 𝑏 𝑏 𝑇 +𝜏−1 𝑇 +𝜏−1 𝑇 +𝜏−1 𝑐 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 1 2 3 𝑡=𝑇+𝜏 󵄨 𝑡󵄨 𝑡−𝑠+1󵄨 𝑚 𝑚 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑡=𝑠 󵄨𝑎 󵄨 −𝑓 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 󵄨 𝑗󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨 󵄨 −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 ∞ ∞ 1t 𝑘𝑡 󵄨 1 󵄨 𝑚 𝑚 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨𝑎 󵄨 𝑇1+𝜏−1 𝑇2+𝜏−1 ∞ 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 𝑡−𝑠+1󵄨 𝑚 𝑚 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑇 +𝜏 󵄨𝑎 󵄨 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 3 󵄨 𝑗󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 󵄨 󵄨 󵄨 𝑚 𝑚 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 𝑜1𝑡 𝑜𝑘𝑡 󵄨 󵄨 𝑐1𝑡 𝑐𝑘𝑡 󵄨

󵄨 𝑇1+𝜏−1 ∞ ∞ −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )󵄨] 𝑡−𝑠+1󵄨 𝑚 𝑚 1𝑡 𝑘𝑡 󵄨 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡=𝑠 󵄨𝑎 󵄨 ∞ ∞ ∞ 𝑗=𝑇+𝜏 𝑠=𝑇2+𝜏 󵄨 𝑗󵄨 𝑡−𝑠+1󵄨 𝑚 𝑚 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑡=𝑠 󵄨𝑎 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 󵄨 𝑗󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 Abstract and Applied Analysis 13

∞ ∞ ∞ 𝑡−𝑠+1󵄨 𝑚 𝑚 2𝜀 1(𝑇 −𝑇) 2𝜀 1(𝑇 −𝑇)(𝑇2 −𝑇) + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) + + 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡=𝑠 󵄨𝑎 󵄨 18𝑇1 18𝑇1𝑇2 𝑗=𝑇1+𝜏 𝑠=𝑗 󵄨 𝑗󵄨 2𝜀 (𝑇 −𝑇) 𝜀 󵄨 + 1 + <𝜀, ∀𝑚≥𝑇, −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 ) 4 1𝑡 𝑘𝑡 󵄨 18𝑇1 18 (47) 𝑇 +𝜏−1 1 1 󵄨 󵄨 𝑚 𝑚 ≤ ( ∑ 𝐴 󵄨𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑐 󵄨 1𝑡 𝑘𝑡 𝑡=𝑇+𝜏 which means that 𝑆𝐿 is continuous in Ω2𝑇(𝑁, 𝑀).Itfollows 󵄨 from (13)thatforgiven𝜀>0, there exists 𝑉>𝑇satisfying −𝑓 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨

𝑇 +𝜏−1 𝑇 +𝜏−1 ∞ 1 2 󵄨 𝑊 󵄨 𝑚 𝑚 ∑ 𝑡 + ∑ ∑ [𝐵 󵄨ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨 󵄨 󵄨 1𝑡 𝑘𝑡 󵄨𝑎 󵄨 𝑠=𝑇+𝜏 𝑡=𝑠 𝑡=𝑉+𝜏 󵄨 𝑡󵄨 󵄨 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 ∞ ∞ 1 𝑑1𝑡 𝑑𝑘𝑡 󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨𝑎 󵄨 𝑡 𝑡 (48) 󵄨 𝑚 𝑚 𝑠=𝑉+𝜏𝑡=𝑠 󵄨 𝑠󵄨 +𝐴󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ ∞ 󵄨 𝑡−𝑠+1 󵄨 󵄨 𝜀 −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )󵄨] + ∑ ∑∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨)< . 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 4 𝑗=𝑉+𝜏𝑠=𝑗𝑡=𝑠 󵄨𝑎𝑗󵄨

𝑇1+𝜏−1 𝑇2+𝜏−1 𝑇3+𝜏−1 󵄨 𝑚 𝑚 + ∑ ∑ ∑ 𝐸 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 Next we prove that 𝑆𝐿(Ω2𝑇(𝑁, 𝑀)) is uniformly Cauchy. 𝑥={𝑥𝑛}𝑛∈Z ∈ 󵄨 In view of (42)and(48), we infer that for any 𝛽 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 Ω (𝑁, 𝑀) 𝑚, 𝑛 ≥𝑉 𝑜1𝑡 𝑜𝑘𝑡 󵄨 2𝑇 and ,

∞ 𝑊𝑡 +2 ∑ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨(𝑆𝐿𝑥) −(𝑆𝐿𝑥) 󵄨 󵄨𝑎𝑡󵄨 󵄨 𝑚 𝑛󵄨 𝑡=𝑇1+𝜏 󵄨 󵄨 󵄨 𝑇 +𝜏−1 ∞ 󵄨 ∞ 1 1 󵄨 1 1 +2 ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] = 󵄨 ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨 󵄨 𝑡 𝑡 󵄨𝛾 𝑎 1𝑡 𝑘𝑡 󵄨𝑎𝑠󵄨 󵄨 𝑚+𝜏 𝑡=𝑚+𝜏 𝑡 𝑠=𝑇+𝜏 𝑡=𝑇2+𝜏 󵄨 󵄨

∞ ∞ 1 1 ∞ 1 +2 ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] − ∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 𝑡 𝑡 𝑏1𝑡 𝑏𝑘𝑡 󵄨𝑎𝑠󵄨 𝛾 𝑎 𝑠=𝑇1+𝜏 𝑡=𝑠 󵄨 󵄨 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡

𝑇 +𝜏−1 𝑇 +𝜏−1 1 2 ∞ 𝑡−𝑠+1 1 ∞ ∞ 1 +2 ∑ ∑ ∑ 𝑅 󵄨 󵄨 𝑡 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨𝑎 󵄨 𝛾 𝑎 1𝑡 𝑘𝑡 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑇3+𝜏 󵄨 𝑗󵄨 𝑚+𝜏 𝑠=𝑚+𝜏𝑡=𝑠 𝑠

𝑇 +𝜏−1 ∞ ∞ 1 𝑡−𝑠+1 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] +2 ∑ ∑ ∑ 𝑅 𝑐1𝑡 𝑐𝑘𝑡 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 𝑗=𝑇+𝜏 𝑠=𝑇2+𝜏 󵄨 𝑗󵄨 1 ∞ ∞ 1 ∞ ∞ ∞ − ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑡−𝑠+1 𝑑1𝑡 𝑑𝑘𝑡 +2 ∑ ∑ ∑ 𝑅 ) 𝛾𝑛+𝜏 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 𝑗=𝑇1+𝜏 𝑠=𝑗 󵄨 𝑗󵄨

−𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] 𝜀𝐴 (𝑇 −𝑇) 1𝑡 𝑘𝑡 < 1 18𝑇 𝑇 𝑇 (𝐴+𝐵+𝐸) 1 2 3 1 ∞ ∞ ∞ 𝑡−𝑠+1 + ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡] 𝜀 (𝐵+𝐴) (𝑇 −𝑇)(𝑇 −𝑇) 𝛾 𝑎 1𝑡 𝑘𝑡 + 1 2 𝑛+𝜏 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑗 18𝑇1𝑇2𝑇3 (𝐴+𝐵+𝐸) 󵄨 1 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 𝜀𝐸1 (𝑇 −𝑇)(𝑇2 −𝑇)(𝑇3 −𝑇) 󵄨 + − ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡]󵄨 𝛾 𝑎 1𝑡 𝑘𝑡 󵄨 18𝑇1𝑇2𝑇3 (𝐴+𝐵+𝐸) 𝑚+𝜏 𝑗=𝑚+𝜏𝑠=𝑗𝑡=𝑠 𝑗 󵄨 14 Abstract and Applied Analysis

∞ ∞ ∞ 1 𝑊𝑡 𝑊𝑡 1 𝑊 ≤ ( ∑ 󵄨 󵄨 + ∑ 󵄨 󵄨 ≤ ( ∑ 𝑡 𝑐 󵄨𝑎 󵄨 󵄨𝑎 󵄨 𝑐 󵄨 󵄨 𝑡=𝑚+𝜏 󵄨 𝑡󵄨 𝑡=𝑛+𝜏 󵄨 𝑡󵄨 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨

∞ ∞ ∞ ∞ 1 1 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨𝑎𝑠󵄨 𝑠=𝑚+𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝑠=𝑇+𝜏𝑡=𝑠 󵄨 󵄨

∞ ∞ ∞ ∞ ∞ 1 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨)) + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 󵄨 󵄨 󵄨𝑎 󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 s=𝑛+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 󵄨 󵄨 1 𝑀 ∞ ∞ ∞ < {𝑀 − 𝐿, 𝐿 −𝑁 } 𝑡−𝑠+1 󵄨 󵄨 𝑐 min 𝑐 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗=𝑛+𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 𝑀 < , ∀𝑥={𝑥𝑛}𝑛∈Z ∈Ω2𝑇 (𝑁,) 𝑀 , 𝑐 𝛽 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 (50) + ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨)) 󵄨𝑎 󵄨 𝑗=𝑚+𝜏 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 which gives that 𝑆𝐿(Ω2𝑇(𝑁, 𝑀)) is bounded. Thus Lemma 3 𝑥={𝑥} ∈Ω (𝑁, 𝑀) means that there exists 𝑛 𝑛∈Z𝛽 2𝑇 such that 2 ∞ 𝑊 𝑈𝐿𝑥+𝑆𝐿𝑥=𝑥, which is a bounded positive solution of (2)in ≤ ( ∑ 𝑡 ∞ 󵄨 󵄨 Ω2𝑇(𝑁, 𝑀)𝛽 ⊂𝑙 . 𝑐 𝑡=𝑉+𝜏 󵄨𝑎𝑡󵄨 Let 𝐿1,𝐿2 ∈ (𝑁 + 𝑀/𝑐, 𝑀) and 𝐿1 =𝐿̸ 2.Forany𝑗∈ {1, 2}, we deduce similarly that there exist a positive integer ∞ ∞ 1 𝑇𝑗 ≥𝑛0 +𝑛1 +𝜏+|𝛽|, a closed bounded and convex + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] ∞ 󵄨𝑎 󵄨 𝑡 𝑡 Ω (𝑁, 𝑀) 𝑙 𝑈 𝑆 𝑠=𝑉+𝜏𝑡=𝑠 󵄨 𝑠󵄨 subset 2𝑇𝑗 of 𝛽 ,andtwomappings 𝐿𝑗 and 𝐿𝑗 satisfying (40)–(42), where 𝐿 and 𝑇 are replaced by 𝐿𝑗 and 𝑇𝑗,respectively,and𝑈𝐿 +𝑆𝐿 possesses a fixed point 𝑧𝑗 = ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 𝑗 𝑗 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨)) {𝑧 } ∈Ω (𝑁, 𝑀) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗𝑛 𝑛∈Z𝛽 2𝑇𝑗 , which is a bounded positive solution 󵄨𝑎 󵄨 𝑗=𝑉+𝜏 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 of (2); that is, 𝑧 <𝜀. 𝑗𝑛 (49) 1 ∞ 1 = (𝐿 −𝑧 + ∑ 𝑓(𝑡,𝑧 ,...,𝑧 ) 𝑗 𝑗𝑛+𝜏 𝑗𝑏1𝑡 𝑗𝑏𝑘𝑡 𝛾𝑛+𝜏 𝑡=𝑛+𝜏 𝑎𝑡 Note that (40)and(42)yieldthat ∞ ∞ 󵄩 󵄩 1 󵄩𝑆 𝑥󵄩 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑧 ,...,𝑧 ) 󵄩 𝐿 󵄩 𝑗𝑑1𝑡 𝑗𝑑𝑘𝑡 𝑠=𝑛+𝜏 𝑡=𝑠 𝑎𝑠 󵄨 󵄨 = sup 󵄨(𝑆𝐿𝑥) 󵄨 −𝑔 (𝑡, 𝑧 ,...,𝑧 )] 󵄨 𝑛󵄨 𝑗𝑐1𝑡 𝑗𝑐𝑘𝑡 𝑛∈Z𝛽 ∞ ∞ ∞ 𝑡−𝑠+1 − ∑ ∑∑ [𝑝 (𝑡, 𝑧 ,...,𝑧 )−𝑟 ]) , 𝑗𝑜1𝑡 𝑗𝑜𝑘𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 = max { sup 󵄨(𝑆𝐿𝑥)𝑛󵄨 , sup 󵄨(𝑆𝐿𝑥)𝑛󵄨} 𝑇>𝑛≥𝛽 𝑛≥𝑇 ∀𝑛 ≥𝑗 𝑇 ,𝑗∈{1, 2} . 󵄨 󵄨 ∞ (51) 󵄨 1 1 = sup 󵄨 ( ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨𝛾 𝑎 1𝑡 𝑘𝑡 Observe that (13) implies that there exists 𝑇3 ∈ N with 𝑇3 > 𝑛≥𝑇 󵄨 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 max{𝑇1,𝑇2} satisfying

∞ ∞ ∞ 1 𝑊𝑡 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) ∑ 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 󵄨 𝑎 󵄨𝑎𝑡󵄨 𝑠=𝑛+𝜏𝑡=𝑠 𝑠 𝑡=𝑇3+𝜏 󵄨 󵄨 ∞ ∞ 1 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 󵄨 𝑡 𝑡 (52) 󵄨𝑎𝑠󵄨 𝑠=𝑇3+𝜏 𝑡=𝑠 󵄨 󵄨 󵄨 ∞ ∞ ∞ 󵄨 ∞ ∞ ∞ 󵄨 󵄨 𝑡−𝑠+1 󵄨 𝑡−𝑠+1 󵄨𝐿1 −𝐿2󵄨 − ∑ ∑∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟 ])󵄨 + ∑ ∑ ∑ 𝑅 < , 𝑜1𝑡 𝑜𝑘𝑡 𝑡 󵄨 󵄨 󵄨 𝑡 𝑡=𝑠 𝑎𝑗 󵄨 𝑡=𝑠 󵄨𝑎 󵄨 4 𝑗=𝑛+𝜏𝑠=𝑗 󵄨 𝑗=𝑇3+𝜏 𝑠=𝑗 󵄨 𝑗󵄨 Abstract and Applied Analysis 15 󵄨 󵄨 1 󵄨𝐿 −𝐿 󵄨 which together with (51)yieldsthatforeach𝑛≥𝑇3 󵄨 󵄨 󵄨 1 2󵄨 ≥ (󵄨𝐿1 −𝐿2󵄨 − ) 𝛾𝑛+𝜏 2 󵄨 󵄨 󵄨 𝑧1𝑛+𝜏 −𝑧2𝑛+𝜏 󵄨 󵄨𝑧1𝑛 −𝑧2𝑛 + 󵄨 󵄨 󵄨 󵄨 𝛾 󵄨 󵄨𝐿 −𝐿 󵄨 𝑛+𝜏 > 󵄨 1 2󵄨, 󵄨 󵄨 ∞ 2𝛾𝑛+𝜏 󵄨 1 1 = 󵄨 (𝐿1 −𝐿2 + ∑ 𝑓(𝑡,𝑧1𝑏 ,...,𝑧1𝑏 ) 󵄨𝛾 𝑎 1𝑡 𝑘𝑡 (53) 󵄨 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 ∞ 1 − ∑ 𝑓(𝑡,𝑧 ,...,𝑧 ) which implies that 2𝑏1𝑡 2𝑏𝑘𝑡 𝑡=𝑛+𝜏 𝑎𝑡 ∞ ∞ 󵄨 󵄨 1 󵄨 𝑧1𝑛+𝜏 −𝑧2𝑛+𝜏 󵄨 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑧1𝑑 ,...,𝑧1𝑑 ) 󵄨𝑧1𝑛 −𝑧2𝑛 + 󵄨 >0, ∀𝑛≥𝑇3; 1𝑡 𝑘𝑡 󵄨 󵄨 (54) 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 󵄨 𝛾𝑛+𝜏 󵄨 −𝑔 (𝑡, 𝑧 ,...,𝑧 )] 1𝑐1𝑡 1𝑐𝑘𝑡 that is, 𝑧1 =𝑧̸ 2.Consequently,(2) has uncountably many ∞ ∞ ∞ 1 bounded positive solutions in 𝑙𝛽 .Thiscompletesthe − ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑧2𝑑 ,...,𝑧2𝑑 ) 1𝑡 𝑘𝑡 proof. 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 −𝑔 (𝑡, 𝑧 ,...,𝑧 )] Theorem 10. 𝑛 ∈ N 𝑀 2𝑐1𝑡 2𝑐𝑘𝑡 Assume that there exist constants 1 𝑛 , , 𝑁 𝑐 𝑑 𝑑>𝑐>1 𝑀(1 − 1/𝑐) >0 𝑁(1 −1/𝑑) ∞ ∞ ∞ , ,and with and 𝑡−𝑠+1 and nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N , − ∑ ∑∑ [𝑝 (𝑡, 𝑧 ,...,𝑧 )−𝑟] 𝑛0 𝑛0 𝑛0 1𝑜1𝑡 1𝑜𝑘𝑡 𝑡 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 and {𝑅𝑛}𝑛∈N satisfying (13): 𝑛0 󵄨 ∞ ∞ ∞ 󵄨 𝑡−𝑠+1 󵄨 + ∑ ∑∑ [𝑝 (𝑡,2𝑜 𝑧 ,...,𝑧2𝑜 )−𝑟𝑡])󵄨 𝑐≤𝛾 ≤𝑑, ∀𝑛≥𝑛, 𝑎 1𝑡 𝑘𝑡 󵄨 𝑛 1 (55) 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑗 󵄨 ∞ 󵄨 󵄨 󵄨 󵄨 1 󵄨 󵄨 1 󵄨 󵄨𝑓(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑊𝑛, 󵄨𝑔(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑃𝑛, ≥ (󵄨𝐿 −𝐿 󵄨 − ∑ 󵄨𝑓(𝑡,𝑧 ,...,𝑧 ) 󵄨 1 2󵄨 󵄨 1𝑏1𝑡 1𝑏𝑘𝑡 𝛾𝑛+𝜏 𝑡=𝑛+𝜏 𝑎𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨ℎ(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑄𝑛, 󵄨𝑝(𝑛,𝑢1,𝑢2,...,𝑢𝑘)󵄨 ≤𝑅𝑛, −𝑓 (𝑡, 𝑧 ,...,𝑧 )󵄨 2𝑏1𝑡 2𝑏𝑘𝑡 󵄨 ∞ ∞ 𝑘 1 󵄨 𝑁 𝑀 − ∑ ∑ [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑧 ,...,𝑧 ) ∀(𝑛,𝑢1,𝑢2,...,𝑢𝑘)∈N𝑛 ×[ , ] . 󵄨 1𝑑1𝑡 1𝑑𝑘𝑡 0 𝑑 𝑐 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 (56) 󵄨 −ℎ (𝑡, 𝑧 ,...,𝑧 )󵄨 2𝑑1𝑡 2𝑑𝑘𝑡 󵄨 󵄨 Then (2) possesses uncountably many bounded positive solu- + 󵄨𝑔(𝑡, ,...,𝑧 ) ∞ 󵄨 z1𝑐1𝑡 1𝑐𝑘𝑡 tions in 𝑙𝛽 . 󵄨 −𝑔 (𝑡, 𝑧 ,...,𝑧 )󵄨] 2𝑐1𝑡 2𝑐𝑘𝑡 󵄨 Proof. Let 𝐿∈(𝑁+𝑀/𝑐,𝑀+𝑁/𝑑). It follows from (13)that ∞ ∞ ∞ 𝑡−𝑠+1󵄨 there exists an integer 𝑇≥𝑛0 +𝑛1 + 𝜏 + |𝛽| satisfying − ∑ ∑ ∑ 󵄨𝑝(𝑡,𝑧 ,...,𝑧 ) 󵄨 1𝑜1𝑡 1𝑜𝑘𝑡 𝑗=𝑛+𝜏 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 ∞ 𝑊 ∞ ∞ 1 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 −𝑝 (𝑡, 𝑧 ,...,𝑧 )󵄨 ) 󵄨𝑎𝑡󵄨 𝑡=𝑠 󵄨𝑎𝑠󵄨 2𝑜1𝑡 2𝑜𝑘𝑡 󵄨 𝑡=𝑇 𝑠=𝑇

∞ ∞ ∞ ∞ 𝑡−𝑠+1 1 󵄨 󵄨 𝑊 󵄨 󵄨 ≥ (󵄨𝐿 −𝐿 󵄨 −2 ∑ 𝑡 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) (57) 󵄨 1 2󵄨 󵄨 󵄨 󵄨𝑎 󵄨 𝛾𝑛+𝜏 󵄨𝑎𝑡󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 𝑡=𝑇3+𝜏 󵄨 󵄨

∞ ∞ 1 𝑁 𝑀 −2 ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 < min {𝑀 + −𝐿, 𝐿−𝑁− }. 󵄨𝑎𝑠󵄨 𝑑 𝑐 𝑠=𝑇3+𝜏 𝑡=𝑠 󵄨 󵄨

∞ ∞ ∞ 𝑡−𝑠+1 ∞ −2 ∑ ∑ ∑ 𝑅 ) Let the mappings 𝑈𝐿 and 𝑆𝐿 :Ω2𝑇(𝑁, 𝑀)𝛽 →𝑙 be defined 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 𝑗=𝑇3+𝜏 𝑠=𝑗 󵄨 𝑗󵄨 by (41)and(42), respectively. Using (13)and(41), (42)and 16 Abstract and Applied Analysis

1 𝑀 𝑁 𝑀 (57), we obtain that for any 𝑥={𝑥𝑛}𝑛∈Z , 𝑦={𝑦𝑛}𝑛∈Z ∈ > (𝐿 − − {𝑀 + −𝐿, 𝐿−𝑁− }) 𝛽 𝛽 𝛾 𝑐 min 𝑑 𝑐 Ω2𝑇(𝑁, 𝑀) and 𝑛≥𝑇, 𝑛+𝜏 𝑁 ≥ , (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 𝛾𝑛+𝜏 1 ∞ 1 (58) = (𝐿 −𝑛+𝜏 𝑥 + ∑ 𝑓(𝑡,𝑦𝑏 ,...,𝑦𝑏 ) 𝛾 𝑎 1𝑡 𝑘𝑡 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 which imply that 𝑈𝐿𝑥+𝑆𝐿𝑦∈Ω2𝑇(𝑁, 𝑀) for any 𝑥, 𝑦 ∈ Ω2𝑇(𝑁, 𝑀).Therestoftheproofissimilartothatof ∞ ∞ 1 Theorem 9 and is omitted. This completes the proof. + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) 𝑑1𝑡 𝑑𝑘𝑡 𝑠=𝑛+𝜏 𝑡=𝑠 𝑎𝑠 Theorem 11. 𝑛 ∈ N 𝑀 Assume that there exist constants 1 𝑛0 , , −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑁,and𝑐 with 𝑀>𝑁>0and 𝑐 > 𝑀/(𝑀 −𝑁) and 𝑐1𝑡 𝑐𝑘𝑡 nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N ,and 𝑛0 𝑛0 𝑛0 ∞ ∞ ∞ {𝑅𝑛}𝑛∈N satisfying (13), (39),and 𝑡−𝑠+1 𝑛0 − ∑ ∑∑ [𝑝 (𝑡, 𝑦 ,...,𝑦 )−𝑟]) 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 𝛾𝑛 ≤−𝑐, ∀𝑛≥𝑛1. (59) Then (2) possesses uncountably many bounded positive solu- 1 𝑁 ∞ 𝑊 𝑙∞ ≤ (𝐿 − + ∑ 𝑡 tions in 𝛽 . 𝛾 𝛾 󵄨 󵄨 𝑛+𝜏 𝑛+𝜏 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨 Proof. Let 𝐿 ∈ (𝑁, 𝑀(1−1/𝑐)).Itfollowsfrom(13)thatthere ∞ ∞ 1 exists an integer 𝑇≥𝑛0 +𝑛1 + 𝜏 + |𝛽| satisfying + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨𝑎 󵄨 𝑡 𝑡 ∞ 𝑊 ∞ ∞ 1 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ (𝑅𝑡 + 󵄨𝑟𝑡󵄨)) ∞ ∞ ∞ 𝑎 𝑡−𝑠+1 󵄨 󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 (60) 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 1 𝑁 𝑁 𝑀 󵄨 󵄨 < (𝐿 − + min {𝑀 + −𝐿, 𝐿−𝑁− }) 𝛾𝑛+𝜏 𝑑 𝑑 𝑐 𝑀 < {𝐿 − 𝑁, 𝑀− −𝐿}. min 𝑐 𝑀 ≤ , ∞ Define two mappings 𝑈𝐿 and 𝑆𝐿 :Ω3𝑇(𝑁, 𝑀)𝛽 →𝑙 by (42) 𝛾𝑛+𝜏 and (𝑈 𝑥) +(𝑆 𝑦) 1 𝐿 𝑛 𝐿 𝑛 { (−𝐿 − 𝑥 ), 𝑛≥𝑇, 𝛾 𝑛+𝜏 ∞ (𝑈𝐿𝑥)𝑛 = { 𝑛+𝜏 (61) 1 1 (𝑈 𝑥) , 𝛽≤𝑛<𝑇, = (𝐿 −𝑛+𝜏 𝑥 + ∑ 𝑓(𝑡,𝑦𝑏 ,...,𝑦𝑏 ) { 𝐿 𝑇 𝛾 𝑎 1𝑡 𝑘𝑡 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 𝑥={𝑥} ∈Ω (𝑁, 𝑀) for each 𝑛 𝑛∈Z𝛽 3𝑇 . It follows from (42) ∞ ∞ 1 and (59)–(61)thatforany𝑥={𝑥𝑛}𝑛∈Z , 𝑦={𝑦𝑛}𝑛∈Z ∈ + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) 𝛽 𝛽 𝑑1𝑡 𝑑𝑘𝑡 Ω3𝑇(𝑁, 𝑀),and𝑛≥𝑇, 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑐1𝑡 𝑐𝑘𝑡 1 ∞ 1 = (−𝐿 −𝑛+𝜏 𝑥 + ∑ 𝑓(𝑡,𝑦𝑏 ,...,𝑦𝑏 ) ∞ ∞ ∞ 𝑡−𝑠+1 𝛾 𝑎 1𝑡 𝑘𝑡 − ∑ ∑∑ [𝑝 (𝑡, 𝑦 ,...,𝑦 )−𝑟 ]) 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 ∞ ∞ 1 ∞ + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦𝑑 ,...,𝑦𝑑 ) 1 𝑀 𝑊 1𝑡 𝑘𝑡 𝑡 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 ≥ (𝐿 − − ∑ 󵄨 󵄨 𝛾𝑛+𝜏 𝛾𝑛+𝜏 󵄨𝑎𝑡󵄨 𝑡=𝑇+𝜏 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ 1 ∞ ∞ ∞ − ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑡−𝑠+1 󵄨𝑎 󵄨 𝑡 𝑡 − ∑ ∑∑ [𝑝 (𝑡, 𝑦 ,...,𝑦 ) 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗

∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 − ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨)) −𝑟 ]) 𝑎 𝑡 󵄨 𝑡󵄨 𝑡 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 Abstract and Applied Analysis 17

∞ 1 𝑀 𝑊𝑡 Then (2) possesses uncountably many bounded positive solu- ≤ (−𝐿 + − ∑ ∞ 𝛾 𝛾 󵄨 󵄨 𝑙 𝑛+𝜏 𝑛+𝜏 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨 tions in 𝛽 . ∞ ∞ 1 Proof. Let 𝐿 ∈ (𝑁(1 + 1/𝑑), 𝑀(1.Itfollowsfrom( −1/𝑐)) 13) − ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 that there exists an integer 𝑇≥𝑛0 +𝑛1 +𝜏+|𝛽|satisfying 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨

∞ ∞ ∞ ∞ 𝑡−𝑠+1 𝑊𝑡 󵄨 󵄨 ∑ 󵄨 󵄨 − ∑ ∑ ∑ (𝑅𝑡 + 󵄨𝑟𝑡󵄨)) 󵄨 󵄨 󵄨 󵄨 󵄨𝑎𝑡󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 𝑎𝑗 𝑡=𝑇 1 𝑀 𝑀 ∞ ∞ 1 < (−𝐿 − − {𝐿 − 𝑁, 𝑀− −𝐿}) + ∑∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝛾 𝑐 min 𝑐 󵄨 󵄨 𝑡 𝑡 𝑛+𝜏 𝑠=𝑇𝑡=𝑠 󵄨𝑎𝑠󵄨 −𝑀 (64) ≤ , ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 𝛾𝑛+𝜏 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 𝑁 𝑀 1 ∞ 1 < min {𝐿 − −𝑁,𝑀− −𝐿}. = (−𝐿 − 𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) 𝑑 𝑐 𝑛+𝜏 𝑏1𝑡 𝑏𝑘𝑡 𝛾𝑛+𝜏 𝑡=𝑛+𝜏 𝑎𝑡 𝑈 𝑆 :Ω (𝑁, 𝑀) →𝑙∞ ∞ ∞ 1 Let the mappings 𝐿 and 𝐿 3𝑇 𝛽 be defined + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦 ,...,𝑦 ) by (42)and(61), respectively. By means of (42), (56), (61), and 𝑑1𝑡 𝑑𝑘𝑡 𝑠=𝑛+𝜏𝑡=𝑠 𝑎𝑠 𝑥={𝑥} 𝑦={𝑦} ∈ (64), we deduce that for any 𝑛 𝑛∈Z𝛽 , 𝑛 𝑛∈Z𝛽 Ω (𝑁, 𝑀) 𝑛≥𝑇 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 3𝑇 ,and , 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ ∞ 𝑡−𝑠+1 (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 − ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 ) 𝑎 1𝑡 𝑘𝑡 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑗 1 = (−𝐿−𝑥𝑛+𝜏 𝛾𝑛+𝜏 −𝑟𝑡]) ∞ 1 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) 𝑏1𝑡 𝑏𝑘𝑡 ∞ 𝑡=𝑛+𝜏 𝑎𝑡 1 𝑊𝑡 ≥ (−𝐿 −𝑛+𝜏 𝑥 + ∑ 󵄨 󵄨 𝛾 󵄨𝑎 󵄨 ∞ ∞ 𝑛+𝜏 𝑡=𝑇+𝜏 󵄨 𝑡󵄨 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦𝑑 ,...,𝑦𝑑 ) 𝑎 1𝑡 𝑘𝑡 ∞ ∞ 1 𝑠=𝑛+𝜏𝑡=𝑠 𝑠 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨𝑎 󵄨 𝑡 𝑡 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨 𝑠󵄨 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑐1𝑡 𝑐𝑘𝑡 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨)) ∞ ∞ ∞ 𝑎 𝑡 󵄨 𝑡󵄨 𝑡−𝑠+1 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 − ∑ ∑∑ [𝑝 (𝑡, 𝑦 ,...,𝑦 )−𝑟 ]) 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 1 𝑀 > (−𝐿 + {𝐿 − 𝑁, 𝑀− −𝐿}) 𝛾 min 𝑐 ∞ 𝑛+𝜏 1 𝑀 𝑊𝑡 ≤ (−𝐿 + − ∑ 󵄨 󵄨 −𝑁 𝛾 𝛾 󵄨𝑎 󵄨 ≥ , 𝑛+𝜏 𝑛+𝜏 𝑡=𝑇+𝜏 󵄨 𝑡󵄨 𝛾𝑛+𝜏 ∞ ∞ 1 − ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] (62) 󵄨 󵄨 𝑡 𝑡 𝑠=𝑇+𝜏𝑡=𝑠 󵄨𝑎𝑠󵄨 which yield that 𝑈𝐿𝑥+𝑆𝐿𝑦∈Ω3𝑇(𝑁, 𝑀).Therestofthe proof is similar to that of Theorem 9 and hence is omitted. ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 This completes the proof. − ∑ ∑∑ (𝑅𝑡 + 󵄨𝑟𝑡󵄨)) 𝑗=𝑇+𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 Theorem 12. 𝑛 ∈ N 𝑀 Assume that there exist constants 1 𝑛0 , , 𝑁, 𝑐,and𝑑 with 𝑀>𝑁>0and 𝑀(1 − 1/𝑐) > 𝑁(1 +1/𝑑), 1 𝑀 𝑁 𝑀 < (−𝐿 − − min {𝐿 − −𝑁,𝑀− −𝐿}) 𝑑>𝑐>1, and nonnegative sequences {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , 𝑛0 𝑛0 𝛾𝑛+𝜏 𝑐 𝑑 𝑐 {𝑄𝑛}𝑛∈N ,and{𝑅𝑛}𝑛∈N satisfying (13), (56),and 𝑛0 𝑛0 −𝑀 ≤ , −𝑑 ≤ 𝛾 ≤−𝑐, ∀𝑛≥𝑛. 𝑛 1 (63) 𝛾𝑛+𝜏 18 Abstract and Applied Analysis

∞ (𝑈𝐿𝑥)𝑛 +(𝑆𝐿𝑦)𝑛 Define a mapping 𝑆𝐿 :Ω1(𝑁, 𝑀)𝛽 →𝑙 by 1 ∞ 1 ∞ 𝑛+2𝑖𝜏−1 1 = (−𝐿 − 𝑥 + ∑ 𝑓(𝑡,𝑦 ,...,𝑦 ) { 𝑛+𝜏 𝑏 𝑏 {𝐿+∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝛾 𝑎 1𝑡 𝑘𝑡 { 𝑎 1𝑡 𝑘𝑡 𝑛+𝜏 𝑡=𝑛+𝜏 𝑡 { 𝑖=1 𝑡=𝑛+(2𝑖−1)𝜏 𝑡 { { ∞ 𝑛+2𝑖𝜏−1 ∞ 1 ∞ ∞ { +∑ ∑ ∑ [ (𝑡−𝑠+1) 1 { + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑦𝑑 ,...,𝑦𝑑 ) { 𝑡=𝑠 𝑎𝑠 𝑎 1𝑡 𝑘𝑡 { 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 𝑠=𝑛+𝜏 𝑡=𝑠 𝑠 { { ×ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) { 1𝑡 𝑘𝑡 { −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] −𝑔 (𝑡,𝑐 𝑦 ,...,𝑦𝑐 )] (𝑆 𝑥) = 1𝑡 𝑘𝑡 1𝑡 𝑘𝑡 𝐿 𝑛 { ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ { 𝑡−𝑠+1 { −∑ ∑ ∑ ∑ ∞ ∞ ∞ { 𝑎 𝑡−𝑠+1 { 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 − ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑦 ,...,𝑦𝑜 )−𝑟𝑡]) { 𝑎 1𝑡 𝑘𝑡 { ×[𝑝(𝑡,𝑥 ,...,𝑥 ) 𝑗=𝑛+𝜏𝑠=𝑗𝑡=𝑠 𝑗 { 𝑜1𝑡 𝑜𝑘𝑡 { { −𝑟𝑡], { 1 𝑁 ∞ 𝑊 { 𝑛≥𝑇, ≥ (−𝐿 + −𝑥 + ∑ 𝑡 { 𝑛+𝜏 󵄨 󵄨 (𝑆 𝑥) , 𝛽≤𝑛<𝑇, 𝛾𝑛+𝜏 𝑑 𝑡=𝑇+𝜏 󵄨𝑎𝑡󵄨 { 𝐿 𝑇 (68) ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] for each 𝑥={𝑥𝑛}𝑛∈Z ∈Ω1(𝑁, 𝑀).Inviewof(12), (67), and 󵄨 󵄨 𝑡 𝑡 𝛽 𝑠=𝑇+𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝑥={𝑥} ∈Ω(𝑁, 𝑀) (68), we deduce that for every 𝑛 𝑛∈Z𝛽 1 and 𝑛≥𝑇, ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 󵄨 󵄨 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨)) 󵄨(𝑆 𝑥) −𝐿󵄨 𝑎 𝑡 󵄨 𝑡󵄨 󵄨 𝐿 𝑛 󵄨 𝑗=𝑇+𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 󵄨 󵄨 ∞ 𝑛+2𝑖𝜏−1 󵄨 1 1 𝑁 𝑁 𝑀 = 󵄨∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨 𝑎 1𝑡 𝑘𝑡 > (−𝐿 + + min {𝐿 − −𝑁,𝑀− −𝐿}) 󵄨𝑖=1 𝑡=𝑛+(2𝑖−1)𝜏 𝑡 𝛾𝑛+𝜏 𝑑 𝑑 𝑐 ∞ 𝑛+2𝑖𝜏−1 ∞ 1 −𝑁 + ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) ≥ , 𝑑1𝑡 𝑑𝑘𝑡 𝑖=1 𝑡=𝑠 𝑎𝑠 𝛾𝑛+𝜏 𝑠=𝑛+(2𝑖−1)𝜏 (65) −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡

𝑈 𝑥+𝑆𝑦∈Ω (𝑁, 𝑀) ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 which mean that 𝐿 𝐿 3𝑇 .Therestofthe − ∑ ∑ ∑ ∑ 𝑎 proof is similar to that of Theorem 9 and hence is omitted. 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 This completes the proof. 󵄨 󵄨 Theorem 13. 𝑛 ∈ N 𝑀 ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟] 󵄨 Assume that there exist constants 1 𝑛0 , , 𝑜1𝑡 𝑜𝑘𝑡 𝑡 󵄨 󵄨 and 𝑁 with 𝑀>𝑁>0and nonnegative sequences {𝑊𝑛}𝑛∈N , 𝑛0 {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N ,and{𝑅𝑛}𝑛∈N satisfying (12), (13),and ∞ 𝑛+2𝑖𝜏−1 𝑛0 𝑛0 𝑛0 𝑊𝑡 ≤ ∑ ∑ 󵄨 󵄨 󵄨𝑎 󵄨 𝑖=1 𝑡=𝑛+(2𝑖−1)𝜏 󵄨 𝑡󵄨 𝛾𝑛 =1, ∀𝑛≥𝑛1. (66) ∞ 𝑛+2𝑖𝜏−1 ∞ 1 + ∑ ∑ ∑󵄨 󵄨 [(t −𝑠+1) 𝑄𝑡 +𝑃𝑡] 󵄨𝑎 󵄨 Then (2) possesses uncountably many bounded positive solu- 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 𝑡=𝑠 󵄨 𝑠󵄨 tions in Ω1(𝑁, 𝑀) ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) Proof. Let 𝐿 ∈ (𝑁,.Equation( 𝑀) 13) ensures that there exists 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 𝑇≥𝑛0 +𝑛1 +𝜏+|𝛽|sufficiently large such that ∞ ∞ ∞ 𝑊𝑡 1 ≤ ∑ 󵄨 󵄨 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] ∞ 𝑊 ∞ ∞ 1 󵄨𝑎 󵄨 󵄨𝑎 󵄨 𝑡 𝑡 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑡=𝑇 󵄨 𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨 𝑠󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 𝑡=𝑇 󵄨𝑎𝑡󵄨 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 󵄨𝑎 󵄨 + ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) (67) 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 󵄨 󵄨 < min {𝑀−𝐿,𝐿−𝑁} , < min {𝑀−𝐿, }𝐿−𝑁 . (69) Abstract and Applied Analysis 19

𝑆 (Ω (𝑁, 𝑀)) ⊆Ω (𝑁, 𝑀) ‖𝑆 𝑥‖ ≤ ∞ 𝑊 ∞ ∞ 1 which means that 𝐿 1 1 and 𝐿 ≤2∑ 𝑡 +2∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑀 𝑥∈Ω(𝑁, 𝑀) 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 for all 1 .Itfollowsfrom(13)thatforeach 𝑡=𝑉 󵄨𝑎𝑡󵄨 𝑠=𝑉 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝜀>0, there exists 𝑉>𝑇satisfying (22). Using (22)and (68), we obtain that for any 𝑥={𝑥𝑛}𝑛∈Z ∈Ω1(𝑁, 𝑀) and ∞ ∞ ∞ 𝛽 𝑡−𝑠+1 󵄨 󵄨 𝑚, 𝑛 >𝑉 +2∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) , 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑗=𝑉 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 󵄨 󵄨 󵄨(𝑆 𝑥) −(𝑆 𝑥) 󵄨 󵄨 𝐿 𝑚 𝐿 𝑛󵄨 <𝜀, 󵄨 󵄨 ∞ 𝑚+2𝑖𝜏−1 (70) 󵄨 1 = 󵄨∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨 𝑎 1𝑡 𝑘𝑡 󵄨𝑖=1 𝑡=𝑚+(2𝑖−1)𝜏 𝑡 which yields that 𝑆𝐿(Ω1(𝑁, 𝑀)) is uniformly Cauchy. ∞ 𝑛+2𝑖𝜏−1 1 Now we prove that 𝑆𝐿 is continuous in Ω1(𝑁, 𝑀).Sup- 𝑚 − ∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑎 1𝑡 𝑘𝑡 pose that {𝑥 }𝑚∈𝑁 is an arbitrary sequence in Ω1(𝑁, 𝑀) 𝑖=1 𝑡=𝑛+(2𝑖−1)𝜏 𝑡 𝑚 𝑚 and 𝑥∈Ω1(𝑁, 𝑀) with lim𝑚→∞𝑥 =𝑥,where𝑥 = 𝑚 ∞ 𝑚+2𝑖𝜏−1 ∞ {𝑥𝑛 } for each 𝑚∈𝑁and 𝑥={𝑥𝑛}𝑛∈Z .Using(12), 1 𝑛∈𝑍𝛽 𝛽 𝑚 + ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 𝑥 =𝑥 𝑓, 𝑔, ℎ 𝑝 𝑎 1𝑡 𝑘𝑡 (13), lim𝑚→∞ ,andthecontinuityof ,and , 𝑖=1 𝑠=𝑚+(2𝑖−1)𝜏 𝑡=𝑠 𝑠 we conclude that for given 𝜀>0, there exist 𝑇1,𝑇2, T3,and 𝑇4 ∈ N with 𝑇4 >𝑇3 >𝑇2 >𝑇1 >𝑇satisfying (19)and(20). −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡 It follows from (19), (20), and (68)that

∞ 𝑛+2𝑖𝜏−1 ∞ 1 󵄩 𝑚 󵄩 − ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄩𝑆𝐿𝑥 −𝑆𝐿𝑥󵄩 𝑑1𝑡 𝑑𝑘𝑡 𝑡=𝑠 𝑎𝑠 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 󵄨 𝑚 󵄨 = sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨 𝑛∈Z −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝛽 𝑐1𝑡 𝑐𝑘𝑡 󵄨 󵄨 = { 󵄨(𝑆 𝑥𝑚) −(𝑆 𝑥) 󵄨 , ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 max sup 󵄨 𝐿 𝑛 𝐿 𝑛󵄨 + ∑ ∑ ∑ ∑ 𝑇>𝑛≥𝛽 𝑎 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 󵄨 𝑚 󵄨 sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨} ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟] 𝑛≥𝑇 𝑜1𝑡 𝑜𝑘𝑡 𝑡 󵄨 󵄨 ∞ 𝑛+2𝑖𝜏−1 1 ∞ 𝑚+2𝑖𝜏−1 ∞ ∞ 󵄨 𝑚 𝑚 𝑡−𝑠+1 = sup 󵄨𝐿+∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) − ∑ ∑ ∑ ∑ 󵄨 𝑎 1𝑡 𝑘𝑡 𝑎 𝑛≥𝑇 󵄨 𝑖=1𝑡=𝑛+(2𝑖−1)𝜏 𝑡 𝑖=1 𝑗=𝑚+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 ∞ 𝑛+2𝑖𝜏−1 ∞ 1 󵄨 + ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑚 ,...,𝑥𝑚 ) 󵄨 𝑑1𝑡 𝑑𝑘𝑡 ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟] 󵄨 𝑖=1 𝑡=𝑠 𝑎𝑠 𝑜1𝑡 𝑜𝑘𝑡 𝑡 󵄨 𝑠=𝑛+(2𝑖−1)𝜏 󵄨 −𝑔 (𝑡,𝑚 𝑥 ,...,𝑥𝑚 )] ∞ 𝑚+2𝑖𝜏−1 ∞ 𝑛+2𝑖𝜏−1 𝑐1𝑡 𝑐𝑘𝑡 W𝑡 𝑊𝑡 ≤ ∑ ∑ 󵄨 󵄨 + ∑ ∑ 󵄨 󵄨 ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 󵄨𝑎 󵄨 󵄨𝑎 󵄨 𝑡−𝑠+1 𝑖=1 𝑡=𝑚+(2𝑖−1)𝜏 󵄨 𝑡󵄨 𝑖=1 𝑡=𝑛+(2𝑖−1)𝜏 󵄨 𝑡󵄨 𝑚 𝑚 −∑ ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡] 𝑎 1𝑡 𝑘𝑡 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗𝑡=𝑠 𝑗 ∞ 𝑚+2𝑖𝜏−1 ∞ 1 + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] ∞ 𝑛+2𝑖𝜏−1 1 󵄨𝑎 󵄨 𝑡 𝑡 𝑖=1 𝑠=𝑚+(2𝑖−1)𝜏 𝑡=𝑠 󵄨 𝑠󵄨 −(𝐿+∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑎 1𝑡 𝑘𝑡 𝑖=1𝑡=𝑛+(2𝑖−1)𝜏 𝑡 ∞ 𝑛+2𝑖𝜏−1 ∞ 1 ∞ 𝑛+2𝑖𝜏−1 ∞ 1 + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨𝑎 󵄨 𝑡 𝑡 + ∑ ∑ ∑ [ (𝑡−𝑠+1) 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 𝑡=𝑠 󵄨 𝑠󵄨 𝑎 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 𝑡=𝑠 𝑠

∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 ×ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨 󵄨 1𝑡 𝑘𝑡 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) 󵄨𝑎 󵄨 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡

∞ 𝑚+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) − ∑ ∑ ∑∑ 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑎 𝑖=1 𝑗=𝑚+(2𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗𝑡=𝑠 𝑗 20 Abstract and Applied Analysis

󵄨 ∞ 𝑛+2𝑖𝜏−1 ∞ 󵄨 1 ×[𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡])󵄨 + ∑ ∑ ∑ [ (𝑡−𝑠+1) 1𝑡 𝑘𝑡 󵄨 𝑎 󵄨 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 𝑡=𝑠 𝑠 ∞ 𝑛+2𝑖𝜏−1 1 󵄨 󵄨 𝑚 𝑚 ≤ ∑ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) ×ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 1𝑡 𝑘𝑡 𝑖=1𝑡=𝑛+(2𝑖−1)𝜏 󵄨 𝑡󵄨

󵄨 −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] −𝑓 (𝑡, 𝑥 ,...,𝑥 )󵄨 1𝑡 𝑘𝑡 𝑏1𝑡 𝑏𝑘𝑡 󵄨 ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 ∞ 𝑛+2𝑖𝜏−1 ∞ 1 󵄨 󵄨 𝑚 𝑚 − ∑ ∑ ∑∑ + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 𝑎 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 𝑖=1𝑗=𝑛+(2𝑖−1)𝜏𝑠=𝑗𝑡=𝑠 𝑗 𝑖=1𝑠=𝑛+(2𝑖−1)𝜏𝑡=𝑠 󵄨 𝑠󵄨

󵄨 ×[𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 )−𝑟𝑡], −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 1𝑡 𝑘𝑡 𝑑1𝑡 𝑑𝑘𝑡 󵄨 ∀𝑛 ≥ 𝑇 + 𝜏, 󵄨 𝑚 𝑥 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 ∞ 𝑛+(2𝑖−1)𝜏−1 󵄨 1 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑥𝑛−𝜏 =𝐿+∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑐1𝑡 𝑐𝑘𝑡 󵄨 𝑎 1𝑡 𝑘𝑡 𝑖=1𝑡=𝑛+(2𝑖−2)𝜏 𝑡 ∞ 𝑛+2𝑖𝜏−1 ∞ ∞ 𝑡−𝑠+1 ∞ 𝑛+(2𝑖−1)𝜏−1 ∞ + ∑ ∑ ∑ ∑ 󵄨 󵄨 1 𝑡=𝑠 󵄨𝑎 󵄨 + ∑ ∑ ∑ [ (𝑡−𝑠+1) 𝑖=1 𝑗=𝑛+(2𝑖−1)𝜏 𝑠=𝑗 󵄨 𝑗󵄨 𝑎 𝑖=1 𝑠=𝑛+(2𝑖−2)𝜏 𝑡=𝑠 𝑠 󵄨 󵄨 𝑚 𝑚 × 󵄨𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 ) 󵄨 1𝑡 𝑘𝑡 ×ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑑1𝑡 𝑑𝑘𝑡 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡

∞ 1 󵄨 󵄨 ∞ 𝑛+(2𝑖−1)𝜏−1 ∞ ∞ 󵄨 𝑚 𝑚 󵄨 𝑡−𝑠+1 ≤ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )−𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )󵄨 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 1𝑡 𝑘𝑡 󵄨 − ∑ ∑ ∑ ∑ 𝑡=𝑇 󵄨 𝑡󵄨 𝑎 𝑖=1 𝑗=𝑛+(2𝑖−2)𝜏 𝑠=𝑗 𝑡=𝑠 𝑗

∞ ∞ 1 󵄨 𝑚 𝑚 + ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟], 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑠=𝑇 𝑡=𝑠 󵄨𝑎𝑠󵄨 󵄨 ∀𝑛 ≥ 𝑇 + 𝜏, −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 1𝑡 𝑘𝑡 󵄨 (72)

󵄨 𝑚 𝑥 󵄨 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 )−𝑔(𝑡,𝑥 ,...,𝑥 )󵄨] 󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑐1𝑡 𝑐𝑘𝑡 󵄨 which imply that

∞ ∞ ∞ 𝑥 +𝑥 𝑡−𝑠+1󵄨 𝑚 𝑚 𝑛 𝑛−𝜏 + ∑ ∑ ∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑗=𝑇 𝑠=𝑗 𝑡=𝑠 󵄨𝑎 󵄨 󵄨 𝑗󵄨 ∞ 1 =2𝐿+∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨 𝑎 1𝑡 𝑘𝑡 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑡=𝑛 𝑡 𝑜1𝑡 𝑜𝑘𝑡 󵄨 ∞ ∞ 1 + ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) <𝜀, 𝑑1𝑡 𝑑𝑘𝑡 𝑠=𝑛 𝑡=𝑠 𝑎𝑠 (71) −𝑔 (𝑡, 𝑥 ,..., )] 𝑐1𝑡 x𝑐𝑘𝑡 which implies that S𝐿 is continuous in Ω1(𝑁, 𝑀).Thus Lemma 4 means that 𝑆𝐿 possesses a fixed point 𝑥={𝑥𝑛}𝑛∈Z ∈ ∞ ∞ ∞ 𝛽 𝑡−𝑠+1 Ω (𝑁, 𝑀) − ∑ ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟], 1 ;thatis, 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑗=𝑛 𝑠=𝑗 𝑡=𝑠 𝑎𝑗

∞ 𝑛+2𝑖𝜏−1 1 ∀𝑛 ≥ 𝑇 + 𝜏, 𝑥𝑛 =𝐿+∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑎 1𝑡 𝑘𝑡 𝑖=1 𝑡=𝑛+(2𝑖−1)𝜏 𝑡 (73) Abstract and Applied Analysis 21 which means that Proof. Let 𝐿 ∈ (𝑁,.Itfollowsfrom( 𝑀) 76) that there exists 𝑇≥𝑛0 +𝑛1 + 𝜏 + |𝛽| sufficiently large such that ∞ ∞ 𝑊𝑡 ∑ ∑ 󵄨 󵄨 Δ(𝑎𝑛Δ(𝑥𝑛 +𝑥𝑛−𝜏)) + Δ𝑓 (𝑛,𝑏 𝑥 ,...,𝑥𝑏 ) 󵄨 󵄨 1𝑛 𝑘𝑛 𝑖=1 𝑡=𝑇+𝑖𝜏 󵄨𝑎𝑡󵄨

∞ ∞ ∞ +𝑔(𝑛,𝑥𝑐 ,...,𝑥𝑐 ) 1 1𝑛 𝑘𝑛 + ∑ ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 𝑡=𝑠 󵄨𝑎𝑠󵄨 ∞ 𝑖=1 𝑠=𝑇+𝑖𝜏 (77) = ∑ℎ(𝑛,𝑥 ,...,𝑥 ) 𝑑1𝑛 𝑑𝑘𝑛 𝑡=𝑛 ∞ ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 ∞ 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 − ∑ (𝑡−𝑛+1) [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟], 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑡=𝑛 (74) < min {𝑀−𝐿,𝐿−𝑁} . ∀𝑛 ≥ 𝑇 + 𝜏, ∞ Define a mapping 𝑆𝐿 :Ω1(𝑁, 𝑀)𝛽 →𝑙 by

3 3 Δ (𝑎 Δ(𝑥 +𝑥 )) + Δ 𝑓(𝑛,𝑥 ,...,𝑥 ) (𝑆𝐿𝑥)𝑛 𝑛 𝑛 𝑛−𝜏 𝑏1𝑛 𝑏𝑘𝑛 ∞ ∞ 1 2 {𝐿−∑ ∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) +Δ 𝑔(𝑛,𝑥𝑐 ,...,𝑥𝑐 )+Δℎ(𝑛,𝑥𝑑 ,...,𝑥𝑑 ) { 𝑏1𝑡 𝑏𝑘𝑡 1𝑛 𝑘𝑛 1𝑛 𝑘𝑛 { 𝑎 { 𝑖=1 𝑡=𝑛+𝑖𝜏 𝑡 { ∞ ∞ ∞ 1 { +𝑝(𝑛,𝑥 ,...,𝑥 )=𝑟,∀𝑛≥𝑇+𝜏, { −∑ ∑ ∑ 𝑜1𝑛 𝑜𝑘𝑛 𝑛 { 𝑎 { 𝑖=1𝑠=𝑛+𝑖𝜏 𝑡=𝑠 𝑠 { { ×[(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) { 1𝑡 𝑘𝑡 (78) = −𝑔 (𝑡, 𝑥 ,...,𝑥 )] { 𝑐1𝑡 𝑐𝑘𝑡 𝑥={𝑥} ∈Ω(𝑁, 𝑀) { ∞ ∞ ∞ ∞ which yields that 𝑛 𝑛∈Z𝛽 1 is bounded { 𝑡−𝑠+1 { −∑ ∑ ∑ ∑ positive solution of (2). The rest of the proof is similar to that { 𝑎 { 𝑖=1𝑗=𝑛+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 of Theorem 6 and is omitted. This completes the proof. { { ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟], { 𝑜1𝑡 𝑜𝑘𝑡 𝑡 { { 𝑛≥𝑇, Theorem 14. Assume that there exist constants 𝑛1 ∈ N𝑛 , { 0 (𝑆 𝑥) ,𝛽≤𝑛<𝑇, 𝑀,and𝑁 with 𝑀>𝑁>0and nonnegative sequences { 𝐿 𝑇 {𝑊𝑛}𝑛∈N , {𝑃𝑛}𝑛∈N , {𝑄𝑛}𝑛∈N ,and{𝑅𝑛}𝑛∈N satisfying (12) 𝑛0 𝑛0 𝑛0 𝑛0 𝑥={𝑥} ∈Ω(𝑁, 𝑀) for all 𝑛 𝑛∈Z𝛽 1 .Byvirtueof(12), (77), and and 𝑥={𝑥} ∈Ω(𝑁, 𝑀) (78), we know that for every 𝑛 𝑛∈Z𝛽 1 and 𝑛≥𝑇, 󵄨 󵄨 󵄨(𝑆𝐿𝑥)𝑛 −𝐿󵄨 𝛾𝑛 = −1, ∀𝑛 ≥1 𝑛 ; (75) 󵄨 󵄨 ∞ ∞ 󵄨 1 ∞ ∞ = 󵄨−∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑊 󵄨 𝑎 1𝑡 𝑘𝑡 𝑡 󵄨 𝑖=1𝑡=𝑛+𝑖𝜏 𝑡 max {∑ ∑ 󵄨 󵄨, 𝑖=1 𝑡=𝑛 +𝑖𝜏 󵄨𝑎𝑡󵄨 0 ∞ ∞ ∞ 1 − ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑑1𝑡 𝑑𝑘𝑡 𝑖=1 𝑠=𝑛+𝑖𝜏 𝑡=𝑠 𝑎𝑠 ∞ ∞ ∞ 1 ∑ ∑ ∑ [𝑃 + (𝑡−𝑠+1) 𝑄 ], 󵄨 󵄨 𝑡 𝑡 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 󵄨𝑎 󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑖=1 𝑠=𝑛0+𝑖𝜏 𝑡=𝑠 󵄨 𝑠󵄨 (76) ∞ ∞ ∞ ∞ 𝑡−𝑠+1 − ∑ ∑ ∑ ∑ ∞ ∞ ∞ ∞ 𝑡−𝑠+1 𝑎 ∑ ∑ ∑ ∑ 󵄨 󵄨 𝑖=1 𝑗=𝑛+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 󵄨 󵄨 𝑖=1 𝑗=𝑛 +𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 󵄨 0 󵄨 󵄨 󵄨 󵄨 ×[𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 )−𝑟𝑡] 󵄨 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 󵄨 ×(𝑅 + 󵄨𝑟 󵄨)}<+∞. 𝑡 󵄨 𝑡󵄨 ∞ ∞ 𝑊𝑡 ≤ ∑ ∑ 󵄨 󵄨 𝑖=1𝑡=𝑛+𝑖𝜏 󵄨𝑎𝑡󵄨 ∞ ∞ ∞ 1 Then (2) possesses uncountably many bounded positive solution + ∑ ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 in Ω1(𝑁, 𝑀). 𝑖=1 𝑠=𝑛+𝑖𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 22 Abstract and Applied Analysis

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 𝑡−𝑠+1 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) + ∑ ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡] 󵄨 󵄨 𝑎 1𝑡 𝑘𝑡 𝑖=1𝑗=𝑛+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 𝑖=1𝑗=𝑛+𝑖𝜏𝑠=𝑗𝑡=𝑠 𝑗 ∞ ∞ 𝑊 ∞ ∞ ∞ ∞ 𝑡−𝑠+1 ≤ ∑ ∑ 𝑡 − ∑ ∑ ∑ ∑ 󵄨 󵄨 𝑎 𝑖=1𝑡=𝑇+𝑖𝜏 󵄨𝑎𝑡󵄨 𝑖=1 𝑗=𝑚+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 𝑗

∞ ∞ ∞ 1 󵄨 + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨𝑎 󵄨 𝑡 𝑡 ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟] 󵄨 𝑖=1 𝑠=𝑇+𝑖𝜏 𝑡=𝑠 󵄨 𝑠󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡 󵄨 󵄨 ∞ ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) ∞ ∞ ∞ ∞ 󵄨 󵄨 󵄨 󵄨 𝑊𝑡 𝑊𝑡 𝑡=𝑠 󵄨𝑎 󵄨 𝑖=1𝑗=𝑇+𝑖𝜏 𝑠=𝑗 󵄨 𝑗󵄨 ≤ ∑ ∑ 󵄨 󵄨 + ∑ ∑ 󵄨 󵄨 𝑖=1 𝑡=𝑛+𝑖𝜏 󵄨𝑎𝑡󵄨 𝑖=1 𝑡=𝑚+𝑖𝜏 󵄨𝑎𝑡󵄨 < min {𝑀−𝐿,𝐿−𝑁} , ∞ ∞ ∞ 1 (79) + ∑ ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 𝑖=1 𝑠=𝑛+𝑖𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 which means that 𝑆𝐿(Ω1(𝑁, 𝑀))1 ⊆Ω (𝑁, 𝑀) and ‖𝑆𝐿𝑥‖ ≤ ∞ ∞ ∞ 𝑀 𝑥∈Ω(𝑁, 𝑀) 1 for each 1 . + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨𝑎 󵄨 𝑡 𝑡 Next we prove that 𝑆𝐿(Ω1(𝑁, 𝑀)) is uniformly Cauchy. It 𝑖=1 𝑠=𝑚+𝑖𝜏 𝑡=𝑠 󵄨 𝑠󵄨 ∗ follows from (76)thatforanygiven𝜀>0there exists 𝑇 ≥𝑇 with ∞ ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑖=1 𝑗=𝑛+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 ∞ ∞ 𝑊𝑡 ∑ ∑ 󵄨 󵄨 󵄨𝑎 󵄨 ∞ ∞ ∞ ∞ 𝑖=1 𝑡=𝑇∗+𝑖𝜏 󵄨 𝑡󵄨 𝑡−𝑠+1 󵄨 󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅𝑡 + 󵄨𝑟𝑡󵄨) 󵄨𝑎 󵄨 ∞ ∞ ∞ 1 𝑖=1 𝑗=𝑚+𝑖𝜏 𝑠=𝑗 𝑡=s 󵄨 𝑗󵄨 + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨𝑎 󵄨 𝑡 𝑡 𝑖=1 𝑠=𝑇∗+𝑖𝜏 𝑡=𝑠 󵄨 𝑠󵄨 ∞ ∞ 𝑊 (80) ≤2∑ ∑ 𝑡 ∞ ∞ ∞ ∞ 󵄨 󵄨 𝑡−𝑠+1 󵄨 󵄨 𝑖=1 𝑡=𝑇∗+𝑖𝜏 󵄨𝑎𝑡󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 (𝑅 + 󵄨𝑟 󵄨) 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝑖=1𝑗=𝑇∗+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 ∞ ∞ ∞ 1 𝜀 +2∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 𝑄 +𝑃] < . 󵄨𝑎 󵄨 𝑡 𝑡 2 𝑖=1 𝑠=𝑇∗+𝑖𝜏 𝑡=𝑠 󵄨 𝑠󵄨

∞ ∞ ∞ ∞ 𝑡−𝑠+1 󵄨 󵄨 +2∑ ∑ ∑ ∑ (𝑅 + 󵄨𝑟 󵄨) It follows from (12), (78), and (80)thatforany𝑥={𝑥𝑛}𝑛∈Z ∈ 󵄨 󵄨 𝑡 󵄨 𝑡󵄨 𝛽 ∗ 󵄨𝑎 󵄨 ∗ 𝑖=1 𝑗=𝑇 +𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨 𝑗󵄨 Ω1(𝑁, 𝑀) and 𝑚, 𝑛 ≥𝑇 , <𝜀, 󵄨 󵄨 󵄨(𝑆𝐿𝑥)𝑚 −(𝑆𝐿𝑥)𝑛󵄨 (81) 󵄨 󵄨 ∞ ∞ 󵄨 1 𝑆 (Ω (𝑁, 𝑀)) = 󵄨∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) which yields that 𝐿 1 is uniformly Cauchy. 󵄨 𝑎 1𝑡 𝑘𝑡 󵄨𝑖=1𝑡=𝑛+𝑖𝜏 𝑡 Now we prove that 𝑆𝐿 is continuous in Ω1(𝑁, 𝑀).Sup- {𝑥𝑚} Ω (𝑁, 𝑀) ∞ ∞ pose that 𝑚∈N is an arbitrary sequence in 1 and 1 𝑥∈Ω(𝑁, 𝑀) 𝑥𝑚 =𝑥 𝑥𝑚 ={𝑥𝑚} − ∑ ∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) 1 with lim𝑚→∞ ,where 𝑛 𝑛∈Z𝛽 𝑏1𝑡 𝑏𝑘𝑡 𝑚 𝑖=1 𝑡=𝑚+𝑖𝜏 𝑎𝑡 𝑚∈N 𝑥={𝑥} 𝑥 = for each and 𝑛 𝑛∈Z𝛽 .By(12), (76), lim𝑚→∞ ∞ ∞ ∞ 𝑥,andthecontinuityof𝑓, 𝑔, ℎ,and𝑝, we get that for given 1 𝜀>0 𝑇 ,𝑇 ,𝑇 ,𝑇 𝑇 ∈ N 𝑇 >𝑇 > + ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) , there exist 1 2 3 4,and 5 with 5 4 𝑎 1𝑡 𝑘𝑡 𝑖=1 𝑠=𝑛+𝑖𝜏 𝑡=𝑠 𝑠 𝑇3 >𝑇2 >𝑇1 >𝑇satisfying

−𝑔 (𝑡, 𝑥 ,...,𝑥 )] ∞ ∞ 𝑐1𝑡 𝑐𝑘𝑡 𝑊 ∑ ∑ 𝑡 ∞ ∞ ∞ 󵄨 󵄨 󵄨𝑎𝑡󵄨 1 𝑖=𝑇1 𝑡=𝑇+𝑖𝜏 󵄨 󵄨 − ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 𝑎 1𝑡 𝑘𝑡 𝑖=1 𝑠=𝑚+𝑖𝜏 𝑡=𝑠 𝑠 ∞ ∞ ∞ 1 + ∑ ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 −𝑔 (𝑡, 𝑦 ,...,𝑦 )] 𝑡=𝑠 󵄨𝑎𝑠󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑖=𝑇1 𝑠=𝑇+𝑖𝜏 󵄨 󵄨 Abstract and Applied Analysis 23

∞ ∞ ∞ ∞ 𝑡−𝑠+1 𝜀 𝑡−𝑠+1 + ∑ ∑ ∑ ∑ 𝑅 < ; 𝐵= { :𝑇≤𝑠≤𝑇+𝑇𝜏, 󵄨 󵄨 𝑡 max 󵄨 󵄨 2 𝑡=𝑠 󵄨𝑎 󵄨 18 󵄨𝑎𝑠󵄨 𝑖=𝑇1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 󵄨 𝑗󵄨 ∞ 𝑊 ∞ ∞ 1 ∑ 𝑡 + ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 𝑠≤𝑡≤𝑇+𝑇3𝜏} , 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨𝑎𝑡󵄨 󵄨𝑎𝑠󵄨 𝑡=𝑇+𝑇2𝜏 󵄨 󵄨 𝑠=𝑇+𝑇2𝜏 𝑡=𝑠 󵄨 󵄨 𝑡−𝑠+1 ∞ ∞ ∞ 𝑡−𝑠+1 𝜀 𝐸= { :𝑇≤𝑗≤𝑇+𝑇𝜏, + ∑ ∑ ∑ 𝑅 < ; max 󵄨 󵄨 2 󵄨 󵄨 𝑡 󵄨𝑎𝑗󵄨 𝑡=𝑠 󵄨𝑎 󵄨 18𝑇1 󵄨 󵄨 𝑗=𝑇+𝑇2𝜏 𝑠=𝑗 󵄨 𝑗󵄨

{ ∞ 1 𝑗≤𝑠≤𝑇+𝑇3𝜏, 𝑠 ≤ 𝑡 ≤4 𝑇+ 𝜏} . ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] max { 󵄨 󵄨 𝑡 𝑡 𝑡=𝑇+𝑇 𝜏 󵄨𝑎𝑠󵄨 { 3 (83) ∞ ∞ 𝑡−𝑠+1 + ∑ ∑ 𝑅 In terms of (12), (78), and (83), we have 󵄨 󵄨 𝑡 𝑠=𝑇+𝑇 𝜏 𝑡=𝑠 󵄨𝑎 󵄨 3 󵄨 𝑗󵄨 󵄩 𝑚 󵄩 󵄩𝑆𝐿𝑥 −𝑆𝐿𝑥󵄩 } 󵄨 𝑚 󵄨 = sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨 :𝑇+𝜏≤𝑠≤𝑇+𝑇2𝜏, 𝑇 + 𝜏 ≤ 𝑗2 ≤𝑇+ 𝜏 } 𝑛∈Z𝛽 } 𝜀 󵄨 󵄨 < ; = { 󵄨(𝑆 𝑥𝑚) −(𝑆 𝑥) 󵄨 , 18𝑇 𝑇 𝜏 max sup 󵄨 𝐿 𝑛 𝐿 𝑛󵄨 1 2 𝛽≥𝑛>𝑇

{ ∞ 𝑡−𝑠+1 󵄨 𝑚 󵄨 max ∑ 󵄨 󵄨 𝑅𝑡 󵄨 󵄨 { 󵄨 󵄨 sup 󵄨(𝑆𝐿𝑥 )𝑛 −(𝑆𝐿𝑥)𝑛󵄨} 𝑡=𝑇+𝑇 𝜏 󵄨𝑎 󵄨 { 4 󵄨 𝑗󵄨 𝑛≥𝑇 󵄨 󵄨 ∞ ∞ } 󵄨 1 𝑚 𝑚 = sup 󵄨𝐿−∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) :𝑇+𝜏≤𝑗≤𝑇+𝑇𝜏, 𝑗 ≤ 𝑠 ≤ 𝑇+ 𝜏 󵄨 𝑎 1𝑡 𝑘𝑡 2 3 } 𝑛≥𝑇 󵄨 𝑖=1𝑡=𝑛+𝑖𝜏 𝑡 } ∞ ∞ ∞ 1 𝜀 − ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥𝑚 ,...,𝑥𝑚 ) < ; 𝑑1𝑡 𝑑𝑘𝑡 2 𝑖=1𝑠=𝑛+𝑖𝜏𝑡=𝑠 𝑎𝑠 18𝑇1𝑇2𝑇3𝜏 𝑚 𝑚 𝑇1−1 𝑇+𝑇2𝜏−1 −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] 󵄨 𝑚 𝑚 󵄨 1𝑡 𝑘𝑡 ∑ ∑ 𝐴 󵄨𝑓(𝑡,𝑥 ,...,𝑥 )−𝑓(𝑡,𝑥 ,...,𝑥 )󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑏1𝑡 𝑏𝑘𝑡 󵄨 ∞ ∞ ∞ ∞ 𝑖=1 𝑡=𝑇+𝑖𝜏 𝑡−𝑠+1 𝑚 𝑚 − ∑ ∑ ∑∑ [𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )−𝑟𝑡] 𝑎 1𝑡 𝑘𝑡 𝑇1−1 𝑇+𝑇2𝜏−1 𝑇+𝑇3𝜏−1 𝑖=1𝑗=𝑛+𝑖𝜏𝑠=𝑗𝑡=𝑠 𝑗 󵄨 𝑚 𝑚 + ∑ ∑ ∑ [𝐵 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑑1𝑡 𝑑𝑘𝑡 ∞ ∞ 𝑖=1 𝑠=𝑇+𝑖𝜏 𝑡=𝑠 1 −(𝐿−∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 󵄨 𝑎 1𝑡 𝑘𝑡 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑖=1𝑡=𝑛+𝑖𝜏 𝑡 𝑑1𝑡 𝑑𝑘𝑡 󵄨 ∞ ∞ ∞ 󵄨 𝑚 𝑚 1 +𝐴󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) − ∑ ∑ ∑ [(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑑1𝑡 𝑑𝑘𝑡 𝑖=1𝑠=𝑛+𝑖𝜏𝑡=𝑠 𝑎𝑠 󵄨 −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )󵄨] 1𝑡 𝑘𝑡 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡 𝑇1−1 𝑇+𝑇2𝜏−1 𝑇+𝑇3𝜏−1 𝑇+𝑇4𝜏−1 󵄨 𝑚 𝑚 ∞ ∞ ∞ ∞ + ∑ ∑ ∑ ∑ 𝐸 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 𝑡−𝑠+1 󵄨 𝑜1𝑡 𝑜𝑘𝑡 − ∑ ∑ ∑∑ 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 𝑖=1𝑗=𝑛+𝑖𝜏𝑠=𝑗𝑡=𝑠 𝑎𝑗 󵄨 𝜀 󵄨 −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 < , 󵄨 1𝑡 𝑘𝑡 󵄨 18 󵄨 ×[𝑝(𝑡,𝑥𝑜 ,...,𝑥𝑜 )−𝑟𝑡])󵄨 1𝑡 𝑘𝑡 󵄨 (82) 󵄨 ∞ ∞ 1 󵄨 𝑚 𝑚 󵄨 ≤ ∑ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 )−𝑓(𝑡,𝑥 ,...,𝑥 )󵄨 where 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑏1𝑡 𝑏𝑘𝑡 󵄨 𝑖=1𝑡=𝑇+𝑖𝜏 󵄨𝑎𝑡󵄨 ∞ ∞ ∞ 1 󵄨 1 󵄨 𝑚 𝑚 𝐴= { :𝑇≤𝑠≤𝑇+𝑇𝜏} , + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) max 󵄨 󵄨 2 󵄨𝑎 󵄨 󵄨 1𝑡 𝑘𝑡 󵄨𝑎𝑠󵄨 𝑖=1𝑠=𝑇+𝑖𝜏𝑡=𝑠 󵄨 𝑠󵄨 24 Abstract and Applied Analysis

󵄨 󵄨 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑑1𝑡 𝑑𝑘𝑡 󵄨 𝑐1𝑡 𝑐𝑘𝑡 󵄨 󵄨 󵄨 𝑚 𝑚 + 󵄨𝑔(𝑡,𝑥𝑐 ,⋅⋅⋅ ,𝑥𝑐 ) ∞ ∞ ∞ 󵄨 1𝑡 𝑘𝑡 1 󵄨 𝑚 𝑚 + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑖=𝑇 𝑠=𝑇+𝑖𝜏𝑡=𝑠 󵄨𝑎𝑠󵄨 𝑐1𝑡 𝑐𝑘𝑡 󵄨 1 ∞ ∞ ∞ ∞ 󵄨 𝑡−𝑠+1󵄨 𝑚 𝑚 󵄨 + ∑ ∑ ∑∑ 󵄨 󵄨 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 1𝑡 𝑘𝑡 󵄨 𝑡=𝑠 󵄨𝑎 󵄨 𝑖=𝑇1𝑗=𝑇+𝑖𝜏𝑠=𝑗 󵄨 𝑗󵄨 󵄨 𝑚 𝑚 󵄨 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑇 −1 𝑇+𝑇 𝜏−1 󵄨 1 2 󵄨 1 󵄨 𝑚 𝑚 −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )󵄨] ≤ ∑ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 ) 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑖=1 𝑡=𝑇+𝑖𝜏 󵄨𝑎𝑡󵄨 𝑇 −1 𝑇+𝑇 𝜏−1 𝑇+𝑇 𝜏−1 𝑇+𝑇 𝜏−1 󵄨 1 2 3 4 𝑡−𝑠+1 −𝑓 (𝑡,𝑏 𝑥 ,...,𝑥𝑏 )󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 󵄨𝑎𝑗󵄨 𝑇1−1 ∞ 1 󵄨 𝑚 𝑚 + ∑ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨 𝑚 𝑚 󵄨𝑎𝑡󵄨 × 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 𝑖=1 𝑡=𝑇+𝑇2𝜏 󵄨 󵄨 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 −𝑓 (𝑡, 𝑥 ,...,𝑥 )󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 ∞ ∞ 1 󵄨 𝑚 𝑚 𝑇 −1 𝑇+𝑇 𝜏−1 𝑇+𝑇 𝜏−1 + ∑ ∑ 󵄨 󵄨 󵄨𝑓(𝑡,𝑥 ,...,𝑥 ) 1 2 3 ∞ 󵄨 󵄨 󵄨 𝑏1𝑡 𝑏𝑘𝑡 𝑡−𝑠+1 𝑖=𝑇 𝑡=𝑇+𝑖𝜏 󵄨𝑎𝑡󵄨 + ∑ ∑ ∑ ∑ 󵄨 󵄨 1 󵄨𝑎 󵄨 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 𝑡=𝑇+𝑇4𝜏 󵄨 𝑗󵄨 󵄨 −𝑓 (𝑡,𝑏 𝑥 ,...,𝑥𝑏 )󵄨 1𝑡 𝑘𝑡 󵄨 󵄨 𝑚 𝑚 × 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑇 −1𝑇+𝑇 𝜏−1𝑇+𝑇 𝜏−1 1 2 3 1 + ∑ ∑ ∑ 󵄨 󵄨 󵄨 󵄨 󵄨 −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 𝑖=1 𝑠=𝑇+𝑖𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 1𝑡 𝑘𝑡 󵄨 󵄨 𝑇 −1 𝑇+𝑇 𝜏−1 ∞ ∞ 󵄨 𝑚 𝑚 1 2 𝑡−𝑠+1 ×[(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨 1𝑡 𝑘𝑡 + ∑ ∑ ∑ ∑ 󵄨 󵄨 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑇+𝑇 𝜏 𝑡=𝑠 󵄨𝑎 󵄨 󵄨 3 󵄨 𝑗󵄨 −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 1𝑡 𝑘𝑡 󵄨 󵄨 𝑚 𝑚 × 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑚 𝑚 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 󵄨 −𝑝 (𝑡,𝑜 𝑥 ,...,𝑥𝑜 )󵄨 󵄨 1𝑡 𝑘𝑡 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 𝑇 −1 ∞ ∞ ∞ 1 𝑡−𝑠+1 + ∑ ∑ ∑ ∑ 𝑇 −1𝑇+𝑇 𝜏−1 ∞ 󵄨 󵄨 1 2 1 𝑡=𝑠 󵄨𝑎 󵄨 𝑖=1 𝑗=𝑇+𝑇2𝜏 𝑠=𝑗 󵄨 𝑗󵄨 + ∑ ∑ ∑ 󵄨 󵄨 𝑖=1 𝑠=𝑇+𝑖𝜏 𝑡=𝑇+𝑇 𝜏 󵄨𝑎𝑠󵄨 󵄨 𝑚 𝑚 3 × 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑚 𝑚 ×[(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 󵄨 ∞ ∞ ∞ ∞ 󵄨 𝑡−𝑠+1 −ℎ (𝑡,𝑑 𝑥 ,...,𝑥𝑑 )󵄨 + ∑ ∑ ∑ ∑ 1𝑡 𝑘𝑡 󵄨 󵄨 󵄨 𝑡=𝑠 󵄨𝑎 󵄨 󵄨 𝑖=𝑇1𝑗=𝑇+𝑖𝜏 𝑠=𝑗 󵄨 𝑗󵄨 󵄨 𝑚 𝑚 + 󵄨𝑔(𝑡,𝑥𝑐 ,...,𝑥𝑐 ) 󵄨 1𝑡 𝑘𝑡 󵄨 𝑚 𝑚 × 󵄨𝑝(𝑡,𝑥 ,...,𝑥 ) 󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] 󵄨 𝑐1𝑡 𝑐𝑘𝑡 󵄨 −𝑝 (𝑡, 𝑥 ,...,𝑥 )󵄨 𝑜1𝑡 𝑜𝑘𝑡 󵄨

𝑇 −1 𝑇 −1𝑇+𝑇 𝜏−1 1 ∞ ∞ 1 2 󵄨 󵄨 1 󵄨 𝑚 𝑚 󵄨 𝑚 𝑚 󵄨 + ∑ ∑ ∑󵄨 󵄨 [(𝑡−𝑠+1) 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) ≤ ∑ ∑ 𝐴 󵄨𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )−𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 )󵄨 󵄨 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 󵄨 1𝑡 𝑘𝑡 1𝑡 𝑘𝑡 󵄨 󵄨𝑎𝑠󵄨 𝑖=1 𝑡=𝑇+𝑖𝜏 𝑖=1 𝑠=𝑇+𝑇2𝜏 𝑡=𝑠 󵄨 󵄨

𝑇1−1𝑇+𝑇2𝜏−1 𝑇+𝑇3𝜏−1 󵄨 󵄨 𝑚 𝑚 −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 + ∑ ∑ ∑ [𝐵 󵄨ℎ(𝑡,𝑥 ,...,𝑥 ) 𝑑1𝑡 𝑑𝑘𝑡 󵄨 󵄨 𝑑1𝑡 𝑑𝑘𝑡 𝑖=1 𝑠=𝑇+𝑖𝜏 𝑡=𝑠 󵄨 𝑚 𝑚 󵄨 + 󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) −ℎ (𝑡, 𝑥 ,...,𝑥 )󵄨 󵄨 𝑐1𝑡 𝑐𝑘𝑡 𝑑1𝑡 𝑑𝑘𝑡 󵄨 Abstract and Applied Analysis 25

󵄨 𝑚 𝑚 +𝐴󵄨𝑔(𝑡,𝑥 ,...,𝑥 ) ×ℎ(𝑡,𝑥𝑑 ,...,𝑥𝑑 ) 󵄨 𝑐1𝑡 𝑐𝑘𝑡 1𝑡 𝑘𝑡 󵄨 −𝑔 (𝑡, 𝑥 ,...,𝑥 )󵄨] −𝑔 (𝑡, 𝑥 ,...,𝑥 )] 𝑐1𝑡 𝑐𝑘𝑡 󵄨 𝑐1𝑡 𝑐𝑘𝑡

𝑇1−1𝑇+𝑇2𝜏−1𝑇+𝑇3𝜏−1𝑇+𝑇4𝜏−1 ∞ ∞ ∞ ∞ 󵄨 𝑚 𝑚 𝑡−𝑠+1 + ∑ ∑ ∑ ∑ 𝐸 󵄨𝑝(𝑡,𝑥 ,⋅⋅⋅ ,𝑥 ) + ∑ ∑ ∑ ∑ 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑎 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 𝑖=1𝑗=𝑛+𝑖𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 󵄨 −𝑝 (𝑡, 𝑥 ,⋅⋅⋅ ,𝑥 )󵄨 ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟], 𝑜1𝑡 𝑜𝑘𝑡 󵄨 𝑜1𝑡 𝑜𝑘𝑡 𝑡

𝑇1−1 ∞ ∞ ∞ ∀𝑛 ≥ 𝑇 + 𝜏, 𝑊𝑡 𝑊𝑡 +2∑ ∑ 󵄨 󵄨 +2∑ ∑ 󵄨 󵄨 󵄨𝑎𝑡󵄨 󵄨𝑎𝑡󵄨 ∞ ∞ 𝑖=1 𝑡=𝑇+𝑇2𝜏 󵄨 󵄨 𝑖=𝑇1 𝑡=𝑇+𝑖𝜏 󵄨 󵄨 1 𝑥𝑛−𝜏 =𝐿−∑ ∑ 𝑓(𝑡,𝑥𝑏 ,...,𝑥𝑏 ) 𝑇 −1𝑇+𝑇 𝜏−1 𝑎 1𝑡 𝑘𝑡 1 2 ∞ 1 𝑖=1 𝑡=𝑛+(𝑖−1)𝜏 𝑡 +2∑ ∑ ∑ 󵄨 󵄨 [(𝑡−𝑠+1) 𝑄𝑡 +𝑃𝑡] 󵄨𝑎 󵄨 ∞ ∞ ∞ 𝑖=1 𝑠=𝑇+𝑖𝜏 𝑡=𝑇+𝑇 𝜏 󵄨 𝑠󵄨 1 3 − ∑ ∑ ∑ 𝑇 −1 𝑎 1 ∞ ∞ 1 𝑖=1 𝑠=𝑛+(𝑖−1)𝜏 𝑡=𝑠 𝑠 +2∑ ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] 󵄨 󵄨 𝑡 𝑡 𝑖=1 𝑠=𝑇+𝑇 𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 ×[(𝑡−𝑠+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 2 𝑑1𝑡 𝑑𝑘𝑡 ∞ ∞ ∞ 1 +2∑ ∑ ∑ [(𝑡−𝑠+1) 𝑄 +𝑃] −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] 󵄨 󵄨 𝑡 𝑡 1𝑡 𝑘𝑡 𝑖=𝑇 𝑠=𝑇+𝑖𝜏 𝑡=𝑠 󵄨𝑎𝑠󵄨 1 ∞ ∞ ∞ ∞ 𝑡−𝑠+1 𝑇 −1 𝑇+𝑇 𝜏−1 𝑇+𝑇 𝜏−1 + ∑ ∑ ∑ ∑ 1 2 3 ∞ 𝑡−𝑠+1 𝑎 +2∑ ∑ ∑ ∑ 󵄨 󵄨 𝑅 𝑖=1𝑗=𝑛+(𝑖−1)𝜏 𝑠=𝑗 𝑡=𝑠 𝑗 󵄨𝑎 󵄨 𝑡 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑗 𝑡=𝑇+𝑇4𝜏 󵄨 𝑗󵄨 ×[𝑝(𝑡,𝑥 ,...,𝑥 )−𝑟], 𝑜1𝑡 𝑜𝑘𝑡 𝑡 𝑇 −1 𝑇+𝑇 𝜏−1 1 2 ∞ ∞ 𝑡−𝑠+1 +2∑ ∑ ∑ ∑ 𝑅 󵄨 󵄨 𝑡 ∀𝑛 ≥ 𝑇 + 𝜏, 𝑖=1 𝑗=𝑇+𝑖𝜏 𝑠=𝑇+𝑇 𝜏 𝑡=𝑠 󵄨𝑎 󵄨 3 󵄨 𝑗󵄨 (85) 𝑇 −1 1 ∞ ∞ ∞ 𝑡−𝑠+1 +2∑ ∑ ∑ ∑ 𝑅 which imply that 󵄨 󵄨 𝑡 𝑡=𝑠 󵄨𝑎 󵄨 𝑖=1 𝑗=𝑇+𝑇2𝜏 𝑠=𝑗 󵄨 𝑗󵄨 Δ(𝑥𝑛 −𝑥𝑛−𝜏) ∞ ∞ ∞ ∞ 𝑡−𝑠+1 +2∑ ∑ ∑ ∑ 𝑅 󵄨 󵄨 𝑡 1 𝑠=𝑗 𝑡=𝑠 󵄨𝑎 󵄨 =− 𝑓(𝑛,𝑥 ,...,𝑥 ) 𝑖=𝑇1 𝑗=𝑇+𝑖𝜏 󵄨 𝑗󵄨 𝑏1𝑛 𝑏𝑘𝑛 𝑎𝑛 𝜀 2(𝑇 −1)(𝑇 −1)𝜏𝜀 < + 1 2 ∞ 1 − ∑ [(𝑡−𝑛+1) ℎ(𝑡,𝑥 ,...,𝑥 ) 18 18𝑇1𝑇2𝜏 𝑑1𝑡 𝑑𝑘𝑡 𝑡=𝑛 𝑎𝑛 (86) 2(𝑇1 −1)𝜀 + −𝑔 (𝑡,𝑐 𝑥 ,...,𝑥𝑐 )] 18𝑇1 1𝑡 𝑘𝑡 ∞ ∞ 2 𝑡−𝑠+1 2(𝑇1 −1)(𝑇2 −1)(𝑇3 −1)𝜏 𝜀 + ∑ ∑ [𝑝 (𝑡, 𝑥 ,...,𝑥 )−𝑟], + 𝑜1𝑡 𝑜𝑘𝑡 𝑡 2 𝑠=𝑛 𝑡=𝑠 𝑎𝑛 18𝑇1𝑇2𝑇3𝜏 2𝜀 ∀𝑛 ≥ 𝑇 + 𝜏, + <𝜀, ∀𝑚≥𝑇, 18 5 (84) which yields that 𝑆 Ω (𝑁, 𝑀) 3 3 which yields that 𝐿 is continuous in 1 .Itfollows Δ (𝑎𝑛Δ(𝑥𝑛 −𝑥𝑛−𝜏)) + Δ 𝑓(𝑛,𝑥𝑏 ,...,𝑥𝑏 ) 𝑆 𝑥={𝑥} ∈ 1𝑛 𝑘𝑛 from Lemmas 2 and 4 that 𝐿 has a fixed point 𝑛 𝑛∈Z𝛽 Ω (𝑁, 𝑀) +Δ2𝑔(𝑛,𝑥 ,...,𝑥 )+Δℎ(𝑛,𝑥 ,...,𝑥 ) 1 ;thatis, 𝑐1𝑛 𝑐𝑘𝑛 𝑑1𝑛 𝑑𝑘𝑛 (87) ∞ ∞ 1 +𝑝(𝑛,𝑥𝑜 ,...,𝑥𝑜 )=𝑟𝑛,∀𝑛≥𝑇+𝜏; 𝑥 =𝐿−∑ ∑ 𝑓(𝑡,𝑥 ,...,𝑥 ) 1𝑛 𝑘𝑛 𝑛 𝑏1𝑡 𝑏𝑘𝑡 𝑖=1 𝑡=𝑛+𝑖𝜏 𝑎𝑡 that is, 𝑥={𝑥𝑛}𝑛∈Z ∈Ω1(𝑁, 𝑀) is a bounded positive ∞ ∞ ∞ 1 𝛽 − ∑ ∑ ∑ [ (𝑡−𝑠+1) solution of (2). The rest of the proof is similar to that of 𝑖=1 𝑠=𝑛+𝑖𝜏 𝑡=𝑠 𝑎𝑠 Theorem 6 and is omitted. This is completes proof. 26 Abstract and Applied Analysis

4. Applications Example 3. Consider the fourth order nonlinear neutral delay difference equation: Now we display nine examples as applications of the results presented in Section 3.

3 7 3 Example 1. Consider the fourth order nonlinear neutral delay Δ ((𝑛 −𝑛 +2) difference equation: 2 4 2 4𝑛 cos (3𝑛 +2) (−1)𝑛𝑛2 ×Δ (𝑥 − 𝑥 )) Δ3 ((𝑛5 +2𝑛)Δ(𝑥 + 𝑥 )) 𝑛 5𝑛2 +2𝑛 𝑛−𝜏 𝑛 4𝑛2 +𝑛+3 𝑛−𝜏 +Δ3 (√𝑛−9𝑥3 𝑥5 ) 3 17 √ 5 √ 𝑛3−5 𝑛+11 +Δ (𝑛𝑥𝑛2−𝑛 + 𝑛𝑥𝑛2−𝑛 − 𝑛−3) √ 21 √ 12 𝑛 21 2 2 𝑛+3𝑥𝑛3−4 − 2𝑛 + 5𝑥𝑛2−1 2 (−1) 𝑥 +5𝑛 +1 +Δ ( ) +Δ ( 𝑛−3 ) 𝑛5 +3𝑛𝑥2 𝑥2 √𝑛7 −17(𝑥4 +3𝑛2 +2) 𝑛3−4 𝑛2−1 𝑛−3 (90) 4 15 2 3 5 (𝑥𝑛−7 +𝑥𝑛4−1 +𝑛) 12√𝑛−𝑥2𝑛−1cos (𝑛 𝑥2𝑛−1) +Δ( ) +Δ( 󵄨 󵄨 ) (88) 󵄨 5 3󵄨 5 󵄨 󵄨 󵄨(𝑛 +𝑛−7 𝑥 ) (𝑛 +𝑛 𝑥 4−1) 󵄨 +1 2+𝑛 󵄨𝑥2𝑛−1󵄨 󵄨 󵄨 2 3 5 𝑛−1 11 9 𝑛 𝑥𝑛2−3𝑥𝑛−4 + ln 𝑛 (−1) 𝑛 sin (𝑛𝑥𝑛3−3𝑛) + + 2 4 6 𝑛 +𝑥 +(𝑛+𝑥 2 ) √𝑛2 +2𝑛+5(1+𝑥22 )+𝑛5 𝑛−4 𝑛 −3 𝑛3−3𝑛 𝑛+1 5 3 (−1) (4𝑛 −3) 4𝑛 +2𝑛−1 = , = , 9 2 (𝑛+1)12√3𝑛2 +7 𝑛 +4𝑛 +5 ∀𝑛 ≥ 9. ∀𝑛 ≥ 7.

It follows from Theorem 6 that (88) possesses uncountably Theorem 8 implies that (90) possesses uncountably many Ω (𝑁, 𝑀) manyboundedpositivesolutionsinΩ1(𝑁, 𝑀). bounded positive solutions in 1 .

Example 2. Consider the fourth order nonlinear neutral Example 4. Consider the fourth order nonlinear neutral delay difference equation: delay difference equation:

3 𝑛+1 5/2 6𝑛 − 3 Δ ((−1) 𝑛 Δ(𝑥 + 𝑥 )) Δ3 (𝑛3Δ(𝑥 + √ (3𝑛3 − 134𝑛)𝑥 )) 𝑛 10𝑛 + 2 𝑛−𝜏 𝑛 ln 𝑛−𝜏

5 3 3 3 𝑥𝑛3−5 +2𝑛 2𝑛 + 𝑥 √𝑛 +Δ ( ) +Δ3 ( 2𝑛−3 cos ) 𝑥8 + (𝑛+3)4 4 5 𝑛3−5 (𝑛 +2𝑛−3 𝑥 ) (𝑛+1) +72 𝑛−1 2 2 (−1) 𝑛 7 3 2 3 +Δ ( ) 𝑛𝑥 5 sin (𝑛 𝑥 5 ) 14 5 󵄨 2 3 󵄨 +Δ2 ( 𝑛 𝑛 ) 𝑥2𝑛−1 +𝑛 + 󵄨cos (𝑥2𝑛−1𝑛 )󵄨 6 (√𝑛+2+3) 𝑥4 √𝑛+𝑥2 −𝑛4 𝑛−1 sin 𝑛−1 2 𝑛+1 (91) +Δ( ) (89) 𝑥𝑛+6 − (−1) 𝑛 6 󵄨 3 󵄨 +Δ( ) 𝑛7 +𝑥 4√󵄨𝑥 󵄨 +𝑛 √ 2 10 𝑛−1cos 󵄨 𝑛−1󵄨 𝑛 +1+(𝑥𝑛+6 + √𝑛) 𝑥8 𝑛6 𝑛2 +𝑥27 + 5𝑛−9 + 2𝑛−18 9 󵄨 3 2 󵄨 13 2 󵄨 3 4 󵄨 𝑛 + 󵄨𝑛𝑥5𝑛−9sin (𝑛 +1)󵄨 𝑛 +𝑥2𝑛−18 󵄨cos (𝑛𝑥2𝑛−18)󵄨 (𝑛3 −2𝑛+3) 𝑛7 −𝑛11 −2 cos = , = , 15 𝑛4 + √3𝑛 − 1 𝑛 +36 ∀𝑛 ≥ 15. ∀𝑛 ≥ 1.

It follows from Theorem 7 that (89) possesses uncountably Theorem 9 yields that (91) has uncountably many bounded ∞ manyboundedpositivesolutionsinΩ1(𝑁, 𝑀). positive solutions in 𝑙𝛽 . Abstract and Applied Analysis 27

Example 5. Consider the fourth order nonlinear neutral Example 7. Consider the fourth order nonlinear neutral delay delay difference equation: difference equation:

𝑛 3 8 9 3 𝑛 2 1 Δ (−𝑛 Δ(𝑥 −(3− 𝑛) 𝑥 )) Δ ((−1) 𝑛 𝑛Δ (𝑥 +(1+ ) 𝑥 )) 𝑛 sin 𝑛−𝜏 ln 𝑛 𝑛 𝑛−𝜏 5 2 3 𝑛 ln (𝑛 + 𝑥 )−𝑥 √𝑛𝑥3 −𝑥2 +1 +Δ3 ( 𝑛−30 𝑛−30 ) +Δ3 ( 𝑛−4 𝑛−4 ) 𝑛25 +𝑥8 5 2 𝑛−30 𝑛 + ln (1 + 𝑥𝑛−4) 3 3 2 5 2 3 𝑛 𝑥 −𝑛𝑥 − (𝑛 ) 𝑛 𝑥 −𝑛𝑥 − √𝑛 2 𝑛2−1 𝑛2−1 cos 2 𝑛−9 𝑛−9 +Δ ( ) +Δ ( ) 2 8 4 4 𝑛81 +(𝑛𝑥3 −1) 𝑛 +(𝑥𝑛−9 −𝑛) 𝑛2−1 (94) 5 𝑛 𝑥 − (−1) 𝑛 (92) 𝑛6𝑥3 −𝑛+3 +Δ( 𝑛−6 ) +Δ( 𝑛−2 ) 𝑛6 + 3 (𝑛 + 𝑥8 ) 16 3 7 2 ln 𝑛−6 𝑛 +(𝑛 −𝑥𝑛−2) 2 cos (𝑛 − ln (1 + 𝑥 )) 𝑛8 +3𝑛5𝑥2 −𝑥3 + 𝑛−7 + 𝑛−4 𝑛−4 5 3 7 58 22 5 3 𝑛 + cos (𝑛𝑥𝑛−7)+1 𝑛 +𝑛 − sin (𝑛 𝑥𝑛−4) 𝑛2 −𝑛+(−1)𝑛 𝑛25 + (−1)𝑛𝑛11 −1 = , = ,∀𝑛≥1. 𝑛9 +3𝑛+1 𝑛91 + 33𝑛 + 5 ∀𝑛 ≥ 10.

Theorem 12 guarantees that (94) possesses uncountably many ∞ bounded positive solutions in 𝑙𝛽 . Theorem 10 guarantees that (92) possesses uncountably many 𝑙∞ bounded positive solutions in 𝛽 . Example 8. Consider the fourth order nonlinear neutral delay difference equation: Example 6. Consider the fourth order nonlinear neutral delay difference equation:

3 𝑛−1 3 Δ ((−1) (𝑛+2) Δ(𝑥𝑛 +𝑥𝑛−𝜏)) 4 3 𝑛 +2𝑛 3 𝑛4𝑥5 𝑥2 Δ ( Δ(𝑥 − (𝑛−16) 𝑥 )) 3 𝑛2−9 𝑛−4 (𝑛+5) 𝑛 𝑛−𝜏 +Δ ( 󵄨 󵄨) ln 𝑛3 + 󵄨𝑥25 −𝑛𝑥37 󵄨 󵄨 𝑛2−9 𝑛−4󵄨 +Δ3 (√𝑛2 −2𝑛 2 (𝑛𝑥2 )) cos 𝑛−5 𝑛4 +𝑥2 𝑥3 +Δ2 ( 𝑛−1 𝑛3−3 ) 3 2 𝑛9 + 3 (𝑛2 +𝑥6 ) 𝑥 2 ln (𝑛−13) ln 𝑛3−3 +Δ2 ( 𝑛 −2𝑛+3 ) 11 󵄨 󵄨 𝑛 + 󵄨𝑥 2 +1󵄨 9 8 9 󵄨 𝑛 −2𝑛+3 󵄨 sin (𝑛 𝑥 𝑥 ) +Δ( 𝑛−2 5𝑛−6 ) (95) 4 2 2 𝑛𝑥 sin (𝑥 +𝑛) (93) 𝑛5 +(𝑛+2𝑥82 𝑥19 ) +Δ( 𝑛−19 𝑛−19 ) 𝑛−2 5𝑛−6 9 6 𝑛 +2𝑛+𝑥𝑛−19 2 6 14 ln (𝑛 +𝑥 2 𝑥 ) + 𝑛 −17 𝑛−6 2 16 4 16 2 𝑛 sin (𝑛 + 𝑥 ) 𝑛 +(2𝑥 2 +𝑥 ) + 𝑛−13 𝑛 −17 𝑛−6 2 8 (𝑛 + 𝑥 ) 𝑛(𝑛+1)/2 7 5 𝑛−13 (−1) cos (3𝑛 +1) = , (−1)𝑛+1 (𝑛3 + 11) 𝑛5 +𝑛3 +3 = , ∀𝑛 ≥ 18. (𝑛+2)5(𝑛+5)4 ∀𝑛 ≥ 8.

Theorem 11 ensures that (93) possesses uncountably many It follows from Theorem 13 that (95) possesses uncountably ∞ bounded positive solutions in 𝑙𝛽 . many bounded positive solutions in Ω(𝑁, 𝑀). 28 Abstract and Applied Analysis

Example 9. Consider the fourth order nonlinear neutral [7] W.T. Li and S. S. Cheng, “Oscillation criteria for a nonlinear dif- delay difference equation: ference equation,” Computers & Mathematics with Applications, vol. 36, no. 8, pp. 87–94, 1998. 3 17 Δ (𝑛 Δ(𝑥𝑛 −𝑥𝑛−𝜏)) [8] W.-T. Li and S.-S. Cheng, “Remarks on two recent oscillation theorems for second-order linear difference equations,” Applied 3 √ 3 2 3 2 +Δ ( 𝑛𝑥9𝑛−5𝑥2𝑛−1 − sin (arctan (𝑛 +1))) Mathematics Letters,vol.16,no.2,pp.161–163,2003. [9] W.-T. Li, X.-L. Fan, and C.-k. Zhong, “On unbounded positive 3 4 5 2 𝑛 +𝑥𝑛−7𝑥𝑛2−2 solutions of second-order difference equations with a singular +Δ ( 󵄨 󵄨) nonlinear term,” Journal of Mathematical Analysis and Applica- (𝑛 + 𝑥6 )13 + 󵄨𝑥 −𝑛𝑥2 󵄨 𝑛−7 󵄨 𝑛−7 𝑛2−2󵄨 tions, vol. 246, no. 1, pp. 80–88, 2000. [10] W.-T. Li and S. H. Saker, “Oscillation of second-order sublinear 3 4 2 2 𝑥 ln (𝑛 +𝑥 3 ) neutral delay difference equations,” Applied Mathematics and +Δ( 𝑛−10 𝑛 −3 ) 9 (96) Computation,vol.146,no.2-3,pp.543–551,2003. (𝑛 +2 𝑥 𝑥2 ) (𝑛+2)31 𝑛−10 𝑛3−3 [11] H.-J. Li and C.-C. Yeh, “Oscillation criteria for second-order neutral delay difference equations,” Computers & Mathematics (3 − 𝑛2)𝑥2 −𝑛𝑥9 + 3𝑛−2 𝑛−9 with Applications, vol. 36, no. 10-12, pp. 123–132, 1998. 60 󵄨 󵄨 𝑛 + 󵄨𝑥3𝑛−2𝑥𝑛−9󵄨 [12]X.LiandD.Zhu,“Newresultsfortheasymptoticbehaviorof a nonlinear second-order difference equation,” Applied Mathe- 𝑛 2 5 (−1) (𝑛 −1)cos (𝑛 − √𝑛) matics Letters,vol.16,no.5,pp.627–633,2003. = , 𝑛19 + 15𝑛8 +1 [13] H. Liang and P. Weng, “Existence and multiple solutions for a second-order difference boundary value problem via ∀𝑛 ≥ 11. critical point theory,” Journal of Mathematical Analysis and Applications,vol.326,no.1,pp.511–520,2007. Theorem 14 means that (96) possesses uncountably many [14] Z. Liu, S. M. Kang, and J. S. Ume, “Existence of uncountably bounded positive solutions in Ω(𝑁, 𝑀). many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay dif- ference equations,” Applied Mathematics and Computation,vol. Conflict of Interests 213, no. 2, pp. 554–576, 2009. The authors declare that there is no conflict of interests [15]Z.Liu,L.Wang,G.I.Kim,andS.M.Kang,“Existenceof regarding the publication of this paper. uncountably many bounded positive solutions for a third order nonlinear neutral delay difference equation,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2399–2416, Acknowledgments 2010. [16] Z. Liu, Y. Xu, and S. M. Kang, “Global solvability for a second The authors would like to express their thanks to the order nonlinear neutral delay difference equation,” Computers anonymous referee for her/his valuable suggestions and com- & Mathematics with Applications,vol.57,no.4,pp.587–595, ments. This research was supported by the Science Research 2009. Foundation of Educational Department of Liaoning Province [17] J. W. Luo and D. D. Bainov, “Oscillatory and asymptotic (L2012380). behavior of second-order neutral difference equations with maxima,” Journal of Computational and Applied Mathematics, References vol. 131, no. 1-2, pp. 333–341, 2001. [18] M. Ma, H. Tang, and W. Luo, “Periodic solutions for nonlinear [1] M. H. Abu-Risha, “Oscillation of second-order linear difference second-order difference equations,” Applied Mathematics and equations,” Applied Mathematics Letters,vol.13,no.1,pp.129– Computation,vol.184,no.2,pp.685–694,2007. 135, 2000. [19] Q. Meng and J. Yan, “Bounded oscillation for second-order [2] R. P. Agarwal, Difference Equations and Inequalities,Marcel nonlinear delay differential equation in a critical state,” Journal Dekker, New York, NY, USA, 2nd edition, 2000. of Computational and Applied Mathematics,vol.197,no.1,pp. [3]R.P.Agarwal,S.R.Grace,andD.O’Regan,“Nonoscillatory 204–211, 2006. solutions for discrete equations,” Computers & Mathematics [20] M. Migda and J. Migda, “Asymptotic properties of solutions of with Applications,vol.45,no.6–9,pp.1297–1302,2003. second-order neutral difference equations,” Nonlinear Analysis: [4] R.P.Agarwal,M.M.S.Manuel,andE.Thandapani,“Oscillatory Theory, Methods and Applications,vol.63,no.5–7,pp.789–799, and nonoscillatory behavior of second-order neutral delay 2005. difference equations,” Applied Mathematics Letters,vol.10,no. [21] S. H. Saker, “Oscillation of third-order difference equations,” 2, pp. 103–109, 1997. Portugaliae Mathematica,vol.61,no.3,pp.249–257,2004. [5]S.S.ChengandW.T.Patula,“Anexistencetheoremfor [22] S. H. Saker, “New oscillation criteria for second-order nonlinear a nonlinear difference equation,” Nonlinear Analysis: Theory, neutral delay difference equations,” Applied Mathematics and Methods & Applications,vol.20,no.3,pp.193–203,1993. Computation, vol. 142, no. 1, pp. 99–111, 2003. [6]S.S.Cheng,H.J.Li,andW.T.Patula,“Boundedandzero [23] M.-C. Tan and E.-H. Yang, “Oscillation and nonoscillation convergent solutions of second-order difference equations,” theorems for second order nonlinear difference equations,” Journal of Mathematical Analysis and Applications,vol.141,no. JournalofMathematicalAnalysisandApplications,vol.276,no. 2, pp. 463–483, 1989. 1, pp. 239–247, 2002. Abstract and Applied Analysis 29

[24] X. H. Tang, “Bounded oscillation of second-order delay dif- ference equations of unstable type,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1147–1156, 2002. [25] E. Thandapani, M. M. S. Manuel, J. R. Graef, and P. W. Spikes, “Monotone properties of certain classes of solutions of second- order difference equations,” Computers & Mathematics with Applications, vol. 36, pp. 291–297, 1998. [26] Y. Tian and W. Ge, “Multiple positive solutions of boundary value problems for second-order discrete equations on the half- line,” Journal of Difference Equations and Applications,vol.12, no. 2, pp. 191–208, 2006. [27]J.YanandB.Liu,“Asymptoticbehaviorofanonlineardelay difference equation,” Applied Mathematics Letters,vol.8,no.6, pp. 1–5, 1995. [28] B. G. Zhang, “Oscillation and asymptotic behavior of second order difference equations,” Journal of Mathematical Analysis and Applications,vol.173,no.1,pp.58–68,1993. [29] Z. Zhang, J. Chen, and C. Zhang, “Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term,” Computers & Mathematics with Applications,vol. 41,no.12,pp.1487–1494,2001. [30] Z. Zhang and Q. Li, “Oscillation theorems for second-order advanced functional difference equations,” Computers & Math- ematics with Applications, vol. 36, no. 6, pp. 11–18, 1998. [31] Z. Zhang and J. Zhang, “Oscillation criteria for second-order functional difference equations with “summation small” coeffi- cient,” Computers & Mathematics with Applications,vol.38,no. 1,pp.25–31,1999. [32] M. Pinto and D. Sepulveda,´ “ℎ-asymptotic stability by fixed point in neutral nonlinear differential equations with delay,” Nonlinear Analysis: Theory, Methods & Applications,vol.74,no. 12, pp. 3926–3933, 2011. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 502756, 9 pages http://dx.doi.org/10.1155/2014/502756

Research Article Three Solutions Theorem for 𝑝-Laplacian Problems with a Singular Weight and Its Application

Yong-Hoon Lee,1 Seong-Uk Kim,2 and Eun Kyoung Lee3

1 Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea 2 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA 3 Department of Mathematics Education, Pusan National University, Busan 609-735, Republic of Korea

Correspondence should be addressed to Eun Kyoung Lee; [email protected]

Received 3 September 2013; Revised 18 December 2013; Accepted 1 January 2014; Published 18 February 2014

Academic Editor: Adem Kılıc¸man

Copyright © 2014 Yong-Hoon Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove Amann type three solutions theorem for one dimensional p-Laplacian problems with a singular weight function. To prove this theorem, we define a strong upper and lower solutions and compute the Leray-Schauder degree on a newly established weighted solution space. As an application, we consider the combustion model and show the existence of three positive radial solutions on an exterior domain.

1 1. Introduction solutions of (𝑃) may not be in 𝐶0[0, 1]; for example, take −1 𝑝−2 ℎ(𝑡) = (𝑝 − 1)𝑡 |1 + ln 𝑡| , 𝑝>2,and𝑓≡1,then Let us consider the following 𝑝-Laplacian problem with a 1 ℎ∈H\𝐿 (0, 1) and the solution 𝑢 for corresponding problem sign-changing singular weight: 1 of (𝑃) is given by 𝑢(𝑡) = −𝑡 ln 𝑡 which is not in 𝐶 [0, 1].Thus 1 󸀠 󸀠 if ℎ∈H \𝐿(0, 1), the three solutions theorem in [1]can 𝜑𝑝(𝑢 (𝑡)) +ℎ(𝑡) 𝑓 (𝑢 (𝑡)) =0, 𝑡∈(0, 1) , (𝑃) not be applied. Our main interest in this paper is to establish 𝑢 (0) =0, 𝑢(1) =0, three solutions theorem for problem (𝑃) for those weights ℎ 1 satisfying ℎ∈H \𝐿 (0, 1). 𝑝−2 1 where 𝜑𝑝(𝑠) = |𝑠| 𝑠, 𝑝>1, 𝑓∈𝐶(R, R),andℎ∈𝐿loc((0, Mainstepfortheproofofthreesolutionstheorem,in 1), R) may change sign. Moreover, ℎ∈H and satisfies |ℎ(𝑡)| > general, is to compute the Leray-Schauder degree on a sector 0,for𝑡 ∈ (0, 𝛿) ∪ (1 − 𝛿, 1) for some 𝛿>0,whereaclassH of in solution space made from strong sense of upper and weight functions is given as lower solutions. Since the sector needs to be open in the space, strong sense of ordering and the sector made from the 1/2 1/2 1 −1 ordering are closely related to the topology of solution space. H ={ℎ∈𝐿loc (0, 1) | ∫ 𝜑𝑝 (∫ |ℎ (𝜏)| 𝑑𝜏) 𝑑𝑠 0 𝑠 A typical situation in application usually happens as follows. (1) It is comparatively easy to find a lower solution 𝛼 and an 1 𝑠 upper solution 𝛽 of (𝑃) satisfying 𝛼(𝑡) < 𝛽(𝑡),forall𝑡 ∈ (0, 1) + ∫ 𝜑−1 (∫ |ℎ (𝜏)| 𝑑𝜏) 𝑑𝑠 <∞}. 𝑝 and 𝛼(0) = 𝛼(1) = 𝛽(0) = 𝛽(1).Denote =0 Ω={𝑢∈ 1/2 1/2 1 𝑋 | 𝛼(𝑡) < 𝑢(𝑡) <𝛽(𝑡),forall𝑡 ∈ (0, 1)}.In𝐶 -analysis, 1 1 It is well known that 𝐿 (0, 1) ⫋ H. that is, 𝑋=𝐶0[0, 1],weseethatΩ is nonempty and open 1 1 󸀠 󸀠 If ℎ∈𝐿(0, 1),thenallsolutionsof(𝑃) are in 𝐶0[0, 1] in 𝑋 by providing additional conditions like 𝛼 (0) < 𝛽 (0) 1 󸀠 󸀠 and based on the Leray-Schauder degree argument on 𝐶 - and 𝛼 (1) > 𝛽 (1) which implies a strong sense of ordering. space: Ben-Naoum and de Coster [1]provedthreesolutions On the other hand, in 𝐶-analysis, that is, 𝑋=𝐶0[0, 1],we 1 theorem for (𝑃). On the other hand, if ℎ∉𝐿(0, 1),then see int Ω=0so that the Leray-Schauder degree on Ω is not 2 Abstract and Applied Analysis

1 even defined. Three solutions theorem with no use of such Ω If ℎ∈H \𝐿 (0, 1),then0<𝑤(𝑡)≤1,for𝑡 ∈ (0, 1), isveryrestrictiveinapplication.Toovercomethisdifficulty lim 𝑤 (𝑡) =0 or lim 𝑤 (𝑡) =0, in our problem, we introduce a weighted solution space, say 𝑡→0+ 𝑡→1− (5) 𝐶𝑤[0, 1],whichisfinerthan𝐶[0, 1] andalsointroducea −1 1 𝐶 and 𝑤 is integrable on (0, 1).Moreprecisely,ifℎ∉𝐿(0, strong sense of ordering suitable to 𝑤-space which makes 1 1/2] + 𝑤(𝑡) =0 ℎ∉𝐿[1/2, 1) the degree computation effective (see Section 2 in detail). ,thenlim𝑡→0 and if ,then − 𝑤(𝑡) =0 As to a question of triple multiplicity of solutions, besides lim𝑡→1 . 𝑢∈𝐶 [0, 1] 𝑤𝑢󸀠 ∈𝐶[0,1] the Amann type three solutions theorems, many works are For 𝑤 , define by done by using the variational method (see [2–5]andthe 󸀠 { lim 𝑤 (𝑡) 𝑢 (𝑡) ,𝑡=0, references therein) and by using several extensions of the {𝑡→0+ 󸀠 󸀠 Liggett-Williams fixed point theorem and Guo-Krasnoselskii 𝑤𝑢 (𝑡) = {𝑤 (𝑡) 𝑢 (𝑡) ,𝑡∈(0, 1) , (6) { 󸀠 fixed point theorem, especially for positive solutions (see lim 𝑤 (𝑡) 𝑢 (𝑡) ,𝑡=1. { − [6–8] and the references therein). For the problem we are 𝑡→1 󸀠 concerned with in this paper, the variational setup is not And also define ‖𝑢‖𝑤 =‖𝑢‖∞ +‖𝑤𝑢 ‖∞;then(𝐶𝑤[0, 1], ‖ ⋅ ‖𝑤) possibleduetolackofregularityofsolutions.Threesolutions is a Banach space. We give a proof for reader’s convenience. theorem proved in this paper is for the case that ℎ is not only ℎ∉𝐿1(0, 1) 1 but also possibly sign-changing. By this aspect, it Lemma 1. Let ℎ∈H \𝐿(0, 1);then𝐶𝑤[0, 1] is a Banach ℎ≥0 󸀠 is new as far as the authors know. For the case ,onemay space with a norm ‖𝑢‖𝑤 =‖𝑢‖∞ +‖𝑤𝑢‖∞. refer to [9]forapartialresultaboutthetheorem. As an application, we study the existence of triple positive Proof. Let {𝑢𝑛} be a Cauchy sequence in (𝐶𝑤[0, 1], ‖ ⋅ ‖𝑤). 󸀠 solutions for certain nonlinear 𝑝-Laplacian problems with Then {𝑢𝑛} and {𝑤𝑢𝑛} are Cauchy sequences in 𝐶[0, 1] so that 󸀠 positive singular coefficient function and give an example ofa there exist 𝑢,V ∈𝐶[0,1]such that 𝑢𝑛 →𝑢and 𝑤𝑢𝑛 → V in 󸀠 combustion model defined on an exterior domain. Applying 𝐶[0, 1].Itissufficienttoshowthat𝑤𝑢 ≡ V on [0, 1].Since this three solutions theorem to the case having sign-changing 𝑤(𝑡) >0 for 𝑡 ∈ (0, 1), there exists 𝑧 ∈ 𝐶(0, 1) such that 󸀠 coefficient function could be an interesting and challenging 𝑧(𝑡) = V(𝑡)/𝑤(𝑡) for all 𝑡 ∈ (0, 1).For𝛿>0,weknow𝑢𝑛 →𝑧 󸀠 problem. in 𝐶[𝛿, 1 − 𝛿]. This implies 𝑧≡𝑢 in [𝛿, 1 − 𝛿].Since𝛿>0 We organize this paper as follows. In Section 1,weintro- 󸀠 󸀠 is arbitrary, 𝑤𝑢𝑛 →𝑤𝑢pointwise in 𝑡 ∈ (0, 1). Therefore, duce a weighted solution space 𝐶𝑤[0, 1],thestrongupper 󸀠 󸀠 𝑤𝑢 ≡ V on (0, 1).Since𝑤𝑢 converges uniformly to V on and lower solutions of (𝑃), and prove three solutions theorem 𝑛 [0, 1],wehave for problem (𝑃).InSection 3, we prove a multiplicity result 󸀠 󸀠 for certain nonlinear 𝑝-Laplacian problems by using three V (0) = lim 𝑤𝑢𝑛 (0) = lim lim 𝑤 (𝑡) 𝑢𝑛 (𝑡) 𝑛→∞ 𝑛→∞ 𝑡→0+ solutions theorem introduced in Section 2.InSection 4,we (7) apply the result in Section 3 to a combustion model to show 󸀠 󸀠 = lim lim 𝑤 (𝑡) 𝑢𝑛 (𝑡) = lim 𝑤 (𝑡) 𝑢 (𝑡) . the existence of three positive radial solutions on exterior 𝑡→0+ 𝑛→∞ 𝑡→0+ domain. 󸀠 Thus 𝑢∈𝐶𝑤[0, 1] and 𝑤𝑢 ≡ V on [0, 1]. This implies 𝑢𝑛 →𝑢 in 𝐶𝑤[0, 1] and the proof is completed. 2. Preliminaries Define In this section, we introduce a weighted solution space 𝑋={𝑢∈𝐶𝑤 [0, 1] |𝑢(0) =0=𝑢(1)}. (8) 𝐶𝑤[0, 1] and prove three solutions theorem for (𝑃) on 𝐶 [0, 1] 1 𝑤 .Let Then we see that for given ℎ∈H\𝐿 (0, 1), 𝑢 being a solu- 1/2 (𝑃) 𝑢∈𝑋 + ∫ ℎ(𝑠)𝑑𝑠 =∞ 𝐶𝑤 [0, 1] tion of implies . In fact, if lim𝑡→0 𝑡 and 𝑢 is a solution of (𝑃),thenfor𝑡 ∈ (0, 1/2), 1 󸀠 ={𝑢∈𝐶[0, 1] ∩𝐶 (0, 1) |−∞< lim 𝑤 (𝑡) 𝑢 (𝑡) <∞, 1/2 𝑡→0+ 󸀠 −1 󸀠 1 𝑢 (𝑡) =𝜑𝑝 (∫ ℎ (𝑠) 𝑓 (𝑢 (𝑠)) 𝑑𝑠𝑝 +𝜑 (𝑢 ( ))) , 𝑡 2 󸀠 −∞ < lim 𝑤 (𝑡) 𝑢 (𝑡) <∞}, (9) 𝑡→1− (2) and by using L’Hospital’s rule, we have 󸀠 where lim 𝑤 (𝑡) 𝑢 (𝑡) 𝑡→0+ 𝑤 (𝑡) = {𝑤 (𝑡) ,1}, min ℎ (3) 1/2 ∫ ℎ (𝑠) 𝑓 (𝑢 (𝑠)) 𝑑𝑠 +𝜑 (𝑢󸀠 (1/2)) −1 𝑡 𝑝 with = lim 𝜑𝑝 ( ) 𝑡→0+ 1/2 −1 ∫ ℎ (𝑠) 𝑑𝑠 { 1/2 1 𝑡 {(𝜑−1 (∫ |ℎ (𝑠)| 𝑑𝑠)) ,0<𝑡≤, { 𝑝 2 𝑤 (𝑡) = 𝑡 −1 ℎ (𝑡) 𝑓 (𝑢 (𝑡)) −1 ℎ { 𝑡 −1 (4) = lim 𝜑𝑝 ( lim )=𝜑𝑝 (𝑓 (0)). { −1 1 𝑡→0+ 𝑡→0+ ℎ (𝑡) (𝜑𝑝 (∫ |ℎ (𝑠)| 𝑑𝑠)) , ≤𝑡<1. { 1/2 2 (10) Abstract and Applied Analysis 3

1/2 + ∫ ℎ(𝑠)𝑑𝑠 =−∞ − (0, 𝑡) Forthecasesthatlim𝑡→0 𝑡 and lim𝑡→1 so that we may integrate on . Using a boundary condition 𝑡 𝑢(0) = 0 ∫ ℎ(𝑠)𝑑𝑠 =∞ −∞ ,weget 1/2 (or ), by the same argument, we have 𝑡 1/2 −1 1 󸀠 𝑢 (𝑡) = ∫ 𝜑𝑝 (𝜌 + ∫ 𝑔 (𝑟) 𝑑𝑟) 𝑑𝑠, for 𝑡∈[0, ]. −∞ < lim 𝑤 (𝑡) 𝑢 (𝑡) <∞, 0 𝑡 2 𝑡→0+ (11) (18) 󸀠 −∞ < lim 𝑤 (𝑡) 𝑢 (𝑡) <∞. 𝑡→1− We note that 𝑢 in (18) is a solution of (𝐴) only on the interval [0, 1/2]. Doing similar computation on the interval [1/2, 1], Example 2. Let us consider the following example: we get 1 𝑡 󸀠 󸀠 𝛼𝑢 1 𝜑 (𝑢 (𝑡)) +ℎ(𝑡) ( )=0, 𝑡∈(0, 1] , 𝑢 (𝑡) = ∫ 𝜑−1 (−𝜌 + ∫ 𝑔 (𝑟) 𝑑𝑟) 𝑑𝑠, 𝑡∈[ ,1). 𝑝 exp 𝛼+𝑢 𝑝 for (12) 𝑡 1/2 2 𝑢 (0) =0, 𝑢(1) =0, (19) ItisknownbyLemma 2.2in[10]thattheequation where 𝛼, 𝜆, >0 𝑝>3/2and ℎ:(0,1)→ R is given by 1/2 1/2 −1 −3/2 1 ∫ 𝜑𝑝 (𝜌 + ∫ 𝑔 (𝑟) 𝑑𝑟) 𝑑𝑠 {𝑡 ,0<𝑡≤, 0 𝑠 2 ℎ (𝑡) = (13) (20) { 1 1 𝑠 −1, <𝑡<1. −1 { 2 = ∫ 𝜑𝑝 (−𝜌 + ∫ 𝑔 (𝑟) 𝑑𝑟) 𝑑𝑠 1/2 1/2 ℎ∈H \𝐿1(0, 1] Then we see that sign-changing and every has a unique zero 𝜌≜𝜌(𝑔)in R for each 𝑔∈H. Therefore 𝑢 𝑢󸀠(0+)=∞ solution satisfies by using (9). We also see by it is natural to paste 𝑢’s in (18)and(19) in a continuous way. 𝑤 calculation that canbegivenas Now let us define a function 𝑢 by

1/(𝑝−1) 𝑡 1/2 { √ √ −1 1 { 𝑡 36 − 16 2 {∫ 𝜑 (𝜌 (𝑔) + ∫ 𝑔 (𝑟) 𝑑𝑟) 𝑑𝑠, 0≤𝑡 , {( ) , if 𝑡∈(0, ], { 𝑝 2 { 2(1−√2𝑡) 49 𝑢 (𝑡) = 0 𝑠 𝑤 (𝑡) = { { 1 𝑠 1 { 36 − 16√2 {∫ 𝜑−1 (−𝜌 (𝑔) + ∫ 𝑔 (𝑟) 𝑑𝑟) 𝑑𝑠, ≤𝑡≤1. {1, 𝑡∈( ,1], { 𝑝 2 if 49 𝑡 1/2 { (21) (14) 1 Then 𝑢 satisfies 𝑢 ∈ 𝐶[0, 1] ∩𝐶 (0, 1) and 𝑢 is a unique 󸀠 + and 𝑤𝑢 (0 )=1by using (10). solution of problem (𝐴). Basedonthissetup,wenowintroducecorresponding To establish corresponding integral operator for problem integral operator for problem (𝑃).For𝑢 ∈ 𝐶[0, 1], define (𝑃), let us first consider the problem 𝑇𝑢 (𝑡) 󸀠 󸀠 −𝜑𝑝(𝑢 ) =𝑔, 𝑡∈(0, 1) , 𝑡 (𝐴) {∫ 𝜑−1 (𝜌 (ℎ𝑓 (𝑢)) { 𝑝 𝑢 (0) =0=𝑢(1) , { 0 { 1/2 { 1 { +∫ ℎ (𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏) 𝑑𝑠, 0≤𝑡 , where 𝑔∈H. We remind the reader that 𝑔 needs not be = 𝑠 2 𝑡=0 1 [𝑡, 1/2] 𝑡∈ { 1 integrable near or .Integratingon for {∫ 𝜑−1 (−𝜌(ℎ𝑓 𝑢 ) (0, 1/2] { 𝑝 ( ) ,wehave { 𝑡 { 𝑠 { 1 1 1/2 +∫ ℎ (𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏) 𝑑𝑠, ≤𝑡≤1. 󸀠 󸀠 { 1/2 2 𝜑𝑝 (𝑢 (𝑡)) =𝜑𝑝 (𝑢 ( )) +∫ 𝑔 (𝑟) 𝑑𝑟. (15) 2 𝑡 (22) 󸀠 𝑢=𝑇𝑢 𝐶[0, 1] 𝑢 (𝑃) Denoting 𝜌=𝜑𝑝(𝑢 (1/2)), Then in if and only if is a solution of .

1/2 Remark 3. We understand the number 𝜌(ℎ𝑓(𝑢)) in the above 󸀠 −1 𝑢 𝐶[0, 1] 𝜌 : 𝐶[0, 1]→ 𝑢 (𝑡) =𝜑𝑝 (𝜌 + ∫ 𝑔 (𝑟) 𝑑𝑟) . (16) as a function of defined on .Thatis, 𝑡 R.Itisknownthat𝜌 maps bounded sets in 𝐶[0, 1] into bounded sets in R ([10, Lemma 3.1]). It is also known that 𝑔∈H 𝜌 Since and is a fixed constant, we can see that 𝑇 is completely continuous on 𝐶[0, 1] ([10,Theorem3.4]). 1/2 −1 1 1 As mentioned in Introduction, the regularity of solutions 𝜑𝑝 (𝜌 + ∫ 𝑔 (𝑟) 𝑑𝑟) ∈𝐿 (0, ), (17) 𝑡 2 of problem (𝑃) sensitively depends on the shape of nonlinear 4 Abstract and Applied Analysis

1 + − term 𝑓 even if ℎ∈H\𝐿 (0, 1), and we are concerned with the taking limt 𝑡→0 and 𝑡→1 in (27), we have the same 1 󸀠 󸀠 case that problem (𝑃) does not have 𝐶 -solutions. Therefore, upper bound of |𝑤(𝐺𝑢𝑛) (0)| and |𝑤(𝐺𝑢𝑛) (1)| as in (27). This 󸀠 it is interesting to consider operator 𝑇 restricted on 𝑋 to proves that {𝑤(𝐺𝑢𝑛) } is bounded in 𝐶[0, 1]. complete three solutions theorem for those problems with no 󸀠 1 {𝑤(𝐺𝑢 ) } [0, 1] 𝐶 -solutions. Claim 2. 𝑛 is equicontinuous on . ℎ∈𝐿1(0, 1) ‖𝑤𝑢󸀠 ‖ <𝑀 𝑛 Define 𝐺 the restriction of 𝑇 on 𝑋. If ,thensince 𝑛 ∞ 𝐵,forall and 1 −1 1 Inwhatistofollow,weassumeℎ∈H \𝐿 (0, 1) and we 𝑤 ∈𝐿(0, 1),weget nowprovethecompletecontinuityof𝐺 on the solution space 󵄨 󵄨 󵄨𝑢󸀠 󵄨 <𝑀 (𝑤)−1 ∈𝐿1 (0, 1) , 𝑋. Before doing that, we give a remark useful to calculate 󵄨 𝑛󵄨 𝐵 (28) 𝑝-Laplacians. for all 𝑛. This implies that {𝑢𝑛} is equicontinuous in 𝐶[0, 1] {𝑢 } Remark 4. If 𝑎, 𝑏,then >0 and by Arzela-Ascoli theorem, there exist a subsequence 𝑛𝑘 of {𝑢𝑛} and V ∈𝐶[0,1]such that 𝑢𝑛𝑘 converges uniformly −1 −1 −1 𝜑𝑝 (𝑎+𝑏) ≤𝐶𝑝 (𝜑𝑝 (𝑎) +𝜑𝑝 (𝑏)), (23) to V on [0, 1] as 𝑘→∞. Thus using Lebesgue Dominated Convergence theorem, we obtain where 1/2 −1 1, 𝑝 > 2, 𝑤 (𝑡) 𝜑𝑝 (𝜌 (ℎ𝑓𝑛𝑘 (𝑢 )) + ∫ ℎ (𝑠) 𝑓(𝑢𝑛𝑘 (𝑠))𝑑𝑠) 𝐶𝑝 := { (2−𝑝)/(𝑝−1) (24) 𝑡 2 , 1<𝑝≤2. 1/2 −1 󳨀→ 𝑤 (𝑡) 𝜑𝑝 (𝜌 (ℎ𝑓 (V))+∫ ℎ (𝑠) 𝑓 (V (𝑠)) 𝑑𝑠) , Theorem 5. 𝐺:𝑋is →𝑋 completely continuous. 𝑡 (29) Proof. Let 𝐵 be a bounded subset of 𝑋. Then for any sequence (𝑢 )⊂𝐵 (𝐺𝑢 ) 󸀠 𝑛 ,weneedtoshowtherelativecompactnessof 𝑛 uniformly on [0, 1]. This implies that {𝑤(𝐺𝑢𝑛) } is equicon- with respect to ‖⋅‖𝑤-norm. We know by Remark 3 that 𝐺 is tinuous in 𝐶[0, 1]. 1 completely continuous on 𝐶[0, 1] so that there exists 𝑢0 ∈ On the other hand, for the case of ℎ∈H \𝐿(0, 1), 𝐶[0, 1] (𝑢 ) (𝑢 ) 󸀠 andasubsequenceof 𝑛 ,sayagain 𝑛 such that suppose that {𝑤(𝐺𝑢𝑛) } is not equicontinuous on [0, 1].Then 𝐺𝑢𝑛 →𝑢0 in 𝐶[0, 1]. Tocomplete the proof, we need to show there exists 𝜀>0such that we may choose a subsequence the following. {𝑢𝑛𝑘} of {𝑢𝑛} and sequences {𝑡𝑘}, {𝑠𝑘} ⊂ (0, 1) satisfying 𝑢0 ∈𝑋and there is a subsequence (𝐺𝑢𝑛𝑙) of (𝐺𝑢𝑛) such 󵄨 󵄨 1 that 𝐺𝑢𝑛𝑙 →𝑢0 as 𝑙→∞in 𝑋 and 𝐺 is continuous on 𝑋. 󵄨𝑡 −𝑠 󵄨 < , 󵄨 𝑘 𝑘󵄨 𝑘 󸀠 (30) {𝑤(𝐺𝑢 ) } 𝐶[0, 1] 󵄨 󵄨 Claim 1. 𝑛 is uniform bounded in . 󵄨 󸀠 󸀠 󵄨 󵄨𝑤(𝐺𝑢𝑛𝑘) (𝑡𝑘) − 𝑤(𝐺𝑢𝑛𝑘) (𝑠𝑘)󵄨 ≥𝜀. Since 𝐵 is bounded in 𝑋, there exists 𝑀𝐵 >0such that ‖𝑢‖ <𝑀 ‖𝑤𝑢󸀠‖ <𝑀 𝑢∈𝐵 ∞ 𝐵 and ∞ 𝐵,forall .Wealsoknow As for sequences {𝑡𝑘} and {𝑠𝑘},itiseasytoseethat 𝑁 >0 |𝜌(ℎ𝑓(𝑢))| <𝑁 by Remark 3 that there is 𝐵 such that 𝐵, lim𝑘→∞𝑡𝑘 = lim𝑘→∞𝑠𝑘.Weshowthatlim𝑘→∞𝑡𝑘 =0or 𝑢∈𝐵 𝑡 ∈ (0, 1/2) for all .For ,byusingRemark 4,weget 1. Suppose it is not true so let lim𝑘→∞𝑡𝑘 =𝑡0 ∈ (0, 1). 𝜂 0<𝜂< {𝑡 ,1−𝑡 } 1/2 Then taking satisfying min 0 0 ,weseethat 󵄨 󸀠 󵄨 −1 󵄨 󵄨 󵄨 󵄨 1 󵄨(𝐺𝑢 ) (𝑡)󵄨 ≤𝜑 (󵄨𝜌(ℎ𝑓(𝑢))󵄨 + ∫ |ℎ (𝑠)| 󵄨𝑓(𝑢 (𝑠))󵄨 𝑑𝑠) ℎ∈𝐿[𝜂, 1 − 𝜂] and 𝑢𝑛𝑘 → V uniformly on [𝜂, 1 − 𝜂].Bythe 󵄨 𝑛 󵄨 𝑝 󵄨 󵄨 󵄨 𝑛 󵄨 1 𝑡 same argument of the above case of ℎ∈𝐿(0, 1),wecanprove 󸀠 1/2 that {𝑤(𝐺𝑢𝑛) } is equicontinuous on [𝜂, 1 − 𝜂].Thusthereis −1 𝑁∈N ≤𝜑𝑝 (𝑁𝐵 + 𝑓 ∫ |ℎ (𝑠)| 𝑑𝑠) sufficiently large such that 𝑡 󵄨 󵄨 󵄨𝑤(𝐺𝑢 )󸀠 (𝑡 ) − 𝑤(𝐺𝑢 )󸀠 (𝑠 )󵄨 <𝜀, 1/2 󵄨 𝑛𝑁 𝑁 𝑛𝑁 𝑁 󵄨 (31) −1 −1 −1 ≤𝐶𝑝 (𝜑 𝑝 (𝑁𝐵)+𝜑𝑝 (𝑓)𝑝 𝜑 (∫ |ℎ (𝑠)| 𝑑𝑠)), 𝑡 and this contradicts with (30). Now we consider the case (25) lim𝑘→∞𝑡𝑘 =0=lim𝑘→∞𝑠𝑘.Theargumentforthecase lim𝑘→∞𝑡𝑘 =1=lim𝑘→∞𝑠𝑘 is similar. It is easy to see where 𝑓=max𝑠∈[−𝑀 ,𝑀 ]|𝑓(𝑠)|.Fromthefactthat0≤𝑤(𝑡)≤ that this case implies lim𝑘→∞𝑤(𝑡𝑘)=0.Since𝑤(𝑡𝑘)≥0, 𝐵 𝐵 −1 𝑝−1 1 and the definition of 𝑤(𝑡),wesee 𝑤(𝑡𝑘)=𝜑𝑝 (𝑤 (𝑡𝑘)),forall𝑘 and we get 1/2 −1 𝑤(𝐺𝑢 )󸀠 (𝑡 ) 𝑤 (𝑡) 𝜑𝑝 (∫ |ℎ (𝑠)| 𝑑𝑠) ≤ 1, (26) 𝑛𝑘 𝑘 𝑡 =𝜑−1 (𝑤𝑝−1 (𝑡 )𝜌(ℎ𝑓(𝑢 )) thus we have 𝑝 𝑘 𝑛𝑘 (32) 󵄨 󵄨 󵄨 󸀠 󵄨 −1 −1 1/2 𝑤 (𝑡) 󵄨(𝐺𝑢𝑛) (𝑡)󵄨 ≤𝐶𝑝 (𝜑𝑝 (𝑁𝐵)+𝜑𝑝 (𝑓)) , (27) 𝑝−1 󵄨 󵄨 +𝑤 (𝑡𝑘) ∫ ℎ (𝑠) 𝑓(𝑢𝑛𝑘 (𝑠))𝑑𝑠). 𝑡𝑘 for 𝑡 ∈ (0, 1/2].For𝑡∈[1/2,1), by similar calculation, we 󸀠 󸀠 get the same upper bound of 𝑤(𝑡)|(𝐺𝑢𝑛) (𝑡)| as in (27)andby Now we want to calculate lim𝑘→∞𝑤(𝐺𝑢𝑛𝑘) (𝑡𝑘). Abstract and Applied Analysis 5

Since 𝜌(ℎ𝑓(𝑢𝑛𝑘)) is bounded and lim𝑘→∞𝑤(𝑡𝑘)=0,we Therefore we get have 󸀠 𝑓 (0) , if ℎ>0near 0, 𝑝−1 lim 𝑤(𝐺𝑢𝑛𝑘) (𝑡𝑘)={ (38) lim 𝑤 (𝑡𝑘)𝜌(ℎ𝑓(𝑢𝑛𝑘)) = 0. 𝑘→∞ −𝑓 (0) , ℎ<0 0. 𝑘→∞ (33) if near 󸀠 Bythesameargument,𝑤(𝐺𝑢𝑛𝑘) (𝑠𝑘) also has the same limit On the other hand, 󸀠 and this contradicts with (30). Consequently, {𝑤(𝐺𝑢𝑛) } is 1/2 𝑝−1 equicontinuous in 𝐶[0, 1]. By Arzela-Ascoli theorem, there 𝑤 (𝑡𝑘) ∫ ℎ (𝑠) 𝑓(𝑢𝑛𝑘 (𝑠))𝑑𝑠 exists a subsequence {𝑢𝑛𝑙} of {𝑢𝑛} and 𝑧 ∈ 𝐶[0, 1] such that 𝑡𝑘 󸀠 1/2 𝑤(𝐺𝑢𝑛𝑙) 󳨀→ 𝑧 as 𝑙󳨀→∞, in 𝐶 [0, 1] . (39) 𝑝−1 =𝑤 (𝑡𝑘) ∫ ℎ (𝑠) [𝑓 𝑛𝑘(𝑢 (𝑠))−𝑓(V (𝑠))]𝑑𝑠 (34) 𝑡 Claim 3. 𝑢0 ∈𝑋and 𝐺𝑢𝑛𝑙 →𝑢0 in 𝑋. 𝑘 󸀠 It is enough to show that 𝑧≡𝑤𝑢0.Since𝑤(𝑡) >0 for 1/2 𝑡 ∈ (0, 1) 𝑟(𝑡) := 𝑧(𝑡)/𝑤(𝑡) 𝑟 ∈ 𝐶((0, 1)) +𝑤𝑝−1 (𝑡 ) ∫ ℎ (𝑠) 𝑓 (V (𝑠)) 𝑑𝑠. ,let ;then and for 𝑘 𝛿>0 (𝐺𝑢 )󸀠 →𝑟 𝐶[𝛿, 1 − 𝛿] 𝑢󸀠 ≡𝑟 𝑡𝑘 , 𝑛𝑙 uniformly in and thus 0 󸀠 on [𝛿, 1 − 𝛿].Since𝛿 is arbitrary, 𝑢0 ≡𝑟on (0, 1) and from 𝑝−1 1/2 We show lim 𝑘→∞𝑤 (𝑡𝑘)∫ ℎ(𝑠)[𝑓(𝑢𝑛𝑘(𝑠)) − 𝑓(V(𝑠))]𝑑𝑠 = (39), we have 𝑡𝑘 0 󸀠 󸀠 . 𝑧 (0) = lim 𝑤(𝐺𝑢𝑛𝑙) (0) = lim lim𝑤 (𝑡) (𝐺𝑢𝑛𝑙) (𝑡) 𝑝−1 𝑙→∞ 𝑙→∞𝑡→0 Indeed, 𝑤 (𝑡𝑘) is close to 0 for sufficiently large 𝑘, (40) since 𝑝−1 > 0and lim𝑘→∞𝑤(𝑡𝑘)=0. There- 󸀠 󸀠 = lim lim 𝑤 (𝑡) (𝐺𝑢𝑛𝑙) (𝑡) = lim𝑤 (𝑡) 𝑢0 (𝑡) . fore, without loss of generality, we may assume that 𝑡→0𝑙→∞ 𝑡→0 −1 1/2 −1 𝑝−1 1/2 𝑤(𝑡𝑘)=(𝜑𝑝 (∫ |ℎ(𝑠)|𝑑𝑠)) so that 𝑤 (𝑡𝑘)∫ |ℎ(𝑠)|𝑑𝑠 = 󸀠 𝑡𝑘 𝑡𝑘 Therefore 𝑤𝑢0 ≡𝑧on [0, 1). By similar argument near 1,we 1/2 −1 1/2 𝑤𝑢󸀠 ≡𝑧 [0, 1] 𝑢 ∈𝑋 (∫ |ℎ(𝑠)|𝑑𝑠) ∫ |ℎ(𝑠)|𝑑𝑠. =1 Thus using the fact 𝑢𝑛𝑘 → get on and thus 0 . 𝑡𝑘 𝑡𝑘 V in 𝐶[0, 1],wehave Claim 4. 𝐺 is continuous on 𝑋. 󵄨 󵄨 󵄨 1/2 󵄨 Assume that 𝑢𝑛 → 𝑢̃ in X. By compactness of 𝐺,thereisa 󵄨𝑤𝑝−1 (𝑡 ) ∫ ℎ (𝑠) [𝑓 (𝑢 (𝑠))−𝑓(V (𝑠))]𝑑𝑠󵄨 lim 󵄨 𝑘 𝑛𝑘 󵄨 subsequence {𝑢𝑛𝑗 } of {𝑢𝑛} and V ∈𝑋such that 𝐺(𝑢𝑛𝑗 )→V in 𝑘→∞󵄨 𝑡 󵄨 󵄨 𝑘 󵄨 𝑋.Itissufficetoseethat𝐺𝑢(𝑡)̃ = V(𝑡),forall𝑡 ∈ [0, 1].Since 1/2 𝐺 is continuous in 𝐶[0, 1] and 𝑢𝑛𝑗 → 𝑢̃ in 𝐶[0, 1],wehave 𝑝−1 󵄩 󵄩 (35) ≤ lim 𝑤 (𝑡𝑘) ∫ |ℎ (𝑠)| 𝑑𝑠󵄩𝑓(𝑢𝑛𝑘)−𝑓(V)󵄩 𝐺𝑢 →𝐺𝑢̃ 𝐶[0, 1] 𝐺𝑢≡̃ V 𝑘→∞ 󵄩 󵄩∞ 𝑛𝑗 in .Thus and by standard limit 𝑡𝑘 argument, we see that 𝐺(𝑢𝑛)→V =𝐺𝑢̃.Thiscompletesthe 󵄩 󵄩 = lim 󵄩𝑓(𝑢𝑛𝑘)−𝑓(V)󵄩 =0. proof. 𝑘→∞󵄩 󵄩∞ Now we define a strong sense of ordering in 𝐶𝑤[0, 1]. Next we show 𝑢, V ∈𝐶 [0, 1] 𝑢≺V 1/2 Definition 6. For 𝑤 ,onesaysthat if and 𝑝−1 only if lim 𝑤 (𝑡𝑘) ∫ ℎ (𝑠) 𝑓 (V (𝑠)) 𝑑𝑠 𝑘→∞ 𝑡 𝑘 (i) 𝑢(𝑡) < V(𝑡) for all 𝑡 ∈ (0, 1), (36) 󸀠 󸀠 𝑓 (0) , if ℎ>0near 0, (ii) either 𝑢(0) < V(0) or 𝑤𝑢 (0) < 𝑤V (0), ={ 󸀠 󸀠 −𝑓 (0) , if ℎ<0near 0. (iii) either 𝑢(1) < V(1) or 𝑤𝑢 (1) > 𝑤V (1).

1/2 𝛼 (𝑃) 𝑤𝑝−1(𝑡 )=(∫ |ℎ(𝑠)|𝑑𝑠)−1 Definition 7. One says that is a lower solution of if and Indeed, using the fact 𝑘 𝑡 for 󸀠 1 𝑘 only if 𝛼∈𝐶𝑤[0, 1], 𝜑𝑝(𝛼 )∈𝐶(0, 1) and sufficiently large 𝑘,weget 󸀠 𝜑 (𝛼󸀠 (𝑡)) +𝑓(𝑡, 𝛼 (𝑡)) ≥0, 𝑡∈(0, 1) , 1/2 𝑝 𝑝−1 (41) lim 𝑤 (𝑡𝑘) ∫ ℎ (𝑠) 𝑓 (V (𝑠)) 𝑑𝑠 𝑘→∞ 𝛼 (0) ≤0, 𝛼(1) ≤0. 𝑡𝑘 1/2 (37) 𝛽 (𝑃) 𝛽∈ ∫ ℎ (𝑠) 𝑓 (V (𝑠)) 𝑑𝑠 We also say that is an upper solution of if and only if = 𝑡 . 𝐶 [0, 1] 𝜑 (𝛽󸀠)∈𝐶1(0, 1) lim 1/2 𝑤 with 𝑝 and it satisfies the reverse of 𝑡→0+ ∫ |ℎ (𝑠)| 𝑑𝑠 𝑡 the above inequalities.

1 𝛼 (𝑃) If ℎ𝑓(V)∈𝐿(0, 1/2),thenwecaneasilyverify𝑓(0) = 0. Definition 8. One says that is a strict lower solution of 1/2 if and only if 𝛼 is a lower solution of (𝑃) and satisfies 𝛼≺𝑢 + ∫ |ℎ(𝑠)|𝑑𝑠 =∞ 0 Since lim𝑡→0 𝑡 , we see that the limit is . 𝑢 (𝑃) 𝑢(𝑡) ≥ 𝛼(𝑡) 𝑡∈[0,1] 1 where is a solution of such that , . On the other hand, if ℎ𝑓(V)∉𝐿(0, 1/2).Wenotethatℎ∈ We say that 𝛽 is a strict upper solution of (𝑃) if and only 1 𝐿 [𝑡, 1/2],forgiven𝑡>0.ByusingL’Hospital’srule,weget if 𝛽 is an upper solution of (𝑃) and satisfies 𝑢≺𝛽where 𝑢 is the conclusion. asolutionof(𝑃) such that 𝛽(𝑡) ≥ 𝑢(𝑡), 𝑡∈[0,1]. 6 Abstract and Applied Analysis

Theorem 9. Assume that there exist a strict lower solution 𝛼 where 𝛾1,2 : (0, 1) × R → R is defined by and a strict upper solution 𝛽 of (𝑃) such that 𝛼≺𝛽.Then (𝑃) 𝑢 𝛼≺𝑢≺𝛽 𝛽 (𝑡) ,𝑢>𝛽(𝑡) , problem has at least one solution such that . { 2 2 Moreover, for 𝑅>0largeenough,theLeray-Schauderdegree 𝛾1,2 (𝑡, 𝑢) = {𝑢,1 𝛼 (𝑡) ≤𝑢≤𝛽2 (𝑡) , (48) can be computed as { {𝛼1 (𝑡) ,𝑢<𝛼1 (𝑡) . 𝑑𝐿𝑆 (𝐼−𝐺,Ω,0) =1, (42) For any 𝜀>0, 𝛼1 −𝜀and 𝛽2 +𝜀are strict lower solution where Ω={𝑢∈𝑋|𝛼≺𝑢≺𝛽,‖𝑢‖𝑤 <𝑅}. and strict upper solution of (𝑀1,2). In fact, if 𝑢 is a solution of (𝑀1,2),thenwehave𝛼1(𝑡) − 𝜀1 <𝛼 (𝑡) ≤ 𝑢(𝑡)2 ≤𝛽 (𝑡) < Proof. Consider the modified problem 𝛽2(𝑡) + 𝜀.ByTheorem 5,thereisasufficientlarge𝑅>0: 󸀠 𝜑 (𝑢󸀠 (𝑡)) +ℎ(𝑡) 𝑓(𝛾(𝑡, 𝑢 (𝑡)))=0, 𝑡∈(0, 1) , 𝑝 d (𝐼 −1,2 𝐺 ,Ω1,1,0)=1, (𝑀) LS 𝑢 (0) =0, 𝑢(1) =0, (𝐼 − 𝐺 ,Ω ,0)=1, dLS 1,2 2,2 (49) 𝛾:(0,1)×R → R (𝐼 − 𝐺 ,Ω ,0)=1, where is defined by dLS 1,2 1,2 𝛽 (𝑡) ,𝑢>𝛽(𝑡) , { where 𝐺1,2 :𝑋 →is 𝑋 defined by 𝐺1,2(𝑢)(𝑡) =1,2 𝐺(𝛾 (𝑡, 𝑢(𝑡)) 𝛾 (𝑡, 𝑢) = 𝑢, 𝛼 (𝑡) ≤𝑢≤𝛽(𝑡) , { (43) and {𝛼 (𝑡) ,𝑢<𝛼(𝑡) . Ω1,1 ={𝑢∈𝑋𝛼1 −𝜀≺𝑢≺𝛽1, ‖𝑢‖𝑤 <𝑅}, Then it is well known that if 𝑢 is a solution of (𝑀),then𝛼(𝑡) ≤ Ω2,2 ={𝑢∈𝑋𝛼2 ≺𝑢≺𝛽2 +𝜀,‖𝑢‖𝑤 <𝑅}, (50) 𝑢(𝑡) ≤ 𝛽(𝑡) and thus 𝑢 is solution of (𝑃).Define𝐺:𝑋 →𝑋 𝐺(𝑢)(𝑡) = 𝐺(𝛾(𝑡, 𝑢(𝑡))) 𝐺 by .Then is bounded and thus there Ω1,2 ={𝑢∈𝑋|𝛼1 −𝜀≺𝑢≺𝛽2 +𝜀,‖𝑢‖𝑤 <𝑅}. exists 𝑅≫1such that ‖𝐺𝑢‖𝑤 <𝑅for all 𝑢∈𝑋.Bythe homotopy invariance property of degree, we have Then by excision and additive property, we have

d (𝐼 − 𝐺,𝑅 𝐵 (0) ,0)=d (𝐼,𝑅 𝐵 (0) ,0)=1, (44) (𝐼 − 𝐺 ,Ω \(Ω ∪ Ω ),0)=−1. LS LS dLS 1,2 1,2 1,1 2,2 (51) 𝐵 (0) = {𝑢 ∈ 𝑋 | ‖𝑢‖ <𝑅} (𝑀) where 𝑅 𝑤 .Thus has a solution (𝑀 ) 𝑢 ∈ Ω 𝑢 ∈ Ω and (𝑃) has a solution 𝑢 satisfying 𝛼(𝑡) ≤ 𝑢(𝑡) ≤𝛽(𝑡).Since Thus there are three solutions of 1,2 , 1 1,1, 2 2,2, 𝑢 ∈Ω \(Ω ∪ Ω ) 𝑢 (𝑀 ) 𝛼 and 𝛽 are strict lower and upper solutions, respectively, by and 3 1,2 1,1 2,2 . Since all solutions of 1,2 𝑢∈[𝛼,𝛽 ] (𝑃) the definition of strict lower and upper solution, we have 𝛼≺ satisfy 1 2 ,theyaresolutionsof and the proof is 𝑢≺𝛽. Moreover, by using the fact that 𝐺=𝐺on Ω,(44)and done. excision property, we conclude that there exists 𝑅>0large enough such that 3. Application (𝐼−𝐺,Ω,0) = (𝐼 − 𝐺, Ω, 0) dLS dLS In this section, we prove the existence of triple positive solu- (45) tions for a problem of the form = (𝐼 − 𝐺, 𝐵 (0) ,0)=1. dLS 𝑅 󸀠 󸀠 𝜑𝑝(𝑢 (𝑡)) +𝜆ℎ(𝑡) 𝑓 (𝑢 (𝑡)) =0, 𝑡∈(0, 1) , (𝑃𝜆) 𝑢 (0) =0=𝑢(1) , Theorem 10 (three solutions theorem). Assume that there 𝛼 𝛽 exist a lower solution 1,anuppersolution 2, a strict lower where 𝜆>0and 𝑓 ∈ 𝐶([0, ∞), (0,. ∞)) Let us assume solution 𝛼2, and a strict upper solution 𝛽1 of (𝑃) such that 1 (𝐻1)ℎ∈H \𝐿 (0, 1) with ℎ≥0, 𝛼1 ≤𝛽1 ≤𝛽2,𝛼1 ≤𝛼2 ≤𝛽2, (46) (𝐻2) min𝑡∈(0,1)ℎ(𝑡) = ℎ >0, and there exists 𝑡0 ∈ [0, 1] with 𝛽1(𝑡0)<𝛼2(𝑡0).Thenproblem (𝐹1) lim𝑢→∞(𝑓(𝑢)/𝜑𝑝(𝑢)) =, 0 (𝑃) has at least three solutions 𝑢1, 𝑢2,and𝑢3 such that (𝐹2)𝑓is nondecreasing. 𝛼1 ≤𝑢1 ≺𝛽1,𝛼2 ≺𝑢2 ≤𝛽2, (47) The existence of two positive solutions for problem (𝑃𝜆) 𝑢3 ∈[𝛼1,𝛽2]\([𝛼1,𝛽1]∪[𝛼2,𝛽2]) . was proved in [11] under a stronger condition on ℎ such as 1 ∫ 𝑠𝛿(1 − 𝑠)𝛾ℎ(𝑠)𝑑𝑠 <∞ 𝛿, 𝛾 < 𝑝 −1 Proof. Consider the modified problem, 0 for some .Since 𝑓(0) > 0,wecaneasilyseethatanysolution𝑢 of problem 󸀠 󸀠 (𝑃 ) 𝐶 [0, 1] 𝐶1[0, 1] 𝜑𝑝(𝑢 (𝑡)) +ℎ(𝑡) 𝑓(𝛾1,2 (𝑡, 𝑢 (𝑡)))=0, 𝑡∈(0, 1) , 𝜆 is in 0 but not in so that with the aid of (𝑀1,2) three solutions theorem in Section 2, we prove the following 𝑢 (0) =0, 𝑢(1) =0, theorem. Abstract and Applied Analysis 7

󸀠 󸀠 Theorem 11. Assume (𝐻1), (𝐻2), (𝐹1) and (𝐹2),andalso and thus 𝑢 (1) > 𝛽 (1) and assume that there exist 𝑎>0and 𝑏>0such that 𝑎<𝑏and 𝑤𝑢󸀠 (1) >𝑤𝛽󸀠 (1) . 𝑎𝑝−1 𝑏𝑝−1 (60) >𝐶 , (52) 𝑓 (𝑎) 𝑓 (𝑏) 𝑡 − ∫ ℎ(𝑠)𝑑𝑠 =∞ For the case that lim𝑡→1 1/2 , 𝑝 𝑝−1 where 𝐶=4 (‖𝑒‖∞ /ℎ) and 𝑒 theuniquesolutionof 󸀠 󸀠 𝑤𝛽1 (1) = lim 𝑤 (𝑡) 𝛽1 (𝑡) 󸀠 󸀠 𝑡→1− 𝜑𝑝(𝑒 (𝑡)) +ℎ(𝑡) =0, 𝑡∈(0, 1) , (53) 1 = 𝑒 (0) =0=𝑒(1) . lim 𝑡 𝑡→1− 𝜑−1 (∫ ℎ (𝑠) 𝑑𝑠) 𝑝 1/2 Then for 𝜆>0satisfying 𝑎𝑝−1 𝑎𝑝−1 𝑡 𝑏𝑝−1𝐶 𝑎𝑝−1 ×[−𝜑−1 (−𝛼 ( ℎ) + ∫ ℎ (𝑠) 𝑑𝑠)] <𝜆< , 𝑝 𝑝−1 𝑝−1 𝑝−1 𝑝−1 (54) ‖𝑒‖∞ ‖𝑒‖∞ 1/2 𝑓 (𝑏) ‖𝑒‖∞ 𝑓 (𝑎) ‖𝑒‖∞ 𝑝−1 𝑝−1 𝑝−1 𝛼((𝑎 /‖𝑒‖∞ )ℎ) 𝑎 problem (𝑃𝜆) hasatleastthreepositivesolutions. −1 = lim −𝜑𝑝 (− 𝑡 + ) 𝑡→1− ∫ ℎ (𝑠) 𝑑𝑠 ‖𝑒‖𝑝−1 𝑝−1 𝑝−1 𝑝−1 𝑝−1 1/2 ∞ Proof. For 𝜆∈(𝑏 𝐶/𝑓(𝑏)‖𝑒‖∞ , 𝑎 /𝑓(𝑎)‖𝑒‖∞ ),itis 𝑝−1 trivial that 𝛼1 ≡0is a lower solution of (𝑃𝜆).Let𝛽1 = 𝑎 =𝜑−1 (− ). 𝑎(𝑒/‖𝑒‖∞).Then 𝑝 𝑝−1 ‖𝑒‖∞ 𝑝−1 𝑝−1 󸀠 𝑎 󸀠 𝑎 (61) −𝜑 (𝛽󸀠 (𝑡)) =− 𝜑 (𝑒󸀠 (𝑡)) = ℎ (𝑡) 𝑝 1 ‖𝑒‖𝑝−1 𝑝 ‖𝑒‖𝑝−1 ∞ ∞ (55) From the monotonicity of 𝑓 and choice of 𝜆 with help of >𝜆ℎ(𝑡) 𝑓 (𝑎) ≥𝜆ℎ(𝑡) 𝑓(𝛽1 (𝑡)). L’Hospital’s rule, we have 󸀠 󸀠 Thus 𝛽1 is an upper solution of (𝑃𝜆).Toshowthat𝛽1 is a strict 𝑤𝑢 (1) = lim 𝑤 (𝑡) 𝑢 (𝑡) 𝑡→1− upper solution of (𝑃𝜆), assume that 𝑢 is a solution of (𝑃𝜆) such that 𝑢≤𝛽1.Wefirstshowthat𝑢(𝑡)1 <𝛽 (𝑡),forall𝑡 ∈ (0, 1). 1 = lim Suppose it is not true, then there exist 𝑡0 and 𝑡1 with 𝑡0 <𝑡1 in 𝑡→1− −1 𝑡 󸀠 󸀠 󸀠 󸀠 𝜑 (∫ ℎ (𝑠) 𝑑𝑠) 𝑝 1/2 (0, 1) such that 𝑢 (𝑡0)=𝛽1(𝑡0) and 𝑢 (𝑡1)<𝛽1(𝑡1).Integrate 󸀠 󸀠 󸀠 󸀠 𝜑 (𝑢 (𝑡)) −𝜑 (𝛽 (𝑡)) 𝑡 𝑡 𝑡 𝑝 𝑝 1 from 0 to 1,by(55) and monotonicity −1 of 𝑓, we have the following contradiction: × [−𝜑𝑝 (−𝛼 (𝜆ℎ𝑓 (𝑢))+∫ 𝜆ℎ (𝑠) 𝑓 (𝑢 (𝑠)) 𝑑𝑠)] 1/2 󸀠 󸀠 0>𝜑𝑝 (𝑢 (𝑡1)) − 𝜑𝑝 (𝛽 (𝑡1)) 𝑡 1 𝛼 (𝜆ℎ𝑓 (𝑢)) ∫ 𝜆ℎ (𝑠) 𝑓 (𝑢 (𝑠)) 𝑑𝑠 = −𝜑−1 (− + 1/2 ) 𝑡1 lim 𝑝 𝑡 𝑡 󸀠 󸀠 󸀠 󸀠 𝑡→1− ∫ ℎ (𝑠) 𝑑𝑠 ∫ ℎ (𝑠) 𝑑𝑠 = ∫ 𝜑𝑝(𝑢 (𝑠)) −𝜑𝑝(𝛽1 (𝑠)) 𝑑𝑠 1/2 1/2 𝑡0 (56) 𝑡 𝑡 ∫ 𝜆ℎ (𝑠) 𝑓 (𝑢 (𝑠)) 𝑑𝑠 1 −1 1/2 > ∫ −𝜆ℎ (𝑠) 𝑓 (𝑢 (𝑠)) +𝜆ℎ(𝑠) 𝑓(𝛽1 (𝑠))≥0. =𝜑𝑝 (− lim 𝑡 ) 𝑡 𝑡→1− ∫ ℎ (𝑠) 𝑑𝑠 0 1/2 𝑢(0) = 𝛽 (0) = 𝑢(1) = 𝛽 (1) = 0 Since 1 1 ,itsufficestoshow 𝜆ℎ (𝑡) 𝑓 (𝑢 (𝑡)) 𝑤𝑢󸀠(1) > 𝑤𝛽󸀠 (1) 𝑤𝑢󸀠(0) < 𝑤𝛽󸀠 (0) −1 that 1 .Theinequality 1 can be =𝜑𝑝 (− lim ) 𝑡 𝑡→1− ℎ (𝑡) − ∫ ℎ(𝑠)𝑑𝑠 <∞ proved similarly. For the case that lim𝑡→1 1/2 , 𝑢󸀠(1) 𝛽󸀠(1) −1 −1 we know and exist. Since =𝜑𝑝 (− lim 𝜆𝑓 (𝑢 (𝑡)))=𝜑𝑝 (−𝜆𝑓 (0)) 𝑡→1− 󸀠 󸀠 𝜑 (𝑢󸀠 (𝑡)) −𝜑 (𝛽󸀠 (𝑡)) >0, 𝑡∈(0, 1) , 𝑝 𝑝 1 (57) 𝑎𝑝−1𝑓 (0) >𝜑−1 (− ) 󸀠 󸀠 𝑝 𝑓 (𝑎) ‖𝑒‖𝑝−1 we know that there exists 𝑑 ∈ (0, 1) such that 𝑢 (𝑑) > 𝛽 (𝑑). ∞ 𝑢󸀠(𝑡) ≤ 𝛽󸀠(𝑡) 𝑡 ∈ (0, 1) Indeed, otherwise, for all ;thenby 𝑎𝑝−1 integrating this from 𝑡 to 1, we have the contradiction >𝜑−1 (− )=𝑤𝛽󸀠 (1) . 𝑝 ‖𝑒‖𝑝−1 1 𝑢 (𝑡) ≥𝛽(𝑡) ,𝑡∈(0, 1) . ∞ (58) (62) Integrating (57)from𝑑 to 1, we have Thus we proved that 𝛽1 is a strict upper solution of (𝑃𝜆).Now, 󸀠 󸀠 󸀠 󸀠 𝑝−1 𝑝−1 ∗ 𝜑𝑝 (𝑢 (1))−𝜑𝑝 (𝛽1 (1))>𝜑𝑝 (𝑢 (𝑑))−𝜑𝑝 (𝛽1 (𝑑))>0, since 𝜆>𝑏 𝐶/𝑓(𝑏)‖𝑒‖∞ ,wemaychoose𝜆 satisfying 𝑝−1 𝑝−1 ∗ ∗ 𝑝 𝑝−1 (59) 𝑏 𝐶/𝑓(𝑏)‖𝑒‖∞ <𝜆 <𝜆;thensince𝜆 ℎ𝑓(𝑏)/4 𝑏 >1, 8 Abstract and Applied Analysis

𝑝−1 ∗ 󸀠 󸀠 we may choose 𝑘, 𝑗 >1 such that 1<(𝑘𝑗) <𝜆ℎ𝑓(𝑏)/ Similarly, we can prove 𝑤𝑢 (1) < 𝑤𝛼2(1) and thus 𝑢≻𝛼2 for 𝑝 𝑝−1 𝑢 (𝑃 ) 𝑢≥𝛼 𝛼 4 𝑏 .Let𝛼2 be the solution of all solution of 𝜆 such that 2. This implies that 2 is a strict lower solution of (𝑃𝜆).Since‖𝛼2‖∞ ≥‖V‖∞ =𝑏> 󸀠 󸀠 ∗ 𝜑𝑝(𝛼2 (𝑡)) +𝜆 ℎ𝑓 (V (𝑡)) =0, 𝑡∈(0, 1) , 𝑎=‖𝛽1‖∞, there exists 𝑡0 ∈ (0, 1) such that 𝛼2(𝑡0)>𝛽1(𝑡0). (63) Define 𝛼 (0) =0=𝛼 (1) , 2 2 𝑒 𝛽2 =𝜆𝑀 . (68) where V(𝑡) = 𝑏𝛾(𝑡) when ‖𝑒‖∞

𝑘 𝑗 1 {1−(1−(4𝑡) ) , 0≤𝑡≤ , Then from (𝐹1), there exists sufficiently large 𝑀≫1such { if 4 𝛾 (𝑡) = that { 1 1 (64) {1, if ≤𝑡≤ , 𝑓 (𝜆𝑀) 𝜆 { 4 2 < (𝜆𝑀)𝑝−1 ‖𝑒‖𝑝−1 (69) 󸀠 ∞ and 𝛾(𝑡) = 𝛾(1−𝑡),for𝑡 ∈ (1/2, 1].Wenotethat|V (𝑡)| ≤ 4𝑘𝑗𝑏 and let us show that V(𝑡) ≤2 𝛼 (𝑡) for 0≤𝑡≤1/2.Itisclearthat and 𝛽2 >𝛼2, 𝛽2 >𝛽1.Thuswehave 󸀠 󸀠 󸀠 𝛼2(𝑡) ≥ 0 = V (𝑡) for 1/4 ≤ 𝑡 ≤ 1/2.Wenotethat𝛼2(1/2) = 0 𝑝−1 󸀠 󸀠 from the symmetry of V.For0≤𝑡≤1/4,byintegrating(63) 󸀠 (𝜆𝑀) 𝜑𝑝(𝑒 (𝑡)) ∗ −𝜑 (𝛽󸀠 (𝑡)) =− from 𝑡 to 1/2, from the choice of 𝜆 and 𝐶,wehave 𝑝 2 𝑝−1 ‖𝑒‖∞ 1/2 (70) 󸀠 −1 ∗ 𝛼2 (𝑡) =𝜑𝑝 (∫ 𝜆 ℎ𝑓 (V (𝑠)) 𝑑𝑠) 𝑡 >𝜆ℎ(𝑡) 𝑓 (𝜆𝑀) ≥𝜆ℎ(𝑡) 𝑓(𝛽2 (𝑡)).

1/2 −1 ∗ This implies that 𝛽2 is an upper solution of (𝑃𝜆) and the proof ≥𝜑𝑝 (∫ 𝜆 ℎ𝑓 (𝑤 (𝑠)) 𝑑𝑠) (65) 1/4 is complete by three solutions theorem. 1 1/(𝑝−1) =(𝜆∗ℎ𝑓 (𝑏) ) >4𝑘𝑗𝑏≥V󸀠 (𝑡) . 4. Example 4 As an example, let us consider the following combustion Thus 𝛼2(𝑡) > V(𝑡) for 0<𝑡≤1/2and it is clear that 𝛼2(𝑡) > model defined on an exterior domain: V(𝑡) for 0<𝑡≤1,bythesymmetryof𝛼2 and V.Fromthe 𝑓 𝛼𝑢 monotonicity of ,wehave −Δ 𝑢=𝜆𝑘(|𝑥|) ( ), 𝑥∈Ω, 𝑝 exp 𝛼+𝑢 󸀠 󸀠 ∗ (𝐸 ) −𝜑𝑝(𝛼2 (𝑡)) =𝜆 ℎ𝑓 (V (𝑡)) <𝜆ℎ(𝑡) 𝑓 (V (𝑡)) 𝜆 (66) 𝑢| =0, 𝑢󳨀→0 |𝑥| 󳨀→ ∞, |𝑥|=𝑟0 as ≤𝜆ℎ(𝑡) 𝑓(𝛼2 (𝑡)). Δ 𝑢= (|∇𝑢|𝑝−2∇𝑢) Ω={𝑥∈R𝑁 |𝑟<|𝑥|<∞} This implies that 𝛼2 is a lower solution of (𝑃𝜆) and by using where 𝑝 div , , 1 the similar argument as of 𝛽1,wecanshowthat𝛼2(𝑡) < 𝑢(𝑡) 1<𝑝<𝑁,and𝛼, 𝜆.Moreover >0 𝑘∈𝐿loc([𝑟0, ∞), (0, ∞)). for all solution 𝑢 of (𝑃𝜆) such that 𝑢≥𝛼2. Now to show For the radial solutions of (𝐸𝜆),bychangesofvariables, 𝛼 (𝑃 ) (−𝑁+𝑝)/(𝑝−1) that 2 is a strict lower solution of 𝜆 , we need to show 𝑟 = |𝑥|, 𝑢(𝑟) = 𝑢(|𝑥|), 𝑡 = (𝑟/𝑟0) , (𝐸𝜆) can be 󸀠 󸀠 󸀠 󸀠 that 𝑤𝑢 (0) > 𝑤𝛼2(0) and 𝑤𝑢 (1) < 𝑤𝛼2(1).Forthecase transformed into (𝑃𝜆) with 1/2 󸀠 󸀠 + ∫ ℎ(𝑠)𝑑𝑠 <∞ 𝑤𝑢 (1) < 𝑤𝛼 (1) that lim𝑡→0 𝑡 , 2 can be 𝑝−1 𝑝 𝛽 𝑝 −𝑝(𝑁−1)/(𝑁−𝑝) −(𝑝−1)/(𝑁−𝑝) provedbysimilarargumentasof 1.Now,forthecaseof ℎ (𝑡) =( ) 𝑟0 𝑡 𝑘(𝑟0𝑡 ). 1/2 𝑁−𝑝 + ∫ ℎ(𝑠)𝑑𝑠 =∞ + 𝑤(𝑡) =0 lim𝑡→0 𝑡 ,from(65)andlim𝑡→0 , 󸀠 󸀠 (71) we have 𝑤𝛼2(0) =lim𝑡→0+ 𝑤(𝑡)𝛼2(𝑡) = 0 and 󸀠 󸀠 Let us define 𝑤𝑢 (0) = lim 𝑤 (𝑡) 𝑢 (𝑡) 𝑡→0+ ∞ 𝜏 1 −1 1−𝑁 𝑁−1 K={𝑘∈𝐿loc ([𝑟0,∞))|∫ 𝜑𝑝 (𝜏 ∫ 𝑟 𝑘 (𝑟) 𝑑𝑟)𝑑𝜏 1 𝑟 𝑟 = 0 0 lim 1/2 𝑡→0+ 𝜑−1 (∫ ℎ (𝑠) 𝑑𝑠) 𝑝 𝑡 <∞}, 1/2 −1 ×[𝜑𝑝 (𝛼 (𝜆ℎ𝑓 (𝑢))+∫ 𝜆ℎ (𝑠) 𝑓 (𝑢 (𝑠)) 𝑑𝑠)] 𝑡 ∞ K ={𝑘∈𝐿1 ([𝑟 ,∞))|∫ 𝑟𝑁−1𝑘 (𝑟) 𝑑𝑟 < ∞} ; 𝜆ℎ (𝑡) 𝑓 (𝑢 (𝑡)) 1 loc 0 −1 𝑟0 =𝜑𝑝 ( lim )=𝜆𝑓(0) >0 𝑡→0+ ℎ (𝑡) (72)

󸀠 =𝑤𝛼2 (0) . then it is easy to check K1 ⫋ K and if 𝑘∈K \ K1,then 1 (67) corresponding ℎ in (71) satisfies ℎ∈H \𝐿 (0, 1). Abstract and Applied Analysis 9

𝛿 As an example of 𝑘,take𝑘(𝑟) =𝑟 for −𝑁<𝛿<−𝑝; [5] B. Ricceri, “On a three critical points theorem,” Archiv der then 𝑘∈K \ K1.Moreover,ℎ in (71)canbecalculatedas Mathematik,vol.75,no.3,pp.220–226,2000. 𝜌 ℎ(𝑡) =2 𝐶 𝑡 ,forsome𝐶2 >0and 𝜌 given as [6] G. L. Karakostas, “Triple positive solutions for the Φ-Laplacian when Φ is a sup-multiplicative-like function,” Electronic Journal −𝑝 (𝑁+𝛿) +(𝑝+𝛿) of Differential Equations,vol.69,pp.1–13,2004. 𝜌= , (73) 𝑁−𝑝 [7] J. Li and J. Wang, “Triple positive solutions for a type of second- order singular boundary problems,” Boundary Value Problems, 1 and we see that ℎ∈H \𝐿 (0, 1),when−𝑁 < 𝛿 <. −𝑝 vol.2010,ArticleID376471,2010. Finally, we have a multiplicity result of positive solutions [8]J.Ren,W.Ge,andB.Ren,“Existenceofthreepositivesolutions for quasi-linear boundary value problem,” Acta Mathematicae for combustion model (𝐸𝜆). Applicatae Sinica. English Series,vol.21,no.3,pp.353–358,2005.

Corollary 12. Assume 𝑘∈K \ K1.If𝛼 is sufficiently large, [9] E.K.LeeandY.H.Lee,“Aresultonthreesolutionstheoremand then (𝐸𝜆) hasatleastthreepositiveradialsolutionsfor𝜆∈ its application to p-Laplacian systems with singular weights,” Boundary Value Problems,vol.2012,article63,20pages,2012. (𝐼1,𝐼2),where [10] I. Sim and Y. H. Lee, “A new solution operaor of one 4𝑝𝛼𝑝−1 1 dimensional p-Laplacian with a sign-changing weight and its 𝐼1 = ,𝐼2 = . (74) ℎ (𝛼/2) (𝛼/ (𝛼+1)) ‖𝑒‖𝑝−1 application,” Abstract and Applied Analysis,vol.2012,ArticleID exp exp ∞ 243740, 15 pages, 2012. [11] E. Ko, E. K. Lee, and R. Shivaji, “Multiplicity results for classes Proof. Let 𝑓(𝑢) = exp(𝛼𝑢/(𝛼 + 𝑢)) and 𝑎=1, 𝑏=𝛼.Then 𝑓 of singular problems on an exterior domain,” Discrete and since is nondecreasing and Continuous Dynamical Systems,vol.33,no.11/12,pp.5153–5166, 1/𝑓 (1) 1 𝑓 (𝛼) 1 𝛼 𝛼 2013. = = ( − )󳨀→∞ 𝛼𝑝−1/𝑓 (𝛼) 𝛼𝑝−1 𝑓 (1) 𝛼𝑝−1 exp 2 𝛼+1 (75) as 𝛼→∞, all hypotheses of Theorem 11 are satisfied for sufficiently large 𝛼 andwegettheconclusion.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments ThisresearchwassupportedbyBasicScienceResearch Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012005767). This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A1011225).

References

[1] A. K. Ben-Naoum and C. de Coster, “On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem,” Differential and Integral Equations, vol. 10, pp. 1093–1112, 1997. [2] G. Bonannao, “Existence of three solutions for a two point boundary value problem,” Applied Mathematics Letters,vol.13, no. 5, pp. 53–57, 2000. [3]J.R.Graef,S.Heidarkhani,andL.Kong,“Acriticalpoints approach for the existence of multiple solutions of a Dirichlet quasilinear system,” Journal of Mathematical Analysis and Appli- cations, vol. 388, no. 2, pp. 1268–1278, 2012. [4] B. Ricceri, “A three critical points theorem revisited,” Nonlinear Analysis:Theory,MethodsandApplications,vol.70,no.9,pp. 3084–3089, 2009. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 291397, 7 pages http://dx.doi.org/10.1155/2014/291397

Research Article Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces

Zunwei Fu,1 Shanzhen Lu,2 Yibiao Pan,3 and Shaoguang Shi1

1 Department of Mathematics, Linyi University, Linyi 276005, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 3 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Correspondence should be addressed to Shaoguang Shi; [email protected]

Received 6 October 2013; Accepted 9 December 2013; Published 3 February 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 Zunwei Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.

1. Introduction and Main Results Throughout this paper, the letter 𝐶 will denote a positive constant which may vary from line to line but will remain Oscillatory integrals in one form or another have been an independent of the relevant quantities. essential part of harmonic analysis from the very beginnings We state a celebrated result of Ricci and Stein on oscilla- of that subject; three chapters are devoted to them in the tory integrals as follows. celebrated Stein’s book [1]. Many operators in harmonic analysis or partial differential equations are related to some Theorem 1 (see [8]). Let 1<𝑝<∞, 𝐾 satisfy (2) and (3). versions of oscillatory integrals, such as the Fourier trans- Then for any real-valued polynomial 𝑃(𝑥,𝑦),theoscillatory 𝑝 𝑝 form, the Bochner-Riesz means, and the Radon transform integral operator 𝑇 is of type (𝐿 ,𝐿 ) and its norm depends which has important applications in the CT technology. on the total degree of 𝑃, but not on the coefficients of 𝑃 in other Among numerous papers dealing with oscillatory singular respects. integral operators in some function spaces, we refer to [2– 7] and the references therein. More generally, let us now Weighted inequalities arise naturally in harmonic analy- consider a class of oscillatory integrals defined by Ricci and sis, but their use is best justified by the variety of applications Stein [8]: in which they appear. It is worth pointing out that many authors are interested in the inequalities when the weight 𝑖𝑃(𝑥,𝑦) 𝑇𝑓 (𝑥) = p.v. ∫ 𝑒 𝐾 (𝑥−𝑦) 𝑓 (𝑦) 𝑑𝑦, (1) functions belong to the Muckenhoupt classes ([9]), which R are denoted by 𝐴𝑝 (1<𝑝<∞)classes for simplicity. This where 𝑃(𝑥,𝑦)is a real-valued polynomial defined on R × R class consists of positive locally integrable functions (weight 1 𝑤 and the function 𝐾∈𝐶(R \{0})is a Calderon-Zygmund´ functions) for which kernel. That means 𝐾 satisfies 𝑝−1 𝐶 𝐶 1 1 1−𝑝󸀠 |𝐾 (𝑥)| ≤ , |∇𝐾 (𝑥)| ≤ ,𝑥=0,̸ sup ( ∫ 𝑤 (𝑥) 𝑑𝑥)( ∫ 𝑤(𝑥) 𝑑𝑥) <∞, (4) |𝑥| |𝑥|2 (2) 𝐼 |𝐼| 𝐼 |𝐼| 𝐼

∫ 𝐾 (𝑥) 𝑑𝑥 = 0 ∀𝑎, 𝑏 (0<𝑎<𝑏) . where the supremum is taken over all intervals 𝐼⊂R and (3) 󸀠 𝑎<|𝑥|<𝑏 1/𝑝 + 1/𝑝 =1. 2 Abstract and Applied Analysis

In 1992, Lu and Zhang [10]establishedtheweighted The one-sided weight classes are of interest, not only version of Theorem 1. because they control the boundedness of the one-sided Hardy-Littlewood maximal operator, but also because they Theorem 2. Let 𝑝, 𝑃 (𝑥, 𝑦) and 𝐾 be as in Theorem 1.Then 𝑝 aretherightclassesfortheweightedestimatesofone-sided the oscillatory singular integral operator 𝑇 is of type (𝐿 (𝑤), 𝑝 Calderon-Zygmund´ singular integral operators [15], which 𝐿 (𝑤)) with 𝑤∈𝐴𝑝. Here its operator norm is bounded by are defined by 𝑃 a constant depending on the total degree of ,butnotonthe ∞ 𝑃 ̃+ coefficients of in other respects. 𝑇 𝑓 (𝑥) = lim ∫ 𝐾 (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦, 𝜀→0+ 𝑥+𝜀 We point out that Theorems 1 and 2 also hold for 𝑥−𝜀 (9) ̃− dimension 𝑛≥2. We choose the results for 𝑛=1here 𝑇 𝑓 (𝑥) = lim ∫ 𝐾 (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦, in order to introduce the one-sided operators which were 𝜀→0+ −∞ R defined on .Theorems1 and 2 arealsotrueformore where 𝐾 is the one-sided Calderon-Zygmund´ kernel with 𝐿2 − + general kernels, that is, nonconvolution kernels, under the - support in R =(−∞,0)and R =(0,+∞),respectively.We boundedness assumption on the corresponding Calderon-´ say a function 𝐾 is a one-sided Calderon-Zygmund´ kernel if Zygmund singular integral operators: 𝐾 satisfies (2)and 󵄨 󵄨 ̃ 󵄨 󵄨 𝑇 (𝑥) = p.v. ∫ 𝐾 (𝑥−𝑦) 𝑓 (𝑦) 𝑑𝑦. (5) 󵄨∫ 𝐾 (𝑥) 𝑑𝑥󵄨 ≤𝐶, 0<𝑎<𝑏 R𝑛 󵄨 󵄨 (10) 󵄨 𝑎<|𝑥|<𝑏 󵄨 However,thistopicexceedsthescopeofthispaper.Formore R− =(−∞,0) R+ =(0,+∞) information about this work, see [8, 10], for example. with support in or . An example The study of weights for one-sided operators was moti- of such a kernel is vated not only by the generalization of the theory of both- sin (log |𝑥|) 𝐾 (𝑥) = 𝜒(−∞,0) (𝑥) , (11) sided ones, but also by their natural appearance in harmonic (𝑥 log |𝑥|) analysis; for example, they are required when we treat the one-sided Hardy-Littlewood maximal operator [11]: where 𝜒𝐸 denotes the characteristic function of a set 𝐸. 𝑥+ℎ + 1 󵄨 󵄨 Theorem 4 1<𝑝<∞ 𝐾 𝑀 𝑓 (𝑥) = sup ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑦, (see [15]). Let and be a one-sided ℎ>0 ℎ 𝑥 Calderon-Zygmund´ kernel. Then (6) 1 𝑥 ̃+ 𝑝 + − 󵄨 󵄨 (1) 𝑇 is bounded in 𝐿 (𝑤) ifandonlyif𝑤∈𝐴𝑝; 𝑀 𝑓 (𝑥) = sup ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑦, ℎ>0 ℎ 𝑥−ℎ ̃− 𝑝 − (2) 𝑇 is bounded in 𝐿 (𝑤) ifandonlyif𝑤∈𝐴𝑝. arising in the ergodic maximal function. Sawyer first intro- + duced the classical one-sided weight 𝐴𝑝 classes in [11]. The Theorem 4 is the one-sided version of weighted norm + − general definitions of 𝐴𝑝 and 𝐴𝑝 were introduced in [12]as inequality of singular integral due to Coifman and Fefferman [9]. 𝑏 𝑐 𝑝−1 + 1 1−𝑝󸀠 Highly inspired by the above statements for oscillatory 𝐴𝑝:sup 𝑝 ∫ 𝑤 (𝑥) 𝑑𝑥(∫ 𝑤(𝑥) 𝑑𝑥) ≤𝐶, 𝑎<𝑏<𝑐 (𝑐−𝑎) 𝑎 𝑏 singular integral operators and one-sided operator theory, in [16], the authors had introduced the one-sided oscillatory 𝑐 𝑏 𝑝−1 − 1 1−𝑝󸀠 singular integral operators and studied the weighted weak 𝐴𝑝:sup 𝑝 ∫ 𝑤 (𝑥) 𝑑𝑥(∫ 𝑤(𝑥) 𝑑𝑥) ≤𝐶, type (1, 1) norm inequalities for these operators. In this paper, (𝑐−𝑎) 𝑏 𝑎 𝑎<𝑏<𝑐 we will further study the one-sided Muckenhoupt weight (7) classes and give the one-sided version of Theorem 2.Itis 󸀠 where 1<𝑝<∞, 1/𝑝 + 1/𝑝 =1;also,for𝑝=1, well known that the property of the one-sided Muckenhoupt + − − + weight classes is worse than the Muckenhoupt weight classes 𝐴1: 𝑀 𝑤≤𝐶𝑤,1 𝐴 : 𝑀 𝑤≤𝐶𝑤. (8) (see also [17]). For example, both the reverse Holder¨ inequal- ity and the doubling condition are not true for the one-sided The smallest constant 𝐶 for which the above inequalities + − case. Therefore, some new methods are needed to deal with are satisfied will be denoted by 𝐴 (𝑤) and 𝐴 (𝑤), 𝑝≥1. 𝑝 𝑝 some new difficulties. 𝐴+ (𝑤) 𝐴− (𝑤) 𝐴+ 𝐴− 𝑝 ( 𝑝 )willbecalledthe 𝑝 (resp., 𝑝)constant We first recall the definition of one-sided oscillatory of 𝑤. By Lebesgue’s differentiation theorem, we can easily + − + integral operator as prove 𝐴1(𝑤) (resp., 𝐴1(𝑤)) ≥.In[ 1 13], the class 𝐴∞ was + + ∞ 𝐴 = ⋃ 𝐴 + 𝑖𝑃(𝑥,𝑦) introduced as ∞ 𝑝<∞ 𝑝 (see also [14]). It is easy to see 𝑇 𝑓 (𝑥) = lim ∫ 𝑒 𝐾(𝑥−𝑦)𝑓(𝑦)𝑑𝑦, + − + − 𝜀→0+ 𝑥+𝜀 that for 1≤𝑝≤∞, 𝐴𝑝 ⊂𝐴𝑝, 𝐴𝑝 ⊂𝐴𝑝,and𝐴𝑝 =𝐴𝑝 ⋂ 𝐴𝑝. 𝑥−𝜀 (12) − 𝑖𝑃(𝑥,𝑦) Theorem 3 (see [11]). Let 1<𝑝<∞.Then 𝑇 𝑓 (𝑥) = lim ∫ 𝑒 𝐾(𝑥−𝑦)𝑓(𝑦)𝑑𝑦, 𝜀→0+ −∞ + 𝑝 + (1) 𝑀 is bounded in 𝐿 (𝑤) if and only if 𝑤∈𝐴 ; 𝑝 where 𝑃(𝑥,𝑦)is a real-valued polynomial defined on R × R − 𝑝 − (2) 𝑀 is bounded in 𝐿 (𝑤) if and only if 𝑤∈𝐴𝑝. and the kernel 𝐾 is a one-sided Calderon-Zygmund´ kernel Abstract and Applied Analysis 3

− + 󸀠 󸀠 󸀠 󸀠 with support in R and R ,respectively.Now,weformulate For 𝜆>0, 𝑎 =𝜆𝑎, 𝑏 =𝜆𝑏, 𝑐 =𝜆𝑐,and𝑑 =𝜆𝑑,wehave our results as follows. 𝑏 𝑐 𝑝−1 1 1−𝑝󸀠 Theorem 5. 1<𝑝<∞ 𝐾 𝑝 ∫ 𝑤 (𝜆𝑥) 𝑑𝑥(∫ 𝑤(𝜆𝑥) 𝑑𝑥) Let and be a one-sided Calderon-´ (𝑐−𝑎) 𝑎 𝑏 Zygmund kernel. Then for any real-valued polynomial 𝑃(𝑥,𝑦), 𝑏𝜆 𝑐𝜆 𝑝−1 1 󸀠 = ∫ 𝑤 (𝑥) 𝜆−1𝑑𝑥(∫ 𝑤(𝑥)1−𝑝 𝜆−1𝑑𝑥) (𝑐−𝑎)𝑝 (1) there exists constant 𝐶>0such that 𝑎𝜆 𝑏𝜆 𝑏𝜆 𝑐𝜆 𝑝−1 1 1−𝑝󸀠 󵄩 + 󵄩 󵄩 󵄩 = 𝑝 ∫ 𝑤 (𝑥) 𝑑𝑥(∫ 𝑤(𝑥) 𝑑𝑥) 󵄩𝑇 𝑓󵄩𝐿𝑝(𝑤) ≤𝐶󵄩𝑓󵄩𝐿𝑝(𝑤), (13) (𝜆 (𝑐−𝑎)) 𝑎𝜆 𝑏𝜆

𝑝−1 𝑏󸀠 𝑐󸀠 1 󸀠 + 1−𝑝 𝑤∈𝐴 = 𝑝 ∫ 𝑤 (𝑥) 𝑑𝑥(∫ 𝑤(𝑥) 𝑑𝑥) where 𝑝 and the operator norm depend on the (𝑐󸀠 −𝑎󸀠) 𝑎󸀠 𝑏󸀠 + total degree of 𝑃 and 𝐴𝑝(𝑤), but not on the coefficients of 𝑃 in other respects; ≤𝐶. (16) (2) there exists constant 𝐶>0such that The proof is complete. 󵄩 󵄩 󵄩 󵄩 󵄩𝑇−𝑓󵄩 ≤𝐶󵄩𝑓󵄩 , We say a weight 𝑤 satisfies the one-sided reverse Holder¨ 󵄩 󵄩𝐿𝑝(𝑤) 󵄩 󵄩𝐿𝑝(𝑤) (14) + 𝑅𝐻𝑟 condition [18] if there exists 𝐶>0such that for any 𝑎<𝑏and 1<𝑟<∞, 𝑤∈𝐴− where 𝑝 and the operator norm depend on the 𝑏 𝑏 − 𝑟 𝑟−1 total degree of 𝑃 and 𝐴𝑝(𝑤), but not on the coefficients ∫ 𝑤(𝑥) 𝑑𝑥≤𝐶(𝑀(𝑤𝜒(𝑎,𝑏)) (𝑏)) ∫ 𝑤 (𝑥) 𝑑𝑥, (17) 𝑎 𝑎 of 𝑃 in other respects. where 𝑀 is the classical Hardy-Littlewood maximal operator. The rest of this paper is devoted to the argument for + The smallest such constant will be called the 𝑅𝐻𝑟 constant Theorem 5. Section 2 contains some preliminaries which are + of 𝑤 and will be denoted by 𝑅𝐻 (𝑤). Corresponding to the essential to our proof. In Section 3,wewillgivetheproofof 𝑟 classical reverse Holder¨ inequality, (17)isnamedtheweak Theorem 5. reverse Holder¨ inequality. For 𝑟=∞,wesayaweight𝑤 + satisfies the one-sided reverse Holder¨ 𝑅𝐻∞ condition if there + exists 𝐶>0such that 𝑤(𝑥)≤𝐶𝑚 𝑤(𝑥) for almost all 𝑥∈R 2. Preliminaries + where 𝑚 is the one-sided minimal operator defined as Lemma 6 1<𝑝<∞ 𝑤≥0 (see [11, 18]). Let and be locally 𝑥+ℎ + 1 󵄨 󵄨 integrable. Then the following statements are equivalent: 𝑚 𝑓 (𝑥) = inf ∫ 󵄨𝑓󵄨 𝑑𝑦. (18) ℎ>0 ℎ 𝑥 𝑤∈𝐴+ + (1) 𝑝; The smallest such constant will be called the 𝑅𝐻∞ constant of + + 𝑤 andwillbedenotedby𝑅𝐻∞(𝑤).Itisclearthat𝑅𝐻∞(𝑤) ≥ + 󸀠 1 𝑅𝐻 𝑤1−𝑝 ∈𝐴− .In[18], the authors give several characterizations of 𝑟 (2) 𝑝󸀠 ; where the constants 𝐶 are not necessary the same.

+ − Lemma 8. 𝑎<𝑏<𝑐<𝑑 1<𝑟<∞ 𝑤≥0 (3) there exist 𝑤1 ∈𝐴1 and 𝑤2 ∈𝐴1 such that 𝑤= Let , ,and be locally 1−𝑝 integrable. Then the following statements are equivalent: 𝑤1(𝑤2) . 𝑏 𝑏 + ∫ 𝑤(𝑥)𝑟𝑑𝑥 ≤ 𝐶(𝑀(𝑤𝜒 )(𝑏))𝑟−1 ∫ 𝑤(𝑥)𝑑𝑥 According to the definition of 𝐴𝑝,wecaneasilyobtain (1) 𝑎 (𝑎,𝑏) 𝑎 ; 𝑏 𝑐 𝑟 the following lemma. (1/(𝑏 − 𝑎)) ∫ 𝑤(𝑥)𝑟𝑑𝑥 ≤ 𝐶((1/(𝑐 −𝑏))∫ 𝑤(𝑥)𝑑𝑥) (2) 𝑎 𝑏 + + 𝜆 with 𝑏−𝑎=2(𝑐−𝑏); Lemma 7. Let 1<𝑝<∞and 𝑤∈𝐴𝑝.Then𝐴𝑝(𝛿 (𝑤)) = 𝑏 𝑑 𝑟 𝐴+ (𝑤) 𝛿𝜆(𝑤)(𝑥) = 𝑤(𝜆𝑥) 𝜆>0 (1/(𝑏 − 𝑎)) ∫ 𝑤(𝑥)𝑟𝑑𝑥≤𝐶((1/(𝑑−𝑐))∫ 𝑤(𝑥)𝑑𝑥) 𝑝 ,where for all . (3) 𝑎 𝑐 𝑏−𝑎=𝑑−𝑏=2(𝑑−𝑐) + with ; Proof. For 1<𝑝<∞,if𝑤∈𝐴 ,then 𝑏 𝑐 𝑟 𝑝 (1/(𝑏 − 𝑎)) ∫ 𝑤(𝑥)𝑟𝑑𝑥 ≤ 𝐶((1/(𝑐 −𝑏))∫ 𝑤(𝑥)𝑑𝑥) (4) 𝑎 𝑏 with 𝑏−𝑎=𝑐−𝑏; 𝑏 𝑐 𝑝−1 𝑟 1 󸀠 𝑏 𝑟 𝑑 ∫ 𝑤 (𝑥) 𝑑𝑥(∫ 𝑤(𝑥)1−𝑝 𝑑𝑥) ≤𝐶. (5) (1/(𝑏 − 𝑎)) ∫ 𝑤(𝑥) 𝑑𝑥≤𝐶((1/(𝑑−𝑐))∫ 𝑤(𝑥)𝑑𝑥) sup (𝑐−𝑎)𝑝 (15) 𝑎 𝑐 𝑎<𝑏<𝑐 𝑎 𝑏 with 𝑏−𝑎=𝑑−𝑐=𝛾(𝑑−𝑎), 0<𝛾≤1/2. 4 Abstract and Applied Analysis

+ Lemma 9 (see [18]). Aweight𝑤∈𝐴𝑝 for 𝑝>1if and only The inequality 𝐶𝐻 ≤𝐶implies 𝐶𝐻/(𝐶𝐻 −1)≥𝐶/(𝐶−1). 𝑟 ≤ 𝐶/(𝐶 −1) if there exist 0<𝛾<1/2and a constant 𝐶𝛾 such that for Therefore, if ,thenwehave 𝑏−𝑎=𝑑−𝑐=𝛾(𝑑−𝑎)with 𝑎<𝑏<𝑐<𝑑,thefollowing 𝑏 𝑟 𝑟−1 inequality holds: ∫ 𝑤𝐻 (𝑥) 𝑑𝑥=𝐶𝐻(𝑀 𝐻(𝑤 𝜒𝐼) (𝑏)) ∫ 𝑤𝐻 (𝑥) 𝑑𝑥 𝐼 𝑎 𝑝−1 (25) 𝑏 𝑑 󸀠 𝑏 ∫ 𝑤 (𝑥) 𝑑𝑥(∫ 𝑤(𝑥)1−𝑝 𝑑𝑥) ≤𝐶(𝑏−𝑎)𝑝. 𝑟−1 𝛾 (19) =𝐶(𝑀(𝑤1𝜒(𝑎,𝑏)) (𝑏)) ∫ 𝑤𝐻 (𝑥) 𝑑𝑥. 𝑎 𝑐 𝑎 + Hence 𝑤1 ∈𝑅𝐻𝑟 by the monotone convergence theorem. Combining the results in [12, 15, 18, 19], we can deduce − 1−𝑝 + Lemma 10. In what follows, we will include its proof with Since 𝑤2 ∈𝐴1, we next claim that 𝑤2 ∈𝑅𝐻∞. In fact, for slight modifications for the sake of completeness. any interval 𝐼 = (𝑎, 𝑏),wehave + 1 1−𝑝 1 Lemma 10. Let 𝑤∈𝐴. Then there exists 𝜀>0such that 1−𝑝 𝑝 ( ∫ 𝑤2 (𝑥) 𝑑𝑥) ≤ ∫ 𝑤2(𝑥) 𝑑𝑥 (26) 1+𝜀 + |𝐼| 𝐼 |𝐼| 𝐼 𝑤 ∈𝐴𝑝. − by Holder’s¨ inequality and the 𝐴1 condition. For almost every 1−𝑝 + − Proof. By Lemma 6,wehave𝑤=𝑤1𝑤2 with 𝑤1 ∈𝐴1 𝑥∈𝐼 =(2𝑎−𝑏,𝑎),wehave − and 𝑤2 ∈𝐴1.Forfixedinterval𝐼 = (𝑎, 𝑏), we next claim + 1 that 𝑤1 ∈𝑅𝐻𝑟 for all 1 < 𝑟 < 𝐶/(𝐶 −1) with 𝐶= 𝐶𝑤2 ≥ ∫ 𝑤2 (𝑥) 𝑑𝑥. (27) + − |𝐼| 𝐼 max{𝐴1(𝑤1), 𝐴1(𝑤1)} > 1. In fact, we consider the truncation of 𝑤 at height 𝐻 defined by 𝑤𝐻 = min{𝑤1,𝐻} which also + Thus, satisfies 𝐴 condition (with a constant 𝐶𝐻 ≤𝐶). Therefore, if 1 1 1−𝑝 𝜆𝐼 =𝑀(𝑤𝐻𝜒𝐼)(𝑏) and 𝑆𝜆 ={𝑥∈𝐼:𝑤𝐻(𝑥) > ,thenwe𝜆} 1−𝑝 𝑤2(𝑥) ≤𝐶( ∫ 𝑤2 (𝑥) 𝑑𝑥) have |𝐼| 𝐼 1 󵄨 󵄨 ≤𝐶 ∫ 𝑤 (𝑥)1−𝑝𝑑𝑥 ∫ 𝑤𝐻 (𝑥) 𝑑𝑥≤𝐶𝐻𝜆 󵄨𝑆𝜆󵄨 ,𝜆≥𝜆𝐼. 2 (28) 󵄨 󵄨 (20) |𝐼| 𝐼 𝑆𝜆 𝑏 𝑆 =𝐼 1 1−𝑝 Indeed, it is straightforward if 𝜆 since ≤𝐶 ∫ 𝑤2(𝑥) 𝑑𝑥, 𝑏−𝑥 𝑥 𝑏 󵄨 󵄨 which implies our claim. Hence, 𝑤𝐻 (𝑆𝜆) =∫ 𝑤𝐻 (𝑥) 𝑑𝑥≤𝜆𝐼 (𝑏−𝑎) ≤𝐶𝐻𝜆 󵄨𝑆𝜆󵄨 . (21) 𝑎 1 1 ∫ 𝑤𝑟 ≤ ∫ 𝑤𝑟 (𝑤−𝑟(𝑝−1)) 𝐼 𝐼 1sup 2 We now assume 𝑆𝜆 =𝐼̸ and fix 𝜀>0andanopenset𝑂 such | | 𝐼 | | 𝐼 𝐼 that 𝑆𝜆 ⊂𝑂⊂𝐼with |𝑂| ≤ 𝜀 𝜆+ |𝑆 |.Let𝑂𝑖 =(𝑐,𝑑),which 𝑟 𝑟 𝑎≤𝑐<𝑑<𝑏 1 1 1−𝑝 is connected. There are two cases; that is, ≤𝐶( ∫ 𝑤1) ( ∫ 𝑤2 ) 𝐼 𝐼 𝐼 𝐼 and 𝑎≤𝑐<𝑑=𝑏.Inthefirstcase,itiseasytocheck 1 1 1 1 + that 𝑑 is not contained in 𝑆𝜆. By the definition of 𝑆𝜆, 𝑤1 ,we 𝑟 𝑟 𝑑 1−𝑝 ∫ 𝑤 (𝑥) 𝑑𝑥 ≤𝐶 𝜆(𝑑 − 𝑐) ≤𝐶(inf𝑤1) (sup𝑤 ) have 𝑐 𝐻 𝐻 , while the second case is 𝐼 2 (29) 𝑑 1 𝐼1 handled as the case 𝑆𝜆 =𝐼since ∫ 𝑤𝐻 (𝑥) 𝑑𝑥 ≤ 𝐶(𝑏. −𝑐) 𝑐 1 𝑟 Thus 𝑤𝐻 (𝑂𝑖)≤𝐶𝐻𝜆|𝑂𝑖|. Adding up with 𝑖,weget 𝑟 1−𝑝 ≤𝐶(inf 𝑤1) ( ∫ 𝑤2 ) 𝐼 𝐼 󵄨 󵄨 󵄨 󵄨 2 2 𝑤𝐻 (𝑆𝜆) ≤𝑤𝐻 (𝑂) ≤𝐶𝐻𝜆 󵄨𝑂𝑖󵄨 ≤𝐶𝐻𝜆 (𝜀+󵄨𝑆𝜆󵄨) . (22) 1 𝑟 ≤𝐶( ∫ 𝑤) , 𝜃>−1 𝐼 Therefore, we obtain (20). For fixed , multiply both 2 𝐼2 𝜃 sides of (20)by𝜆 and integrate from 𝜆𝐼 to infinity; we can where 𝐼1 =(𝑏,2𝑏−𝑎)and 𝐼2 = (2𝑏 − 𝑎, 3𝑏 − 2𝑎).ByLemma 8, obtain + 1−𝑝󸀠 − we obtain 𝑤∈𝑅𝐻𝑟 .Hence,𝑤 ∈𝑅𝐻𝑟 for all 1<𝑟< 1 𝜃+2 𝜃+1 𝐶𝐻 𝜃+2 𝐶/(𝐶 − 1) by Lemma 6. ∫ (𝑤𝐻 −𝜆𝐼 ) (𝑥) 𝑑𝑥 ≤ ∫ 𝑤𝐻 (𝑥) 𝑑𝑥. (23) 𝜃+1 𝐼 𝜃+2 𝐼 Let us fix 𝑎<𝑑and choose 𝑏, 𝑐 such that 𝑏−𝑎=𝑑−𝑐= (𝑑 − 𝑎)/4 (e.g., we choose 𝑏 = (𝑑 + ,3𝑎)/4 𝑐 = (3𝑑 +). 𝑎)/4 Now if 𝑟=𝜃+2<𝐶𝐻/(𝐶𝐻−1),then1/(𝜃+1)−𝐶𝐻/(𝜃+2) > 0, Following from the five points 𝑎, 𝑏, (𝑏 + 𝑐)/2, 𝑐,𝑑,wehave which implies four intervals, namely, (𝑏+𝑐) 𝑟 𝑟−1 𝐼 = (𝑎,) 𝑏 ,𝐼= (𝑏, ), ∫ 𝑤𝐻 (𝑥) 𝑑𝑥≤𝐶𝐻𝜆𝐼 ∫ 𝑤𝐻 (𝑥) 𝑑𝑥 1 2 𝐼 𝐼 2 (24) (30) 𝑟−1 (𝑏+𝑐) =𝐶𝐻(𝑀 𝐻(𝑤 𝜒𝐼) (𝑏)) ∫ 𝑤𝐻 (𝑥) 𝑑𝑥. 𝐼3 =( ,𝑐),4 𝐼 = (𝑐, 𝑑) . 𝐼 2 Abstract and Applied Analysis 5

By Lemma 8,wehave If 𝑑𝑥(𝑃) = 𝑘 and 𝑑𝑦(𝑃) =,then 0

𝑝−1 𝑘 1 𝑟 1 𝑟(1−𝑝󸀠) 𝑃(𝑥,𝑦)=𝑎 𝑥 +𝑄(𝑥,𝑦) (37) 󵄨 󵄨 ∫ 𝑤 (󵄨 󵄨 ∫ 𝑤 ) 𝑘0 󵄨𝐼 󵄨 󵄨𝐼 󵄨 󵄨 1󵄨 𝐼 󵄨 4󵄨 𝐼4 𝑖𝑎 𝑥𝑘 (31) 𝑑 (𝑄)≤ 𝑘−1 𝑒 𝑘0 𝑟 𝑟(𝑝−1) with 𝑥 .Bytakingthefactor out of the 1 1 (1−𝑝󸀠) 𝑟 ≤(󵄨 󵄨 ∫ 𝑤) (󵄨 󵄨 ∫ 𝑤 ) ≤𝐶. integral sign, we see that this case follows from the above 󵄨𝐼 󵄨 󵄨𝐼 󵄨 󵄨 2󵄨 𝐼2 󵄨 3󵄨 𝐼3 inductive hypothesis. Suppose 𝑙≥1and the desired bound holds when 𝑑𝑥(𝑃) = 𝑟 + Thus, 𝑤 ∈𝐴𝑝 by Lemma 9.Choosing0<𝜀=𝑟−1<1/ 𝑘 and 𝑑𝑦(𝑃) ≤ 𝑙. −1 (𝐶 − 1), then we complete the proof of the lemma. Now, let 𝑃(𝑥,𝑦) be a polynomial with 𝑑𝑥(𝑃) = 𝑘 and 𝑑𝑦(𝑃) = 𝑙,asgivenin(34). To prove Theorem 5, we still need a celebrated interpola- tion theorem of operators with change of measures. Case 1 (|𝑎𝑘𝑙|=1). Write

1+𝑥 Lemma 11 (see [20]). Suppose that 𝑢0, V0,𝑢1, V1 are positive + 𝑖𝑃(𝑥,𝑦) 1<𝑝 𝑝 <∞ 𝑇 𝑓 (𝑥) = ∫ 𝑒 𝐾 (𝑥 − 𝑦) 𝑓 (𝑦)𝑑𝑦 weight functions and 0, 1 . Assume sublinear 𝑥 operator 𝑆 satisfies ∞ 2𝑗+𝑥 󵄩 󵄩 󵄩 󵄩 + ∑ ∫ 𝑒𝑖𝑃(𝑥,𝑦)𝐾(𝑥−𝑦)𝑓(𝑦)𝑑𝑦 󵄩𝑆𝑓󵄩 𝑝 ≤𝐶󵄩𝑓󵄩 𝑝 , 󵄩 󵄩𝐿 0 (𝑢 ) 0󵄩 󵄩𝐿 0 (V ) 𝑗−1 (38) 0 0 𝑗=1 2 +𝑥 󵄩 󵄩 󵄩 󵄩 (32) 󵄩𝑆𝑓󵄩 ≤𝐶󵄩𝑓󵄩 . 󵄩 󵄩𝐿𝑝1 (𝑢 ) 1󵄩 󵄩𝐿𝑝1 (V ) ∞ 1 1 + + =: 𝑇0 𝑓 (𝑥) + ∑𝑇𝑗 𝑓 (𝑥) . Then, 𝑗=1 󵄩 󵄩 󵄩 󵄩 󵄩𝑆𝑓󵄩 ≤𝐶󵄩𝑓󵄩 󵄩 󵄩𝐿𝑝(𝑢) 󵄩 󵄩𝐿𝑝(V) (33) Take any ℎ∈R,andwrite

0<𝜃<1 1/𝑝 = 𝜃/𝑝 +(1−𝜃)/𝑝 𝑙 holds for any and 0 1,where 𝑃(𝑥,𝑦)=𝑎 (𝑥−ℎ)𝑘(𝑦 − ℎ) +𝑅(𝑥,𝑦,ℎ), (39) 𝑝𝜃/𝑝0 𝑝(1−𝜃)/𝑝1 𝑝𝜃/𝑝0 𝑝(1−𝜃)/𝑝1 𝜃 1−𝜃 𝑘𝑙 𝑢=𝑢0 𝑢1 , V = V0 V1 ,and𝐶≤𝐶0𝐶1 . where the polynomial 𝑅(𝑥,𝑦,ℎ) satisfies the induction Lemmas 10 and 11 are the main tools in proving assumption and the coefficients of 𝑅(𝑥,𝑦,ℎ)depend on ℎ. + Theorem 5. We consider first the estimates for 𝑇0 .Itiseasytocheck that 3. Proof of Theorem 5 + 𝑇0 𝑓 (𝑥) In this section, we will prove Theorem 5 by induction, which 1+𝑥 𝑖(𝑅(𝑥,𝑦,ℎ)+𝑎 (𝑦−ℎ)𝑘+𝑙) is partly motivated by [8, 10].Webeginwiththeproofof(1). = ∫ 𝑒 𝑘𝑙 𝐾(𝑥−𝑦)𝑓(𝑦)𝑑𝑦 For any nonzero real polynomial 𝑃(𝑥,𝑦)in 𝑥 and 𝑦,thereare 𝑥 𝑘, 𝑙, 𝑚≥0 such that 1+𝑥 𝑖𝑃(𝑥,𝑦) 𝑖(𝑅(𝑥,𝑦,ℎ)+𝑎 (𝑦−ℎ)𝑘+𝑙) + ∫ {𝑒 −𝑒 𝑘𝑙 } (40) 𝑘 𝑙 𝑥 𝑃 (𝑥,) 𝑦 =𝑎𝑘𝑙𝑥 𝑦 +𝑅(𝑥,) 𝑦 (34) ×𝐾(𝑥−𝑦)𝑓(𝑦)𝑑𝑦 with 𝑎𝑘𝑙 =0̸ and =: 𝑇+ 𝑓 (𝑥) +𝑇+ 𝑓 (𝑥) . 𝑅(𝑥,𝑦)= ∑ 𝑎 𝑥𝛼𝑦𝛽 + ∑ 𝑎 𝑥𝑘𝑦𝛽. 01 02 𝛼𝛽 𝑘𝛽 (35) 0≤𝛼<𝑘,0≤𝛽≤𝑚 0≤𝛽<𝑙 Now we split 𝑓 into three parts as 𝑑 (𝑃) = 𝑘 𝑑 (𝑃) = 𝑙 We will write 𝑥 and 𝑦 . Below we will carry 𝑓(𝑦)=𝑓(𝑦)𝜒 (𝑦) out the argument by using a double induction on 𝑘 and 𝑙. {|𝑦−ℎ|<1/2} If 𝑑𝑥(𝑃) = 0 and 𝑑𝑦(𝑃) is arbitrary, then 𝑃 (𝑥, 𝑦) = 𝑃(𝑦) +𝑓(𝑦)𝜒 (𝑦) + {1/2≤|𝑦−ℎ|<5/4} and 𝑇 𝑓 canbewrittenas (41) +𝑓(𝑦)𝜒 (𝑦) ∞ {|𝑦−ℎ|≥5/4} 𝑇+𝑓 (𝑥) = ∫ 𝐾 (𝑥 − 𝑦) 𝑔 (𝑦) 𝑑𝑦, lim+ (36) 𝜀→0 𝑥+𝜀 =: 𝑓1 (𝑦) +2 𝑓 (𝑦) +3 𝑓 (𝑦) .

𝑖𝑃(𝑦) where 𝑔(𝑦) =𝑒 𝑓(𝑦). Therefore, the conclusion of Observe that if |𝑥−ℎ|<1/4,then Theorem 5 follows from Theorem 4. + Let 𝑘≥1and assume that the conclusion of Theorem 5 𝑇01𝑓1 (𝑥) holds for all 𝑃(𝑥,𝑦)with 𝑑𝑥(𝑃) ≤ 𝑘 −1 and 𝑑𝑦(𝑃) arbitrary. 1+𝑥 𝑘+𝑙 (42) 𝑖(𝑅(𝑥,𝑦,ℎ)+𝑎𝑘𝑙(𝑦−ℎ) ) Wewill now prove that the conclusion of Theorem 5 holds = ∫ 𝑒 𝐾(𝑥−𝑦)𝑓1 (𝑦) 𝑑𝑦. for all 𝑃(𝑥,𝑦)with 𝑑𝑥(𝑃) = 𝑘 and 𝑑𝑦(𝑃) arbitrary. 𝑥 6 Abstract and Applied Analysis

Thus, it follows from the induction assumption that From (47)and(50), it follows that the inequality

𝑝 󵄨 󵄨𝑝 󵄨 + 󵄨 ∫ 󵄨𝑇+𝑓 (𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥 ∫ 󵄨𝑇01𝑓1 (𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥 󵄨 0 󵄨 |𝑥−ℎ|<1/4 |𝑥−ℎ|<1/4 (43) (51) 󵄨 󵄨𝑝 󵄨 󵄨𝑝 ≤𝐶∫ 󵄨𝑓(𝑦)󵄨 𝑤(𝑦)𝑑𝑦, ≤𝐶∫ 󵄨𝑓(𝑦)󵄨 𝑤(𝑦)𝑑𝑦 |𝑦−ℎ|<1/2 |𝑦−ℎ|<5/4

+ where 𝐶 is independent of ℎ and the coefficients of 𝑃(𝑥,𝑦). holds uniformly in ℎ∈R , which implies Notice that if |𝑥 − ℎ| < 1/4, 1/2 ≤ |𝑦 − ℎ| <5/4,then 󵄩 󵄩 󵄩 󵄩 󵄩𝑇+𝑓󵄩 ≤𝐶󵄩𝑓󵄩 , 𝑦−𝑥>1/4.Thus, 󵄩 0 󵄩𝐿𝑝(𝑤) 󵄩 󵄩𝐿𝑝(𝑤) (52)

󵄨 󵄨 𝑥+1 󵄨 󵄨 where 𝐶 is independent of the coefficients of 𝑃(𝑥,𝑦)and 𝑤∈ 󵄨𝑇+ 𝑓 (𝑥)󵄨 ≤𝐶∫ 󵄨𝐾(𝑥−𝑦)𝑓 (𝑦)󵄨 𝑑𝑦 + 󵄨 01 2 󵄨 󵄨 2 󵄨 𝐴 . 𝑥+1/4 𝑝 (44) We proceed with the proof of Theorem 5 with the esti- + + ≤𝐶𝑀 (𝑓2) (𝑥) . mates for 𝑇𝑗 𝑓. Because of the size condition (2), we observe that for 𝑗≥1 So we have 2𝑗+𝑥 󵄨 󵄨 󵄨 + 󵄨 󵄨𝑓(𝑦)󵄨 + 󵄨 + 󵄨𝑝 󵄨𝑇 𝑓 (𝑥)󵄨 ≤ ∫ 𝑑𝑦 ≤ 𝐶𝑀 (𝑓) (𝑥) , ∫ 󵄨𝑇 𝑓 𝑥 󵄨 𝑤 𝑥 𝑑𝑥 󵄨 𝑗 󵄨 󵄨 󵄨 (53) 󵄨 01 2 ( )󵄨 ( ) 2𝑗−1+𝑥 󵄨𝑥−𝑦󵄨 |𝑥−ℎ|<1/4 󵄨 󵄨 (45) 𝐶 𝑗 󵄨 󵄨𝑝 where is independent of .ByLemma 10,weknowthat ≤𝐶∫ 󵄨𝑓(𝑦)󵄨 𝑤(𝑦)𝑑𝑦, 𝜀>0 𝑤1+𝜀 ∈𝐴+ |𝑦−ℎ|<5/4 there exists such that 𝑝.Thuswehave 󵄩 󵄩 󵄩 󵄩 𝐶 ℎ 𝑃(𝑥,𝑦) 󵄩𝑇+𝑓󵄩 ≤𝐶󵄩𝑓󵄩 , where is independent of and the coefficients of . 󵄩 𝑗 󵄩𝐿𝑝(𝑤1+𝜀) 󵄩 󵄩𝐿𝑝(𝑤1+𝜀) (54) Again observe that if |𝑥−ℎ|<1/4and |𝑦 − ℎ| ≥ 5/4,then 𝑦−𝑥>1.Thus, where 𝐶 is independent of 𝑗. We now only need to recall Lemma 3.7 in [16]toseethat 𝑇+ 𝑓 (𝑥) =0. 01 3 (46) 󵄩 󵄩 󵄩 󵄩 󵄩𝑇+𝑓󵄩 ≤𝐶2−𝑗𝛿󵄩𝑓󵄩 , 󵄩 𝑗 󵄩𝐿𝑝 󵄩 󵄩𝐿𝑝 (55) Combining (43), (45), and (46), we get where 𝐶 depends only on the total degree of 𝑃(𝑥,𝑦)and 𝛿> 󵄨 + 󵄨𝑝 0 ∫ 󵄨𝑇01𝑓 (𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥 . It follows from (54), (55), and Lemma 11 that |𝑥−ℎ|<1/4 󵄩 󵄩 󵄩 󵄩 (47) 󵄩𝑇+𝑓󵄩 ≤𝐶2−𝑗𝜃𝛿󵄩𝑓󵄩 , 󵄩 𝑗 󵄩 𝑝 󵄩 󵄩𝐿𝑝(𝑤) (56) 󵄨 󵄨𝑝 󵄩 󵄩𝐿 (𝑤) ≤𝐶∫ 󵄨𝑓(𝑦)󵄨 𝑤(𝑦)𝑑𝑦, |𝑦−ℎ|<5/4 where 0<𝜃<1, 𝜃 is independent of 𝑗,and𝐶 depends only 𝐶 ℎ 𝑃(𝑥,𝑦) on the total degree of 𝑃(𝑥,𝑦). where is independent of and the coefficients of . 𝑤∈𝐴+ Evidently, if |𝑥 − ℎ| < 1/4 and 0<𝑦−𝑥<1,then From (52)and(56), it is clear that when 𝑝, 󵄩 + 󵄩 󵄩 󵄩 󵄨 𝑘+𝑙 󵄨 󵄩 󵄩 󵄩 󵄩 󵄨 𝑖𝑃(𝑥,𝑦) 𝑖(𝑅(𝑥,𝑦,ℎ)+𝑎 (𝑦−ℎ) )󵄨 󵄩𝑇 𝑓󵄩 𝑝 ≤𝐶󵄩𝑓󵄩 𝑝 , 󵄨𝑒 −𝑒 𝑘𝑙 󵄨 󵄩 󵄩𝐿 (𝑤) 󵄩 󵄩𝐿 (𝑤) (57) 󵄨 󵄨 (48) 𝐶 𝑃(𝑥,𝑦) 󵄨 󵄨 󵄨 󵄨 where depends only on the total degree of . ≤𝐶󵄨𝑎𝑘𝑙󵄨 󵄨𝑥−𝑦󵄨 =𝐶(𝑦−𝑥). 1/(𝑘+𝑙) Case 2 (|𝑎𝑘𝑙| =1̸). In this case, we write 𝜆=|𝑎𝑘𝑙| and Therefore, when |𝑥 − ℎ| < 1/4,wehave 𝜆𝑥 𝜆𝑦 𝑥+1 −(𝑘+𝑙) 𝑘 𝑙 󵄨 󵄨 󵄨 󵄨 𝑃(𝑥,𝑦)=𝜆 𝑎𝑘𝑙(𝜆𝑥) (𝜆𝑦) +𝑅( , ) 󵄨𝑇+ 𝑓 (𝑥)󵄨 ≤𝐶∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑥 𝜆 𝜆 󵄨 02 󵄨 󵄨 󵄨 (58) 𝑥 (49) + =𝑄(𝜆𝑥,𝜆𝑦). ≤𝐶𝑀 (𝑓 (⋅) 𝜒𝐵(ℎ,5/4) (⋅)) (𝑥) . Therefore, It follows from Theorem 3 that 𝑇+𝑓 (𝑥) = . . ∫ 𝑒𝑖𝑄(𝜆𝑥,𝜆𝑦)𝐾(𝑥−𝑦)𝑓(𝑦)𝑑𝑦 󵄨 + 󵄨𝑝 p v ∫ 󵄨𝑇02𝑓 (𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥 |𝑥−ℎ|<1/4 𝜆𝑥 𝑦 𝑦 (50) = . . ∫ 𝑒𝑖𝑄(𝜆𝑥,𝑦)𝐾( − )𝑓( )𝜆−1𝑑𝑦 󵄨 󵄨𝑝 p v 𝜆 𝜆 𝜆 (59) ≤𝐶∫ 󵄨𝑓(𝑦)󵄨 𝑤(𝑦)𝑑𝑦, |𝑦−ℎ|<5/4 + ⋅ =𝑇𝜆 (𝑓 ( )) (𝜆𝑥) , where 𝐶 is independent of ℎ and the coefficients of 𝑃(𝑥,𝑦). 𝜆 Abstract and Applied Analysis 7

−1 where 𝐾𝜆(𝑥 − 𝑦) =𝜆 𝐾(𝑥/𝜆 − 𝑦/𝜆) and [4] S. Lu, “Multilinear oscillatory integrals with Calderon-Zyg-´ mund kernel,” Science in China A,vol.42,no.10,pp.1039–1046, + 𝑖𝑄(𝑥,𝑦) 1999. 𝑇𝜆 𝑓 (𝑥) = p.v. ∫ 𝑒 𝐾𝜆 (𝑥−𝑦)𝑓(𝑦)𝑑𝑦. (60) 𝑝 [5] S. Lu and Y. Zhang, “Criterion on 𝐿 -boundedness for a class of oscillatory singular integrals with rough kernels,” Revista It is easy to check that 𝐾𝜆 satisfies (2)and(10). We have thus Matematica´ Iberoamericana,vol.8,no.2,pp.201–219,1992. established that [6] Y.Pan, “Hardy spaces and oscillatory singular integrals,” Revista 󵄩 󵄩 󵄩 󵄩 Matematica´ Iberoamericana,vol.7,no.1,pp.55–64,1991. 󵄩𝑇+𝑓󵄩 ≤𝐶󵄩𝑓󵄩 󵄩 𝜆 󵄩𝐿𝑝(𝑤) 󵄩 󵄩𝐿𝑝(𝑤) (61) [7] D. H. Phong and E. M. Stein, “Singular integrals related to the Radon transform and boundary value problems,” Proceedings of with similar statements as in Case 1.ByLemma 7,wehave the National Academy of Sciences of the United States of America, vol.80,no.24,pp.7697–7701,1983. 󵄨 + 󵄨𝑝 ∫ 󵄨𝑇 𝑓 (𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥 [8] F. Ricci and E. M. Stein, “Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals,” Journal 󵄨 󵄨𝑝 of Functional Analysis,vol.73,no.1,pp.179–194,1987. 󵄨 + ⋅ 󵄨 = ∫ 󵄨𝑇 𝑓( ) (𝜆𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥 󵄨 𝜆 𝜆 󵄨 [9] R. R. Coifman and C. Fefferman, “Weighted norm inequalities for maximal functions and singular integrals,” Studia Mathe- 󵄨 󵄨𝑝 −1 󵄨 + ⋅ 󵄨 𝑥 matica, vol. 51, pp. 241–250, 1974. =𝜆 ∫ 󵄨𝑇 𝑓( ) (𝑥)󵄨 𝑤( )𝑑𝑥 (62) 󵄨 𝜆 𝜆 󵄨 𝜆 [10] S. Lu and Y. Zhang, “Weighted norm inequality of a class of oscillatory singular operators,” Chinese Science Bulletin,vol.37, 󵄨 𝑥 󵄨𝑝 𝑥 󵄨 󵄨 pp. 9–13, 1992. ≤𝐶∫ 󵄨𝑓( )󵄨 𝑤( )𝑑𝑥 󵄨 𝜆 󵄨 𝜆 [11] E. Sawyer, “Weighted inequalities for the one-sided Hardy- Littlewood maximal functions,” Transactions of the American 󵄨 󵄨𝑝 =𝐶∫ 󵄨𝑓 (𝑥)󵄨 𝑤 (𝑥) 𝑑𝑥; Mathematical Society,vol.297,no.1,pp.53–61,1986. [12] F. J. Mart´ın-Reyes, P. Ortega Salvador, and A. de la Torre, that is, “Weighted inequalities for one-sided maximal functions,” Transactions of the American Mathematical Society,vol.319,no. 󵄩 󵄩 󵄩 󵄩 󵄩𝑇+𝑓󵄩 ≤𝐶󵄩𝑓󵄩 , 2,pp.517–534,1990. 󵄩 󵄩𝐿𝑝(𝑤) 󵄩 󵄩𝐿𝑝(𝑤) (63) + [13] F. J. Mart´ın-Reyes, L. Pick, and A. de la Torre, “𝐴∞ condition,” Canadian Journal of Mathematics,vol.45,no.6,pp.1231–1244, where 𝐶 depends on the total degree of 𝑃(𝑥,𝑦)but not on the 1993. coefficients of 𝑃(𝑥,𝑦). [14] F. J. Mart´ın-Reyes and A. de la Torre, “One-sided BMO spaces,” (2) We omit the details, since they are very similar to that Journal of the London Mathematical Society,vol.49,no.3,pp. − + 529–542, 1994. of the proof of (1) with 𝑤∈𝐴𝑝 instead of 𝑤∈𝐴𝑝. [15]H.Aimar,L.Forzani,andF.J.Mart´ın-Reyes, “On weighted inequalities for singular integrals,” Proceedings of the American Conflict of Interests Mathematical Society,vol.125,no.7,pp.2057–2064,1997. [16] Z. Fu, S. Lu, S. Sato, and S. Shi, “On weighted weak type norm The authors declare that there is no conflict of interests inequalities for one-sided oscillatory singular integrals,” Studia regarding the publication of this paper. Mathematica,vol.207,no.2,pp.137–151,2011. [17] F. J. Mart´ın-Reyes, P. Ortega Salvador, and A. de la Torre, Acknowledgments “Weights for one-sided operators,” in Recent Development in Real and Harmonic Analysis, Applied and Numerical Harmonic This work was partially supported by NSF of China (Grant Analysis, Birkhauser,¨ Boston, Mass, USA, 2009. 𝐴+ nos. 11271175, 10931001, and 11301249), NSF of Shandong [18] M. S. Riveros and A. de la Torre, “On the best ranges for 𝑝 + Province (Grant no. ZR2012AQ026), the AMEP and DYSP and 𝑅𝐻𝑟 ,” Czechoslovak Mathematical Journal,vol.51,no.2,pp. of Linyi University, and the Key Laboratory of Mathematics 285–301, 2001. and Complex System (Beijing Normal University), Ministry [19] F. J. Mart´ın-Reyes, “New proofs of weighted inequalities for the of Education, China. one-sided Hardy-Littlewood maximal functions,” Proceedings of the American Mathematical Society,vol.117,no.3,pp.691–698, 1993. References [20] E. M. Stein and G. Weiss, “Interpolation of operators with [1] E. M. Stein, Harmonic Analysis (Real-variable methods, orthogo- change of measures,” Transactions of the American Mathemat- nality, and oscillatory integrals),vol.43ofPrinceton Mathemati- ical Society,vol.87,pp.159–172,1958. cal Series, Princeton University Press, Princeton, NJ, USA, 1993. [2] L. Grafakos, Classical and Modern Fourier Analysis,Pearson Education, Upper Saddle River, NJ, USA, 2004. [3] Y. Hu and Y. Pan, “Boundedness of oscillatory singular integrals on Hardy spaces,” Arkiv for¨ Matematik,vol.30,no.2,pp.311– 320, 1992. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 942092, 14 pages http://dx.doi.org/10.1155/2014/942092

Research Article Multiple Solutions for a Class of 𝑁-Laplacian Equations with Critical Growth and Indefinite Weight

Guoqing Zhang1,2 and Ziyan Yao1

1 College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China 2 Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA

Correspondence should be addressed to Ziyan Yao; [email protected]

Received 7 September 2013; Revised 10 December 2013; Accepted 20 December 2013; Published 23 January 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 G. Zhang and Z. Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using the suitable Trudinger-Moser inequality and the Mountain Pass Theorem, we prove the existence of multiple solutions for 𝑁−2 𝑁−2 𝑁−2 𝛽 aclassof𝑁-Laplacian equations with critical growth and indefinite weight −div(|∇𝑢| ∇𝑢) + 𝑉(𝑥)|𝑢| 𝑢 = 𝜆(|𝑢| 𝑢/|𝑥| )+ 𝛽 𝑁 𝑁 𝑁 (𝑓(𝑥, 𝑢)/|𝑥| )+𝜀ℎ(𝑥), 𝑥∈R , 𝑢 =0̸, 𝑥∈R ,where0<𝛽<𝑁, 𝑉(𝑥) is an indefinite weight, 𝑓:R × R → R behaves like 𝑁/(𝑁−1) 1,𝑁 𝑁 ∗ exp(𝛼|𝑢| ) and does not satisfy the Ambrosetti-Rabinowitz condition, and ℎ∈(𝑊 (R )) .

󸀠 󸀠 1. Introduction where 1/𝑞 + 1/𝑞 =1, 𝑆𝑟(𝑟 = 𝑁, 𝑁𝑞 ) is the best constant In this paper, we consider the existence of multiple solutions 𝑁 𝑁 1,𝑁 𝑁 for the 𝑁-Laplacian elliptic equations with critical growth 𝑆𝑟‖𝑢‖𝐿𝑟(R𝑁) ≤ ∫ |∇𝑢| 𝑑𝑥, ∀𝑢 ∈𝑊 (R ); (2) R𝑁 and singular potentials that is, 𝑁−2 𝑁−2 − div (|∇𝑢| ∇𝑢) + 𝑉 (𝑥) |𝑢| 𝑢 𝑁 1,𝑁 𝑁 𝑆𝑟 = inf {∫ |∇𝑢| 𝑑𝑥 : 𝑢 ∈𝑊 (R ) , ‖𝑢‖𝐿𝑟(R𝑁) =1} . R𝑁 𝑁−2 |𝑢| 𝑢 𝑓 (𝑥, 𝑢) 𝑁 (3) =𝜆 + +𝜀ℎ(𝑥) ,𝑥∈R , (1) |𝑥|𝛽 |𝑥|𝛽 Note that if 𝑉 is a measurable function which satisfies 𝑁 − 𝑢 =0,̸ 𝑥∈R , (H2), there exists 𝛿0 >0such that ‖𝑉0 ‖𝐿𝑞(R𝑁) ≤(1−𝛿0)𝑆𝑁𝑞󸀠 . Recently, 𝑁-Laplacian equations had been studied by 𝑁 many authors. Marcos do O[´ 1] studied the existence of where 𝑁≥2, 0<𝜆<𝜆1, 𝜆1 = inf{∫ 𝑁 (|∇𝑢| + R nontrivial solutions for the following 𝑁-Laplacian equations 𝑉(𝑥)|𝑢|𝑁)𝑑𝑥 : 𝑢 ∈𝑊1,𝑁(R𝑁), ∫ (|𝑢|𝑁/|𝑥|𝛽)𝑑𝑥 = 1} R𝑁 , with critical growth: 1,𝑁 𝑁 ∗ 𝑁−2 0<𝛽<𝑁, ℎ∈(𝑊 (R )) , Δ 𝑁𝑢=div(|∇𝑢| ∇𝑢) is the 1,𝑁 𝑢∈𝑊0 (Ω) ,𝑢≥0,−Δ𝑁𝑢=𝑓(𝑥, 𝑢) ,𝑥∈Ω, 𝑁-Laplacian, the indefinite weight 𝑉(𝑥) ∈0 𝑅(𝑉 ),and𝑅(𝑉0) (4) is the classes of rearrangement of 𝑉0; 𝑉0 satisfies the following 𝑁 conditions: where Ω isboundedsmoothdomaininR (𝑁 ≥ 2).

𝑞 𝑁 Adimurthi and Sandeep [2] proved that the singular (H1) 𝑉0 ∈𝐿(R ), ∀𝑞 ≥ 1, Trudinger-Moser inequality − 𝑁/(𝑁−1) (H2) ‖𝑉0 ‖ 𝑞 𝑁 <𝑆𝑁𝑞󸀠 ,or𝑉0 ≥−𝑆𝑁 +𝛿,forsome𝛿>0, exp (𝛼|𝑢| ) 𝐿 (R ) ∫ 𝑑𝑥<+∞ sup 𝛽 (5) 1 𝑁 𝑢∈𝑊1,𝑁(Ω) Ω |𝑥| (H3) (1/𝑉0)∈𝐿(R ), 0 2 Abstract and Applied Analysis

holds if and only if 𝛼/𝛼𝑁 + 𝛽/𝑁,where ≤1 𝛼𝑁 = Theorem 1. Supposethat(H1)–(H3)and(f1)–(f4)aresatisfied 1/(𝑁−1) 0<𝜆<𝜆 𝑁𝑤𝑁−1 , 𝛼>0, 0≤𝛽<𝑁,and‖∇𝑢‖𝐿𝑁(Ω) ≤1,and and 1.Furthermore,assumethat 𝑁 studied the corresponding critical exponent problem. For the (f5) lim sup𝑠→0+ (𝑁𝐹(𝑥, 𝑠)/|𝑠| )=0,uniformlyon𝑥∈ unbounded domain, Li and Ruf [3] proved that, if we replace R𝑁 𝐿𝑁 ∇𝑢 , the -norm of in the supermum by the standard Sobolev 𝑟>0 norm, the supermum can still be finite. Adimurthi and Yang and there exists such that [4] obtained the following Trudinger-Moser inequality (f6) 𝑁/(𝑁−1) 𝑁/(𝑁−1) lim𝑠𝑓 (𝑥,) 𝑠 exp (−𝛼0|𝑠| ) exp (𝛼|𝑢| −𝑆𝑁−2 (𝛼, 𝑢)) 𝑠→0 ∫ 𝑑𝑥<+∞, 𝛽 (6) R𝑁 |𝑥| 2 > (𝛼 𝑑(𝑁−𝛽)/𝑁) 𝑁−𝛽 𝑁−𝛽 𝑒 𝑁 +𝐶𝑟 −(𝑟 /(𝑁−𝛽)) (10) 1,𝑁 𝑁 where 𝛼>0, 0≤𝛽<𝑁, 𝑢∈𝑊(R ), 𝑆 (𝛼, 𝑢) = ∑𝑁−2(𝛼𝑘/𝑘!)|𝑢|𝑁𝑘/(𝑁−1) 𝑁−𝛽 𝑁−1 and 𝑁−2 𝑘=0 ,andstudiedthe ×( ) , 𝑁 existence of nontrivial solution for the corresponding - 𝛼0 Laplacian equations with critical growth. In particular, using 𝑁 inequality (6)andtheMountainPassTheorem,LamandLu uniformly on compact subsets of R ,where𝑑 and 𝐶 are defined [5] studied the following nonuniformly elliptic equations of in Section 3. Then there exists 𝜀1 >0such that, for each 0<𝜀< 𝑁-Laplacian type of the form 𝜀1,problem(1) has at least two nontrivial weak solutions.

𝑁−2 𝑓 (𝑥, 𝑢) In this paper, as the function 𝑉(𝑥) is an indefinite − div (𝑎 (𝑥, ∇𝑢)) +𝑉(𝑥) |𝑢| 𝑢= +𝜀ℎ(𝑥) , |𝑥|𝛽 weight, we establish a singular Trudinger-Moser inequality (7) (see Lemma 8) and investigate the eigenvalue problem cor- 𝑥∈R𝑁, responding to problem (1). Using the singular Trudinger- Moser inequality, the eigenvalue problem and the Mountain where 𝑉(𝑥)0 >𝑉 >0, and obtained the existence and Pass Theorem, we prove the multiplicity result for problem multiplicity results of problem (7). (1). Furthermore, condition (f2) is used by Lam and Lu [5], and it implies that the function 𝑓(𝑥, 𝑢) does not satisfy the On the other hand, some authors have studied the Ambrosetti-Rabinowitz condition. case for the nonlinear term which does not satisfy the The paper is organized as follows. In Section 2,we Ambrosetti-Rabinowitz condition. Lam and Lu [6, 7]studied recall some important lemmas and consider the eigenvalue the existence of nontrivial solutions for the 𝑁-Laplacian problem corresponding to problem (1). Section 3 is devote to equations and systems and polyharmonic equations without prove Theorem 1. Ambrosetti-Rabinowitz conditions, respectively. Miyagaki and Souto [8] discussed a class of superlinear problems for the polynomial case without Ambrosetti-Rabinowitz conditions. 2. Preliminary Results Motivated by a suitable Trudinger-Moser inequality, we assume the following growth conditions on the nonlinearity 2.1. Key Lemmas. Now, we define the following Sobolev space 𝑓(𝑥, 𝑢) : 𝐸= {𝑢∈𝑊1,𝑁 (R𝑁):∫ |∇𝑢|𝑁𝑑𝑥 𝑁 R𝑁 (f1) the function 𝑓:R ×R → R is continuous, for some (11) 𝑁 𝑁 constants 𝛼0,𝑏1,𝑏2 >0and for all (𝑥, 𝑠) ∈ R × R, + ∫ 𝑉 (𝑥) |𝑢| 𝑑𝑥<+∞}, R𝑁 󵄨 󵄨 𝑁−1 󵄨𝑓 (𝑥,) 𝑠 󵄨 ≤𝑏1|𝑠| and the corresponding norm, (8) 1/𝑁 𝑁/(𝑁−1) 𝑁 𝑁 +𝑏2 [exp (𝛼0|𝑠| )−𝑆𝑁−2 (𝛼0,𝑠)]; ‖𝑢‖𝐸 =(∫ (|∇𝑢| +𝑉(𝑥) |𝑢| )𝑑𝑥) . (12) R𝑁

𝑁 From the Radial Lemma [9, 10], we have (f2) 𝐻(𝑥, 𝑡) ≤ 𝐻(𝑥, 𝑠),forall0<𝑡<𝑠, ∀𝑥 ∈ R ,where 1/𝑁 −1 𝑁 𝐻 (𝑥,) 𝑠 =𝑠𝑓(𝑥,) 𝑠 −𝑁𝐹(𝑥,) 𝑠 , |𝑢 (𝑥)| ≤ |𝑥| ( ) ‖𝑢‖𝐿𝑁(R𝑁),∀𝑥=0,̸ (13) 𝑤𝑁−1 1,𝑁 𝑁 𝑠 (9) for all 𝑢∈𝑊 (R ) being radially symmetric, where 𝐹 (𝑥,) 𝑠 = ∫ 𝑓 (𝑥, 𝜏) 𝑑𝜏; 𝑁 0 𝑤𝑁−1 is the surface area of the unit sphere in R . 𝑉(𝑥) is a rearrangement of 𝑉0 if 𝑁 + 󵄨 󵄨 (f3) there exists 𝑐>0such that for all (𝑥, 𝑠) ∈ R × R , 󵄨{𝑥 ∈ R𝑁 :𝑉(𝑥) ≥𝛼}󵄨 𝑁 󵄨 󵄨 0 < 𝐹(𝑥, 𝑠) ≤ 𝑐|𝑠| +𝑐𝑓(𝑥,𝑠); 󵄨 󵄨 (14) 𝑁 𝑁 = 󵄨{𝑥 ∈ R𝑁 :𝑉 (𝑥) ≥𝛼}󵄨 ,∀𝛼∈R𝑁, (f4) lim𝑠→∞(𝐹(𝑥, 𝑠)/|𝑠| )=∞,uniformlyon𝑥∈R . 󵄨 0 󵄨 We state our main result in this paper. where |⋅|denotes the Lebesgue measure. Abstract and Applied Analysis 3

1 𝑁 Lemma 2 (see [11]). Let 𝑉0 satisfy (𝐻1) and (𝐻2). Then there From (19), we have 𝑢𝑘 →𝑢in 𝐿 (𝐵𝑅(0)) and 𝐵𝑅(0) ⊂ R exists 𝛿0 >0such that is the ball centered at 0 with radius 𝑅. This together with21 ( ) ∫ |𝑢 −𝑢|𝑑𝑥≤𝐶𝜀 𝜀 𝑁 𝑁 leads to lim sup𝑘→+∞ R𝑁 𝑘 .Since is arbitrary, 𝛿0 ∫ |∇𝑢| 𝑑𝑥 ≤ ‖𝑢‖𝐸 ,∀𝑉∈𝑅(𝑉0). (15) R𝑁 we have − 󵄨 󵄨 ‖𝑉 ‖ <𝑆 󸀠 󵄨 󵄨 Proof. Assume that 0 𝐿𝑁(R𝑁) 𝑁𝑞 .Since lim ∫ 󵄨𝑢𝑘 −𝑢󵄨 𝑑𝑥 = 0. (22) 𝑘→+∞ R𝑁 𝑁 𝑁 − 𝑁 ‖𝑢‖𝐸 ≥ ∫ (|∇𝑢| +𝑉 (𝑥) |𝑢| )𝑑𝑥, 𝑞≥𝑁 R𝑁 Hence, for every ,wehave 󵄨 󵄨𝑞 󵄨 󵄨1/2󵄨 󵄨𝑞−1/2 − 𝑁 󵄩 − 󵄩 𝑁 ∫ 󵄨𝑢 −𝑢󵄨 𝑑𝑥 ≤ ∫ 󵄨𝑢 −𝑢󵄨 󵄨𝑢 −𝑢󵄨 𝑑𝑥 ∫ 𝑉 (𝑥) |𝑢| 𝑑𝑥 ≤ 󵄩𝑉 (𝑥)󵄩 ‖𝑢‖ 󸀠 (16) 󵄨 𝑘 󵄨 󵄨 𝑘 󵄨 󵄨 𝑘 󵄨 󵄩 󵄩𝐿𝑞(R𝑁) 𝑁𝑞 𝑁 𝑁 𝑁 R𝑁 𝐿 (R ) R R 󵄩 − 󵄩 𝑁 󵄨 󵄨 1/2 = 󵄩𝑉 (𝑥)󵄩 𝑞 𝑁 ‖𝑢‖ 󸀠 . 󵄨 󵄨 󵄩 0 󵄩𝐿 (R ) 𝐿𝑁𝑞 (R𝑁) ≤(∫ 󵄨𝑢𝑘 −𝑢󵄨 𝑑𝑥) R𝑁 𝛿0 Then, by (H2), there exists such that 1/2 (23) 󵄨 󵄨2𝑞−1 󵄩 − 󵄩 ×(∫ 󵄨𝑢 −𝑢󵄨 𝑑𝑥) 󵄩𝑉0 (𝑥)󵄩𝐿𝑞(R𝑁) ≤(1−𝛿0)𝑆𝑁𝑞󸀠 . (17) 󵄨 𝑘 󵄨 R𝑁 Therefore, we have 1/2 󵄨 󵄨 𝑁 𝑁 ≤𝐶(∫ 󵄨𝑢𝑘 −𝑢󵄨 𝑑𝑥) 󳨀→ 0. ‖𝑢‖𝐸 ≥𝛿0 ∫ |∇𝑢| 𝑑𝑥. (18) R𝑁 R𝑁

Lemma 6. 𝐸 is a reflexive Banach space. Remark 3. In this paper, we denote 𝐶 as positive (possibly different) constants. Proof. Suppose that ∀𝑢1 ∈𝐸, ∀𝑢2 ∈𝐸,wehave 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑢 󵄩 ≤1, 󵄩𝑢 󵄩 ≤1, 󵄩𝑢 −𝑢 󵄩 >𝜀, Remark 4. If 𝑉 ∈ 𝑅(𝑉0),then𝑉 satisfies (H1)–(H3). 󵄩 1󵄩𝐸 󵄩 2󵄩𝐸 󵄩 1 2󵄩𝐸 (24)

Lemma 5. 𝑁 𝑁 If (H1)–(H3) are satisfied, then and there exists 𝛿=1−√1 − (𝜀/4) , using the following 1,𝑁 𝑁 𝑞 𝑁 (1) the embedding 𝐸󳨅→𝑊 (R )󳨅→𝐿(R ) is inequality continuous, for all 1≤𝑞<∞; 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨𝑎+𝑏󵄨 󵄨𝑎+𝑏󵄨 1 𝑞 𝑁 󵄨 󵄨 + 󵄨 󵄨 ≤ (|𝑎|𝑝 + |𝑏|𝑝), (2) the embedding 𝐸󳨅→𝐿(R ) is compact, for all 𝑞≥𝑁. 󵄨 2 󵄨 󵄨 2 󵄨 2 󵄨 󵄨 󵄨 󵄨 (25) Proof. (1) From Lemma 2 and Sobolev-Poincare inequality, ∀𝑎, 𝑏, 𝑝∈ R, we obtain the conclusion. such that (2) Let {𝑢𝑘}⊂𝐸satisfy ‖𝑢𝑘‖𝐸 ≤𝐶for all 𝑘, and we assume 󵄩𝑢 +𝑢 󵄩𝑁 𝐸, 󵄩 1 2 󵄩 𝑢𝑘 ⇀𝑢, weakly in 󵄩 󵄩 󵄩 2 󵄩𝐸 𝑢 󳨀→ 𝑢, 𝐿𝑞 (R𝑁), ∀𝑞≥1, 𝑘 strongly in loc (19) 󵄩 󵄩𝑁 󵄩 󵄩𝑁 󵄩 󵄩𝑁 󵄩𝑢1 +𝑢2 󵄩 󵄩𝑢1 −𝑢2 󵄩 󵄩𝑢1 −𝑢2 󵄩 𝑢 󳨀→ 𝑢, 𝑁 = 󵄩 󵄩 − 󵄩 󵄩 + 󵄩 󵄩 𝑘 a.e. in R . 󵄩 2 󵄩𝐸 󵄩 2 󵄩𝐸 󵄩 2 󵄩𝐸 In view of (H3), for every 𝜀→0, there exists 𝑅>0such 1 󵄨 󵄨𝑁 󵄨 󵄨𝑁 ≤ (∫ 󵄨∇𝑢1󵄨 𝑑𝑥 + ∫ 󵄨∇𝑢2󵄨 𝑑𝑥) (26) that 2 R𝑁 R𝑁 1−1/𝑁 𝑁 1 1 󵄨 󵄨𝑁 󵄨 󵄨𝑁 𝜀 (∫ 𝑑𝑥) <𝜀. + ∫ 𝑉 (𝑥) (󵄨𝑢 󵄨 + 󵄨𝑢 󵄨 )𝑑𝑥−( ) 1/(𝑁−1) (20) 󵄨 1󵄨 󵄨 2󵄨 |𝑥|>𝑅 2 R𝑁 2 𝑉0 𝜀 𝑁 Hence, we have ≤1−( ) < (1−𝛿)𝑁. 1/𝑁 2 󵄨 󵄨 𝑉 󵄨 󵄨 ∫ 󵄨𝑢 −𝑢󵄨 𝑑𝑥 = ∫ 0 󵄨𝑢 −𝑢󵄨 𝑑𝑥 󵄨 𝑘 󵄨 1/𝑁 󵄨 𝑘 󵄨 |𝑥|>𝑅 |𝑥|>𝑅 Hence, 𝐸 is uniformly convex. We obtain that 𝐸 is a reflexive 𝑉0 Banach space. 1−1/𝑁 𝐼:𝐸 → R 1 Now, we define the functional ≤(∫ 𝑑𝑥) 1/(𝑁−1) |𝑥|>𝑅 1 𝑁 1 𝑁 𝑉0 (21) 𝐼 (𝑢) = ∫ |∇𝑢| 𝑑𝑥 + ∫ 𝑉 (𝑥) |𝑢| 𝑑𝑥 𝑁 R𝑁 𝑁 R𝑁 1/𝑁 󵄨 󵄨𝑁 ×(∫ 𝑉 󵄨𝑢 −𝑢󵄨 𝑑𝑥) 𝐹 (𝑥, 𝑢) 𝜆 |𝑢|𝑁 0󵄨 𝑘 󵄨 − ∫ 𝑑𝑥 − ∫ 𝑑𝑥 −𝜀 ∫ ℎ𝑢 𝑑𝑥; |𝑥|>𝑅 𝛽 𝛽 R𝑁 |𝑥| 𝑁 R𝑁 |𝑥| R𝑁 󵄩 󵄩 ≤𝜀󵄩𝑢𝑘 −𝑢󵄩𝐸 ≤𝐶𝜀. (27) 4 Abstract and Applied Analysis

1/𝑁 −1/𝑁 ∗ then the functional 𝐼(𝑢) is well defined by Lemma 5.More- sufficiently large; that is, 𝑟>𝑁 𝑤𝑁−1 ‖𝑢 ‖𝐿𝑁(R𝑁).Bythe 1 ∗ over, 𝐼(𝑢) is the 𝐶 functional on 𝐸 and ∀𝑢, V ∈𝐸;we radial lemma, for all |𝑥| ≥,wehave 𝑟 𝑢 (𝑥) < 1 and have 𝑅(𝛼∗,𝑢) 𝑅(𝛼∗,𝑢∗) ∫ 𝑑𝑥 = ∫ 𝑑𝑥 𝛽 𝛽 |𝑥|≥𝑟 |𝑥| |𝑥|≥𝑟 |𝑥| 𝐷𝐼 (𝑢) V = ∫ |∇𝑢|𝑁−2∇𝑢∇V 𝑑𝑥 + ∫ 𝑉 (𝑥) |𝑢|𝑁−2𝑢V 𝑑𝑥 R𝑁 R𝑁 𝑁−1󵄨 󵄨𝑁 1 (𝛼∗) 󵄨𝑢∗󵄨 ≤ ∫ ( 󵄨 󵄨 𝛽 𝑓 (𝑥, 𝑢) V |𝑢|𝑁−2𝑢V 𝑟 |𝑥|>𝑟 (𝑁−1)! − ∫ 𝑑𝑥 −𝜆 ∫ 𝑑𝑥 𝛽 𝛽 R𝑁 |𝑥| R𝑁 |𝑥| 𝑚󵄨 󵄨𝑚𝑁/(𝑁−1) (33) ∞ (𝛼∗) 󵄨𝑢∗󵄨 + ∑ 󵄨 󵄨 )𝑑𝑥 𝑚! −𝜀∫ ℎV 𝑑𝑥. 𝑚=𝑁 𝑁 R 󵄩 󵄩𝑁 󵄩𝑢∗󵄩 ∞ ∗ 𝑚 (28) 󵄩 󵄩𝐿𝑁(R𝑁) (𝛼 ) ≤ ( ∑ )≤𝐶. 𝛽 𝑟 𝑚=𝑁−1 𝑚! Hence, the critical point of the functional 𝐼(𝑢) is the weak solution of problem (1). Define the set 𝑆={𝑥∈B𝑟(0) : |𝑢(𝑥) − 𝑢(𝑟)| > 2|𝑢(𝑟)|}. Assume that 𝑆 is nonempty; then for all 𝑥∈𝑆and 𝜀>0we Lemma 7. Let 0 < 𝛼 ≤ (1 − 𝛽/𝑁)𝛼𝑁, 0<𝛽<𝑁, 𝑢∈𝐸and have ‖𝑢‖ ≤1 𝑞>𝑁 𝛼/𝛼 +𝛽/𝑁+1/𝑞≤1 𝐸 ;thenforsome and 𝑁 , 𝑁/(𝑁−1) one has |𝑢 (𝑥)| = |𝑢 (𝑥) −𝑢(𝑟) +𝑢(𝑟)|𝑁/(𝑁−1) 𝑁/(𝑁−1) [exp (𝛼|𝑢| )−𝑆𝑁−2 (𝛼, 𝑢)] |𝑢| ∫ 𝑑𝑥≤𝐶‖𝑢‖ 𝑞 𝑁 . 𝑁/(𝑁−1) 𝛽 𝐿 (R ) 𝑢 (𝑟) R𝑁 |𝑥| = |𝑢 (𝑥) −𝑢(𝑟)|𝑁/(𝑁−1)(1 + ) (29) |𝑢 (𝑥) −𝑢(𝑟)| 1/(𝑁−1) 𝑁/(𝑁−1) 𝑁 3 𝑁/(𝑁−1) ∗ ≤ |𝑢 (𝑥) −𝑢(𝑟)| + ( ) (34) Proof. Let 𝑅(𝛼, 𝑢) = exp(𝛼|𝑢| )−𝑆𝑁−2(𝛼, 𝑢); 𝑢 is the 𝑁−1 2 𝑢 Schwarz symmetrization of ;wecanconcludethat × |𝑢 (𝑟)||𝑢 (𝑥) −𝑢(𝑟)|1/(𝑁−1)

∗ 󵄨 ∗󵄨 𝑁/(𝑁−1) 𝑅 (𝛼, 𝑢) |𝑢| 𝑅(𝛼,𝑢 ) 󵄨𝑢 󵄨 ≤ (1+𝜀) |𝑢 (𝑥) −𝑢(𝑟)| ∫ 𝑑𝑥 = ∫ 𝑑𝑥. (30) 𝑁 𝛽 𝑁 𝛽 R |𝑥| R |𝑥| 2 1/𝑁−1 3 𝑁/(𝑁−1) |𝑢 (𝑟)|𝑁 +( ) ( ) . 2 (𝑁−1) 𝜀 󸀠 Let 𝑢 =𝑢/‖𝑢‖𝐸.Itiseasytoobtainthat𝑅(𝛼, 𝑢) is increasing with respect to |𝑢|.If‖𝑢‖𝐸 ≤1; then there holds Since

‖𝑢‖𝑁 = ‖∇𝑢‖𝑁 + ∫ 𝑉 (𝑥) |𝑢|𝑁𝑑𝑥=1, 𝑁/(𝑁−1) 𝐸 𝐿𝑁(R𝑁) (35) exp (𝛼|𝑢| )−𝑆𝑁−2 (𝛼, 𝑢) R𝑁 ∫ 𝑑𝑥 𝛽 R𝑁 |𝑥| so we have (31) 󵄨 󵄨𝑁/(𝑁−1) (𝛼󵄨𝑢󸀠󵄨 )−𝑆 (𝛼,󸀠 𝑢 ) ‖∇𝑢‖𝑁 =1−∫ 𝑉 (𝑥) |𝑢|𝑁𝑑𝑥, exp 󵄨 󵄨 𝑁−2 𝐿𝑁(R𝑁) ≤ ∫ 𝑑𝑥. R𝑁 𝑁 𝛽 R |𝑥| 1/(𝑁−1) (36) ‖∇𝑢‖𝑁/(𝑁−1) =(1−∫ 𝑉 (𝑥) |𝑢|𝑁𝑑𝑥) . 𝐿𝑁 R𝑁 ( ) R𝑁 Now, we prove that there exists a uniform constant 𝐶 such that,forallradiallydecreasingsymmetricfunctions𝑢∈ Thus, we obtain 𝑊1,𝑁(R𝑁) ‖𝑢‖ =1 and 𝐸 , 1/(𝑁−1) 1 1 =( ) . (37) ∗ 𝑁/(𝑁−1) ∗ ‖∇𝑢‖𝑁/(𝑁−1) 1−∫ 𝑉 (𝑥) |𝑢|𝑁𝑑𝑥 exp (𝛼 |𝑢| )−𝑆𝑁−2 (𝛼 ,𝑢) 𝐿𝑁(R𝑁) R𝑁 ∫ 𝑑𝑥≤𝐶, 𝛽 (32) R𝑁 |𝑥| From Hardy-Littlewood inequality, we have

∗ 𝑁 where 𝛼 = (1 − 𝛽/𝑁)𝛼𝑁. In the following, assume that 𝑢 󵄨 ∗󵄨 𝑁 ∫ 𝑉 (𝑥) 󵄨𝑢 󵄨 𝑑𝑥 ≥ ∫ 𝑉0 (𝑥) |𝑢| 𝑑𝑥. (38) 𝑁 𝑁 𝑁 is radially decreasing function in R and ‖𝑢‖𝐸 =1.Take𝑟 R R Abstract and Applied Analysis 5

∗ 𝑁 󸀠 𝑉(𝑥) ∈ 𝑅(𝑉 ) ∫ 𝑉(𝑥)|𝑢 | 𝑑𝑥 = 𝑢 = (𝑢(𝑥) − 𝑢(𝑟))/‖∇(𝑢(𝑥) −𝑢(𝑟))‖ 𝑁 𝑁 Since 0 ,wehave R𝑁 Let 𝐿 (R );weobtain 𝑁 ∫ 𝑉(𝑥)|𝑢| 𝑑𝑥 ∗ 𝑁/(𝑁−1) R𝑁 .Let 𝑒𝛼 |𝑢| ∫ 𝑑𝑥 𝛽 B (0) 1 𝑟 |𝑥| 1+𝜀= 𝑁/(𝑁−1) ‖∇𝑢‖ 𝛼∗|𝑢|𝑁/(𝑁−1) 𝛼∗|𝑢|𝑁/(𝑁−1) 𝑁 𝑁 𝑒 𝑒 𝐿 (R ) = ∫ 𝑑𝑥 + ∫ 𝑑𝑥 𝛽 𝛽 (46) 1/(𝑁−1) 𝑆 |𝑥| B𝑟(0)\𝑆 |𝑥| 1 =( ) ∗ 󸀠 𝑁/(𝑁−1) 1−∫ 𝑉 (𝑥) |𝑢|𝑁𝑑𝑥 𝑒𝛼 |𝑢 | R𝑁 ≤𝐶∫ 𝑑𝑥 + 𝐶 ≤𝐶. 𝛽 B (0) |𝑥| 1/(𝑁−1) (39) 𝑟 1 ≥( ) Hence, we obtain that (32)holds.For0<𝛼≤(1−𝛽/𝑁)𝛼𝑁, 1−∫ 𝑉 (𝑥) |𝑢|𝑁𝑑𝑥 ‖𝑢‖ ≤1 R𝑁 0 𝐸 ,wehave 𝑁/(𝑁−1) 1/(𝑁−1) exp (𝛼|𝑢| )−𝑆𝑁−2 (𝛼, 𝑢) 1 ∫ 𝑑𝑥≤𝐶. 𝛽 (47) ≥( ) . R𝑁 |𝑥| 1−(𝛿−𝑆 )∫ |𝑢|𝑁𝑑𝑥 𝑁 R𝑁 󸀠 Since 𝛼/𝛼𝑁 + 𝛽/𝑁 + 1/𝑞 ≤1 and 1/𝑞 + 1/𝑞 =1, 𝑞>𝑁,that 𝛼𝑞󸀠 <𝛼 𝛼𝑞󸀠/𝛼 +𝛽𝑞󸀠/𝑁 ≤ 1 Applying the mean value theorem to the function 𝜓(𝑡) = is, 𝑁, 𝑁 ,wehave 1/(𝑁−1) 𝑅 (𝛼, 𝑢) |𝑢| 𝑡 , we obtain that there exists 𝜉 which satisfies ∫ 𝑑𝑥 𝛽 R𝑁 |𝑥| 𝑁 1−(𝛿−𝑆𝑁) ‖𝑢‖ 𝑁 𝑁 ≤𝜉≤1, (40) (48) 𝐿 (R ) 𝑅(𝛼𝑞󸀠,𝑢) ≤ ∫ 𝑑𝑥‖𝑢‖ 𝑞 𝑁 ≤𝐶‖𝑢‖ 𝑞 𝑁 . 𝛽𝑞󸀠 𝐿 (R ) 𝐿 (R ) such that R𝑁 |𝑥|

1/(𝑁−1) 1−[1−(𝛿−𝑆 ) ‖𝑢‖𝑁 ] As the proof of Lemma 7, we can obtain the following. 𝑁 𝐿𝑁(R𝑁) (41) (𝛿 − 𝑆 ) Lemma 8. For 0 < 𝛼 ≤ (1 − 𝛽/𝑁)𝛼𝑁, 0<𝛽<𝑁, 𝑢∈𝐸and = 𝑁 𝜉(2−𝑁)/(𝑁−1)‖𝑢‖𝑁 . ‖𝑢‖ ≤1 𝑞>𝑁 𝛼/𝛼 + 𝛽/𝑁 + 1/𝑞 ≤1 𝑁−1 𝐿𝑁(R𝑁) 𝐸 , ,and 𝑁 ,onehas 𝑁/(𝑁−1) 𝑞 [exp (𝛼|𝑢| )−𝑆𝑁−2 (𝛼, 𝑢)] |𝑢| ∫ 𝑑𝑥 So we have 𝛽 R𝑁 |𝑥| (49) ‖𝑢‖𝑁 𝑞 𝐿𝑁(R𝑁) ≤𝐶(𝑁, 𝛼) ‖𝑢‖ . 𝜀= 𝐸 1/(𝑁−1) (𝑁−1) 𝜉(𝑁−2)/(𝑁−1)(1 − (𝛿 − 𝑆 ) ‖𝑢‖𝑁 ) 𝑁 𝐿𝑁(R𝑁) 2.2. The Eigenvalue Problem. We consider the following eigenvalue problem: ‖𝑢‖𝑁 𝐿𝑁(R𝑁) 𝑁−2 ≥ . 𝑁−2 𝑁−2 |𝑢| 𝑢 𝑁 𝑁−1 − div (|∇𝑢| ∇𝑢) + 𝑉 (𝑥) |𝑢| 𝑢=𝜆 ,𝑥∈R , |𝑥|𝛽 (42) 𝑢 =0,̸ 𝑥∈R𝑁. 1/𝑁 |𝑢(𝑟)| ≤ (𝑁/𝑤 ) ‖𝑢‖ 𝑁 𝑁 /𝑟 By 𝑁−1 𝐿 (R ) ,wehave (50) 𝑁 𝛽 2 1/(𝑁−1) Now, we denote the set 𝑀={𝑢∈𝐸:∫ 𝑁 (|𝑢| /|𝑥| )𝑑𝑥 = 3 𝑁/(𝑁−1) |𝑢 (𝑟)|𝑁 R ( ) ( ) ≤𝐶, (43) 1}, and define 2 (𝑁−1) 𝜀 𝜆1 = inf {𝐼𝑁 (𝑢) :𝑢∈𝐸\{0}}>0, 0<𝛽<𝑁, 0 =𝑢∈𝑀̸ (51) and ∀𝑥 ∈ 𝑆, 𝐼 (𝑢) = ∫ (|∇𝑢|𝑁 + 𝑉(𝑥)|𝑢|𝑁)𝑑𝑥 where 𝑁 R𝑁 . |𝑢 (𝑥) −𝑢(𝑟)|𝑁/(𝑁−1) Lemma 9 𝑢>0V >0 |𝑢 (𝑥)|𝑁/(𝑁−1) ≤ +𝐶. (see [12]). Let , be two continuous 1/(𝑁−1) (44) Ω (‖∇𝑢‖𝑁 ) functions in differentiable a.e., and 𝐿𝑁(R𝑁) 𝑢𝑝 𝑢𝑝−1 𝐿 (𝑢, V) = |∇𝑢|𝑝 +(𝑝−1) |∇V|𝑝 −𝑝 |∇V|𝑝−2∇V∇𝑢, 𝑝 𝑝−1 1,𝑁 V V Obviously, 𝑢−𝑢(𝑟)∈𝑊 (B𝑟(0)),and 𝑢𝑝 𝑅 (𝑢, V) = |∇𝑢|𝑝 − |∇V|𝑝−2∇( )∇V. 𝑁 𝑁 V𝑝−1 ∫ |∇ (𝑢−𝑢(𝑟))| 𝑑𝑥 ≤ ∫ |∇𝑢| 𝑑𝑥≤1. (45) B𝑟(0) B𝑟(0) (52) 6 Abstract and Applied Analysis

(1) 𝐿(𝑢, V) = 𝑅(𝑢, V)≥0 (2) 𝐿(𝑢, V)=0 Ω ∫ (|𝑢 |𝑁/|𝑥|𝛽)𝑑𝑥=1 Then , a.e. in if and R𝑁 𝑚 . On the other hand, we have and only if 𝑢=𝑘V for some 𝑘>0. ∫ |∇V|𝑁𝑑𝑥 + ∫ 𝑉 (𝑥) V𝑁𝑑𝑥 R𝑁 R𝑁 Proposition 10. Assume that (H1)–(H3) hold; then 𝜆1 >−∞ 𝜆 is the lowest eigenvalue of Problem (50) and 1 is principal. 󵄨 󵄨𝑁 ≤ (∫ 󵄨∇V 󵄨 𝑑𝑥 + ∫ 𝑉 (𝑥) V𝑁𝑑𝑥) =𝜆 , lim𝑛→∞ inf 󵄨 𝑛󵄨 𝑛 1 𝜆 >−∞ R𝑁 R𝑁 Proof. From Lemma 2,wehave 1 .Furthermore,any (59) minimizing sequence {𝑢𝑛} is bounded. Up to a subsequence, 𝑢∈𝐸 ∫ |∇V|𝑁𝑑𝑥 +∫ 𝑉(𝑥)V𝑁𝑑𝑥 =𝜆 >0 there exists such that and R𝑁 R𝑁 1 .Soweconclude that V is an eigenfunction associated with 𝜆1 and V >0. 𝑢 ⇀𝑢, 𝐸, 𝑛 0 in Then we conclude from the convergence in measure of the 󸀠 (53) sequence {V𝑛} towards V that 𝑢 󳨀→ 𝑢 , 𝐿𝑁𝑟 (R𝑁). 𝑛 0 in 󵄨 −󵄨 󵄨Ω𝑛 󵄨 󳨀→ 0, (60) − Hence, we have where Ω𝑛 denotes the negative set of V𝑛,whichcontradicts Proposition 11. 𝐼 (𝑢 )≤ 𝐼 (𝑢 )=𝜆 ,𝑢∈𝑀, 𝑁 0 𝑛→+∞lim 𝑁 𝑛 1 0 (54) Proposition 12. The first eigenvalue 𝜆1 is simple, in the sense and consequently we have that the eigenfunctions associated with it are merely constant multiples of each other.

𝐼𝑁 (𝑢0)=𝜆1. (55) Proof. Let 𝜑, 𝜁 be two eigenfunctions associated with 𝜆1. We assume without restriction that 𝜑>0, 𝜁>0;then𝜑 From Lemma 5,weobtainthat𝑀 is weakly closed in 𝐸. 𝑁−1 𝑁−1 𝛽 satisfies −Δ𝜑+𝑉(𝑥)𝜑 =𝜆1(𝑁)(𝜑 /|𝑥| ).Testingitwith BytheLagrangeMultipliersrule,𝜆1 is an eigenvalue of function 𝜑,weget problem (50). Moreover 𝐼𝑁(|𝑢|) =𝑁 𝐼 (𝑢) for any 𝑢,sothat𝜆1 possesses a nonnegative eigenfunction. We conclude that the 󵄨 󵄨𝑁 𝜆 ∫ 󵄨∇𝜑󵄨 𝑑𝑥 − ∫ [−𝑉 (𝑥) + 1 ]𝜑𝑁𝑑𝑥=0. 󵄨 󵄨 𝛽 (61) eigenvalue is principal from Harnack inequality in [13]. R𝑁 R𝑁 |𝑥|

Proposition 11. The eigenvalue 𝜆1 is isolated. That is, there Let 𝜀→0,fromLemma9,wehave exists 𝜀>0, such that there are no other eigenvalues of problem (𝜆 ,𝜆 +𝜀) 0≤∫ 𝐿 (𝜑, 𝜁 +𝜀)𝑑𝑥 (50) in the interval 1 1 . R𝑁

Proof. Assume by contradiction there exists a sequence of = ∫ 𝑅 (𝜑, 𝜁 +𝜀)𝑑𝑥 eigenvalue 𝜆𝑚 of problem (50)with0<𝜆𝑚 ↘𝜆1.Let{𝑢𝑚} R𝑁 be an eigenfunction associated with 𝜆𝑚.Then{𝑢𝑚} satisfies 𝜆 (62) = ∫ [−𝑉 (𝑥) + 1 ]𝜑𝑁𝑑𝑥 󵄨 󵄨𝑁−2 𝛽 󵄨𝑢 󵄨 𝑢 R𝑁 |𝑥| 󵄨 󵄨𝑁−2 󵄨 𝑚󵄨 𝑚 −Δ 𝑁𝑢𝑚 +𝑉(𝑥) 󵄨𝑢𝑚󵄨 𝑢𝑚 =𝜆𝑚 , |𝑥|𝛽 𝑁 󵄨 󵄨𝑁−2 𝜑 − ∫ 󵄨∇𝜁󵄨 ∇( )∇𝜁𝑑𝑥. 󵄨 󵄨 𝑁−1 󵄨 󵄨𝑁 󵄨 󵄨𝑁 R𝑁 (𝜁+𝜀) ∫ 󵄨∇𝑢𝑚󵄨 𝑑𝑥 + ∫ 𝑉 (𝑥) 󵄨𝑢𝑚󵄨 𝑑𝑥 (56) R𝑁 R𝑁 𝑁 𝑁−1 The function 𝜑 /(𝜁 + 𝜀) ,where𝜀>0,belongsto𝐸 and 𝜆 󵄨 󵄨𝑁 −Δ 𝜁+ − ∫ 𝑚 󵄨𝑢 󵄨 𝑑𝑥=0. then it is admissible for the weak formulation of 𝑁 𝛽 󵄨 𝑚󵄨 𝑁−2 𝑁−2 𝛽 R𝑁 |𝑥| 𝑉(𝑥)|𝜁| 𝜁=𝜆1(|𝜁| 𝜁/|𝑥| ),a.e.,and 𝑁 󵄨 󵄨𝑁−2 𝜑 ∫ 󵄨∇𝜁󵄨 ∇𝜁∇ 𝑑𝑥 We define 󵄨 󵄨 𝑁−1 R𝑁 (𝜁+𝜀) 𝑢𝑚 (63) V = . 𝑁−1 𝑚 1/𝑁 1 𝜁 󵄨 󵄨𝑁 𝛽 (57) − ∫ [−𝑉 (𝑥) +𝜆 ]𝜑𝑁 𝑑𝑥=0. (∫ 𝑁 (󵄨𝑢𝑚󵄨 /|𝑥| )𝑑𝑥) 𝛽 𝑁−1 R R𝑁 |𝑥| (𝜁+𝜀) 𝐼 (𝑢 )=∫ |∇𝑢 |𝑁𝑑𝑥 + It follows from (62)and(63)thatwehave The coercivity of the functional 𝑁 𝑚 R𝑁 𝑚 ∫ 𝑉(𝑥)|𝑢 |𝑁𝑑𝑥 {𝑢 } 0 ≤ 𝐿 (𝜑, 𝜁 +𝜀) R𝑁 𝑚 implies that 𝑚 is a bounded sequence. Hence {V𝑚} is bounded in 𝐸. So there exists a subsequence 𝜁𝑁−1 𝜑𝑁 (still denoted) {V𝑚} and V ∈𝐸such that = ∫ 𝜆 [1 − ] 𝑑𝑥 1 𝑁−1 𝛽 R𝑁 (𝜁+𝜀) |𝑥| (64) V𝑚 ⇀ V, weakly in 𝐸, 𝜁𝑁−1 (58) − ∫ 𝑉 (𝑥) 𝜑𝑁 [1 − ]𝑑𝑥. 𝑁 𝑁 𝑁−1 V𝑚 󳨀→ V, strongly in 𝐿 (R ), R𝑁 (𝜁+𝜀) Abstract and Applied Analysis 7

Let 𝜀→0;wehave𝐿(𝜑, 𝜁).ByLemma =0 9, there exists Proof. Let 𝑢∈𝐸\{0}, 𝑢>0with compact support Ω= 𝑘>0such that 𝜑=𝑘𝜁. supp(𝑢). By (f4), we obtain that for 𝑝>𝑁, there exists a positive constant 𝐶>0such that for every 𝑀>0,

3. The Proof of Theorem 1 𝑁 ∀ |𝑢| >𝐶, ∀𝑥∈Ω,( 𝐹 𝑥, 𝑢) ≥𝑀|𝑢| . (70) 3.1. Palais-Smale Sequence. Now,wecheckthatthefunctional 𝐼 satisfies the geometric conditions of the Mountain Pass Then, we have Theorem. 𝑡𝑁 |𝑢|𝑁 𝐼 (𝑡𝑢) ≤ ‖𝑢‖𝑁 −𝑀𝑡𝑁 ∫ 𝑑𝑥 Lemma 13. 𝐸 𝛽 Supposethat(H1)–(H3)and(f1)–(f5)hold.Then 𝑁 Ω |𝑥| there exists 𝜀2 such that, for 0<𝜀<𝜀2, there exists 𝜌𝜀 >0such 𝐼(𝑢) >0 ‖𝑢‖ =𝜌 𝜌 𝜆 |𝑡𝑢|𝑁 that if 𝐸 𝜀.Furthermore, 𝜀 canbechosensuch −𝜀𝑡∫ ℎ𝑢 𝑑𝑥 − ∫ 𝑑𝑥 𝜌 →0 𝜀→0 𝛽 that 𝜀 ,as . Ω 𝑁 Ω |𝑥| Proof. From (f5), for every 𝜀>0, there exists 𝜎>0such that 𝑡𝑁 ≤ ‖𝑢‖𝑁 (71) |𝑢| ≤ 𝜎 implies 𝑁 𝐸 𝜀 𝐹 𝑥, 𝑢 ≤ 𝑢 𝑁,∀𝑥∈R𝑁. 𝜆 |𝑢|𝑁 ( ) | | (65) −𝑡𝑁 (𝑀 + ) ∫ 𝑑𝑥 − 𝜀𝑡 ∫ ℎ𝑢 𝑑𝑥 𝑁 𝛽 𝑁 Ω |𝑥| Ω Moreover, using (f1), for each 𝑞>𝑁and 𝑘/𝛼𝑁 + 𝛽/𝑁 + 1/𝑞 ≤ ‖𝑢‖𝑁 𝜆 |𝑢|𝑁 1,wefindaconstant𝐶 such that ≤𝑡𝑁 ( 𝐸 −(𝑀+ ) ∫ 𝑑𝑥) . 𝛽 𝑁 𝑁 |𝑥|<𝑅 |𝑥| 𝑞 𝑁/(𝑁−1) 𝐹 (𝑥, 𝑢) ≤𝐶|𝑢| [exp (𝑘|𝑢| )−𝑆𝑁−2 (𝑘, 𝑢)], (66) Choose 𝑅>0and 𝐵𝑅(0) ⊂ Ω and let ∀ |𝑢| ≥𝜎,𝑥∈R𝑁. 𝑅𝛽‖𝑢‖𝑁 𝜆 𝑅𝛽‖𝑢‖𝑁 𝜆 Combining (65)and(66), we have 𝑀> 𝐸 − > 𝐸 − 1 ; 𝑁 𝑁 𝑁 𝑁 (72) 𝑁‖𝑢‖ 𝑁 𝑁 𝑁‖𝑢‖ 𝑁 𝑁 𝜀 𝐿 (R ) 𝐿 (R ) 𝐹 (𝑥, 𝑢) ≤ |𝑢|𝑁 +𝐶|𝑢|𝑞 𝑁 we have 𝐼(𝑡𝑢) →−∞ as 𝑡→∞.Setting𝑒=𝑡𝑢with 𝑡 being 𝑁/(𝑁−1) sufficient large, we obtain the conclusion. ×[exp (𝑘|𝑢| )−𝑆𝑁−2 (𝑘, 𝑢)], (67)

∀ (𝑥, 𝑢) ∈ R𝑁 × R. It is well known that the failure of the (PS) compactness condition creates some difficulties in studying the class of 𝑁 𝑁 Since the embedding 𝐸󳨅→𝐿 (R ) is continuous, we obtain elliptic problems involving critical growth. In Lemma 15, instead of (PS) sequence, we analyze the compactness of 1 𝜀+𝜆 |𝑢|𝑁 Cerami sequences of the functional 𝐼. 𝐼 (𝑢) ≥ ‖𝑢‖𝑁 − ∫ 𝑑𝑥 𝐸 𝛽 𝑁 𝑁 R𝑁 |𝑥| Lemma 15. Let (𝑢𝑛)⊂𝐸be a Cerami sequence of 𝐼;thatis, 𝑞 −𝐶‖𝑢‖ −𝜀‖ℎ‖ ‖𝑢‖ 󵄩 󵄩 󵄩 󵄩 𝐸 ∗ 𝐸 𝐼(𝑢 )󳨀→𝐶 ,(1+󵄩𝑢 󵄩 ) 󵄩𝐷𝐼 (𝑢 )󵄩 󳨀→ 0, (68) 𝑛 𝑀 󵄩 𝑛󵄩𝐸 󵄩 𝑛 󵄩𝐸󸀠 1 𝜆+𝜀 𝑁 (73) ≥ (1 − ) ‖𝑢‖𝐸 as 𝑛󳨀→∞. 𝑁 𝜆1

𝑞 Then there exists a subsequence of (𝑢𝑛) (still denoted by (𝑢𝑛)) −𝐶‖𝑢‖𝐸 −𝜀‖ℎ‖∗‖𝑢‖𝐸. and 𝑢∈𝐸such that Thus, we have 𝑓(𝑥,𝑢 ) 𝑓 (𝑥, 𝑢) 𝑛 󳨀→ , 𝐿1 (R𝑁), 𝛽 𝛽 in loc 1 𝜆+𝜀 𝑁−1 𝑞−1 |𝑥| |𝑥| 𝐼 (𝑢) ≥ ‖𝑢‖𝐸 [ (1 − ) ‖𝑢‖𝐸 −𝐶‖𝑢‖𝐸 −𝜀‖ℎ‖∗]. 𝑁 𝜆1 ∇𝑢 󳨀→ ∇𝑢, . . RN, (69) 𝑛 a e in 󵄨 󵄨𝑁−2 󵄨 󵄨 𝑁−2 𝑁/(𝑁−1) 𝑁 𝑁 Since 𝑞>𝑁and 0<𝜆<𝜆1 and letting 𝜀<𝜆1 −𝜆,wechoose 󵄨∇𝑢𝑛󵄨 󵄨∇𝑢𝑛󵄨 ⇀ |∇𝑢| |∇𝑢| ,(𝐿loc (R )) , 𝑁−1 𝑞−1 𝜌>0such that (1/𝑁)(1−(𝜆+𝜀)/𝜆1)𝜌 −𝐶𝜌 −𝜀‖ℎ‖∗ >0. 𝑢 ⇀𝑢, 𝐸, Thus, if 𝜀 is sufficiently small, we find some 𝜌𝜀 >0such that 𝑛 in 𝐼(𝑢) >0 if ‖𝑢‖𝐸 =𝜌𝜀 and even 𝜌𝜀 →0as 𝜀→0. (74)

Lemma 14. If 0<𝜆<𝜆1, there exists 𝑒∈𝐸,with‖𝑒‖𝐸 >𝜌𝜀, where 𝐶𝑀 ∈ (0, (1/𝑁)(1 − 𝛽/𝑁)(𝛼𝑁/𝛼0)).Furthermore,𝑢 is a 𝐼(𝑒) < 𝐼(𝑢) such that inf‖𝑢‖𝐸=𝜌𝜀 . nontrivial weak solution of problem (1). 8 Abstract and Applied Analysis

Proof. Let 𝑢𝑛 ∈𝐸, V ∈𝐸,as𝑛→∞;wehave Then there exists some constant 𝐶 such that 1 1 󵄩 󵄩 𝜆 󵄨 󵄨𝑁 󵄨 󵄨𝑁 󵄩𝑢 󵄩 [1 − −𝑁𝜀‖ℎ‖ ] ∫ 󵄨∇𝑢𝑛󵄨 𝑑𝑥 + ∫ 𝑉 (𝑥) 󵄨𝑢𝑛󵄨 𝑑𝑥 󵄩 𝑛󵄩𝐸 ∗ 𝑁 R𝑁 𝑁 R𝑁 𝜆1 󵄨 󵄨𝑁 𝐹(𝑥,𝑢 ) 𝐹(𝑥,𝑢𝑛) 𝜆 󵄨𝑢𝑛󵄨 𝑛 − ∫ 𝑑𝑥 − ∫ 𝑑𝑥 ≤𝑁𝐶𝑀 +𝑁∫ 𝑑𝑥+𝑜(1) , as 𝜀󳨀→0, 𝛽 𝛽 (75) 𝑁 𝛽 R𝑁 |𝑥| 𝑁 R𝑁 |𝑥| R |𝑥| (82)

−𝜀∫ ℎ𝑢𝑛 󳨀→ 𝐶 𝑀, R𝑁 which implies that 󵄨 󵄨 𝐹(𝑥,𝑢 ) 󵄨𝐷𝐼 (𝑢𝑛) V󵄨 ∫ 𝑛 𝑑𝑥 󳨀→ ∞, 𝛽 R𝑁 |𝑥| 󵄨 󵄨 󵄨 󵄨𝑁−2 = 󵄨 ∫ 󵄨∇𝑢 󵄨 ∇𝑢 ∇V 𝑑𝑥 󵄨 󵄨 𝑛󵄨 𝑛 𝐹(𝑥,𝑢𝑛) 󵄨 +󵄨𝑁 󵄨 R𝑁 ∫ 󵄨V 󵄨 𝑑𝑥 󵄨 lim inf 󵄨 󵄨𝑁 󵄨 𝑛 󵄨 𝑛→∞ R𝑁 𝛽󵄨 +󵄨 (83) |𝑥| 󵄨𝑢𝑛 󵄨 󵄨 󵄨𝑁−2 𝑓(𝑥,𝑢 ) V + ∫ 𝑉 (𝑥) 󵄨𝑢 󵄨 𝑢 V 𝑑𝑥 − ∫ 𝑛 𝑑𝑥 󵄨 𝑛󵄨 𝑛 𝛽 𝐹(𝑥,𝑢 ) R𝑁 R𝑁 |𝑥| = ∫ 𝑛 𝑑𝑥. lim inf 󵄩 󵄩𝑁 𝑛→∞ R𝑁 |𝑥|𝛽󵄩𝑢+󵄩 󵄨 󵄨𝑁−2 󵄨 󵄩 𝑛 󵄩𝐸 󵄨𝑢 󵄨 𝑢 V 󵄨 󵄨 𝑛󵄨 𝑛 󵄨 −𝜆 ∫ 𝑑𝑥−𝜀∫ ℎV 𝑑𝑥󵄨 Ψ=∫ (𝐹(𝑥, 𝑢 )/|𝑥|𝛽)𝑑𝑥 𝑁 𝛽 𝑁 󵄨 Let 𝑁 𝑛 ;thenwehave R |𝑥| R 󵄨 R

𝐹(𝑥,𝑢𝑛) 󵄨 󵄨𝑁 𝜏𝑛‖V‖𝐸 ∫ 󵄨V+󵄨 𝑑𝑥 ≤ 󵄩 󵄩 , lim𝑛→∞ inf 𝛽󵄨 󵄨𝑁 󵄨 𝑛 󵄨 󵄩 󵄩 R𝑁 |𝑥| 󵄨𝑢+󵄨 (1 + 󵄩𝑢𝑛󵄩𝐸) 󵄨 𝑛 󵄨 (76) Ψ ≤ lim inf . 𝜏 →0 𝑛→∞ V =𝑢 𝑛→∞ 𝑁𝐶 +𝑁Ψ+𝑁𝜀∫ ℎ𝑢+𝑑𝑥 +𝑜 (1) where 𝑛 as .Let 𝑛 in (76); we have 𝑀 R𝑁 𝑛 󵄨 󵄨𝑁 󵄨 󵄨𝑁 (84) − ∫ 󵄨∇𝑢𝑛󵄨 𝑑𝑥 − ∫ 𝑉 (𝑥) 󵄨𝑢𝑛󵄨 𝑑𝑥 R𝑁 R𝑁 So we can conclude that

𝑓(𝑥,𝑢𝑛)𝑢𝑛 𝐹(𝑥,𝑢 ) 󵄨 󵄨𝑁 + ∫ 𝑑𝑥 ∫ 𝑛 󵄨V+󵄨 𝑑𝑥 𝛽 lim inf 󵄨 󵄨𝑁 󵄨 𝑛 󵄨 R𝑁 |𝑥| 𝑛→∞ R𝑁 𝛽󵄨 +󵄨 |𝑥| 󵄨𝑢𝑛 󵄨 󵄨 󵄨𝑁 (77) 󵄨𝑢 󵄨 Ψ 󵄨 𝑛󵄨 ≤ +𝜆∫ 𝑑𝑥 +𝜀 ∫ ℎ𝑢𝑛𝑑𝑥 lim𝑛→∞ inf + 𝑁 𝛽 𝑁 𝑁𝐶 +𝑁Ψ+𝑁𝜀∫ ℎ𝑢 𝑑𝑥 +𝑜 (1) R |𝑥| R 𝑀 R𝑁 𝑛 󵄩 󵄩 𝜏 󵄩𝑢 󵄩 1 𝑛󵄩 𝑛󵄩𝐸 = . ≤ 󵄩 󵄩 󳨀→ 0, as 𝑛󳨀→∞. 𝑁 (1 + 󵄩𝑢𝑛󵄩 ) 𝐸 (85) Suppose that Note that 𝐹(𝑥,𝑛 𝑢 )≥0; by Fatou Lemma, (80), and (85), we 󵄩 󵄩 V ≤0 V+ ⇀0 𝐸 󵄩𝑢𝑛󵄩𝐸 󳨀→ ∞. (78) get a contradiction. So which means that 𝑘 in . Let 𝑡𝑛 ∈ [0, 1],suchthat Set 𝑢 𝐼(𝑡𝑛𝑢𝑛)= max 𝐼(𝑡𝑢𝑛). V = 𝑛 ; 𝑡∈[0,1] (86) 𝑛 󵄩 󵄩 (79) 󵄩𝑢𝑛󵄩 󵄩 󵄩𝐸 (𝑁−1)/𝑁 For any given 𝑅 ∈ (0, (1 − 𝛽/𝑁)𝛼𝑁/𝛼0) ,let𝜀 = (1 − we have ‖V𝑛‖𝐸 =1.From[5], we have 𝑁/(𝑁−1) 𝛽/𝑁)𝛼𝑁/𝑅 −𝛼0 >0, by (f1); there exists 𝐶 = 𝐶(𝑅) >0 𝐹(𝑥,𝑢 ) 󵄨 󵄨𝑁 such that ∫ 𝑛 󵄨V+󵄨 𝑑𝑥=+∞. lim inf 𝛽󵄨 󵄨𝑁 󵄨 𝑛 󵄨 (80) 󵄨 󵄨 R𝑁 𝑛→+∞ |𝑥| 󵄨𝑢+ (𝑥)󵄨 󵄨 󵄨𝑁 󵄨(1 − 𝛽/𝑁)𝑁 𝛼 󵄨 󵄨 𝑛 󵄨 𝐹(𝑥,𝑢 )≤𝐶󵄨𝑢 󵄨 + 󵄨 −𝛼 󵄨 𝑅(𝛼 +𝜀,𝑢 ), 𝑛 󵄨 𝑛󵄨 󵄨 𝑁/(𝑁−1) 0󵄨 0 𝑛 󵄨 𝑅 󵄨 However, since {𝑢𝑛} istheCeramisequenceatthelevel𝐶𝑀, 𝑁 we have that ∀(𝑥,𝑢𝑛)∈Ω×R . 󵄩 󵄩𝑁 𝐹(𝑥,𝑢 ) (87) 󵄩𝑢 󵄩 =𝑁𝐼(𝑢)+𝑁∫ 𝑛 𝑑𝑥 󵄩 𝑛󵄩𝐸 𝑛 𝛽 R𝑁 |𝑥| Since ‖𝑢𝑛‖𝐸 →∞,wehave (81) 󵄨 󵄨𝑁 󵄨𝑢𝑛󵄨 𝑅𝑢𝑛 +𝑁𝜀∫ ℎ𝑢𝑛𝑑𝑥+𝜆∫ 𝑑𝑥 +𝑜 (1) . 𝐼 (𝑡𝑛𝑢𝑛) ≥𝐼(󵄩 󵄩 ) =𝐼(𝑅V𝑛) , (88) 𝑁 𝑁 𝛽 󵄩 󵄩 R R |𝑥| 󵄩𝑢𝑛󵄩𝐸 Abstract and Applied Analysis 9

‖V ‖ =1 ∫ (𝐹(𝑥, V+)/|𝑥|𝛽)𝑑𝑥 = 𝜀 →0 and by (f3), 𝑛 𝐸 ,and R𝑁 𝑛 By (f2) and 𝑛 ,wehave ∫ (𝐹(𝑥, V )/|𝑥|𝛽)𝑑𝑥 R𝑁 𝑛 ,wehave 𝑓(𝑥,𝑡 𝑢 )𝑡 𝑢 𝑁𝐼 (𝑡 𝑢 )= ∫ 𝑛 𝑛 𝑛 𝑛 𝑑𝑥 𝑛 𝑛 𝛽 𝑁𝐼 (𝑅V𝑛) R𝑁 |𝑥| (92) 󵄨 +󵄨𝑁 󵄨V 󵄨 𝐹(𝑥,𝑡𝑛𝑢𝑛) ≥𝑅𝑁 −(𝑁𝐶𝑅𝑁 +𝜆𝑅𝑁) ∫ 󵄨 𝑛 󵄨 𝑑𝑥 −𝑁∫ 𝑑𝑥 +𝑜 (1) . 𝛽 𝑁 𝛽 R𝑁 |𝑥| R |𝑥| 󵄨 󵄨 󵄨 󵄨 󵄨(1 − 𝛽/𝑁) 𝛼 󵄨 𝑅(𝛼 +𝜀,𝑅󵄨V+󵄨) Moreover, we have −𝑁󵄨 𝑁 −𝛼 󵄨 ∫ 0 󵄨 𝑛 󵄨 𝑑𝑥 󵄨 𝑁/(𝑁−1) 0󵄨 𝛽 󵄨 𝑅 󵄨 R𝑁 |𝑥| 𝑓(𝑥,𝑢 )𝑢 𝐹(𝑥,𝑢 ) ∫ 𝑛 𝑛 𝑑𝑥 −𝑁 ∫ 𝑛 𝑑𝑥 󵄨 󵄨 𝛽 𝛽 󵄨(1 − 𝛽/𝑁) 𝛼 󵄨 󵄨 󵄨 R𝑁 |𝑥| R𝑁 |𝑥| −𝑁𝑅󵄨 𝑁 −𝛼 󵄨 ∫ ℎ 󵄨V+󵄨 𝑑𝑥 (93) 󵄨 𝑁/(𝑁−1) 0󵄨 󵄨 𝑛 󵄨 󵄨 𝑅 󵄨 R𝑁 󵄩 󵄩𝑁 󵄩 󵄩𝑁 = 󵄩𝑢𝑛󵄩𝐸 +𝑁𝐶𝑀 − 󵄩𝑢𝑛󵄩𝐸 +𝑜(1) , 󵄨 +󵄨𝑁 𝑁 𝑁 𝑁 󵄨V𝑛 󵄨 ≥𝑅 −(𝑁𝐶𝑅 +𝜆𝑅 ) ∫ 𝑑𝑥 which is a contraction to (75); this proves that {𝑢𝑛} is bounded 𝑁 𝛽 R |𝑥| in 𝐸.Thus,wehave 󵄨 󵄨 󵄨(1 − 𝛽/𝑁)𝑁 𝛼 󵄨 −𝑁󵄨 −𝛼 󵄨 𝑓(𝑥,𝑢𝑛)𝑢𝑛 𝐹(𝑥,𝑢𝑛) 󵄨 𝑁/(𝑁−1) 0󵄨 ∫ 𝑑𝑥≤𝐶, ∫ 𝑑𝑥≤𝐶. 󵄨 𝑅 󵄨 𝛽 𝛽 (94) 󵄨 󵄨 R𝑁 |𝑥| R𝑁 |𝑥| 𝑁/(𝑁−1) 󵄨 +󵄨 𝑅((𝛼0 +𝜀)𝑅 , 󵄨V𝑛 󵄨) 𝑞 𝑁 × ∫ 𝑑𝑥 From Lemma 5, the embedding 𝐸󳨅→𝐿(R ) is compact for 𝑁 𝛽 R |𝑥| all 𝑞≥𝑁.If{𝑢𝑛}∈𝐸,weget 󵄨 󵄨 󵄨(1 − 𝛽/𝑁)𝑁 𝛼 󵄨 󵄨 󵄨 𝑢 ⇀𝑢, 𝐸, −𝑁𝑅󵄨 −𝛼 󵄨 ∫ ℎ 󵄨V+󵄨 𝑑𝑥 𝑛 in 󵄨 𝑁/(𝑁−1) 0󵄨 󵄨 𝑛 󵄨 󵄨 𝑅 󵄨 R𝑁 󵄨 󵄨 𝑞 𝑁 𝑢𝑛 󳨀→ 𝑢, in 𝐿 (R ), 󵄨 󵄨𝑁 loc (95) 󵄨V+󵄨 ≥𝑅𝑁 −(𝑁𝐶𝑅𝑁 +𝜆𝑅𝑁) ∫ 󵄨 𝑛 󵄨 𝑑𝑥 𝑁 𝛽 𝑢 󳨀→ 𝑢, . . R , R𝑁 |𝑥| 𝑛 a e in 󵄨 󵄨 󵄨(1 − 𝛽/𝑁)𝑁 𝛼 󵄨 From (f1), the Trudinger-Moser inequality, and the Holder¨ −𝑁󵄨 −𝛼 󵄨 𝛽 1 𝑁 󵄨 𝑁/(𝑁−1) 0󵄨 𝑓(𝑥, 𝑢 )/|𝑥| ∈𝐿 (R ) 󵄨 𝑅 󵄨 inequality, we have 𝑛 loc .FromLemma 󵄨 󵄨 2.1 in [14], we have 𝑅 ((1 − 𝛽/𝑁) 𝛼 , 󵄨V+󵄨) × ∫ 𝑁 󵄨 𝑛 󵄨 𝑑𝑥 𝛽 𝑓(𝑥,𝑢𝑛) 𝑓 (𝑥, 𝑢) 1 𝑁 R𝑁 |𝑥| 󳨀→ , in 𝐿 (R ) . (96) |𝑥|𝛽 |𝑥|𝛽 loc 󵄨 󵄨 󵄨(1 − 𝛽/𝑁) 𝛼 󵄨 󵄨 󵄨 −𝑁𝑅󵄨 𝑁 −𝛼 󵄨 ∫ ℎ 󵄨V+󵄨 𝑑𝑥. 󵄨 𝑁/(𝑁−1) 0󵄨 󵄨 𝑛 󵄨 For any fixed 𝛿>0,set 󵄨 𝑅 󵄨 R𝑁 (89) 𝑁 󵄨 󵄨𝑁 󵄨 󵄨𝑁 Σ𝛿 ={𝑥∈R : lim lim ∫ (󵄨∇𝑢𝑛󵄨 + 󵄨𝑢𝑛󵄨 )𝑑𝑥≥𝛿}. + 𝑁 𝑁 𝑟→0𝑛→∞ B (𝑥) Since V𝑛 ⇀0in 𝐸 and the embedding 𝐸󳨅→𝐿(R ) 𝑟 is compact from and the Holder¨ inequality, we have (97) ∫ (|V+|𝑁/|𝑥|𝛽)𝑑𝑥 → 0 (𝑛 →∞) R𝑁 𝑛 .ByLemma7,wehave {𝑢 } Σ + 𝛽 Because 𝑛 is bounded, 𝛿 is a finite set. From Lemma 4.4 𝑁 ∫ 𝑁 (𝑅((1 − 𝛽/𝑁)𝛼𝑁,|V |)/|𝑥| )𝑑𝑥. ≤𝐶 R 𝑛 in ([4]), for any compact set 𝐾⊂⊂R \Σ𝛿,wehave 𝑛→∞ 𝑅 → [(1 − 𝛽/𝑁)𝛼 /𝛼 ](𝑁−1)/𝑁 Let in (89)and 𝑁 0 ; 󵄨 󵄨 󵄨𝑓(𝑥,𝑢 )𝑢 −𝑓(𝑥, 𝑢) 𝑢󵄨 we get ∫ 󵄨 𝑛 𝑛 󵄨𝑑𝑥=0. lim 𝛽 (98) 𝑛→∞ 𝐾 |𝑥| 1 (1 − 𝛽/𝑁) 𝛼 𝑁−1 𝐼(𝑡 𝑢 )≥ [ 𝑁 ] >𝐶 . (90) lim𝑛→∞ inf 𝑛 𝑛 𝑀 𝑁 𝛼0 Now, we prove that 𝐼(0) = 0 𝐼(𝑢 )→𝐶 𝑡 ∈ Note that and 𝑛 𝑀;wesupposethat 𝑛 󵄨 󵄨𝑁 (0, 1) 𝐷𝐼(𝑡 𝑢 )𝑡 𝑢 =0 lim ∫ 󵄨∇𝑢𝑛 −∇𝑢󵄨 𝑑𝑥=0. (99) .Since 𝑛 𝑛 𝑛 𝑛 ,wehave 𝑛→∞ 𝐾

𝑁󵄩 󵄩𝑁 𝑓(𝑥,𝑡𝑛𝑢𝑛)𝑡𝑛𝑢𝑛 󸀠 𝑁 󸀠 𝑡 󵄩𝑢 󵄩 = ∫ 𝑑𝑥 It is enough to prove that for any 𝑥 ∈ R \Σ𝛿 and 𝐵𝑟(𝑥 )⊂ 𝑛 󵄩 𝑛󵄩𝐸 𝛽 R𝑁 |𝑥| 𝑁 R \Σ𝛿 there holds 󵄨 󵄨𝑁 (91) 󵄨𝑢 󵄨 𝑁 󵄨 𝑛󵄨 󵄨 󵄨𝑁 +𝜀𝑡𝑛 ∫ ℎ𝑢𝑛𝑑𝑥+𝜆𝑡𝑛 ∫ 𝑑𝑥. lim ∫ 󵄨∇𝑢𝑛 −∇𝑢󵄨 𝑑𝑥 = 0. 𝑁 𝑁 𝛽 𝑛→∞ 󸀠 (100) R R |𝑥| B𝑟/2(𝑥 ) 10 Abstract and Applied Analysis

∞ 󸀠 We take 𝜙∈𝐶0 (𝐵𝑟(𝑥 )) with 0≤𝜙≤1and 𝜙=1on where 󸀠 B (𝑥 ) {𝜙𝑢 } V = 󵄩 󵄩−1/(𝑁−1) 𝑟/2 .Then 𝑛 is a bounded sequence. Choosing 𝑛 −1/(𝑁−1) 󵄩 ̃ 󵄩 𝑑𝑙 =𝑤𝑁−1 log 𝑙(󵄩𝑀𝑙 (𝑥,) 𝑟 󵄩 −1). (107) 𝜙𝑢𝑛 and V =𝜙𝑢in (76), we have 󵄩 󵄩𝐸

󵄨 󵄨𝑁−2 𝑁−2 From (105), we conclude that ‖𝑀𝑙(𝑥, 𝑟)‖𝐸 →1,as𝑙→∞. ∫ 𝜙(󵄨∇𝑢𝑛󵄨 ∇𝑢𝑛 − |∇𝑢| ∇𝑢) (∇𝑢𝑛 − ∇𝑢) 𝑑𝑥 󸀠 𝐵𝑟(𝑥 ) Consequently, we have 𝑑 󵄨 󵄨𝑁−2 𝑙 󳨀→ 0, 𝑙󳨀→∞. ≤ ∫ 󵄨∇𝑢𝑛󵄨 ∇𝑢𝑛∇𝜙 (𝑢𝑛 −𝑢 )𝑑𝑥 as (108) 󸀠 𝑙 𝐵𝑟(𝑥 ) log Lemma 17. 𝑁−2 Supposethat(H1)–(H3)and(f1)–(f6)hold.Then + ∫ 𝜙|∇𝑢| ∇𝑢 (∇𝑢𝑛 −∇𝑢 )𝑑𝑥 𝑘∈N 󸀠 there exists such that 𝐵𝑟(𝑥 ) 𝑁 󵄨 󵄨𝑁 (101) 𝑡 𝐹(𝑥,𝑡𝑀𝑘) 𝜆 󵄨𝑡𝑀𝑘󵄨 𝑓(𝑥,𝑢𝑛) max { − ∫ 𝑑𝑥 − ∫ 𝑑𝑥} + ∫ 𝜙(𝑢 −𝑢) 𝑑𝑥 𝑡≥0 𝑁 𝑁 𝛽 𝑁 𝑁 𝛽 𝑛 𝛽 R |𝑥| R |𝑥| 𝐵 (𝑥󸀠) 𝑥 𝑟 | | (109) 1 𝑁−𝛽𝛼 𝑁−1 󵄩 󵄩 󵄩 󵄩 < ( 𝑁 ) . +𝜏𝑛󵄩𝜙𝑢𝑛󵄩𝐸 +𝜏𝑛󵄩𝜙𝑢󵄩𝐸 +𝜀∫ 𝜙ℎ 𝑛(𝑢 −𝑢)𝑑𝑥 󸀠 𝑁 𝑁 𝛼0 𝐵𝑟(𝑥 ) 󵄨 󵄨 𝑟>0 ( 6) 𝛽 >0 󵄨𝑢 󵄨 𝑢 𝜙 Proof. Choose as in f and 0 such that +𝜆∫ 󵄨 𝑛󵄨 𝑛 (𝑢 −𝑢)𝑑𝑥. 𝛽 𝑛 𝑁/(𝑁−1) 𝐵 (𝑥󸀠) 𝑠𝑓 (𝑥,) 𝑠 (−𝛼 |𝑠| ) 𝑟 |𝑥| 𝑠→∞lim exp 0 Adapting an argument similar to [4], we have 2 ≥𝛽 > 0 (𝛼 𝑑(𝑁−𝛽)/𝑁) 𝑁−𝛽 𝑁−𝛽 󵄨 󵄨𝑁 𝑒 𝑁 +𝐶𝑟 −𝑟 /(𝑁−𝛽) lim ∫ 󵄨∇𝑢𝑛 −∇𝑢󵄨 𝑑𝑥=0. (102) 𝑛→∞ 𝐾 𝑁−𝛽 𝑁−1 Since Σ𝛿 is finite, it follows that ∇𝑢𝑛 →∇𝑢a.e. This implies, ×( ) , 𝑁−2 𝑁−2 𝛼0 up to a subsequence that |∇𝑢𝑛| ∇𝑢𝑛 ⇀ |∇𝑢| ∇𝑢 in 𝑁/(𝑁−1) 𝑁 𝑁 𝛽 (110) (𝐿loc (R )) .Let𝑛→∞in (76), and 𝑓(𝑥,𝑛 𝑢 )/|𝑥| → 𝛽 1 𝑁 where 𝑓(𝑥, 𝑢)/|𝑥| in 𝐿loc(R );weobtain 𝜉 ∞ 𝑁 𝑘 ⟨𝐷𝐼 (𝑢) , V⟩ =0, ∀V ∈𝐶0 (R ). (103) 𝐶= lim 𝜉𝑘 log 𝑘 ∫ exp [(𝑁−𝛽) 𝑘→∞ 0

𝑁/(𝑁−1) × log 𝑘(𝑠 −𝜉𝑘𝑠)] 𝑑𝑠 >0, Remark 16. The idea and proof of Lemma 15 follow as in −(𝑁−𝛽) log 𝑛 Lemma 4.1 in [5]. 󵄩 󵄩 1−𝑒 𝜉 = 󵄩𝑀̃󵄩 ,𝐶≥ . 𝑘 󵄩 𝑘󵄩𝐸 𝑁−𝛽 3.2. Min-Max Value. In order to get a more precise informa- (111) tion about the minimax level obtained by the Mountain Pass Theorem, we consider the following sequence of scale which Suppose, by contradiction, that, for all 𝑘,weget is called the Moser function: 𝑁 󵄨 󵄨𝑁 𝑡 𝐹(𝑥,𝑡𝑀𝑘) 𝜆 󵄨𝑡𝑀𝑘󵄨 (𝑁−1)/𝑁 𝑟 { − ∫ 𝑑𝑥 − ∫ 𝑑𝑥} {( 𝑙) , |𝑥| ≤ , max 𝛽 𝛽 { log if 𝑡≥0 𝑁 R𝑁 |𝑥| 𝑁 R𝑁 |𝑥| { 𝑙 1 { (𝑟/ |𝑥|) 𝑀̃(𝑥,) 𝑟 = log 𝑟 𝑁−1 𝑙 1/𝑁 { , ≤ |𝑥| ≤𝑟, (104) 1 (𝑁 − 𝛽) 𝛼 𝑤 { 1/𝑁 if 𝑙 𝑁 𝑁−1 { (log 𝑙) ≥ ( ) , { 𝑁 𝑁 𝛼0 {0, if |𝑥| ≥𝑟. ̃ 1,𝑁 𝑁 ̃ (112) Hence, we have 𝑀𝑙(𝑥, 𝑟) ∈𝑊 (R ),thesupportof𝑀𝑙(𝑥, 𝑟) 𝑀 (𝑥) = 𝑀 (𝑥, 𝑟) 𝑘 𝑡 >0 is the ball 𝐵𝑟(0),and where 𝑘 𝑘 .Foreach , there exists 𝑘 such that 󵄨 ̃ 󵄨𝑁 ∫ 󵄨∇𝑀 (𝑥,) 𝑟 󵄨 𝑑𝑥=1, 𝑁 󵄨 󵄨𝑁 R𝑁 󵄨 𝑙 󵄨 𝑡 𝐹(𝑥,𝑡 𝑀 ) 𝜆 󵄨𝑡 𝑀 󵄨 𝑘 − ∫ 𝑘 𝑘 𝑑𝑥 − ∫ 󵄨 𝑘 𝑘󵄨 𝑑𝑥 𝛽 𝛽 󵄨 󵄨𝑁 1 (105) 𝑁 𝑁 𝑁 𝑁 󵄨 ̃ 󵄨 R |𝑥| R |𝑥| ∫ 󵄨𝑀𝑙 (𝑥,) 𝑟 󵄨 𝑑𝑥 = 𝑜( ). R𝑁 log 𝑙 𝑁 𝑡 𝐹(𝑥,𝑡𝑀𝑘) 𝑁 = max { − ∫ 𝑑𝑥 (113) ̃ ̃ 𝑡≥0 𝑁 𝑁 𝛽 Let 𝑀𝑙(𝑥, 𝑟) = 𝑀𝑙(𝑥, 𝑟)/‖𝑀𝑙(𝑥, 𝑟)‖𝐸 ;wehave R |𝑥| 𝑁/(𝑁−1) −1/(𝑁−1) 𝑟 󵄨 󵄨𝑁 𝑀 (𝑥,) 𝑟 =𝑤 log 𝑙+𝑑𝑙, for |𝑥| ≤ , 𝜆 󵄨𝑡𝑀 󵄨 𝑙 𝑁−1 𝑙 − ∫ 󵄨 𝑘󵄨 𝑑𝑥} . 𝛽 (106) 𝑁 R𝑁 |𝑥| Abstract and Applied Analysis 11

Thus; we have it follows that 󵄨 󵄨𝑁 𝑡𝑁 𝐹(𝑥,𝑡 𝑀 ) 𝜆 󵄨𝑡 𝑀 󵄨 (𝑁 − 𝛽) 𝛼 𝑁−1 𝑘 − ∫ 𝑘 𝑘 𝑑𝑥 − ∫ 󵄨 𝑘 𝑘󵄨 𝑑𝑥 𝑡𝑁 󳨀→ ( 𝑁 ) . 𝛽 𝛽 𝑘 (122) 𝑁 R𝑁 |𝑥| 𝑁 R𝑁 |𝑥| 𝑁 𝛼0 (114) 1 (𝑁 − 𝛽) 𝛼 𝑁−1 From [4], we have ≥ ( 𝑁 ) . 𝑁 𝑁 𝛼0 󵄨 󵄨𝑁/(𝑁−1) exp (𝛼0󵄨𝑡𝑘𝑀𝑘󵄨 ) ∫ 𝑑𝑥 From 𝐹(𝑥, 𝑢), ≥0 0<𝜆<𝜆1,weobtain 𝛽 |𝑥|≤𝑟 |𝑥| 𝑁−1 𝑁 𝑁−𝛽𝛼𝑁 󵄨 󵄨𝑁/(𝑁−1) 𝑡 ≥ ( ) . (115) (𝛼 󵄨𝑀 󵄨 (𝑁 − 𝛽) /𝑁) 𝑘 𝑁 𝛼 exp 𝑁󵄨 𝑘󵄨 0 ≥ ∫ 𝑑𝑥 𝛽 |𝑥|≤𝑟/𝑘 |𝑥| Let 𝑡=𝑡𝑘,wehave 󵄨 󵄨𝑁/(𝑁−1) 𝛽 󵄨 󵄨 ∫ (𝑡 𝑀 𝑓(𝑥,𝑡 𝑀 )/|𝑥| )𝑑𝑥 exp (𝛼𝑁󵄨𝑀𝑘󵄨 (𝑁 − 𝛽) /𝑁) 𝑡𝑁 = R𝑁 𝑘 𝑘 𝑘 𝑘 + ∫ 𝑑𝑥, 𝑘 󵄨 󵄨𝑁 𝛽 (1 − 𝜆 ∫ (󵄨𝑀 󵄨 /|𝑥|𝛽)𝑑𝑥) 𝑟/𝑘≤|𝑥|≤𝑟 |𝑥| |𝑥|≤𝑟 󵄨 𝑘󵄨 (116) 󵄨 󵄨𝑁/(𝑁−1) (𝛼 󵄨𝑀 󵄨 (𝑁 − 𝛽) /𝑁) ∫ (𝑡 𝑀 𝑓(𝑥,𝑡 𝑀 )/|𝑥|𝛽)𝑑𝑥 exp 𝑁󵄨 𝑘󵄨 |𝑥|≤𝑟 𝑘 𝑘 𝑘 𝑘 ∫ 𝑑𝑥 𝛽 ≥ . |𝑥|≤𝑟/𝑘 |𝑥| 1 + 𝜆/𝜆1 𝑤 𝑁−1 𝑁−𝛽 [𝑁−𝛽+(𝑑𝑘𝛼𝑁/ log 𝑘)(𝑁−𝛽)/𝑁] By (f6), given that 𝜏>0, there exist 𝑅𝜏 >0and |𝑥| ≤;we 𝑟 = 𝑟 𝑘 . 𝑁−𝛽 have 𝑁/(𝑁−1) (123) 𝑢𝑓 (𝑥, 𝑢) ≥(𝛽0 −𝜏)exp (𝛼0|𝑢| ). (117) Now, using the change of variable From (116)and(117), for large 𝑘,weobtain log (𝑟/ |𝑥|) 󵄩 󵄩 󵄨 󵄨𝑁/(𝑁−1) 𝑠= 𝜁 = 󵄩𝑀 󵄩 , 𝜆 (𝛽 −𝜏)∫ ( (𝛼 󵄨𝑡 𝑀 󵄨 )/|𝑥|𝛽)𝑑𝑥 with 𝑘 󵄩 𝑘󵄩𝐸 (124) 1 0 |𝑥|≤𝑟/𝑘 exp 0󵄨 𝑘 𝑘󵄨 𝜁𝑘 log 𝑘 𝑡𝑁 ≥ 𝑘 (𝜆 +𝜆) 1 by straight forward computation, we have −𝑁+𝛽 𝑁−𝛽 (𝛽0 −𝜏)𝑘 𝑤𝑁−1𝑟 󵄨 󵄨𝑁/(𝑁−1) ≥ exp (𝛼𝑁󵄨𝑀𝑘󵄨 (𝑁 − 𝛽) /𝑁) 2(𝑁−𝛽) ∫ 𝑑𝑥 𝛽 𝑟/𝑘≤|𝑥|≤𝑟 |𝑥| 𝑁/(𝑁−1) −1/(𝑁−1) 𝑁/(𝑁−1) × exp (𝛼0𝑡 𝑤 log 𝑘+𝛼0𝑡 𝑑𝑘). 𝑘 𝑁−1 𝑘 =𝑤 𝑟𝑁−𝛽𝜁 𝑘 (125) (118) 𝑁−1 𝑘 log 𝜁−1 Let 𝑘 𝑁/(𝑁−1) × ∫ exp [(𝑁 − 𝛽) log 𝑘(𝑠 −𝜁𝑘𝑠)] 𝑑𝑠, 𝛼0𝑁 log 𝑘 𝑁/(𝑁−1) 𝑁/(𝑁−1) 0 𝐿𝑘 = 𝑡𝑘 +𝛼0𝑡𝑘 𝑑𝑘 −(𝑁−𝛽)log 𝑘; 𝛼𝑁 𝑁−𝛽 which converges to 𝐶𝑤𝑁−1𝑟 as 𝑘→∞,where (119) 𝜉𝑘 we have 𝐶= lim 𝜁𝑘 log 𝑘 ∫ exp [(𝑁−𝛽) 𝑘→∞ 𝑁−𝛽 0 (𝛽0 −𝜏)𝑟 𝑤𝑁−1 1≥ exp 𝐿 (𝑘) . (120) 𝑁/(𝑁−1) 2(𝑁−𝛽) × log 𝑘(𝑠 −𝜉𝑘𝑠)] 𝑑𝑠>0. (126) Hence, the sequence {𝑡𝑘} is bounded. Otherwise, up to subsequences,wehavelim𝑘→∞𝐿(𝑘), =∞ which leads to a Finally, let 𝑘→∞in (118); from (108)and(115), we have contradiction. From (108)and(115)and (𝑁 − 𝛽) 𝛼 𝑁−1 (𝛽 −𝜏) (𝛽 −𝜏)𝑟𝑁−𝛽𝑤 ( 𝑁 ) ≥ 0 𝑡𝑁 ≥ 0 𝑁−1 𝑁 𝛼 2 𝑘 2(𝑁−𝛽) 0 𝑁−𝛽 𝑟 𝑤𝑁−1 (𝛼 𝑑(𝑁−𝛽)/𝑁) 𝑁/(𝑁−1) ×[ 𝑒 𝑁 , 𝛼0𝑡𝑘 × exp [(𝑁 −(𝑁−𝛽))log 𝑘 (121) (𝑁 − 𝛽) 𝛼𝑁 𝑁−𝛽 𝑁−𝛽 𝑟 𝑤𝑁−1 +𝐶𝑟 𝑤𝑁−1 − ] 𝑁/(𝑁−1) 𝑁−𝛽 +𝛼0𝑡𝑘 𝑑𝑘], (127) 12 Abstract and Applied Analysis

which implies that Proposition 24. There exists 𝜀2 >0such that, for each 𝜀 with 0<𝜀<𝜀2, (1) has a minimum type solution 𝑢0 with 𝐼(𝑢0)= 2 𝐶0 <0,where𝐶0 is defined in (130). 𝛽0 ≤ 𝑒(𝛼𝑁𝑑(𝑁−𝛽)/𝑁) +𝐶𝑟𝑁−𝛽 −𝑟𝑁−𝛽/(𝑁−𝛽) (128) Proof. SeeProposition5.1in[5]. 𝑁−𝛽 𝑁−1 ×( ) . Proposition 25. 𝜀 >0 𝛼0 If 2 is sufficiently small, then the solutions of problem (1) obtained in Propositions 23 and 24 are distinct.

Remark 18. The idea and the proof of Lemma 17 come from Proof. By Propositions 23 and 24, there exist sequences {𝑢𝑛}, Lemma 3.6 in [5]. {V𝑛} in 𝐸 such that 𝑢 󳨀→ 𝑢 ,𝐼(𝑢)󳨀→𝐶,𝐷𝐼(𝑢)𝑢 󳨀→ 0, Lemma 19. There exist 𝜏>0and V ∈𝐸with ‖V‖𝐸 =1such 𝑛 0 𝑛 0 𝑛 𝑛 𝐼(𝑡V)<0 0<𝑡<𝜍 𝐼(𝑢) <0 that for all .Inparticular,inf‖V‖𝐸≤𝜍 . V𝑛 ⇀𝑢𝑀,𝐼(V𝑛)󳨀→𝐶𝑀 >0, (134) Proof. See Lemma 3.3 in [10]. 𝑁 𝐷𝐼 (V𝑛) V𝑛 󳨀→ 0 , ∇ V𝑛 󳨀→ ∇ 𝑢 𝑀 a.e. in R . Corollary 20. Under the conditions (H1)–(H3) and (f1)–(f4), 𝑢 =𝑢 if 𝜀→0, then one has Suppose by contradiction that 0 𝑀.Asintheproofof Lemma 15,wehave (𝑁 − 𝛽) 𝑁−1 1 𝛼𝑁 𝑓(𝑥,V ) 𝑓(𝑥,𝑢 ) max𝐼(𝑡𝑀𝑘)< ( ) . (129) 𝑛 0 1 𝑁 𝑡≥0 𝑁 𝑁 𝛼 󳨀→ in 𝐿loc (R ) , as 𝑛󳨀→∞. 0 |𝑥|𝛽 |𝑥|𝛽 (135) From Lemmas 13 and 19,weconcludethat ∞<𝐶 = 𝐼 (𝑢) <0. Hence, by (f2) and (f3) and Generalized Lebesgue’s Domi- 0 inf (130) ‖V‖𝐸≤𝜍 nated Convergence Theorem, we obtain that there exists 𝑅> 0 such that 1,𝑁 𝑁 Corollary 21. There exist 𝜀2 ∈(0,𝜀1] and 𝑢∈𝑊 (R ) with 𝐹(𝑥,V𝑛) 𝐹(𝑥,𝑢0) compact support such that, for all 0<𝜀<𝜀2, 1 󳨀→ in 𝐿 (𝐵𝑅), as 𝑛󳨀→∞. |𝑥|𝛽 |𝑥|𝛽 loc 𝑁−1 1 (𝑁 − 𝛽) 𝛼 (136) 𝐼 (𝑡𝑢) <𝐶 + ( 𝑁 ) ,∀𝑡≥0. (131) 0 𝑁 𝑁 𝛼 0 ∫ (𝐹(𝑥, V )/|𝑥|𝛽)𝑑𝑥 →∫ (𝐹(𝑥, 𝑢 )/|𝑥|𝛽)𝑑𝑥 Claim 1. R𝑁 𝑛 R𝑁 0 ,as 𝑛→∞ Lemma 22. If {𝑢𝑘} is a Cerami sequence for 𝐼(𝑢) at any level .Indeed,by(f2)and(f3),wehave with 𝑁 𝑁 𝐹 (𝑥,) 𝑠 ≤𝐶|𝑠| +𝐶𝑓(𝑥,) 𝑠 ≤𝐶|𝑠| +𝐶𝑅(𝛼0,𝑠)𝑠, (𝑁 − 𝛽) 𝑁−1 󵄩 󵄩 𝛼𝑁 lim inf󵄩𝑢𝑘󵄩 < ( ) , (132) 𝑓(𝑥,V ) V 𝐹(𝑥,V ) 𝑛→∞ 𝐸 𝑁 𝛼 ∫ 𝑛 𝑛 𝑑𝑥≤𝐶, ∫ 𝑛 𝑑𝑥 ≤ 𝐶. 0 𝛽 𝛽 R𝑁 |𝑥| R𝑁 |𝑥| then {𝑢𝑘} possesses a subsequence which converges strongly to a (137) solution 𝑢0 of problem (1). Hence, on the domain {|𝑥| > 𝑅 and |V𝑛|>𝐴},wehave Proof. See Lemma 4.6 in [4]. 𝐹(𝑥,V ) In conclusion, we have ∫ 𝑛 𝑑𝑥 𝛽 {|𝑥|>𝑅,|V𝑛|>𝐴} |𝑥| 1 (𝑁 − 𝛽) 𝛼 𝑁−1 0<𝐶 <𝐶 + ( 𝑁 ) . 󵄨 󵄨𝑁 𝑀 0 (133) 󵄨V 󵄨 𝑓(𝑥,V ) 𝑁 𝑁 𝛼0 ≤𝐶∫ 󵄨 𝑛󵄨 𝑑𝑥 +𝐶 ∫ 𝑛 𝑑𝑥 𝛽 𝛽 |𝑥|>𝑅 |𝑥| {|𝑥|>𝑅,|V𝑛|>𝐴} |𝑥|

𝐶 󵄩 󵄩𝑁 𝐶 𝑓(𝑥,V ) V 3.3. Multiplicity Results. In order to prove the existence of ≤ 󵄩V 󵄩 + ∫ 𝑛 𝑛 𝑑𝑥. 𝛽 󵄩 𝑛󵄩𝐸 𝛽 thesecondsolutionofproblem(1) follows by the minimum 𝑅 𝐴 R𝑁 |𝑥| argument and Ekeland’s Variational Principle. (138) Proposition 23. Under the conditions (H1)–(H3) and (f1)– 𝑁 Since ‖V𝑛‖𝐸 is bounded, and using (137), we have (f6), there exists 𝜀1 >0such that, for each 𝜀 with 0<𝜀<𝜀1, problem (1) has a solution 𝑢𝑀 via Mountain Pass Theorem. 𝐹(𝑥,V ) ∫ 𝑛 𝑑𝑥 ≤ 2𝛿. 𝛽 (139) {|𝑥|>𝑅,|V |>𝐴} |𝑥| Proof. See Proposition 4.1 in [5]. 𝑛 Abstract and Applied Analysis 13

For |𝑠| ≤,wehave 𝐴 On the other hand, since V𝑛 →𝑢0,wehave

𝑁 |𝐹 (𝑥,) 𝑠 | ≤𝐶|𝑠| +𝐶𝑅(𝛼0,𝑠)𝑠 󵄨 󵄨𝑁−2 ∫ 󵄨∇𝑢0󵄨 ∇𝑢0 (∇V𝑛 −∇𝑢0)𝑑𝑥󳨀→0, R𝑁 ∞ 𝑗 (147) 𝑁 𝛼0 (𝑁𝑗/(𝑁−1)+1−𝑁) ≤ |𝑠| [𝐶+𝐶 ∑ 𝐴 ] (140) 󵄨 󵄨𝑁−2 𝑗! ∫ 𝑉 (𝑥) 󵄨𝑢0󵄨 𝑢0 (V𝑛 −𝑢0)𝑑𝑥󳨀→0. [ 𝑗=𝑁−1 ] R𝑁

𝑁 𝑁−2 𝑁−2 2−𝑁 𝑁 ≤𝐶(𝛼0,𝐴)|𝑠| . By the inequality (|𝑥| 𝑥−|𝑦| 𝑦)(𝑥−𝑦)≥2 |𝑥−𝑦| , we have 𝑁 Since ‖V𝑛‖𝐸 is bounded, we have 󵄨 󵄨𝑁 󵄨 󵄨𝑁 ∫ 󵄨∇V𝑛 −∇𝑢0󵄨 𝑑𝑥 + ∫ 𝑉 (𝑥) 󵄨V𝑛 −𝑢0󵄨 𝑑𝑥 𝐹(𝑥,V ) R𝑁 R𝑁 ∫ 𝑛 𝑑𝑥≤𝛿. 𝛽 (141) {|𝑥|>𝑅,|V𝑛|≤𝐴} |𝑥| 󵄨 󵄨𝑁−2 󵄨 󵄨𝑁−2 ≤𝐶1 ∫ (󵄨∇V𝑛󵄨 ∇V𝑛 − 󵄨∇𝑢0󵄨 ∇𝑢0)(V𝑛 −𝑢0)𝑑𝑥 R𝑁 Combining (139)and(141), we have 󵄨 󵄨 󵄨 󵄨𝑁−2 󵄨 𝑁−2󵄨 +𝐶1 ∫ 𝑉 (𝑥) (󵄨V𝑛󵄨 V𝑛 − 󵄨𝑢0 󵄨 𝑢0 (V𝑛 −𝑢0)) 𝑑𝑥. 𝐹(𝑥,V ) 𝑁 󵄨 󵄨 ∫ 𝑛 𝑑𝑥 ≤ 3𝛿. R 𝛽 (142) |𝑥|>𝑅 |𝑥| (148) V →𝑢 𝐸 𝐼(V )→𝐼(𝑢)=𝐶 <0 Similarly, we also have Hence, we have 𝑛 0 in and 𝑛 0 0 . It is a contradiction. The proof is complete. 𝐹(𝑥,𝑢 ) ∫ 0 𝑑𝑥 ≤ 3𝛿. 𝛽 (143) |𝑥|>𝑅 |𝑥| Conflict of Interests

Combining (136), (142), and (143), we obtain The authors declare that there is no conflict of interests 󵄨 󵄨 regarding the publication of this paper. 󵄨 𝐹(𝑥,V ) 𝐹(𝑥,𝑢 ) 󵄨 󵄨∫ 𝑛 𝑑𝑥 − ∫ 0 𝑑𝑥󵄨 󵄨 𝛽 𝛽 󵄨 󵄨 R𝑁 |𝑥| R𝑁 |𝑥| 󵄨 Acknowledgments 󵄨 󵄨 󵄨 𝐹(𝑥,V𝑛) 𝐹(𝑥,𝑢0) 󵄨 ≤ 󵄨∫ 𝑑𝑥 − ∫ 𝑑𝑥󵄨 The authors are grateful to the reviewers for some valuable 󵄨 𝛽 𝛽 󵄨 󵄨 𝐵𝑅 |𝑥| 𝐵𝑅 |𝑥| 󵄨 suggestions. This paper was supported by Shanghai Natural 󵄨 󵄨 Science Foundation Project (no. 11ZR1424500) and Shanghai 󵄨 𝐹(𝑥,V ) 𝐹(𝑥,𝑢 ) 󵄨 + 󵄨∫ 𝑛 𝑑𝑥 − ∫ 0 𝑑𝑥󵄨 ≤𝐶𝛿. Leading Academic Discipline Project (no. XTKX2012). 󵄨 𝛽 𝛽 󵄨 󵄨 |𝑥|>𝑅 |𝑥| |𝑥|>𝑅 |𝑥| 󵄨 (144) References

Hence, the claim is proved. [1] J. Marcos do O,´ “Semilinear Dirichlet problems for the 𝑁- 𝑁 Laplacian in R with nonlinearities in the critical growth Claim 2. 𝐼(V𝑛)→𝐼(𝑢0)=𝐶0 <0. Indeed, we have range,” Differential and Integral Equations,vol.9,no.5,pp.967– 979, 1996. 󵄩 󵄩𝑁 󵄩∇V 󵄩 𝑛→∞lim 󵄩 𝑛󵄩𝐿𝑁(R𝑁) [2] Adimurthi and K. Sandeep, “A singular Moser-Trudinger embedding and its applications,” Nonlinear Differential Equa- 󵄨 󵄨𝑁 tions and Applications, vol. 13, no. 5-6, pp. 585–603, 2007. =𝑁𝐶𝑀 − lim ∫ 𝑉 (𝑥) 󵄨V𝑛󵄨 𝑑𝑥 𝑛→∞ R𝑁 (145) [3] Y. Li and B. Ruf, “A sharp Trudinger-Moser type inequality for R𝑁 󵄨 󵄨𝑁 unbounded domains in ,” Indiana University Mathematics 𝐹(𝑥,𝑢 ) 󵄨V 󵄨 +𝑁∫ 0 𝑑𝑥 +𝜆 ∫ 󵄨 𝑛󵄨 𝑑𝑥. Journal,vol.57,no.1,pp.451–480,2008. 𝛽 𝛽 R𝑁 |𝑥| R𝑁 |𝑥| [4] Adimurthi and Y. Yang, “An interpolation of Hardy inequality 𝑁 and Trundinger-Moser inequality in R and its applications,” From [10], we have International Mathematics Research Notices,vol.13,pp.2394– 2426, 2010. 󵄨 󵄨 󵄨 𝑓(𝑥,V𝑛)(V𝑛 −𝑢0) 󵄨 󵄨∫ 𝑑𝑥󵄨 󳨀→ 0, [5] N. Lam and G. Lu, “Existence and multiplicity of solutions to 󵄨 𝑁 𝛽 󵄨 equations of 𝑁-Laplacian type with critical exponential growth 󵄨 R |𝑥| 󵄨 𝑁 in R ,” Journal of Functional Analysis,vol.262,no.3,pp.1132– 󵄨 󵄨𝑁−2 1165, 2012. ∫ 󵄨∇V𝑛󵄨 ∇V𝑛 (∇V𝑛 −∇𝑢0)𝑑𝑥 (146) R𝑁 [6] N. Lam and G. Lu, “Existence of nontrivial solutions to poly- harmonic equations with subcritical and critical exponential 󵄨 󵄨𝑁−2 + ∫ 𝑉 (𝑥) 󵄨V𝑛󵄨 V𝑛 (V𝑛 −𝑢0)𝑑𝑥󳨀→0. growth,” Discrete and Continuous Dynamical Systems A,vol.32, R𝑁 no. 6, pp. 2187–2205, 2012. 14 Abstract and Applied Analysis

[7] N. Lam and G. Lu, “Elliptic equations and systems with sub- critical and critical exponential growth without the Ambrosetti- Rabinowitz condition,” Journal of Geometric Analysis,vol.24, no. 1, pp. 118–143, 2014. [8] O. H. Miyagaki and M. A. S. Souto, “Superlinear problems without Ambrosetti and Rabinowitz growth condition,” Journal of Differential Equations,vol.245,no.12,pp.3628–3638,2008. [9] H. Brezis´ and E. Lieb, “A relation between pointwise conver- gence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486– 490, 1983. [10] J. Marcos do O,´ E. Medeiros, and U. Severo, “On a quasilinear 𝑁 nonhomogeneous elliptic equation with critical growth in R ,” Journal of Differential Equations,vol.246,no.4,pp.1363–1386, 2009. [11] L. M. Del Pezzo and J. Fernandez´ Bonder, “An optimization problem for the first weighted eigenvalue problem plus a potential,” Proceedings of the American Mathematical Society, vol. 138, no. 10, pp. 3551–3567, 2010. [12] W. Allegretto and Y. X. Huang, “A Picone’s identity for the 𝑝- Laplacian and applications,” Nonlinear Analysis: Theory, Meth- ods & Applications, vol. 32, no. 7, pp. 819–830, 1998. [13] J. Serrin, “Local behavior of solutions of quasi-linear equations,” Acta Mathematica, vol. 111, pp. 247–302, 1964. [14] D. G. de Figueiredo, O. H. Miyagaki, and B. Ruf, “Elliptic 2 equations in R with nonlinearities in the critical growth range,” Calculus of Variations and Partial Differential Equations,vol.3, no. 2, pp. 139–153, 1995. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 872548, 10 pages http://dx.doi.org/10.1155/2014/872548

Research Article The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

Stefan Balint1 and Agneta M. Balint2

1 Department of Computer Science, West University of Timisoara, Boulevard Vasile Parvan4,300223Timisoara,Romaniaˆ 2 Department of Physics, West University of Timisoara, Boulevard Vasile Parvanˆ 4, 300223 Timisoara, Romania

Correspondence should be addressed to Agneta M. Balint; [email protected]

Received 30 August 2013; Revised 8 December 2013; Accepted 15 December 2013; Published 12 January 2014

Academic Editor: Abdullah Alotaibi

Copyright © 2014 S. Balint and A. M. Balint. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

1. Motivation of the duct wall the boundary condition and at 𝑡=0it is equal the Mathematical Considerations to the perturbation in discussion. For describing a source-produced permanent time har- Due to the practical importance of the sound attenuation in monic perturbation propagation, the mathematical objects case of the turbofan aircraft engines, in the last years more describing the perturbation are added as right-hand mem- than six hundred papers, reporting experimental and theoret- bers to the homogeneous linearized Euler equations. It is ical results on the subject, were published. The papers refereed commonly accepted that the so-obtained nonhomogeneous here and those refereed herein concern acoustic perturbation linearized Euler equations govern the propagation of the propagation in a gas flowing through a lined duct and rep- source-produced permanent time harmonic perturbation in resent just a very small part of the literature concerning the discussion. More precisely, it is assumed that if starting at the subject. moment, let us say 𝑡=0, a source begins to produce per- For describing an instantaneous acoustic perturbation manent time harmonic perturbation, then the propagation propagation, the authors consider the solution of the nonlin- of this perturbation is described by that solution of the non- ear Euler equations (without source) governing the gas flow. homogeneous linearized Euler equations which is equal to After that, the nonlinear Euler equations are linearized at zero for 𝑡≤0and verifies on the duct wall the boundary the specified solution and the homogeneous linearized Euler conditions. equations are derived. It is commonly accepted that those It turns that in [1] for a large class of impedance equations govern the propagation of an instantaneous acous- lining models (i.e., boundary conditions) the above pre- tic perturbation (called frequently also initial value perturba- sented initial-boundary value problem was declared “ill- tions).Moreprecisely,itisassumedthatifatthemoment,let posed” because the set of the exponential growth rates of us say 𝑡=0, an instantaneous acoustic perturbation occurs, the solutions of the homogeneous linearized Euler equations, then its propagation is described by that solution of the satisfying the considered boundary conditions, is unbounded homogeneous linearized Euler equations which satisfies on from above. Later in [2–4] considerable efforts were made 2 Abstract and Applied Analysis

for modifying the boundary conditions in order to make the 𝑢≡𝑈0 = const >0, 𝜌≡𝜌0 = const >0, 𝑝≡𝑝0 = const >0 problem “well posed” in the sense defined in [1]. Here it has to be a constant solution of the system of partial differential 󸀠 be mentioned that in [1] a concept of “well posed differential equations (SPDE) (1). According to (2), 𝑝0 =𝜌0 ⋅𝑅⋅𝑇0 and the 2 equation” is defined. This is different from the concept of associated isentropic sound speed 𝑐0 verifies 𝑐0 =𝛾⋅(𝑝0/𝜌0)= 󸀠 “well-posed problem” usually in mathematics [5, 6]and 𝛾⋅𝑅⋅𝑇0,where𝛾=𝑐𝑝/𝑐V. introduced by Hadamard long time ago. Linearizing (1)at𝑢=𝑈0, 𝜌=𝜌0, 𝑝=𝑝0 and using the 󸀠 󸀠 perturbations 𝑝 , 𝜌 of 𝑝0, 𝜌0 satisfying Definition 1. Following Hadamard, for a given class of instan- taneous initial value perturbations (permanent source-pro- 𝜕 𝜕 󸀠 2 󸀠 ( +𝑈0 ⋅ )(𝑝 −𝑐 𝜌 )=0, (3) duced time harmonic perturbations, resp.) one calls the 𝜕𝑡 𝜕𝑥 0 perturbation propagation problem well posed if there is a unique solution to the problem and the solution varies the following system of homogeneous linear partial differ- 𝑢󸀠 𝑝󸀠 𝑈 𝑝 continuously with the initial data (source amplitude, resp.). ential equations, for the perturbations , of 0, 0,is In [7]itwasshownthattheconceptintroducedin[1]is obtained: confusing, because the equation considered in [1], depending 𝜕𝑢󸀠 𝜕𝑢󸀠 1 𝜕𝑝󸀠 on the function space, can be ill-posed or can be well posed. +𝑈0 ⋅ + ⋅ =0, 𝜕𝑡 𝜕𝑥 𝜌0 𝜕𝑥 In fact the concept introduced in [1]isthetranscriptionof (4) the behavior of an example of a semigroup of contractions of 𝜕𝑝󸀠 𝜕𝑝󸀠 𝜕𝑢󸀠 𝐶 +𝑈 ⋅ +𝛾⋅𝑝 ⋅ =0. class 0 which act in a precise function space (see [1, 8]pages 𝜕𝑡 0 𝜕𝑥 0 𝜕𝑥 240-241 referred in [1]). In this situation, according to [8], the resolvent operator of the infinitesimal generator exists in a It is assumed that if at 𝑡=0an acoustic perturbation occurs, half plane of the form Re 𝑧>𝜇and the Laplace transform then its propagation is given by that solution of (4)whichat can be considered for every solution of the homogeneous 𝑡=0is equal to the perturbation in discussion. It is assumed equation. also that if at 𝑡=0a source begins to produce permanent The objective of the present work is to underline that time harmonic acoustic perturbation, then the propagation the precise description of the function space is crucial even of this perturbation is given by that solution of the system of in the case of the 1D gas flow model, where the wall and non-homogeneous linearized Euler equations: the lining effect are absent. For the achievement of this 𝜕𝑢󸀠 𝜕𝑢󸀠 1 𝜕𝑝󸀠 objective the linear stability analysis of the constant 1D gas +𝑈 ⋅ + ⋅ =𝑄, 𝜕𝑡 0 𝜕𝑥 𝜌 𝜕𝑥 1 flow with respect to the initial value and source-produced 0 (5) permanent time harmonic perturbations is presented in four 𝜕𝑝󸀠 𝜕𝑝󸀠 𝜕𝑢󸀠 different function spaces, revealing significant differences. +𝑈 ⋅ +𝛾⋅𝑝 ⋅ =𝑄 𝜕𝑡 0 𝜕𝑥 0 𝜕𝑥 2 For instance, in the topological function space 𝑋4 the propa- gation problem of the initial value perturbation is well posed, which is equal to zero for 𝑡≤0. but that of the source produced perturbation is ill posed; in Here 𝑄1,𝑄2 (functions or distributions) describe the 𝑋 the topological function space 1 in which the origin has source-produced permanent time harmonic acoustic pertur- not absorbing neighborhoods, stability and strictly positive bations, being equal to zero for 𝑡≤0and periodic in 𝑡 for growth rate coexists in contrast to the case of topological 𝑡>0. vector spaces 𝑋2, 𝑋3. Definition 2. The constant solution 𝑢=𝑈0, 𝑝=𝑝0, 𝜌=𝜌0 2. The 1D Gas Flow Model of (1) is linearly stable with respect to the initial value per- turbation (to the source-produced permanent time harmonic 󸀠 󸀠 In the 1D gas flow model the nonlinear Euler equations gov- perturbation, resp.) if the “solution 𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡) of (4)” (of erning the flow of an inviscid, compressible, nonheat con- (5), resp.) “is small = close to zero” all time 𝑡≥0provided 󸀠 󸀠 ducting, isentropic, perfect gas, according to [9], are “𝑢 (𝑥, 0), 𝑝 (𝑥, 0)”((𝑄1,𝑄2), resp.) “is small = close to zero.” 𝜕𝑢 𝜕𝑢 1 𝜕𝑝 In other words, the constant solution of (1)islinearly +𝑢⋅ + ⋅ =0, stable if and only if the null solution of (4)isstable. 𝜕𝑡 𝜕𝑥 𝜌 𝜕𝑥 These stability concepts are not necessarily equivalent to (1) 𝜕𝜌 𝜕𝜌 𝜕𝑢 the hydrodynamic stability defined in10 [ ]. +𝑢⋅ +𝜌⋅ =0. 𝜕𝑡 𝜕𝑥 𝜕𝑥 Theprecisemeaningoftheconcepts:“perturbation,” “small perturbation,” “ solution of the propagation problem,” Here, 𝑡 is time, 𝑢 is velocity along the 𝑂𝑥 axis, 𝑝 is pressure, 󸀠 and “small solution” has to be designed by other definitions and 𝜌 is density. Equation (1) are considered for 𝑥∈𝑅 and 󸀠 and there is some freedom here. 𝑡≥0. It is assumed that 𝑝, 𝜌 and the absolute temperature 𝑇 satisfy the equation of state of the perfect gas: Using this freedom, in the following we present four dif- 󸀠 𝑝=𝜌⋅𝑅⋅𝑇 (2) ferent function spaces in order to reveal that in each of them the meaning of “perturbation,”“small perturbation,”“solution with 𝑅=𝑐𝑝 −𝑐V; 𝑐𝑝,𝑐V being the specific heat capacities at to the propagation problem,” and “small solution” is specific constant pressure and constant volume, respectively. Let andtheresultsandmathematicaltoolsarespecificaswell. Abstract and Applied Analysis 3

3. The First Function Space Hence, by using the method of characteristics, we obtain 𝑋 The set 1 oftheperturbationsoftheinitialvalue(instan- 1 V󸀠 (𝑥,) 𝑡 =𝐹(𝑥−(𝑈 −𝑐)𝑡)− ⋅𝐺(𝑥−(𝑈 −𝑐)𝑡) taneous perturbations) is the topological function space [11] 0 0 𝑐 𝜌 0 0 of the couples 𝐻 = (𝐹, 𝐺) of continuously differentiable 0 0 𝐹, 𝐺 :𝑅󸀠 →𝑅󸀠 functions with respect to the usual algebraic 1 operations and topology generated by the uniform conver- 𝑞󸀠 (𝑥,) 𝑡 =𝐹(𝑥−(𝑈 +𝑐)𝑡)+ ⋅𝐺(𝑥−(𝑈 +𝑐)𝑡). 󸀠 0 0 𝑐 𝜌 0 0 gence on 𝑅 [12]. 0 0 A neighborhood of the origin 𝑂 is a set 𝑉0 of couples from (10) 𝑋1 having the property that there exists 𝜀>0such that if for 𝐻 = (𝐹, 𝐺) ∈𝑋 |𝐹(𝑥)| <𝜀 |𝐺(𝑥)| <𝜀 󸀠 󸀠 󸀠 󸀠 1 we have and for any Taking into account the equalities 𝑢 = (1/2)(V +𝑞), 𝑝 = 𝑥∈𝑅󸀠 𝐻=(𝐹,𝐺)∈𝑉 󸀠 󸀠 󸀠 󸀠 ,then 0. (𝑐0𝜌0/2)(𝑞 − V ),wededucethat𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡) are given 𝜀 󸀠 󸀠 The set 𝑉0 defined by by the formula (7).So,itwasshownthatif(𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)) is a point wise solution of (4)and(8), then necessarily 𝜀 󸀠 󸀠 󸀠 𝑉0 ={𝐻=(𝐹, 𝐺) ∈𝑋1 : |𝐹 (𝑥)| <𝜀,|𝐺 (𝑥)| <𝜀∀𝑥∈𝑅} 𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡) are given by (7). The fact that the couple of (6) functions given by (7) is a solution of (4), (8)isobtainedby verification. is a neighborhood of the origin. The meaning of the concept “the perturbation 𝐻=(𝐹,𝐺) 𝜀 ∈𝑋1 is small” is that there exists 𝜀 small such that 𝐻∈𝑉0 . In other words, when the set of the perturbations of the 󸀠 For a perturbation 𝐻 = (𝐹, 𝐺)1 ∈𝑋 (i.e., 𝑢 (𝑥, 𝑜) = initial value is 𝑋1, then the initial value problem (4)and(8) 󸀠 󸀠 𝐹(𝑥), 𝑝 (𝑥, 𝑜) = )𝐺(𝑥) the couple of functions 𝐻 (𝑥, 𝑡) = has a unique classical (point wise) solution [9]givenby(7). (𝑢󸀠(𝑥, 𝑡),󸀠 𝑝 (𝑥, 𝑡)) given by If a sequence of perturbations 𝐻𝑛 =(𝐹𝑛,𝐺𝑛)∈𝑋1 converges in 𝑋1 to 𝐻 = (𝐹, 𝐺)1 ∈𝑋 ,thenforany𝑡≥0 󸀠 𝐹[𝑥−(𝑈0 −𝑐0)𝑡]+𝐹[𝑥−(𝑈0 +𝑐0)𝑡] 󸀠 𝑢 (𝑥,) 𝑡 = (fixed) the sequence of the corresponding solutions 𝐻𝑛(⋅, 𝑡) 2 󸀠 belongs to 𝑋1 and converges in 𝑋1 to the solution 𝐻 (⋅, 𝑡), corresponding to 𝐻. 𝐺[𝑥−(𝑈 +𝑐)𝑡]−𝐺[𝑥−(𝑈 −𝑐)𝑡] + 0 0 0 0 , This means that when the set of the initial value perturba- 2𝑐 𝜌 0 0 tions is the function space 𝑋1, then the initial value problem (4)and(8)iswellposedinsenseofHadamard[5, 6]on[0, 𝑇], 󸀠 𝐹[𝑥−(𝑈0 +𝑐0)𝑡]−𝐹[𝑥−(𝑈0 −𝑐0)𝑡] for any 𝑇>0. 𝑝 (𝑥,) 𝑡 =𝑐0𝜌0 ⋅ 2 Inthefollowingthecoupleoffunctionsdefinedby(7)is consideredtobethesolutiontotheproblemofpropagation 𝐺[𝑥−(𝑈 +𝑐)𝑡]+𝐺[𝑥−(𝑈 −𝑐)𝑡] 𝑋 + 0 0 0 0 of the initial value perturbation for data in 1. 2 The linear stability of the constant flow 𝑈0, 𝑝0, 𝜌0 would (7) mean that for any 𝜀>0there exists 𝛿=𝛿(𝜀)such that for 𝐻 = (𝐹, 𝐺) ∈𝑉𝛿(𝜀) 𝐻󸀠(𝑥, 𝑡) is continuously differentiable and verifies (4)andtheinitial any 0 the corresponding solution 𝐻󸀠(⋅, 𝑡) ∈ 𝑉𝜀 𝑡≥0 condition: (given by (7)) satisfies 0 for any . Concerning the linear stability the following statement (𝑢󸀠 (𝑥,) 0 ,𝑝󸀠 (𝑥,) 0 )=𝐻󸀠 (𝑥,) 0 =𝐻(𝑥) = (𝐹 (𝑥) ,𝐺(𝑥)) . holds. (8) Proposition 4. For any 𝜀>0and 𝛿(𝜀) = 𝜀/2 max (1+𝑐0𝜌0,1+ Proposition 3. In the class of the continuously differentiable (1/𝑐 𝜌 )) 𝐻∈𝑉𝛿(𝜀) 𝐻󸀠(⋅, 𝑡) ∈ 𝑉𝜀 𝑡≥0 󸀠 󸀠 0 0 if 0 ,then 0 for . functions, the couple of functions (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)),givenby (7), is the unique point wise (classical) solution of the initial 𝛿(𝜀) Proof. If 𝐻 = (𝐹, 𝐺)0 ∈𝑉 ,then|𝐹(𝑥)| < 𝛿(𝜀) and |𝐺(𝑥)| < value problem (4) and (8). 󸀠 𝛿(𝜀) for any 𝑥∈𝑅.Since𝛿(𝜀) ≤ 𝜀/2(10 +𝑐 𝜌0) and 𝛿(𝜀) ≤ 󸀠 󸀠 𝜀/2(1+(1/𝑐 𝜌 )) Proof. If (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡))isapointwise(classical)solution 0 0 it follows that the following inequalities hold: 󸀠 󸀠 󸀠 |𝐹(𝑥)| < 𝜀/2(10 +𝑐 𝜌0), |𝐹(𝑥)| < 𝜀/2(1 0+ (1/𝑐 𝜌0)), |𝐺(𝑥)| < of the initial value problem (4), (8), then V =𝑢 −(1/𝑐0𝜌0)⋅𝑝 󸀠 𝑞󸀠 =𝑢󸀠 +(1/𝑐𝜌 )⋅𝑝󸀠 𝜀/2(1 +0 𝑐 𝜌0) and |𝐺(𝑥)| ≤ 𝜀/2(10 +(1/𝑐 𝜌0)) for any 𝑥∈𝑅. and 0 0 satisfy the equalities 󸀠 By using (7) we obtain that for any 𝑥∈𝑅 and 𝑡≥0the 󸀠 󸀠 (𝜕V /𝜕𝑡) +0 (𝑈 −𝑐0)⋅(𝜕V /𝜕𝑥) = 0, following inequalities hold:

V󸀠 (𝑥,) 0 =𝐹(𝑥) −(1/𝑐𝜌 )⋅𝐺(𝑥) , 󵄨 󵄨 1 𝜀 1 0 0 󵄨𝑢󸀠 (𝑥,) 𝑡 󵄨 < ⋅2⋅ + (9) 󵄨 󵄨 󸀠 󸀠 2 2(1+𝑐0𝜌0) 2𝑐0𝜌0 (𝜕𝑞 /𝜕𝑡) +0 (𝑈 +𝑐0)⋅(𝜕𝑞/𝜕𝑥) = 0, 𝜀 󸀠 ⋅2⋅ =𝜀 𝑞 (𝑥,) 0 =𝐹(𝑥) +(1/𝑐0𝜌0)⋅𝐺(𝑥) . 2(1+(1/𝑐0𝜌0)) 4 Abstract and Applied Analysis

󵄨 󸀠 󵄨 1 𝜀 𝑃(𝐹, 𝐺, 𝜔 ) 󵄨𝑝 (𝑥,) 𝑡 󵄨 < ⋅2𝑐0𝜌0 ⋅ For the perturbation 𝑓 the couple of functions 󵄨 󵄨 2 2(1+(1/𝑐 𝜌 )) 󸀠 󸀠 󸀠 0 0 𝐻 (𝑥, 𝑡) = (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)),givenby 1 2𝜀 𝑢󸀠 (𝑥,) 𝑡 =ℎ(𝑡) + ⋅ =𝜀. 2 2(1+𝑐0𝜌0) 𝑡 (11) ⋅ ∫ [(𝐹[𝑥−(𝑈0 −𝑐0) (𝑡−𝜏)] 0 󸀠 𝜀 −1 Hence, 𝐻 (⋅, 𝑡) ∈0 𝑉 for any 𝑡≥0. +𝐹 [𝑥0 −(𝑈 +𝑐0) (𝑡−𝜏)]) × (2) This means that the null solution4 of( ) is stable; that is, +(𝐺[𝑥−(𝑈0 +𝑐0) (𝑡−𝜏)] the constant solution 𝑢=𝑈0,𝑝 =0 𝑝 ,𝜌 =0 𝜌 of (1)islinearly stable with respect to the initial value perturbation with data −1 −𝐺 [𝑥0 −(𝑈 −𝑐0) (𝑡−𝜏)]) × (2𝑐0𝜌0) ] from 𝑋1. On the other hand, for initial data from 𝑋1 the set of the exponential growth rates of the solutions of the initial value ⋅ sin 𝜔𝑓𝜏𝑑𝜏, problem (4)and(8) is the whole real axis and there exist also 𝑝󸀠 𝑥, 𝑡 =ℎ 𝑡 solutions whose exponential growth rate is equal to +∞ (for ( ) ( ) instance, that of the solution corresponding to the initial data 𝑡 2 𝐹(𝑥) = 𝐺(𝑥) = exp(𝑥 )). ⋅ ∫ [𝑐0𝜌0 ⋅(𝐹[𝑥−(𝑈0 +𝑐0) (𝑡−𝜏)] 󸀠 󸀠 Here, the exponential growth rate of 𝐻 (𝑥, 𝑡) = (𝑢 (𝑥, 𝑡), 0 󸀠 𝑝 (𝑥, 𝑡)) is defined as −1 −𝐹 [𝑥0 −(𝑈 −𝑐0) (𝑡−𝜏)]) × (2) 󵄨 󵄨 󵄨 󵄨 󵄨 󸀠 󵄨 󵄨 󸀠 󵄨 ln 󵄨𝑢 (𝑥,) 𝑡 󵄨 ln 󵄨𝑝 (𝑥,) 𝑡 󵄨 max {sup lim , sup lim }. (12) +(𝐺[𝑥−(𝑈0 +𝑐0) (𝑡−𝜏)] 𝑥 𝑡→∞ 𝑡 𝑥 𝑡→∞ 𝑡 −1 +𝐺 [𝑥0 −(𝑈 −𝑐0) (𝑡−𝜏)]) × (2) ] The stability of the null solution of (4)(i.e.,linearstability oftheconstantsolution)inthepresenceofsolutionshaving ⋅ sin 𝜔𝑓𝜏𝑑𝜏 strictly positive or +∞ growth rate can be surprising. That is because usually the stability of the null solution of the (14) infinite dimensional linear evolutionary equations is analyzed is continuously differentiable and verifies (5)andthecondi- in a function space in which the Hille-Yosida theory can tion be applied [5, 6, 8, 13], that is, Banach space or locally 󸀠 󸀠 󸀠 convex and sequentially complete topological vector spaces. 𝐻 (𝑥,) 𝑡 =(𝑢 (𝑥,) 𝑡 ,𝑝 (𝑥,) 𝑡 )=(0, 0) for 𝑡≤0. (15) In the topological function space 𝑋1 the origin possesses Proposition 5. neighborhoods which are not absorbing. This is the main In the class of the continuously differentiable (𝑢󸀠(𝑥, 𝑡),󸀠 𝑝 (𝑥, 𝑡)) reasonwhilestabilityandstrictlypositivegrowthratecoexist. functions, the couple of functions given by For initial value perturbations from 𝑋1,theverypopular (14), is the unique point wise solution of the problem (5), (13), Briggs-Bers stability analysis [14, 15], in case of the problem and (15). (4), (8), cannot be applied. That is because in this context dispersion relations cannot be derived (for an arbitrary solu- Proof. The proof of this proposition is similar to the proof of tion, the Fourier transform with respect to 𝑥 and the Laplace Proposition 3. 𝑡 transform with respect to in this context do not exist). So, Inthefollowingthecoupleoffunctionsdefinedby(14)is it is impossible to analyze the imaginary parts of the zeros of consideredtobethesolutionoftheproblemofpropagationof the dispersion relations as requires the Briggs-Bers stability the source-produced permanent time harmonic perturbation criterion. in case of the function space 𝑋1. We consider now source-produced permanent time har- monic perturbations whose amplitudes belong to 𝑋1.More Proposition 6. For 𝑇>0, 𝜀>0,and𝛿=𝛿(𝜀,𝑇)= 𝜀/2𝑇 𝛿 precisely, perturbations for which the right-hand members max (1 + 𝑐0𝜌0,1 + (1/𝑐0𝜌0)) > 0 if 𝐻 = (𝐹, 𝐺)0 ∈𝑉 ,then 𝑄1,𝑄2,appearingin(5), are of the form 󸀠 𝜀 𝐻 (∘, 𝑡) ∈0 𝑉 ,forany𝑡∈[0,𝑇]. 𝑄 (𝑥,) 𝑡 =𝐹(𝑥) ⋅ℎ(𝑡) ⋅ 𝜔 𝑡, 1 sin 𝑓 Proof. The proof of this proposition is similar to the proof of (13) Proposition 4. 𝑄2 (𝑥,) 𝑡 =𝐺(𝑥) ⋅ℎ(𝑡) ⋅ sin 𝜔𝑓𝑡. This means that in this function space the propagation Here, (𝐹, 𝐺)1 ∈𝑋 andrepresenttheamplitudeoftheper- problem (5)and(13), of the source-produced permanent time turbation, ℎ(𝑡) is Heaviside function, and 𝜔𝑓 >0is the harmonic perturbation, is well posed on [0, 𝑇],forany𝑇>0. angular frequency. Such a perturbation will be denoted by Concerning the linear stability of the constant flow, the 𝑃(𝐹,𝐺,𝜔𝑓). following statement holds. Abstract and Applied Analysis 5

󸀠 1 Proposition 7. In the case of source produced permanent time Hence, for a given 𝛿,if𝑡 is sufficiently large, then 𝐻 (⋅, 𝑡) ∉0 𝑉 󸀠 󸀠 harmonic perturbations whose amplitude belongs to the func- (that is because (𝑢 (⋅, 𝑡), 𝑝 (⋅, 𝑡)) is as large as we wish). tion space 𝑋1 the constant flow is linearly unstable. The Briggs-Bers stability analysis14 [ , 15], with respect Proof. The linear instability of the constant flow 𝑈0,𝑝0,𝜌0 to source-produced permanent time harmonic perturbation means that there exists 𝜀>0such that for any 𝛿>0there exist whose amplitude belongs to 𝑋1,cannotbeapplied.Thatis 𝛿 𝐻 = (𝐹, 𝐺)0 ∈𝑉 and 𝑡≥0such that the corresponding because in this context dispersion relations cannot be derived 𝐻󸀠(𝑥, 𝑡) 𝐻󸀠(⋅, 𝑡) ∉ 𝑉𝜀 (there exist perturbations whose amplitude has no Fourier solution (given by (14)) satisfies 0 . 𝐹(𝑥) = 𝐺(𝑥) = (𝑥) The instability can be seen considering for instance 𝜀=1, transform; for instance exp ). So, it is impossible to analyze the imaginary parts of the zeros of the perturbation 𝑃𝛿(𝐹, 𝐺,𝑓 𝜔 ) = ℎ(𝑡)⋅(𝛿⋅sin 𝑥⋅sin 𝜔𝑓𝑡, 0) where the dispersion relations as requires the Briggs-Bers stability 𝛿>0is an arbitrary real number and 𝜔𝑓 = 𝑐0 −𝑈0.Notethat by choosing 𝛿 small, the above perturbation can be made as criterion. small as we wish. According to (14), the propagation of this It follows that the above instability cannot be obtained by perturbation is given by Briggs-Bers stability analysis. 𝛿 𝑢󸀠 (𝑥,) 𝑡 =ℎ(𝑡) ⋅ 4. The Second Function Space 4 𝑋 2𝜔 𝑡 The set 2 oftheperturbationsoftheinitialvalueisthe sin 𝑓 normed space [16] of the couples 𝐻 = (𝐹, 𝐺) of continuously ⋅{−𝑡⋅cos (𝑥 +𝑓 𝜔 𝑡) + 󸀠 󸀠 2𝜔𝑓 differentiable and bounded functions 𝐹, 𝐺 :𝑅 →𝑅with respect to the usual algebraic operations and norm defined 2𝜔 𝑡 sin 𝑓 by ⋅ cos (𝑥+𝜔𝑓𝑡) + ⋅ sin (𝑥 +𝑓 𝜔 𝑡) 𝜔𝑓 ‖𝐻‖ = max {sup |𝐹 (𝑥)| , sup |𝐺 (𝑥)|}. (17) sin 2(𝜔𝑓 −𝑐0)𝑡 2𝑐 𝑡 𝑥∈𝑅󸀠 𝑥∈𝑅󸀠 +[ − sin 0 ] 2(𝜔 −𝑐) 2𝑐0 𝜀 𝑓 0 The set 𝑉0 , defined by: ⋅ (𝑥+(𝜔 −2𝑐)𝑡) 𝜀 cos 𝑓 0 𝑉0 = {𝐻=(𝐹, 𝐺) ∈𝑋2 : ‖𝐻‖ <𝜀} , (18) 2 (𝜔 −𝑐)𝑡 2 is a neighborhood of the origin and the meaning of the sin 𝑓 0 sin 𝑐0𝑡 +[ + ] 𝐻 = (𝐹, 𝐺) ∈𝑋 𝑐 concept “the perturbation 1 is small” is that 2(𝜔𝑓 −𝑐0) 0 𝜀 there exists 𝜀>0small such that 𝐻∈𝑉0 . For an initial data 𝐻 = (𝐹, 𝐺)2 ∈𝑋 thecoupleof 󸀠 󸀠 󸀠 functions 𝐻 (𝑥, 𝑡) = (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)),givenby(7), is the ⋅ sin (𝑥+(𝜔𝑓 −2𝑐0)𝑡)}, unique bounded classical solution of (4)and(8). If a sequence of initial data 𝐻𝑛 =(𝐹𝑛,𝐺𝑛)∈𝑋2 converges 󸀠 𝑐0𝜌0𝛿 in 𝑋2 to 𝐻=(𝐹,𝐺)∈𝑋2,thenforany𝑡≥0(fixed) the 𝑝 (𝑥,) 𝑡 =ℎ(𝑡) ⋅ 󸀠 4 sequence of the corresponding solutions 𝐻𝑛(⋅, 𝑡) converges in 󸀠 𝑋2 to the solution 𝐻 (⋅, 𝑡) corresponding to 𝐻. 2𝜔 𝑡 sin 𝑓 This means that for the set of initial data 𝑋2,theinitial ⋅{𝑡⋅cos (𝑥+𝜔𝑓𝑡) − 2𝜔𝑓 value problem (4)and(19) 󸀠 󸀠 󸀠 2 (𝑢 (𝑥,) 0 ,𝑝 (𝑥,) 0 )=𝐻 (𝑥,) 0 =𝐻(𝑥) = (𝐹 (𝑥) ,𝐺(𝑥)) sin 𝜔𝑓𝑡 ⋅ cos (𝑥+𝜔𝑓𝑡) − ⋅ sin (𝑥 +𝑓 𝜔 𝑡) (19) 𝜔𝑓 is well posed in sense of Hadamard [5, 6]on[0, 𝑇],forany sin 2(𝜔𝑓 −𝑐0)𝑡 2𝑐 𝑡 𝑇>0 +[ − sin 0 ] . 2𝑐 Inthefollowingthecoupleoffunctionsdefinedby(7)is 2(𝜔𝑓 −𝑐0) 0 consideredtobethesolutiontotheproblemofpropagation of the initial value perturbation for data in 𝑋2. ⋅ cos (𝑥+(𝜔𝑓 −2𝑐0)𝑡) The linear stability of the constant flow 𝑈0, 𝑝0, 𝜌0 would 2 (𝜔 −𝑐)𝑡 2 mean that for any 𝜀>0there exists 𝛿=𝛿(𝜀)such that for any sin 𝑓 0 sin 𝑐0𝑡 󸀠 +[ + ] 𝐻∈𝑋2 if ‖𝐻‖ < 𝛿(𝜀) the corresponding solution 𝐻 (𝑥, 𝑡) 𝑐 󸀠 2(𝜔𝑓 −𝑐0) 0 (given by (7)) satisfies ‖𝐻 (⋅, 𝑡)‖ < 𝜀 for any 𝑡≥0. Concerning the linear stability, the following statement holds. ⋅ sin (𝑥+(𝜔𝑓 −2𝑐0)𝑡)}. Proposition 8. For any 𝜀>0and 𝐻∈𝑋2 if ‖𝐻‖ < 𝜀/2 max 󸀠 (16) (1 + 𝑐0𝜌0,1+1/𝑐0𝜌0),then‖𝐻 (⋅, 𝑡)‖ < 𝜀 for 𝑡≥0. 6 Abstract and Applied Analysis

Proof. The proof is similar to that of Proposition 4. sin 𝜔𝑓𝑡, 0) has no Fourier transform with respect to 𝑥). So, it is impossible to analyze the imaginary parts of the zeros of This means that the null solution4 of( ) is stable, that is, the dispersion relations as requires the Briggs-Bers stability 𝑢=𝑈 𝑝=𝑝 𝜌=𝜌 the constant solution 0, 0, 0 of (1), is linearly criterion. Therefore, the above instability result cannot be stable,withrespecttotheinitialvalueperturbationwithdata obtained by Briggs-Bers stability analysis. from 𝑋2. For initial data from 𝑋2 the exponential growth rate, given by (12), of the solution of the initial value problem (4) 5. The Third Function Space and (19) is equal to zero. However, for initial values from 𝑋2 In this case the set 𝑋3 of the initial data is the normed space 2 󸀠 the Briggs-Bers stability analysis, in case of the problem (4) of the couples 𝐻=(𝐹,𝐺)of functions 𝐹, 𝐺 ∈𝐿 (𝑅 ) [17] and (19), cannot be applied. That is because, for instance, the 𝑥 𝐹(𝑥) = with respect to the usual algebraic operations and the norm Fourier transform with respect to of the initial value defined by 𝐺(𝑥) ≡1 does not exist. So, dispersion relations cannot be 1/2 1/2 derived and the stability criterion cannot be used. Therefore, 2 2 the above stability results cannot be obtained by Briggs-Bers ‖𝐻‖𝑋 = max {(∫ |𝐹 (𝑥)| 𝑑𝑥) ,(∫ |𝐺 (𝑥)| 𝑑𝑥) }. 3 𝑅󸀠 𝑅󸀠 stability analysis. For source-produced permanent time harmonic pertur- (21) bations, whose amplitudes belong to 𝑋2,theright-hand 2 󸀠 It has to be emphasized that the functions 𝐹, 𝐺 ∈𝐿 (𝑅 ) are members 𝑄1, 𝑄2,appearingin(5), are those given by (13)with defined up to addition of a measure zero function. (𝐹, 𝐺)2 ∈𝑋 . The meaning of the concept “the perturbation 𝐻=(𝐹,𝐺) For the perturbation 𝑃(𝐹, 𝐺,𝑓 𝜔 )((𝐹, 𝐺)2 ∈𝑋 ) the couple 󸀠 󸀠 󸀠 ∈𝑋3 is small” is that there exists 𝜀>0small such that of functions 𝐻 (𝑥, 𝑡) = (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)),givenby(14), is the ‖𝐻‖𝑋 <𝜀. unique solution of the problem (5), (13), and (15)satisfying 3 For an initial data 𝐻 = (𝐹, 𝐺)3 ∈𝑋 the couple of func- 󸀠 󸀠 󸀠 tions 𝐻 (𝑥, 𝑡) = (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)),givenby(7), is called gen- 𝐻󸀠 (𝑥,) 𝑡 =(𝑢󸀠 (𝑥,) 𝑡 ,𝑝󸀠 (𝑥,) 𝑡 )=(0, 0) 𝑡≤0. for (20) eralized solution of (4). A generalized solution is unique up 󸀠 to addition of a measure zero solution. In general, 𝐻 (𝑥, 𝑡) is 󸀠 󸀠 Concerning the continuous dependence on the amplitude of not differentiable but 𝐻 (⋅, 𝑡) ∈3 𝑋 and 𝐻 (𝑥, 0) satisfy the the perturbation, the following statement holds. condition Proposition 9. 𝑇>0 𝜀>0 𝛿 = 𝛿(𝜀, 𝑇) =𝜀/2𝑇 󸀠 󸀠 󸀠 For any , and 𝐻 (𝑥,) 0 =(𝑢 (𝑥,) 0 ,𝑝 (𝑥,) 0 )=(𝐹, 𝐺) . (22) (1 + 𝑐 𝜌 ,1+(1/𝑐 𝜌 )) > 0 ‖𝐻‖ <𝛿 ‖𝐻󸀠(⋅, 𝑡)‖ max 0 0 0 0 if 𝑋2 ,then 𝑋2 <𝜀 𝑡∈[0,𝑇] 󸀠 for any . When 𝐹, 𝐺 are continuously differentiable, then 𝐻 (𝑥, 𝑡) sat- isfies (4)inclassicalsense. Proof. The proof is similar to the proof of Proposition 6. It has to be noted that there exist continuous functions 𝐹, 𝐺 This means that the problem5 ( )and(20)iswellposedon which are not differentiable at every point [17]. The [0, 𝑇],forany𝑇>0. generalized solution which corresponds to an initial data of When the amplitude of the source-produced perturba- this type is continuous but it is not differentiable at every point. tion belongs to the function space 𝑋2 the couple of functions Concerning the continuous dependence on the initial defined by (14) is considered to be the solution of the problem 𝑡 𝐻󸀠(⋅, 𝑡) of propagation of the source-produced permanent time har- data, we remark that for fixed satisfies monic perturbation. 󵄩 󵄩 1 󵄩 󵄩 󵄩 󸀠 󵄩 󵄩 󸀠 󵄩 Concerning the linear stability of the constant flow, the 󵄩𝐻 (⋅, 𝑡)󵄩 ≤ max {1 + 𝑐0𝜌0,1+ }⋅󵄩𝐻 (⋅, 0)󵄩 . 󵄩 󵄩𝑋3 𝑐 𝜌 󵄩 󵄩𝑋3 following statement holds. 0 0 (23) Proposition 10. The constant flow of (1) is linearly unstable Inequality (23) implies that if the sequence of initial data 𝐻𝑛 = with respect to the source-produced permanent time harmonic (𝐹𝑛,𝐺𝑛)∈𝑋3 converges in 𝑋3 to 𝐻=(𝐹,𝐺)∈𝑋3,thenfor perturbations whose amplitude belongs to the function space 󸀠 any 𝑡≥0the sequence of the corresponding solutions 𝐻𝑛(⋅, 𝑡) 𝑋2. 󸀠 converges in 𝑋3 to 𝐻 (⋅, 𝑡), corresponding to 𝐻=(𝐹,𝐺)∈ 𝑋 Proof. It turns that the perturbations 𝑃𝛿(𝐹,𝐺,𝜔𝑓)=ℎ(𝑡)⋅ 3. (𝛿 ⋅ sin 𝑥⋅sin 𝜔𝑓𝑡, 0) considered in the function space 𝑋1 are This means continuous dependence on the initial data. 𝑋 appropriate to show the above statement. Therefore, in the case of the set of initial data 3,the initial value problem (4)and(22)iswellposedinsenseof The Briggs-Bers stability analysis [14, 15], with respect Hadamard on [0, 𝑇] for every 𝑇>0. to source-produced permanent time harmonic perturbation Inthefollowingthecoupleoffunctionsdefinedby(7)up whose amplitude belongs to 𝑋2,cannotbeapplied.That to addition of a measure zero solution, is considered to be is because in this context dispersion relations can not be the solution to the problem of propagation of the initial value derived (there exist perturbations whose amplitude has no perturbation for data in 𝑋3. Fourier transform; for instance, the perturbation ℎ(𝑡) ⋅ (sin 𝑥⋅ As concerns stability, the following statement holds. Abstract and Applied Analysis 7

Proposition 11. For any 𝜀>0and 𝐻∈𝑋3 if ‖𝐻‖ < 𝜀/2 max 6. The Fourth Function Space 󸀠 (1 + 𝑐0𝜌0,1+1/𝑐0𝜌0),then‖𝐻 (⋅, 𝑡)‖ < 𝜀 for 𝑡≥0. The set 𝑋4 of the perturbations of the initial data is the locally convex vector space of the couples 𝐻 = (𝐹, 𝐺) of tempered Proof. The proof is similar to that of Proposition 8. 󸀠 󸀠 distributions 𝐹, 𝐺 ∈𝑆 (𝑅 ) [8] with respect to the usual alge- braic operations and seminorms 𝑞̃𝐵 defined by This means that the null solution4 of( ) is stable; that is, 𝑞̃ (𝐻) = {𝑞 (𝐹) ,𝑞 (𝐺)} , the constant solution =𝑈0, 𝑝=𝑝0, 𝜌=𝜌0 of (1)islinearly 𝐵 max 𝐵 𝐵 (25) stable with respect to the initial value perturbation with data where 𝐵 areboundedsetsinthespaceoftherapidlydecreas- from 𝑋3. 󸀠 󸀠 󸀠 ing functions 𝑆(𝑅 ) and 𝑞𝐵 is the seminorm on 𝑆 (𝑅 ) defined For initial data from 𝑋3 the exponential growth rate, by 𝑞𝐵(𝐹) = sup𝜑∈𝐵 |𝐹(𝜑)|. given by (12), of the solution of the initial value problem (4) 𝜀,𝑞̃ 𝑉 𝐵 and (22), is equal to zero. The set 0 defined by

For every generalized solution the Laplace transform is 𝜀,𝑞̃𝐵 𝑉 ={𝐻=(𝐹, 𝐺) ∈𝑋 : 𝑞̃ (𝐻) <𝜀} (26) defined for Re 𝑧>0and the Fourier transform with respect 0 4 𝐵 to 𝑥 exists. So, dispersion relation can be derived and the is a neighborhood of the origin and the meaning of the con- stability can be analyzed by analyzing the imaginary parts cept “the perturbation 𝐻=(𝐹,𝐺)∈𝑋4 is small” is that there of the zeros of the dispersion relation. This means that both exists 𝜀>0small and 𝐵 bounded such that 𝐻=(𝐹,𝐺)∈ 𝜀,𝑞̃𝐵 steps of the Briggs-Bers stability analysis can be undertaken. 𝑉0 . However, the above stability result was obtained directly not Iftheinitialvalueperturbationisacoupleoftempered by Briggs-Bers stability analysis. distributions 𝐻 = (𝐹, 𝐺)4 ∈𝑋 , then a solution of (4)isa 󸀠 󸀠 󸀠 Whatcanbestrangeforengineersinthisfunctionspace family 𝐻 (𝑡) = (𝑢 (𝑡), 𝑝 (𝑡)) of couples of tempered distribu- is the presence of solutions which are continuous but are not tions which satisfies4 ( )for𝑡≥0and the initial condition differentiable at every point. Which kind of propagation does 𝐻󸀠 (0) =(𝑢󸀠 (0) ,𝑝󸀠 (0))=𝐻=(𝐹, 𝐺) ∈𝑋. represent such a solution? 4 (27) For source-produced permanent time harmonic per- Proposition 13. If the initial value problem (4) and (27) has a turbation, whose amplitudes belong to 𝑋3,theright-hand 𝐻󸀠(𝑡) = (𝑢󸀠(𝑡),󸀠 𝑝 (𝑡)) 𝑄 𝑄 solution , then its Fourier transform with members 1, 2,appearingin(5), are those given by (13)with respect to 𝑥 is a family of couples of tempered distributions (𝐹, 𝐺)3 ∈𝑋 . 󸀠 󸀠 󸀠 𝐻̂ (𝑡) = (𝑢̂ (𝑡), 𝑝̂ (𝑡)), 𝑡≥0which satisfies the following equa- For the perturbation 𝑃(𝐹, 𝐺,𝑓 𝜔 )((𝐹, 𝐺)3 ∈𝑋 ) the couple 󸀠 󸀠 󸀠 tions: of functions 𝐻 (𝑥, 𝑡) = (𝑢 (𝑥, 𝑡), 𝑝 (𝑥, 𝑡)),givenby(14)upto 󸀠 𝜕𝑢̂ 󸀠 1 󸀠 addition of a measure zero solution will be called generalized +𝑖𝑘𝑈0 ⋅ 𝑢̂ +𝑖𝑘 ⋅ 𝑝̂ =0, solution of the propagation problem (5)and(24) 𝜕𝑡 𝜌0 (28) 𝜕𝑝̂󸀠 𝐻󸀠 (𝑥,) 𝑡 =(𝑢󸀠 (𝑥,) 𝑡 ,𝑝󸀠 (𝑥,) 𝑡 )=(0, 0) 𝑡≤0. +𝛾𝑝 𝑖𝑘 ⋅ 𝑢̂󸀠 +𝑖𝑘𝑈 ⋅ 𝑝̂󸀠 =0 for (24) 𝜕𝑡 0 0

Concerning the continuous dependence on the amplitude of and the initial condition 󸀠 󸀠 the perturbation, the following statement holds. 𝑢̂ (0) = 𝐹,̂ 𝑝̂ (0) = 𝐺.̂ (29) Proposition 12. For any 𝑇>0, 𝜀>0and 𝛿 = 𝛿(𝜀, 𝑇) =𝜀/2𝑇 Proof. By computation. 󸀠 max(1+𝑐0𝜌0,1+(1/𝑐0𝜌0)) > 0 if ‖𝐻‖𝑋 <𝛿,then‖𝐻 (⋅, 𝑡)‖𝑋 < 3 3 Remark, that the problem (4)and(27)iswellposedifand 𝜀 for any 𝑡∈[0,𝑇]. only if the problem (28)and(29)iswellposed. Proof. The proof is similar to the proof of Proposition 6. Concerning the problem (28)and(29)thefollowingstate- ment holds.

This means that the problem5 ( )and(24)iswellposedon Proposition 14. The problem (28) and (29) is well posed on [0, 𝑇] for any 𝑇>0. any finite interval and its solutions is given by −𝑖(𝑈 −𝑐 )𝑘𝑡 −𝑖(𝑈 +𝑐 )𝑘𝑡 The linear instability of the constant flow of(1)with 𝑒 0 0 +𝑒 0 0 𝑢̂󸀠 (𝑡) = respect to this kind of perturbations can be shown similarly 2 as in the function space 𝑋1, by considering the perturbations −𝑖(𝑈 +𝑐 )𝑘𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 𝐹(𝑥) =𝛿 𝑥/1 +2 𝑥 𝐺(𝑥) =0 𝜔 =𝑐 −𝑈 𝑒 0 0 −𝑒 0 0 sin , and 𝑓 0 0. ⋅ 𝐹+̂ ⋅ 𝐺,̂ 𝑥 Since the Fourier transform with respect to and the 2𝑐0𝜌0 Laplace transform with respect to 𝑡 of any solution exist, (30) −𝑖(𝑈 +𝑐 )𝑘𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 𝑒 0 0 −𝑒 0 0 dispersion relation can be derived, and the stability analysis 𝑝̂󸀠 (𝑡) =𝑐𝜌 could be undertaken by analyzing the imaginary parts of 0 0 2 zeros of dispersion relations (both steps of the Briggs-Bers −𝑖(𝑈 +𝑐 )𝑘𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 𝑒 0 0 +𝑒 0 0 stability analysis can be undertaken in this function space). ⋅ 𝐹+̂ ⋅ 𝐺.̂ However, in this paper the instability was derived directly. 2 8 Abstract and Applied Analysis 󵄨 󵄨 𝑞 (𝑝̂󸀠 (𝑡))=𝜀⋅ 󵄨𝑝̂󸀠 (𝑡) (𝜑)󵄨 ≥𝜀 Proof. The existence and uniqueness are easy to see. We will 𝐵0 sup 󵄨 󵄨 𝜑∈𝐵 show only the continuous dependence on the initial data 0 in case of the problem (28)and(29). For that, consider 󵄨 󵄨 󵄨 ̂󸀠 󵄨 2 𝑇>0, 𝜀>0and a semi norm 𝑞̃𝐵 in 𝑋4.Afterthat,con- ⋅ 󵄨𝑝 (𝑡) (𝜑0)󵄨 =𝜀𝑐0 𝜌0𝑡, 󸀠 󸀠 󵄨 󵄨 sider a bounded set 𝐵 ⊂𝑆(𝑅), which contains the following bounded sets: 𝑞̃ (𝑢̂󸀠 (𝑡) , 𝑝̂󸀠 (𝑡))≥𝜀𝑡 {𝑈 ,𝑐2𝜌 }. 𝐵0 max 0 0 0 −𝑖(𝑈 −𝑐 )𝑘𝑡 −𝑖(𝑈 +𝑐 )𝑘𝑡 𝑒 0 0 +𝑒 0 0 ⋅𝐵; (32) 2 −𝑖(𝑈 +𝑐 )𝑘𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 󸀠 󸀠 𝑒 0 0 −𝑒 0 0 Therefore, for 𝑡 sufficiently large, we have 𝑞̃𝐵 (𝑢̂ (𝑡), 𝑝̂ (𝑡)) > ⋅𝐵; 0 (31) 1=𝜀0. 2𝑐0𝜌0 Hence, the null solution of (28)isunstable.Sincethe −𝑖(𝑈 +𝑐 )𝑘𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 𝑒 0 0 −𝑒 0 0 inverse Fourier transform is continuous, the null solution of 𝑐0𝜌0 ⋅𝐵 2 (4)isalsounstablein𝑋4. For source-produced permanent time harmonic pertur- 𝑡∈[0,𝑇] 𝑞̃ 󸀠 𝑋 𝛿 = 𝜀/(1 + for any ;theseminorm 𝐵 in 4 and bations, whose amplitudes belong to 𝑋4,theright-hand ̂ ̂ ̂ ̂ 𝑞̃𝐵󸀠 (𝐹, 𝐺)) >. 0 Remark that for any (𝐹, 𝐺) ∈4 𝑋 with 𝑞̃𝐵󸀠 members 𝑄1, 𝑄2,appearingin(5)aregivenby ̂ ̂ 󸀠 󸀠 (𝐹, 𝐺) < 𝛿 we have 𝑞̃𝐵(𝑢̂ (𝑡), 𝑝̂ (𝑡)) < 𝜀 for 𝑡∈[0,𝑇]. 𝑖𝜔 𝑡 𝑖𝜔 𝑡 𝑄 =ℎ(𝑡) ⋅𝑒 𝑓 ⋅𝐹, 𝑄 =ℎ(𝑡) ⋅𝑒 𝑓 ⋅𝐺, (33) Proposition 15. The problem (4) and (27) is well posed. 1 2 󸀠 󸀠 ̂󸀠 (𝑢̂ (𝑡), 𝑝̂ (𝑡)) = 𝐻 (𝑡) where 𝐻=(𝐹,𝐺)∈𝑋4. Proof. The inverse Fourier transform of 󸀠 󸀠 given by (30) is the unique solution of (4)and(27). The The unknown tempered distributions 𝑢 (𝑡), 𝑝 (𝑡) have to continuous dependence of the solutions of (4)and(27)isa satisfy (5) and the condition corollary of the continuous dependence on the initial data 󸀠 󸀠 of the solution of (28)and(29) and of the continuity of the 𝑢 (𝑡) =0, 𝑝 (𝑡) =0, for 𝑡≤0. (34) Fourier and inverse Fourier transforms.

So, the problem (4)and(27)iswellposed. 󸀠 󸀠 If the problem (5)and(34)hasasolution𝐻 (𝑡) = (𝑢 (𝑡), 𝑝󸀠(𝑡)) ∈ 𝑋 𝑢̂󸀠(𝑡) ̂V󸀠(𝑡) Concerning the stability, the following statement holds. 4, then the Fourier transforms , are tem- pered distributions and satisfy Proposition 16. The null solution of (28) is unstable. 󸀠 𝜕𝑢̂ 󸀠 1 󸀠 𝑖𝜔 𝑡 +𝑖𝑘𝑈 ⋅ 𝑢̂ +𝑖𝑘 ⋅ 𝑝̂ =ℎ(𝑡) ⋅𝑒 𝑓 ⋅ 𝐹,̂ Proof. The instability of the null solution of (28)meansthat 𝜕𝑡 0 𝜌 𝜀 >0 𝑞̃ 0 there exist 0 and a semi norm 𝐵0 such that for any real 𝛿>0 𝑞̃ 𝑋 𝐻= 󸀠 number and any semi norm 𝐵 in 4 there exist 𝜕𝑝̂ (35) 𝛿,𝑞̃ 󸀠 󸀠 𝑖𝜔𝑓𝑡 ̂ 𝐵 +𝛾𝑝0𝑖𝑘 ⋅ 𝑢̂ +𝑖𝑘𝑈0 ⋅ 𝑝̂ =ℎ(𝑡) ⋅𝑒 ⋅ 𝐺, (𝐹, 𝐺)0 ∈𝑉 and 𝑡≥0such that the corresponding solution 𝜕𝑡 𝜀 ,𝑞̃ 󸀠 󸀠 0 𝐵0 𝐻 (𝑡) (given by (30)) satisfies 𝐻 (𝑡) ∉ 𝑉 . Let us consider 󸀠 󸀠 0 𝑢̂ (𝑡) =0, 𝑝̂ (𝑡) =0 for 𝑡≤0. now 𝜀0 =1, 𝐵0—a bounded set of rapidly decreasing func- 2 tions which contains the function 𝜑0(𝑘) = exp(−𝑘 ),and ̂ 󸀠 the tempered distribution defined by 𝐹0(𝜑) = 𝜑 (0).Remark Hence, we got that for an arbitrary real number 𝛿>0and an arbitrary 𝐵 ℎ (𝑡) bounded set of rapidly decreasing functions, there exists a 𝑢̂󸀠 (𝑡) = ̂ 𝛿,𝑞𝐵 real number 𝜀>0such that 𝑞𝐵 (𝜀 𝐹0)<𝛿(𝑉0 is absorbing). 2 𝛿,𝑞̃ (𝜀 ⋅ 𝐹̂ ,0) ∈ 𝑉 𝐵 It follows that 0 0 . On the other hand, for the 𝑖𝜔𝑓𝑡 −𝑖(𝑈0−𝑐0)𝑘𝑡 󸀠 󸀠 𝑒 −𝑒 solution (𝑢̂ (𝑡), 𝑝̂ (𝑡)) oftheinitialvalueproblem(28), (29) ×[ ̂ ̂ ̂ 𝑖[(𝑈0 −𝑐0)⋅𝑘+𝜔𝑓] with 𝐹=𝜀𝐹0 and 𝐺=0, the following relations hold: 𝑖𝜔 𝑡 −𝑖(𝑈 +𝑐 )𝑘𝑡 󸀠 󸀠 𝑒 𝑓 −𝑒 0 0 𝑢̂ (𝑡) =−𝑖𝜀𝑈0𝑡𝜑 (0) +𝜑 (0) , + ]⋅𝐹̂ 𝑖[(𝑈0 +𝑐0)⋅𝑘+𝜔𝑓] ̂󸀠 2 𝑝 (𝑡) =−𝑖𝜀𝑐0 𝜌0𝑡𝜑 (0) , 𝑖𝜔 𝑡 −𝑖(𝑈 +𝑐 )𝑘𝑡 ℎ (𝑡) 𝑒 𝑓 −𝑒 0 0 + [ 󵄨 󵄨 2𝑐 𝜌 𝑞 (𝑢̂󸀠 (𝑡))=𝜀⋅ 󵄨𝑢̂󸀠 (𝑡) (𝜑)󵄨 ≥𝜀 0 0 𝑖[(𝑈0 +𝑐0)⋅𝑘+𝜔𝑓] 𝐵0 sup 󵄨 󵄨 𝜑∈𝐵0 𝑖𝜔 𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 𝑒 𝑓 −𝑒 0 0 󵄨 󵄨 − ]⋅𝐺,̂ 󵄨̂󸀠 󵄨 𝑖[(𝑈 −𝑐)⋅𝑘+𝜔 ] ⋅ 󵄨𝑢 (𝑡) (𝜑0)󵄨 =𝜀𝑈0𝑡, 0 0 𝑓 Abstract and Applied Analysis 9

𝑐 𝜌 ⋅ℎ(𝑡) 𝑝̂󸀠 (𝑡) = 0 0 interval of time and stability means continuous dependence 2 on the infinite interval of time [0, +∞). In practice, the 𝑖𝜔 𝑡 −𝑖(𝑈 +𝑐 )𝑘𝑡 duration of the propagation is finite. For the eight situations 𝑒 𝑓 −𝑒 0 0 ×[ considered in this paper, the continuous dependence on any 𝑖[(𝑈0 +𝑐0)⋅𝑘+𝜔𝑓] finite interval of time in seven situations is valid. The stability on [0, +∞) isvalidonlyinthreesituations.So,thecontinuous 𝑖𝜔 𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 𝑒 𝑓 −𝑒 0 0 − ]⋅𝐹̂ dependence on any finite interval of time is less dependent on the function space. Taking into account the above facts, 𝑖[(𝑈0 −𝑐0)⋅𝑘+𝜔𝑓] from practical point of view, the question is: the continuous 𝑖𝜔 𝑡 −𝑖(𝑈 +𝑐 )𝑘𝑡 ℎ (𝑡) 𝑒 𝑓 −𝑒 0 0 dependence or the stability is important? + [ (vi) When theoretical results are tested against experi- 2 𝑖[(𝑈 +𝑐)⋅𝑘+𝜔 ] 0 0 𝑓 mentalresults,thecomputedmathematicalvariablehasto

𝑖𝜔 𝑡 −𝑖(𝑈 −𝑐 )𝑘𝑡 correspond to the experimentally measured quantity. For 𝑒 𝑓 −𝑒 0 0 + ]⋅𝐺.̂ instance, in this paper there are four different mathematical 𝑖[(𝑈0 −𝑐0)⋅𝑘+𝜔𝑓] variables for expressing that the perturbation and the corre- sponding solution of the propagation problem are small, that (36) is, close to zero. Which one of them does correspond to an experimentally measured quantity when the duct is infinitely 𝐹, 𝐺 ∈𝑆󸀠(𝑅󸀠) 𝑢̂󸀠(𝑡) ̂V󸀠(𝑡) Itcanbeseenthatforarbitrary , , , long? given by (36), are not necessarily tempered distributions. (vii) The results obtained could be useful in a better un- This contradiction shows that the problem5 ( )and(34)isill derstanding of some apparently strange results published in posed. the literature concerning the sound propagation in a gas flow- ing through a lined duct, since there are neither set rules nor 7. Conclusions understanding of the “right” way to model the phenomenon.

(i) The obtained results show that the well-posedness and Conflict of Interests linear stability concepts of the constant gas flow in 1D flow model are highly dependent on the function space. There The authors declare that there is no conflict of interests exists function space in which the constant gas flow is stable regarding the publication of this paper. andanotherfunctionspaceinwhichthesameflowisunstable with respect to the initial value perturbations. There exists functionspaceinwhichthepropagationproblemofthe Acknowledgment source produced perturbation is well posed and another This work was supported by a grant of the Romanian National function space in which the propagation problem is ill posed. Authority for Scientific Research, CNCS-UEFISCDI, project (ii)Forthestabilityoftheconstantgasflowwithrespect no. PN-II-ID-PCE-2011-3-0171. The results have been pre- to the initial value perturbations it is not necessary that the sented orally at Sz.-Nagy Centennial Conference, June 24– set of the exponential growth rate of the solutions of the 28, 2013, Szeged, Hungary (http://www.math.u-szeged.hu/ homogeneous linearized Euler equations be bounded from SzNagy100/program.php). above. There exists function space in which the constant gas flowisstablewithrespecttotheinitialvalueperturbations although the set of the exponential growth rate of the References solutionsofthehomogeneousEulerequationsisthewhole [1] E. J. Brambley, “Fundamental problems with the model of real axis. So, the stability of the null solution cannot be denied uniform flow over acoustic linings,” Journal of Sound and just because the set of the exponential growth rate of the Vibration,vol.322,no.4-5,pp.1026–1037,2009. solutions is not bounded from above. [2] E. J. Brambley, “A well-posed modified myers boundary con- (iii) The stability of the constant gas flow with respect to dition,” in Proceedings of the 16th AIAA/CEAS Aeroacoustics the initial value perturbations is not a sufficient condition for Conference (31st AIAA Aeroacoustics Conference),June2010. the stability of the same solution with respect to source pro- [3] Y. Renou and Y. Auregan,´ “On a modified Myers boundary duced permanent time harmonic perturbations. There exists condition to match lined wall impedance deduced from several function space in which the constant gas flow is stable with experimental methods in presence of a grazing flow,”in Proceed- respect to the initial value perturbations and it is unstable ings of the 16th AIAA/CEAS Aeroacoustics Conference,p.14,June with respect to the source produced perturbations. 2010. (iv) In two of the considered function spaces the Briggs- [4] E. J. Brambley, “Well-posed boundary condition for acoustic Bers stability analysis cannot be applied, because dispersion liners in straight ducts with flow,” AIAA Journal,vol.49,no.6, relations cannot be derived. So, the stability or instability pp.1272–1282,2011. resultsobtaineddirectlyinthisfunctionspacescannotbe [5] G. E. Ladas and V. Lakshmikantham, Differential Equations in obtained by Briggs-Bers stability analysis. Abstract Spaces, Academic Press, New York, NY, USA, 1972. (v) Continuous dependence on the initial data (on the [6] S. G. Krein, Differential Equations in Banach Spaces,Izd. source amplitude, resp.) is an expression of stability on a finite “Nauka”, Moskow, 1967 (Russian). 10 Abstract and Applied Analysis

[7]A.M.Balint,St.Balint,andR.Szabo,“Linearstabilityofanon slipping gas flow in a rectangular lined duct with respect to perturbations of the initial value by indefinitely differentiable disturbances having compact support,” AIP Conference Proceed- ings,vol.1493,p.1023,2012. [8] K. Yosida, Functional Analysis, vol. 123, Springer, Berlin, Ger- many, 6th edition, 1980. [9]B.L.RozdjestvenskiiandN.N.Ianenko,Quasilinear Systems of Equations and Their Applications in Gasdynamics,Izd.“Nauka”, Moskow, Russia, 1978 (Russian). [10] P. G. Drazin and W. N. Reid, Hydrodynamic Stability,Cam- bridge University Press, Cambridge, UK, 1995. [11] S. H. Saperstone, Semidynamical Systems in Infinite- Dimensional Spaces,vol.37,Springer,NewYork,NY,USA, 1981. [12] N. Bourbaki, Elements de Mathematique, Premier Partie, Livre III, Topologie Generale, chapitre 1, structure topologique, Her- mann, Paris, France, 1966. [13] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, USA, 1957. [14] A. Bers, “Space-time evolution of plasma-instabilities— absolute and convective,” in Handbook of Plasma Physics I,A. A. Galeev and R. M. Sudan, Eds., pp. 452–484, North Holland, Amsterdam, The Netherlands, 1983. [15]R.J.Briggs,Electron Stream Interaction with Plasmas,Research Monograph no. 29, M.I.T. Press, Cambridge, Mass, USA, 1964. [16] N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, Wiley-Interscience, New York, NY, USA, 1957. [17] R. Riesz and B. Sz-Nagy, Lec¸ons d’Analyse Fonctionnelle, Akademiai´ Kiado,Budapest,Hungary,1953.´ Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 975728, 6 pages http://dx.doi.org/10.1155/2014/975728

Research Article Unique Coincidence and Fixed Point Theorem for 𝑔-Weakly C-Contractive Mappings in Partial Metric Spaces

Saud M. Alsulami

Department of Mathematics, King Abdulaziz University, P.O. Box 138381, Jeddah 21323, Saudi Arabia

Correspondence should be addressed to Saud M. Alsulami; [email protected]

Received 10 September 2013; Accepted 13 December 2013; Published 9 January 2014

Academic Editor: Adem Kılıc¸man

Copyright © 2014 Saud M. Alsulami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove that every map satisfying the 𝑔-weakly C-contractive inequality in partial metric space has a unique coincidence point. Our results generalize several well-known existing results in the literature.

1. Introduction and Preliminaries for all 𝑥, 𝑦∈𝑋,where𝜙 : [0, ∞) × [0, ∞) → [0, ∞) is a continuous mapping such that 𝜙(𝑡,𝑠) =0if and only if 𝑡= The Banach contraction principle is the source of metric 𝑠=0. fixed point theory. This principle had been extended by many authors in different directions (see [1]). Under this kind of contraction, Choudhury [3]estab- Chatterjea [2] introduced the following contraction lished the following fixed point result. which has been named later as C-contraction. Theorem 4 Definition 1 (see [2]). Let (𝑋, 𝑑) be a metric space and 𝑓: (see [3,Theorem2.1]).Every weakly C-con- 𝑋→𝑋amapping.Then𝑓 is called a C-contraction if there traction in a complete metric space has a unique fixed point. exists 𝑘∈[0,1/2)such that Recently, Harjani et al. [4] studied some fixed point results 𝑑 (𝑓𝑥, 𝑓𝑦) ≤ 𝑘 (𝑑 (𝑥, 𝑓𝑦) +𝑑(𝑓𝑥,𝑦)) (1) for weakly C-contractive mappings in a complete metric space endowed with a partial order. Moreover, Shatanawi [5] holds for all 𝑥, 𝑦∈𝑋. proved some fixed point and coupled fixed point theorems Under this kind of contractive inequality, Chatterjea [2] for a nonlinear weakly C-contraction type mapping in metric established the following fixed point result. and ordered metric spaces. Theorem 2 In another aspect, the notion of a partial metric space has (see [2]). Every C-contraction in a complete metric been introduced by Matthews [6]in1994asageneralization space has a unique fixed point. of the usual metric in such a way that each object does As a generalization of C-contractive mapping, Choud- not necessarily have to have a zero distance from itself. hury [3] introduced the concept of weakly C-contractive A motivation behind introducing the concept of a partial mapping and proved that every weakly C-contractive map- metric was to obtain appropriate mathematical models in ping in a complete metric space has a unique fixed point. the theory of computation and, in particular, to give a modified version of the Banach contraction principle (see, Definition 3 (see [3]). Let (𝑋, 𝑑) be a metric space and 𝑓: e.g., [7, 8]). Subsequently, several authors studied the problem 𝑋→𝑋amapping.Then𝑓 is called a weakly C-contractive of existence and uniqueness of a fixed point for mappings if 𝑓 satisfies satisfying different contractive conditions on partial metric 1 𝑑(𝑓𝑥,𝑓𝑦)≤ (𝑑 (𝑥, 𝑓𝑦) + 𝑑 (𝑓𝑥,𝑦)) spaces (e.g., [9–13]). 2 (2) We recall some definitions and properties of partial −𝜙(𝑑(𝑥,𝑓𝑦),𝑑(𝑓𝑥,𝑦)) metric spaces. 2 Abstract and Applied Analysis

Definition 5. A partial metric on a nonempty set 𝑋 is a Definition 8. Let 𝑓, 𝑔:𝑋be →𝑋 two mappings. One says + function 𝑝:𝑋×𝑋 → R such that for all 𝑥, 𝑦, 𝑧∈𝑋, that 𝑥∈𝑋is a coincidence point of 𝑓 and 𝑔 if 𝑓(𝑥) = 𝑔(𝑥). 𝑥 = 𝑦 ⇔ 𝑝(𝑥, 𝑥) = 𝑝(𝑥, 𝑦)=𝑝(𝑦, (p1) ; In this paper, we extend the concept of a weakly C- (p2) 𝑝(𝑥, 𝑥) ≤ 𝑝(𝑥,; 𝑦) contractive mapping to the context of partial metric space (p3) 𝑝(𝑥, 𝑦) = 𝑝(𝑦,; 𝑥) and define 𝑔-weakly C-contractive map. Moreover, we prove that every 𝑔-weakly C-contractive mapping in a complete (p4) 𝑝(𝑥, 𝑧) ≤ 𝑝(𝑥, 𝑦) + 𝑝(𝑦,. 𝑧)−𝑝(𝑦,𝑦) partial metric space has a unique coincidence point. Our A partial metric space is a pair (𝑋, 𝑝) such that 𝑋 is result generalizes several well-known results in the literature. nonempty set and 𝑝 is a partial metric on 𝑋.

From the above definition, if 𝑝(𝑥, 𝑦),then =0 𝑥=𝑦.But 2. Unique Coincidence and Fixed if 𝑥=𝑦, 𝑝(𝑥, 𝑦) may not be 0 in general. A famous example Point Theorem + + of a partial metric space is the pair (R ,𝑝),where𝑝:R × R+ → R+ 𝑝(𝑥, 𝑦) = {𝑥, 𝑦} Definition 9. Let (𝑋, 𝑝) be a partial metric space and 𝑔:𝑋 → is defined as max . For some more 𝑋 𝑓:𝑋 →𝑋 𝑔 examples of partial metric spaces, we refer to [8, 12]. amap.Then,themapping is said to be - weakly C-contractive if Each partial metric 𝑝 on 𝑋 generates a 𝑇0 topology 𝜏𝑝 on 𝑋 which has as a base the family of open 𝑝-balls: {𝐵𝑝(𝑥, 𝜖) : 1 𝑥∈𝑋,𝜖>0} 𝐵 (𝑥, 𝜖) = {𝑦 ∈ 𝑋 : 𝑝(𝑥, 𝑦)< 𝑝(𝑓𝑥,𝑓𝑦)≤ (𝑝 (𝑔𝑥, 𝑓𝑦) + 𝑝 (𝑓𝑥, 𝑔𝑦)) ,where 𝑝 2 𝑝(𝑥, 𝑥) +𝜖} for all 𝑥∈𝑋and 𝜖>0.Asequence{𝑥𝑛} in 𝑋 (5) converges to a point 𝑥∈𝑋,withrespectto𝜏𝑝 if and only if −𝜙(𝑝 (𝑔𝑥,) 𝑓𝑦 ,𝑝(𝑓𝑥, 𝑔𝑦)) 𝑝 (𝑥, 𝑥) = lim𝑛→∞𝑝(𝑥,𝑛 𝑥 ).Asequence{𝑥𝑛} in 𝑋 is called 𝑥 𝑦 𝑢 V ∈𝑋 𝜙 : [0, ∞) × [0, ∞) → [0, ∞) Cauchy sequence if lim𝑛,𝑚 →∞𝑝(𝑥𝑛,𝑥𝑚) exists and is finite. for all , , , ,where is a continuous mapping such that 𝜙(𝑡, 𝑠) =0 if and only if Definition 6 (see [6, 13]). Let (𝑋, 𝑝) be a partial metric space. 𝑡=𝑠=0. Then, Now we state and prove our main result. (i) a sequence {𝑥𝑛} in a partial metric space (𝑋, 𝑝) converges to a point 𝑥∈𝑋if and only if 𝑝(𝑥, 𝑥) = Theorem 10. Let (𝑋, 𝑝) be a complete partial metric space and 𝑝(𝑥, 𝑥 ) lim𝑛→∞ 𝑛 ; 𝑓:𝑋 →𝑋a 𝑔-weakly C-contraction mapping. Suppose that 𝑓(𝑋) ⊂ 𝑔(𝑋) 𝑓 𝑔 (ii) a sequence {𝑥𝑛} in a partial metric space (𝑋, 𝑝) is .Then, and have a unique coincidence point called a Cauchy sequence if there exists (and is finite) in 𝑋. lim𝑛,𝑚 →∞𝑝(𝑥𝑛,𝑥𝑚); Proof. Let 𝑥0 ∈𝑋be arbitrary point in 𝑋.Since𝑓(𝑋) ⊂ (iii) a partial metric space (𝑋, 𝑝) is said to be complete if 𝑔(𝑋), we can construct sequence {𝑔𝑥𝑛} in 𝑋 as everyCauchysequence{𝑥𝑛} in 𝑋 converges to a point 𝑥∈𝑋 𝑝(𝑥, 𝑥) = 𝑝(𝑥 ,𝑥 ) ;thatis, lim𝑛,𝑚 →∞ 𝑛 𝑚 . 𝑔𝑥𝑛+1 =𝑓𝑥𝑛,∀𝑛≥0. (6) 𝑝 𝑋 𝑝𝑠 :𝑋× If is a partial metric on , then the function Set 𝛿𝑛 = 𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+1). 𝑋→R+ given by If there exists 𝑛∈N such that 𝛿𝑛 =0, then by (p1) and 𝑠 𝑔𝑥 =𝑔𝑥 =𝑓𝑥 𝑓 𝑔 𝑝 (𝑥, 𝑦) = 2𝑝 (𝑥, 𝑦)−𝑝 (𝑥, 𝑥) −𝑝(𝑦,𝑦) (3) (p2) we have 𝑛 𝑛+1 𝑛.Hence, and have a coincidence point in 𝑋. Now assume that 𝛿𝑛 =0̸ for all 𝑛≥0. is a metric on 𝑋. Thus by (5), we have

Lemma 7 (see [6, 13]). Let (𝑋, 𝑝) be a partial metric space. 𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+2)=𝑝(𝑓𝑥𝑛,𝑓𝑥𝑛+1) Then, 1 ≤ (𝑝 (𝑔𝑥 ,𝑓𝑥 )+𝑝(𝑓𝑥 ,𝑔𝑥 )) (a) {𝑥𝑛} is a Cauchy sequence in (𝑋, 𝑝) if and only if it is a 𝑛 𝑛+1 𝑛 𝑛+1 𝑠 2 Cauchy sequence in the metric space (𝑋, 𝑝 ); 𝑠 − 𝜙 (𝑝 (𝑔𝑥 ,𝑓𝑥 ),𝑝(𝑓𝑥 ,𝑔𝑥 )) (b) (𝑋, 𝑝) is complete if and only if the metric space (𝑋, 𝑝 ) 𝑛 𝑛+1 𝑛 𝑛+1 𝑠 is complete. Furthermore, lim𝑛→∞𝑝 (𝑥𝑛,𝑥)=0if and 1 = (𝑝 (𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 )) only if 2 𝑛 𝑛+2 𝑛+1 𝑛+1

𝑝 (𝑥, 𝑥) = lim 𝑝(𝑥𝑛,𝑥)= lim 𝑝(𝑥𝑛,𝑥𝑚). 𝑛→∞ 𝑛,𝑚 →∞ (4) − 𝜙 (𝑝𝑛 (𝑔𝑥 ,𝑔𝑥𝑛+2),𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) . (7) Moreover, Bhaskar and Lakshmikantham [14]presented coupled fixed point theorems for contractions in partially By property (p4), we have ordered metric spaces. This concept attracted many math- ematician and for more related work on coupled fixed and 𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2)+𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1) coincidence points results we refer the readers to recent works (8) ≤𝑝(𝑔𝑥,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 ). in [4, 9, 10, 15–19]. 𝑛 𝑛+1 𝑛+1 𝑛+2 Abstract and Applied Analysis 3

Thus from (7), we have and therefore 1 𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+2)≤ (𝑝 (𝑔𝑥𝑛,𝑔𝑥𝑛+2)+𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) 𝜙(𝑝(𝑔𝑥 ,𝑔𝑥 ),𝑝(𝑔𝑥 ,𝑔𝑥 )) = 0. 2 𝑛→∞lim 𝑛 𝑛+2 𝑛+1 𝑛+1 (18)

−𝜙(𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2),𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) By continuity of 𝜙,weconcludethat

(9) lim 𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2)=0, 1 𝑛→∞ ≤ (𝑝 (𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 )) (19) 2 𝑛 𝑛+1 𝑛+1 𝑛+2 𝑝(𝑔𝑥 ,𝑔𝑥 )=0. 𝑛→∞lim 𝑛+1 𝑛+1

−𝜙(𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2),𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) Letting 𝑛→∞in (9)and(15), (19), and the continuity of 𝜙, ≤ max {𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+1),𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+2)} we conclude that 𝛿=0.Thus,

−𝜙(𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2),𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) . 𝑝(𝑔𝑥 ,𝑔𝑥 )=0. (10) 𝑛→∞lim 𝑛 𝑛+1 (20)

From (9), we have either 𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2) =0̸ or 𝑝(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1) =0̸ since 𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+1) =0̸ for all 𝑛∈N. Next, we will prove that If {𝑝 (𝑔𝑥 ,𝑔𝑥 ) ,𝑝(𝑔𝑥 ,𝑔𝑥 )} =𝑝(𝑔𝑥 ,𝑔𝑥 ) , 𝑝(𝑔𝑥 ,𝑔𝑥 )=0. max 𝑛 𝑛+1 𝑛+1 𝑛+2 𝑛+1 𝑛+2 𝑛,𝑚lim →∞ 𝑛 𝑚 (21) (11) then since 𝜙(𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+2), 𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) > 0 and by (10), Suppose the contrary; that is, we have 𝑝(𝑔𝑥 ,𝑔𝑥 )≤𝑝(𝑔𝑥 ,𝑔𝑥 ) 𝑝(𝑔𝑥 ,𝑔𝑥 ) =0.̸ 𝑛+1 𝑛+2 𝑛+1 𝑛+2 𝑛,𝑚lim →∞ 𝑛 𝑚 (22)

−𝜙(𝑝 (𝑔𝑥𝑛,𝑔𝑥𝑛+2) ,𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) Then there exists an 𝜖>0for which we can find <𝑝(𝑔𝑥 ,𝑔𝑥 ), 𝑛+1 𝑛+2 subsequences {𝑔𝑥𝑛(𝑘)}, {𝑔𝑥𝑚(𝑘)} of {𝑔𝑥𝑛} such that 𝑛(𝑘) is the (12) smallest integer for which which is a contradiction. Thus, we have 𝑛 (𝑘) >𝑚(𝑘) ≥𝑘, 𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘))≥𝜖. (23) max {𝑝 (𝑔𝑥𝑛,𝑔𝑥𝑛+1) ,𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+2)} =𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+1) (13) This means that and therefore 𝑝(𝑔𝑥 ,𝑔𝑥 )≤𝑝(𝑔𝑥,𝑔𝑥 ) 𝑛+1 𝑛+2 𝑛 𝑛+1 𝑝(𝑔𝑥𝑛(𝑘)−1,𝑔𝑥𝑚(𝑘))<𝜖. (24)

− 𝜙 (𝑝𝑛 (𝑔𝑥 ,𝑔𝑥𝑛+2),𝑝(𝑔𝑥𝑛+1,𝑔𝑥𝑛+1)) From the above two inequalities and (p4), we have ≤𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+1). (14) 𝜖≤𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘))

Bytheaboveinequalities,wehavethat{𝛿𝑛} = {𝑝(𝑔𝑥𝑛,𝑔𝑥𝑛+1)} ≤𝑝(𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 ) is a non increasing sequence of positive real numbers. 𝑛(𝑘) 𝑛(𝑘)−1 𝑛(𝑘)−1 𝑚(𝑘) Therefore, there is some 𝛿≥0such that −𝑝(𝑔𝑥𝑛(𝑘)−1,𝑔𝑥𝑛(𝑘)−1) (25) 𝑝(𝑔𝑥 ,𝑔𝑥 )=𝛿. 𝑛→∞lim 𝑛 𝑛+1 (15) ≤𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)−1)+𝑝(𝑔𝑥𝑛(𝑘)−1,𝑔𝑥𝑚(𝑘)) Then taking the limit as 𝑛→∞in (10), we have <𝜖+𝑝(𝑔𝑥 ,𝑔𝑥 ). 𝛿≤𝛿− 𝜙(𝑝(𝑔𝑥 ,𝑔𝑥 ),𝑝(𝑔𝑥 ,𝑔𝑥 )) ≤ 𝛿. 𝑛(𝑘) 𝑛(𝑘)−1 𝑛→∞lim 𝑛 𝑛+2 𝑛+1 𝑛+1 (16) Letting 𝑘→∞and using (20), we get Then,

𝛿− 𝜙 (𝑝 (𝑔𝑥 ,𝑔𝑥 ) ,𝑝(𝑔𝑥 ,𝑔𝑥 )) =𝛿 lim 𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘))=𝜖. 𝑛→∞lim 𝑛 𝑛+2 𝑛+1 𝑛+1 (17) 𝑘→∞ (26) 4 Abstract and Applied Analysis

By (p3) and (p4), we have which is a contradiction. Thus, we have

𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘)) 𝑝(𝑔𝑥 ,𝑔𝑥 )=0. 𝑛,𝑚lim → +∞ 𝑛 𝑚 (31) ≤𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)+1)+𝑝(𝑔𝑥𝑛(𝑘)+1,𝑔𝑥𝑚(𝑘)) Therefore, {𝑔𝑥𝑛} isaCauchysequenceinthecomplete −𝑝(𝑔𝑥𝑛(𝑘)+1,𝑔𝑥𝑛(𝑘)+1) partial metric space (𝑋, 𝑝). By Lemma 7,wehavethat ≤𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)+1)+𝑝(𝑔𝑥𝑛(𝑘)+1,𝑔𝑥𝑚(𝑘)) 𝑠 lim 𝑝 (𝑔𝑥𝑛,𝑔𝑥𝑚)=0. ≤𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)+1)+𝑝(𝑔𝑥𝑛(𝑘)+1,𝑔𝑥𝑚(𝑘)+1) 𝑛,𝑚 → +∞ (32)

+𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑚(𝑘))−𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑚(𝑘)+1) Thus, {𝑔𝑥𝑛} is a Cauchy sequence in the complete met- 𝑠 ric space (𝑋, 𝑝 ).Hence,byLemma 7, {𝑔𝑥𝑛} is a Cauchy ≤𝑝(𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 ) 𝑛(𝑘) 𝑛(𝑘)+1 𝑛(𝑘)+1 𝑚(𝑘)+1 sequence in the complete metric space (𝑋, 𝑝). Again, by Lemma 7, there exists 𝑥∈𝑋such that +𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑚(𝑘)) 𝑝𝑠 (𝑔𝑥 ,𝑔𝑥)=0 𝑛→+∞lim 𝑛 (33) ≤2𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)+1)+𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘)+1)

+𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑚(𝑘))−𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)) which implies that ≤2𝑝(𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 ) 𝑝(𝑔𝑥,𝑔𝑥)= 𝑝(𝑔𝑥 ,𝑔𝑥)= 𝑝(𝑔𝑥 ,𝑔𝑥 ). 𝑛(𝑘) 𝑛(𝑘)+1 𝑛(𝑘) 𝑚(𝑘)+1 𝑛→+∞lim 𝑛 𝑛,𝑚lim → +∞ 𝑛 𝑚 (34) +𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑚(𝑘)) 𝑝(𝑓𝑥, 𝑔𝑥 )=𝑝(𝑓𝑥,𝑔𝑥) ≤2𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑛(𝑘)+1)+𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘)) Next, we prove that lim𝑛→+∞ 𝑛+1 . Letting 𝑛→+∞in +2𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑚(𝑘))−𝑝(𝑔𝑥𝑚(𝑘),𝑔𝑥𝑚(𝑘)) 𝑝 (𝑓𝑥, 𝑔𝑥 𝑛+1) ≤𝑝(𝑓𝑥, 𝑔𝑥) +𝑝(𝑔𝑥,𝑛+1 𝑔𝑥 ) −𝑝(𝑔𝑥,) 𝑔𝑥 , ≤2𝑝(𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 ) 𝑛(𝑘) 𝑛(𝑘)+1 𝑛(𝑘) 𝑚(𝑘) (35) +2𝑝(𝑔𝑥 ,𝑔𝑥 ). 𝑚(𝑘)+1 𝑚(𝑘) we have (27) lim 𝑝(𝑓𝑥,𝑔𝑥𝑛+1)≤𝑝(𝑓𝑥,𝑔𝑥). Letting 𝑘→+∞in the above inequalities and using (20) 𝑛→+∞ (36) and (26), we have Also, letting 𝑛→+∞in lim 𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘))= lim 𝑝(𝑔𝑥𝑛(𝑘)+1,𝑔𝑥𝑚(𝑘)) 𝑘→+∞ 𝑘→+∞ 𝑝 (𝑓𝑥, 𝑔𝑥) ≤ 𝑛+1𝑝 (𝑓𝑥,𝑔𝑥 )+𝑝(𝑔𝑥𝑛+1,𝑔𝑥) = lim 𝑝(𝑔𝑥𝑛(𝑘)+1,𝑔𝑥𝑚(𝑘)+1) 𝑘→+∞ (37) −𝑝 (𝑔𝑥𝑛+1,𝑔𝑥𝑛+1), = lim 𝑝(𝑔𝑥𝑛(𝑘),𝑔𝑥𝑚(𝑘)+1)=𝜖. 𝑘→+∞ we have (28) 𝑝(𝑓𝑥,𝑔𝑥)≤ 𝑝(𝑓𝑥,𝑔𝑥 ). Therefore, from (5), we have 𝑛→+∞lim 𝑛+1 (38)

𝑝(𝑔𝑥𝑚(𝑘)+1,𝑔𝑥𝑛(𝑘)+1) From (36)and(38), we have =𝑝(𝑓𝑥𝑚(𝑘),𝑓𝑥𝑛(𝑘)) lim 𝑝(𝑓𝑥,𝑔𝑥𝑛+1)=𝑝(𝑓𝑥,𝑔𝑥). (39) 1 𝑛→+∞ ≤ (𝑝 (𝑔𝑥𝑚(𝑘),𝑓𝑥𝑛(𝑘))+𝑝(𝑓𝑥𝑚(𝑘),𝑔𝑥𝑛(𝑘))) 2 Now, we prove that 𝑓𝑥=𝑔𝑥.By(5), we have

−𝜙(𝑝(𝑔𝑥𝑚(𝑘),𝑓𝑥𝑛(𝑘)),𝑝(𝑓𝑥𝑚(𝑘),𝑔𝑥𝑛(𝑘))) 𝑝(𝑓𝑥,𝑔𝑥𝑛+1)=𝑝(𝑓𝑥,𝑓𝑥𝑛) 1 = (𝑝 (𝑔𝑥 ,𝑔𝑥 )+𝑝(𝑔𝑥 ,𝑔𝑥 )) 1 2 𝑚(𝑘) 𝑛(𝑘)+1 𝑚(𝑘)+1 𝑛(𝑘) ≤ (𝑝 (𝑔𝑥, 𝑓𝑥 )+𝑝(𝑓𝑥,𝑔𝑥 )) 2 𝑛 𝑛 −𝜙(𝑝(𝑔𝑥 ,𝑔𝑥 ),𝑝(𝑔𝑥 ,𝑔𝑥 )) . 𝑚(𝑘) 𝑛(𝑘)+1 𝑚(𝑘)+1 𝑛(𝑘) −𝜙(𝑝(𝑔𝑥,𝑓𝑥 ),𝑝(𝑓𝑥,𝑔𝑥 )) (29) 𝑛 𝑛 (40) 1 Letting 𝑘→+∞intheaboveandusing(28)andthe = (𝑝 (𝑔𝑥,𝑛+1 𝑔𝑥 )+𝑝(𝑓𝑥,𝑔𝑥𝑛)) continuity of 𝜙,weconcludethat 2

𝜖≤𝜖−𝜙(𝜖,) 𝜖 <𝜖 (30) −𝜙(𝑝(𝑔𝑥,𝑔𝑥𝑛+1),𝑝(𝑓𝑥,𝑔𝑥𝑛)) . Abstract and Applied Analysis 5

Letting 𝑛→+∞intheaboveandusing(39)andthe Conflict of Interests continuity of 𝜙,weconcludethat The author declares that there is no conflict of interests 1 𝑝(𝑓𝑥,𝑔𝑥)≤ 𝑝 (𝑓𝑥, 𝑔𝑥) − 𝜙 (0, 𝑝 (𝑓𝑥,𝑔𝑥)) regarding the publication of this paper. 2 (41) 1 ≤ 𝑝(𝑓𝑥,𝑔𝑥). References 2 [1] M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces Hence, 𝑝(𝑓𝑥, 𝑔𝑥). =0 By (p1) and (p2), we have 𝑓𝑥 = 𝑔𝑥. and Fixed Point Theory, Pure and Applied Mathematics, John Thus, 𝑥 is a coincidence point of 𝑓 and 𝑔. Wiley & Sons, New York, NY, USA, 2001. 𝑓 To prove the uniqueness of the coincidence point of and [2] S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de 𝑔,supposethat𝑦 is another coincidence point of 𝑓 and 𝑔. l’Academie´ Bulgare des Sciences, vol. 25, pp. 727–730, 1972. From (5), we have [3] B. S. Choudhury, “Unique fixed point theorem for weakly 1 C-contractive mappings,” Kathmandu University Journal of 𝑝 (𝑔𝑥, 𝑔𝑦) = 𝑝 (𝑓𝑥,𝑓𝑦) ≤ (𝑝 (𝑥, 𝑦) + 𝑝 (𝑥,𝑦)) 2 Science, Engineering and Technology,vol.5,no.1,pp.6–13,2009. (42) [4]J.Harjani,B.Lopez,´ and K. Sadarangani, “Fixed point theorems −𝜙(𝑝(𝑥,𝑦),𝑝(𝑥,𝑦)). for weakly C-contractive mappings in ordered metric spaces,” Computers and Mathematics with Applications,vol.61,no.4,pp. 𝜙(𝑝(𝑥, 𝑦), 𝑝(𝑥, 𝑦))=0 Therefore, we have .Hence, 790–796, 2011. 𝑝(𝑥, 𝑦). =0 By (p1) and (p2), we have 𝑥=𝑦. 𝑓 𝑔 [5] W. Shatanawi, “Fixed point theorems for nonlinear weakly Thus, and have a unique coincidence point. C-contractive mappings in metric spaces,” Mathematical and Computer Modelling,vol.54,no.11-12,pp.2816–2826,2011. As an immediate consequence of the above theorem, we have the following fixed point result. [6] S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, Corollary 11. Let (𝑋, 𝑝) be a complete partial metric space pp. 183–197, Annals of the New York Academy of Sciences, New and 𝑓:𝑋a →𝑋 weakly C-contraction mapping. That is, York, NY, USA, 1994. ´ 𝑇 satisfies [7] L. Ciric,´ B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an 1 𝑝(𝑓𝑥,𝑓𝑦)≤ (𝑝 (𝑥, 𝑓𝑦) + 𝑝 (𝑓𝑥,𝑦)) application,” Applied Mathematics and Computation,vol.218, 2 no. 6, pp. 2398–2406, 2011. (43) −𝜙(𝑝(𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑦)) [8] W. Shatanawi, B. Samet, and M. Abbas, “Coupled fixed point theorems for mixed monotone mappings in ordered partial for all 𝑥, 𝑦∈𝑋,where𝜙 : [0, ∞) × [0, ∞) → [0, ∞) is a metric spaces,” Mathematical and Computer Modelling,vol.55, no. 3-4, pp. 680–687, 2012. continuous mapping such that 𝜙(𝑡, 𝑠) =0 if and only if 𝑡=𝑠= 0. [9] S. M. Alsulami, N. Hussain, and A. Alotaibi, “Coupled fixed and coincidence points for monotone operators in partial metric Then, there exists a unique 𝑥∈𝑋such that 𝑓𝑥 =𝑥. spaces,” Fixed Point Theory and Applications,vol.2012,article 173, 2012. Corollary 12. Let (𝑋, 𝑝) be a complete partial metric space. Suppose that the mapping 𝑓:𝑋satisfies →𝑋 the following [10] H. Aydi, “Some coupled fixed point results on partial metric contractive condition: spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 647091, 11 pages, 2011. 𝑝 (𝑓𝑥, 𝑓𝑦) ≤𝑘𝑝(𝑓𝑥, 𝑦) +𝑙𝑝(𝑥, 𝑓𝑦) , (44) [11] D. Ilic,´ V. Pavlovic,´ and V. Rakoevic,´ “Some new extensions of Banach’s contraction principle to partial metric space,” Applied for all 𝑥, 𝑦, 𝑢, V ∈𝑋,where𝑘, 𝑙 are nonnegative constants with Mathematics Letters, vol. 24, no. 8, pp. 1326–1330, 2011. 𝑘+2𝑙<1 𝑓 .Then, has a unique fixed point. [12] Z. Golubovic,´ Z. Kadelburg, and S. Radenovic,´ “Coupled coin- cidence points of mappings in ordered partial metric spaces,” 𝜙(𝑡, 𝑠) = ((1/2) − 𝑙)𝑡 + ((1/2) −𝑘)𝑠 𝑘 𝑙 Proof. Take ,where , are Abstract and Applied Analysis,vol.2012,ArticleID192581,18 nonnegative constants with 𝑘+2𝑙<1. pages, 2012. [13] S. Oltra and O. Valero, “Banach’s fixed point theorem for Corollary 13(see [10,Corollary2.7]). Let (𝑋, 𝑝) be a complete 𝑓:𝑋 →𝑋 partial metric spaces,” Rendiconti dell’Istituto di Matematica partial metric space. Suppose that the mapping dell’UniversitadiTrieste` ,vol.36,no.1-2,pp.17–26,2004. satisfies the following contractive condition: [14] T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems 𝑘 in partially ordered metric spaces and applications,” Nonlinear 𝑝(𝑓𝑥,𝑓𝑦)≤ (𝑝 (𝑓𝑥,𝑦) + 𝑝 (𝑥, 𝑓𝑦)), (45) 2 Analysis, Theory, Methods and Applications,vol.65,no.7,pp. 1379–1393, 2006. for all 𝑥, 𝑦∈𝑋,where0≤𝑘<2/3.Then,𝑓 has a unique fixed [15] A. Alotaibi and S. M. Alsulami, “Coupled coincidence points for point. monotone operators in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 44, 2011. 𝜙(𝑡, 𝑠) = ((1/2) − (𝑘/2)) (𝑡+𝑠) 0≤𝑘< Proof. Take ,where [16] S. M. Alsulami, “Some coupled coincidence point theorems 2/3. for a mixed monotone operator in a complete metric space 6 Abstract and Applied Analysis

endowed with a partial order by using altering distance func- tions,” Fixed Point Theory and Applications,vol.2013,article194, 2013. [17] S. M. Alsulami and A. Alotaibi, “Coupled coincidence point theorems for compatible mappings in partially ordered metric spaces,” Bulletin of Mathematical Analysis and Applications,vol. 4, no. 2, pp. 129–138, 2012. [18] V. Lakshmikantham and L. Ciri´ c,´ “Coupled fixed point the- orems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 12, pp. 4341–4349, 2009. [19] N. Hussain, A. Latif, and M. H. Shah, “Coupled and tripled coincidence point results without compatibility,” Fixed Point Theory and Applications,vol.2012,article77,2012. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 560258, 9 pages http://dx.doi.org/10.1155/2013/560258

Research Article Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces

Vatan Karakaya,1 Kadri DoLan,1 Faik Gürsoy,2 and Müzeyyen Ertürk2

1 Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yildiz Technical University, DavutpasaCampus,Esenler,34210Istanbul,Turkey 2 Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey

Correspondence should be addressed to Vatan Karakaya; [email protected]

Received 2 September 2013; Accepted 14 October 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 Vatan Karakaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Weintroduce a new three-step iteration scheme and prove that this new iteration scheme is convergent to fixed points of contractive- like operators. Also, by providing an example, we show that our new iteration method is faster than another iteration method due to Suantai (2005). Furthermore, it is shown that this new iteration method is equivalent to some other iteration methods in the sense of convergence. Finally, it is proved that this new iteration method is T-stable.

1. Introduction and Preliminaries The first concept of this work is about convergence of fixed point iteration methods. Fixed point iteration methods 𝑓(𝑥) =𝑦 Most of nonlinear equations appearing in physical may exhibit radically different behaviors for various classes formulations can similarly be transformed into a fixed point of mappings. While a particular fixed point iteration method 𝑥=𝑇𝑥 equation of the form . To obtain results on existence is convergent for an appropriate class of mappings, it could and uniqueness of such equations’ solution, an approximate not be convergent for others. Therefore, it is important to fixed point theorem is applied. That is, this application will determine whether an iteration method converges to fixed bring us to the solution of the original equation via help of pointofamapping.Inthisfield,therearenumerousworks a particular fixed point iteration method. For this reason, regarding convergence of various iterative methods, as one it is crucial to define a new iteration method. To decide can see in [1–14]. whether an iteration method is useful for application, it is of In this work, the second concept is the rate of convergence paramount importance to answer the following questions. of iteration methods. After examining convergence of an iter- (i) Does it converge to fixed point of an operator? ation method, it is important to check whether this iteration method is faster than some well-known iteration methods or (ii) Is it faster than the iterations defined in the existing not. If it is faster than some current iteration methods, then it literature? could be more useful than the others. More details about the (iii) Is its convergence equivalent to the convergence of the rateofconvergencecanbefoundin[10, 15–17]. other iteration methods? The third concept for this work is equivalence among con- (iv) Is it 𝑇-stable? and so forth. vergences of iteration methods. Rhoades and Soltuz [13, 18– 20] showed that the convergence of Mann iteration is equiv- Throughout this paper we examine four essential con- alent to Ishikawa iteration for different classes of operators. cepts based on the above questions for a new three-step Theyalsoshowedin[21]thattheconvergenceofmodified iteration method when applied to contractive-like mapping. Mann iteration is equivalent to modified Ishikawa iteration As a background to our exposition, we now give some under certain mappings. Afterward, Rhoades and Soltuz [14] information about literature of those concepts. studied that Mann and Ishikawa iteration sequences are 2 Abstract and Applied Analysis equivalent to a multistep iteration scheme for various classes Suantai [11] proposed an iterative scheme as follows: of the operators. In addition, Soltuz [22]provedthatthe 𝑥 =𝑥∈𝐶, convergence of Ishikawa iteration is equivalent to that of 1 Mann and Picard iterations for quasicontractive operators. 𝑓(𝑇,𝑥 )=(1−𝛼 −𝛽 )𝑥 +𝛼 𝑇𝑦 +𝛽 𝑇𝑧 , One can find detailed literature concerning this topic in the 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 (7) following list [3–5, 23–25]. 𝑦𝑛 =(1−𝑎𝑛 −𝑏𝑛)𝑥𝑛 +𝑎𝑛𝑇𝑧𝑛 +𝑏𝑛𝑇𝑥𝑛, The final concept in this work is stability of fixed point iteration methods. The topic of stability, as an application 𝑧𝑛 =(1−𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛,𝑛∈N. of the theory of fixed point, has been studied by many authors including Harder and Hicks [26, 27], Rhoades [28, Agarwaletal.establishedanS-iterationmethodin[1]as 29], Osilike [30, 31], Ostrowski [32], Berinde [33], Olantiwo follows: [9] and Singh and Prasad [34]. First stability result in metric 𝑥 =𝑥∈𝐶, spaces is due to Ostrowski [32],whereheestablishedthesta- 1 bility of Picard iteration by employing Banach’s contraction 𝑓(𝑇,𝑥𝑛)=(1−𝛼𝑛)𝑇𝑥𝑛 +𝛼𝑛𝑇𝑦𝑛, (8) condition. Afterward, several authors studied this concept in different ways. 𝑦𝑛 =(1−𝛽𝑛)𝑥𝑛 +𝛽𝑛𝑇𝑥𝑛,𝑛∈N. Throughout this paper, we denote the set of natural numbers by N.Let𝐸 be a normed space, 𝐶 a nonempty convex Thianwan [12] introduced a two-step Mann iteration by subset of a normed space 𝐸,and𝑇 aselfmapof𝐶.Let(𝑎𝑛), 𝑥 =𝑥∈𝐶, (𝑏𝑛), (𝑐𝑛), (𝛼𝑛), (𝛽𝑛)⊂[0,1]be real sequences satisfying 1 (𝑥 )⊂𝐶 certain conditions. Let 𝑛 be a sequence generated by a 𝑓(𝑇,𝑥 )=(1−𝛼)𝑦 +𝛼 𝑇𝑦 , particular iteration process including the operator 𝑇.Thatis, 𝑛 𝑛 𝑛 𝑛 𝑛 (9) 𝑦 =(1−𝛽)𝑥 +𝛽 𝑇𝑥 ,𝑛∈N. 𝑥𝑛+1 =𝑓(𝑇,𝑥𝑛), (1) 𝑛 𝑛 𝑛 𝑛 𝑛 where 𝑓 is suitable function and 𝑥0 ∈𝐶is the initial Recently, Phuengrattana and Suantai [10]definedanSP approximation. Suppose that (𝑥𝑛) converges to a fixed point iteration process as follows: ∗ 𝑥 of 𝑇.Let(𝑦𝑛)⊂𝐶be an arbitrary sequence and set 𝑥1 =𝑥∈𝐶, 󵄩 󵄩 𝜖𝑛 = 󵄩𝑦𝑛+1 −𝑓(𝑇,𝑦𝑛)󵄩 , (2) 𝑓(𝑇,𝑥𝑛)=(1−𝛼𝑛)𝑦𝑛 +𝛼𝑛𝑇𝑦𝑛, for all 𝑛∈N. Then, the iteration algorithm (1)issaidtobe𝑇- (10) 𝑦 =(1−𝛽)𝑧 +𝛽 𝑇𝑧 , stable or stable with respect to 𝑇 if and only if lim𝑛→∞ 𝜖𝑛 =0 𝑛 𝑛 𝑛 𝑛 𝑛 implies that lim𝑛→∞ 𝑦𝑛 =𝑝.If,in(1), 𝑧𝑛 =(1−𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛,𝑛∈N. 𝑥1 =𝑥∈𝐶, (3) Inspired by the above iteration process, we will introduce the 𝑓(𝑇,𝑥𝑛)=𝑇𝑥𝑛,𝑛∈N, following new iterative algorithm: then it is called the Picard iteration process [35]. 𝑥1 =𝑥∈𝐶, The Mann iteration procedure given in[7] is defined by 𝑓(𝑇,𝑥𝑛)=(1−𝛼𝑛 −𝛽𝑛)𝑦𝑛 +𝛼𝑛𝑇𝑦𝑛 +𝛽𝑛𝑇𝑧𝑛, 𝑢1 =𝑢∈𝐶 (11) (4) 𝑦𝑛 =(1−𝑎𝑛 −𝑏𝑛)𝑧𝑛 +𝑎𝑛𝑇𝑧𝑛 +𝑏𝑛𝑇𝑥𝑛, 𝑓(𝑇,𝑢𝑛)=(1−𝛼𝑛)𝑢𝑛 +𝛼𝑛𝑇𝑢𝑛,𝑛∈N. 𝑧𝑛 =(1−𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛,𝑛∈N, The sequence (𝑥𝑛) defined by (𝑎 ) (𝑏 ) (𝑐 ) (𝛼 ) (𝛽 ) [0, 1] 𝑥 =𝑥∈𝐶, where 𝑛 , 𝑛 , 𝑛 , 𝑛 ,and 𝑛 are real sequences in 1 satisfying

𝑓(𝑇,𝑥𝑛)=(1−𝛼𝑛)𝑥𝑛 +𝛼𝑛𝑇𝑦𝑛, (5) ∞ ∞ (𝛼𝑛 +𝛽𝑛)𝑛=0,(𝑎𝑛 +𝑏𝑛)𝑛=0 ∈ [0, 1] ,∀𝑛∈N, 𝑦 =(1−𝛽)𝑥 +𝛽 𝑇𝑥 ,𝑛∈N, 𝑛 𝑛 𝑛 𝑛 𝑛 ∞ (12) ∑ (𝛼𝑛 +𝛽𝑛)=∞. is known as the Ishikawa iteration process [6]. 𝑛=0 The Noor iteration method [8] is defined by 𝑥 =𝑥∈𝐶, Some special cases of the new iteration process given in (11) 1 are as follows. 𝑓(𝑇,𝑥 )=(1−𝛼)𝑥 +𝛼 𝑇𝑦 , 𝑛 𝑛 𝑛 𝑛 𝑛 (i) If 𝑐𝑛 =1and 𝛽𝑛 =𝛼𝑛 =𝑎𝑛 =𝑏𝑛 =0for all 𝑛∈N,then (6) (11) reduces to Picard iteration (3). 𝑦𝑛 =(1−𝛽𝑛)𝑥𝑛 +𝛽𝑛𝑇𝑧𝑛, (ii) If 𝑐𝑛 =𝛽𝑛 =𝑎𝑛 =𝑏𝑛 =0for all 𝑛∈N,then(11)reduces 𝑧𝑛 =(1−𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛,𝑛∈N. to Mann iteration (4). Abstract and Applied Analysis 3

(iii) If 𝑐𝑛 =𝛽𝑛 =𝑎𝑛 =0for all 𝑛∈N,then(11)reducesto 2. Main Results Ishikawa iteration (5). Theorem 3. Let 𝐶 be a nonempty closed convex subset of an 𝑐 =𝑏 =0 𝛼 +𝛽 =1 𝑛∈N (iv) If 𝑛 𝑛 and 𝑛 𝑛 for all ,then(11) arbitrary Banach space 𝐸 and 𝑇:𝐶be →𝐶 a mapping reduces to S-iteration (8). satisfying (13) with 𝐹𝑇 =⌀̸ .Let(𝑥𝑛) a sequence defined by (11) 𝛽 =𝑏 =𝑐 =0 𝑛∈N with real sequences (𝑎𝑛), (𝑏𝑛), (𝑐𝑛), (𝛼𝑛), (𝛽𝑛)⊂[0,1]satisfying (v) If 𝑛 𝑛 𝑛 for all ,then(11)reducesto ∞ ∞ ∞ two-step Mann iteration (9). (𝛼𝑛 +𝛽𝑛)𝑛=0,(𝑎𝑛 +𝑏𝑛)𝑛=0 ⊂ [0, 1],and∑𝑛=0(𝛼𝑛 +𝛽𝑛)=∞. Then the iterative sequence (𝑥𝑛) converges strongly to the fixed (vi) If 𝛽𝑛 =𝑏𝑛 =0for all 𝑛∈N,then(11)reducestoSP point of 𝑇. iteration (10). ∗ Proof. Let 𝑥 be the fixed point of 𝑇.Itcanbeseeneasilyfrom Quite recently, Imoru and Olatinwo [36]introduceda ∗ (13)that𝑥 is the unique fixed point of 𝑇.Toshowthat(𝑥𝑛) class of operators called contractive-like mappings by ∗ ∗ converges to the fixed point 𝑥 =𝑇𝑥 ,weusecondition(13). 󵄩 󵄩 󵄩 󵄩 Hence, we have 󵄩𝑇𝑥−𝑇𝑦󵄩 ≤𝜑(‖𝑥−𝑇𝑥‖) +𝛿󵄩𝑥−𝑦󵄩 ∀𝑥,𝑦∈𝐸, (13) 󵄩 󵄩 󵄩 󵄩 󵄩𝑥 −𝑥∗󵄩 ≤ 󵄩(1 − 𝛼 −𝛽 )𝑦 +𝛼 𝑇𝑦 +𝛽 𝑇𝑧 −𝑥∗󵄩 + + 󵄩 𝑛+1 󵄩 󵄩 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 󵄩 where 𝛿∈[0,1)and 𝜑:R → R is a monotone increasing 𝜑(0) = 0 󵄩 ∗󵄩 󵄩 ∗󵄩 function with . ≤(1−𝛼𝑛 −𝛽𝑛) 󵄩𝑦𝑛 −𝑥 󵄩 +𝛼𝑛 󵄩𝑇𝑦𝑛 −𝑥 󵄩 In inequality (13), if we take 𝜑(𝑡) = 𝐿𝑡, then it is reduced 󵄩 ∗󵄩 to the contractive definition due to Osilike and Udomene31 [ ]. +𝛽𝑛 󵄩𝑇𝑧𝑛 −𝑥 󵄩 Also, by putting 𝐿=2𝛿in (13), the class of quasicontractive 󵄩 󵄩 ≤[(1−𝛼 −𝛽 )+𝛼 𝛿] 󵄩𝑦 −𝑥∗󵄩 operators reduces to class of operators due to Berinde [2]. 𝑛 𝑛 𝑛 󵄩 𝑛 󵄩 In [2] it was shown that the class of these operators is 󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 +𝛽𝑛 󵄩𝑇𝑧𝑛 −𝑥 󵄩 +𝛼𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩) wider than class of Zamfirescu operators given in [37], where 𝛿:= {𝑎, 𝑏/(1 − 𝑏), 𝑐/(1 −𝑐)} 𝛿∈[0,1) 𝑎, 𝑏 󵄩 ∗󵄩 max , ,and ,and ≤[(1−𝛼𝑛 −𝛽𝑛)+𝛼𝑛𝛿] 󵄩𝑦𝑛 −𝑥 󵄩 𝑐 are real numbers satisfying 0<𝑎<1, 0<𝑏,and𝑐< 󵄩 ∗󵄩 1/2. Besides, it is easy to see that special case of Zamfirescu +𝛽𝑛𝛿 󵄩𝑧𝑛 −𝑥 󵄩 operator gives Kannans’ and Chatterjeas’ results given in [38] 󵄩 ∗ ∗󵄩 and [39], respectively. +[𝛽𝑛 +𝛼𝑛]𝜑(󵄩𝑥 −𝑇𝑥 󵄩), In this paper, we will prove that the new iteration (17) method (11) is convergent to fixed point of contractive-like 󵄩 󵄩 󵄩 󵄩 󵄩𝑦 −𝑥∗󵄩 ≤ 󵄩(1 − 𝑎 −𝑏)𝑧 +𝑎 𝑇𝑧 +𝑏𝑇𝑥 −𝑥∗󵄩 mappings satisfying (13). Also, by using a counterexample 󵄩 𝑛 󵄩 󵄩 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 󵄩 󵄩 󵄩 󵄩 󵄩 givenin[17], we compare the rates of convergence between ≤(1−𝑎 −𝑏) 󵄩𝑧 −𝑥∗󵄩 +𝑎 󵄩𝑇𝑧 −𝑥∗󵄩 the new iteration method (11) and the iteration method (7) 𝑛 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 for the same class of mappings satisfying (13). Moreover, 󵄩 ∗󵄩 +𝑏𝑛 󵄩𝑇𝑥𝑛 −𝑥 󵄩 we establish an equivalence among convergences of some 󵄩 ∗󵄩 iteration methods including the new iteration method (11). ≤[(1−𝑎𝑛 −𝑏𝑛)+𝑎𝑛𝛿] 󵄩𝑧𝑛 −𝑥 󵄩 Finally, we prove that the new iteration method (11)is𝑇- 󵄩 ∗ ∗󵄩 󵄩 ∗󵄩 stable. +𝑎𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩)+𝑏𝑛 󵄩𝑇𝑥𝑛 −𝑥 󵄩 We end this section with the following definition and 󵄩 󵄩 ≤[(1−𝑎 −𝑏)+𝑎 𝛿] 󵄩𝑧 −𝑥∗󵄩 lemma which will be useful in proving our main results. 𝑛 𝑛 𝑛 󵄩 𝑛 󵄩 󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 +𝑏𝑛𝛿 󵄩𝑥𝑛 −𝑥 󵄩 +(𝑏𝑛 +𝑎𝑛)𝜑(󵄩𝑥 −𝑇𝑥 󵄩), Definition 1 (see [17]). Assume that (𝑎𝑛)𝑛∈N and (𝑏𝑛)𝑛∈N are two real convergent sequences with limits 𝑎 and 𝑏, (18) respectively. Then (𝑎𝑛)𝑛∈N is said to converge faster than 󵄩 ∗󵄩 󵄩 ∗󵄩 󵄩𝑧𝑛 −𝑥 󵄩 ≤ 󵄩(1 − 𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛 −𝑥 󵄩 (𝑏𝑛)𝑛∈N if 󵄩 ∗󵄩 󵄩 ∗󵄩 󵄨 󵄨 ≤(1−𝑐𝑛) 󵄩𝑥𝑛 −𝑥 󵄩 +𝑐𝑛 󵄩𝑇𝑥𝑛 −𝑥 󵄩 󵄨𝑎 −𝑎󵄨 󵄨 𝑛 󵄨 =0. (19) 𝑛→∞lim 󵄨 󵄨 (14) 󵄩 ∗󵄩 󵄨 𝑏𝑛 −𝑏󵄨 ≤[1−𝑐𝑛 (1−𝛿)] 󵄩𝑥𝑛 −𝑥 󵄩 󵄩 ∗ ∗󵄩 Lemma 2 (see [33]). If 𝜌 is a real number satisfying 0≤ +𝑐𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩). 𝜌<1and (𝜉𝑛)𝑛∈N is a sequence of positive numbers such that lim𝑛→∞𝜉𝑛 =0, then for any sequence of positive numbers By combining (17)–(19), we derive (𝜉 ) 𝑛 𝑛∈N satisfying 󵄩 ∗󵄩 󵄩𝑥𝑛+1 −𝑥 󵄩

𝑎𝑛+1 ≤𝜌𝑎𝑛 +𝜉𝑛, 𝑛=1,2,..., (15) ≤[(1−𝛼𝑛 −𝛽𝑛)+𝛼𝑛𝛿] one has ×[[(1−𝑎𝑛 −𝑏𝑛)+𝑎𝑛𝛿] 𝑎 =0. 󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 𝑛→∞lim 𝑛 (16) ×[[1−𝑐𝑛 (1−𝛿)] 󵄩𝑥𝑛 −𝑥 󵄩 +𝑐𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩)] 4 Abstract and Applied Analysis

󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 +𝑏𝑛𝛿 󵄩𝑥𝑛 −𝑥 󵄩 +(𝑏𝑛 +𝑎𝑛)𝜑(󵄩𝑥 −𝑇𝑥 󵄩)] operators. In the following example, for convenience, we use sequences (V𝑛) and (𝑠𝑛) associated with the iterative methods 󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 +𝛽𝑛𝛿 [[1𝑛 −𝑐 (1−𝛿)] 󵄩𝑥𝑛 −𝑥 󵄩 +𝑐𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩)] (11)and(7), respectively.

󵄩 ∗ ∗󵄩 𝑇 : [0, 1] → [0, 1] +[𝛽𝑛 +𝛼𝑛]𝜑(󵄩𝑥 −𝑇𝑥 󵄩). Example 4 (see [17]). Define a mapping as 𝑇𝑥=𝑥/2 𝛼 =𝛽 =𝑎 =𝑏 =𝑐 =0 𝑛=1,2,...,24 (20) .Let 𝑛 𝑛 𝑛 𝑛 𝑛 ,for , and 𝛼𝑛 =𝛽𝑛 =𝑎𝑛 =𝑏𝑛 =2/√𝑛, 𝑐𝑛 =4/√𝑛,forall𝑛≥24. ∗ ∗ Since 𝜑(‖𝑥 −𝑇𝑥 ‖) = 0,(20)becomes Itcanbeseeneasilythatthemapping𝑇 satisfies condition (13) with the unique fixed point 0∈𝐹𝑇.Furthermore,itiseasy 󵄩 ∗󵄩 󵄩𝑥𝑛+1 −𝑥 󵄩 to see that Example 4 satisfies all the conditions of Theorem 3.

Indeed, let 𝑥0 =0̸ be initial point for iterative methods (11) ≤ [[1 − 𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿] and (7). By using iterative methods (11)and(7), we have

×[[1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿] [1𝑛 −𝑐 (1−𝛿)]+𝑏𝑛𝛿] 2 2 +𝛽𝑛𝛿[1−𝑐𝑛 (1−𝛿)]] V𝑛 =(1− − ) (21) √𝑛 √𝑛 󵄩 ∗󵄩 × 󵄩𝑥𝑛 −𝑥 󵄩 2 2 4 4 × ((1 − − ) ((1 − )𝑥𝑛 + 𝑇𝑥𝑛) ≤([1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿] √𝑛 √𝑛 √𝑛 √𝑛

󵄩 ∗󵄩 ×[1−(𝑎𝑛 +𝑏𝑛) (1−𝛿)]+𝛽𝑛𝛿) 󵄩𝑥𝑛 −𝑥 󵄩 2 4 4 2 + 𝑇((1− )𝑥 + 𝑇𝑥 )+ 𝑇𝑥 ) √𝑛 √𝑛 𝑛 √𝑛 𝑛 √𝑛 𝑛 󵄩 ∗󵄩 ≤(1−(𝛼𝑛 +𝛽𝑛) (1−𝛿)) 󵄩𝑥𝑛 −𝑥 󵄩 . 2 2 2 4 4 + 𝑇 ((1 − − ) ((1 − )𝑥 + 𝑇𝑥 ) By continuing the above processes, we obtain the following √𝑛 √𝑛 √𝑛 √𝑛 𝑛 √𝑛 𝑛 estimates 󵄩 󵄩 󵄩 󵄩 󵄩𝑥 −𝑥∗󵄩 ≤(1−(𝛼+𝛽 ) (1−𝛿)) 󵄩𝑥 −𝑥∗󵄩 2 4 4 2 󵄩 𝑛+1 󵄩 𝑛 𝑛 󵄩 𝑛 󵄩 + 𝑇 ((1 − )𝑥 + 𝑇𝑥 )+ 𝑇𝑥 ) √𝑛 √𝑛 𝑛 √𝑛 𝑛 √𝑛 𝑛 󵄩 ∗󵄩 󵄩 ∗󵄩 󵄩𝑥𝑛 −𝑥 󵄩 ≤(1−(𝛼𝑛−1 +𝛽𝑛−1) (1−𝛿)) 󵄩𝑥𝑛−1 −𝑥 󵄩 2 4 4 (22) + 𝑇 ((1 − )𝑥 + 𝑇𝑥 ) . √𝑛 √𝑛 𝑛 √𝑛 𝑛 .

󵄩 ∗󵄩 󵄩 ∗󵄩 2 2 󵄩𝑥1 −𝑥 󵄩 ≤(1−(𝛼0 +𝛽0) (1−𝛿)) 󵄩𝑥0 −𝑥 󵄩 , =(1− − ) √𝑛 √𝑛 𝑛 󵄩 ∗󵄩 󵄩 ∗󵄩 󵄩𝑥𝑛+1 −𝑥 󵄩 ≤ ∏ [1 − (𝛼𝑖 +𝛽𝑖) (1−𝛿)] 󵄩𝑥0 −𝑥 󵄩 2 2 4 4 1 𝑖=0 × ((1 − − ) ((1 − )𝑥 + 𝑥 ) (23) √𝑛 √𝑛 √𝑛 𝑛 √𝑛 2 𝑛 󵄩 ∗󵄩 (−(1−𝛿)𝑛 ∑ (𝛼 +𝛽 )) ≤ 󵄩𝑥 −𝑥 󵄩 𝑒 𝑖=0 𝑖 𝑖 , 󵄩 0 󵄩 2 1 4 4 1 2 1 + ((1 − )𝑥 + 𝑥 )+ 𝑥 ) ∞ √𝑛 2 √𝑛 𝑛 √𝑛 2 𝑛 √𝑛 2 𝑛 for all 𝑛∈N.Since0<𝛿<1, 𝛼𝑛,𝛽𝑛 ∈ [0, 1] and ∑𝑛=0(𝛼𝑛 + 𝛽𝑛)=∞,wehave 2 1 󵄩 󵄩 + 󵄩𝑥 −𝑥∗󵄩 √𝑛 2 𝑛→∞lim sup 󵄩 𝑛+1 󵄩 (24) 󵄩 ∗󵄩 (−(1−𝛿)𝑛 ∑ (𝛼 +𝛽 )) ≤ (󵄩𝑥 −𝑥 󵄩 𝑒 𝑖=0 𝑖 𝑖 )=0. 2 2 4 4 1 𝑛→∞lim sup 󵄩 0 󵄩 × ((1 − − ) ((1 − )𝑥 + 𝑥 ) √𝑛 √𝑛 √𝑛 𝑛 √𝑛 2 𝑛 ∗ ∗ Therefore lim𝑛→∞‖𝑥𝑛 −𝑥 ‖=0;thatis, 𝑥𝑛 →𝑥 ∈𝐹𝑇 for 𝑛∈N 2 1 4 4 1 2 1 all . + ((1 − )𝑥 + 𝑥 )+ 𝑥 ) √𝑛 2 √𝑛 𝑛 √𝑛 2 𝑛 √𝑛 2 𝑛 Theorem 3 allows us to give the following example which 2 1 4 4 1 compares the rates of convergence between the new iteration + ((1 − )𝑥 + 𝑥 ) method (11) and the iteration method (7) for contractive-like √𝑛 2 √𝑛 𝑛 √𝑛 2 𝑛 Abstract and Applied Analysis 5

6 16 18 (13) with 𝐹𝑇 =⌀̸ .If𝑢1 =𝑥1 ∈𝐶and 𝛼𝑛 +𝛽𝑛 ≥𝐴>0for all =(1− + − )𝑥𝑛 √𝑛 𝑛 𝑛√𝑛 𝑛∈N, then the following statements are equivalent.

. ∗ . (i) Mann iteration (4) convergestofixedpoint𝑥 . 𝑛 6 16 18 ∗ = ∏ (1 − + − )𝑥0, (ii) The new iteration (11) convergestofixedpoint𝑥 . 𝑖=25 √𝑖 𝑖 𝑖√𝑖 Proof. (i)⇒(ii): Suppose that Mann iteration (4)converges (25) ∗ ∗ to fixed point 𝑥 ;thatis,𝑢𝑛 →𝑥 as 𝑛→∞.Wewillshow 2 2 ∗ 𝑠𝑛 =(1− − )𝑥𝑛 that the new iteration (11) converges to the fixed point 𝑥 ;that √𝑛 √𝑛 ∗ is, 𝑥𝑛 →𝑥 as 𝑛→∞.Using(4), (11), and (13), we have 2 1 + √𝑛 2 󵄩 󵄩 󵄩 󵄩𝑢𝑛+1 −𝑥𝑛+1󵄩 = 󵄩(1 − 𝛼𝑛 −𝛽𝑛)𝑢𝑛 +𝛼𝑛𝑇𝑢𝑛 +𝛽𝑛𝑇𝑢𝑛 2 2 × ((1 − − )𝑥 󵄩 √𝑛 √𝑛 𝑛 −(1−𝛼𝑛 −𝛽𝑛)𝑦𝑛 −𝛼𝑛𝑇𝑦𝑛 −𝛽𝑛𝑇𝑧𝑛󵄩 󵄩 󵄩 󵄩 󵄩 2 1 4 4 1 2 1 ≤(1−𝛼𝑛 −𝛽𝑛) 󵄩𝑢𝑛 −𝑦𝑛󵄩 +𝛼𝑛 󵄩𝑇𝑢𝑛 −𝑇𝑦𝑛󵄩 + ((1 − )𝑥𝑛 + 𝑥𝑛)+ 𝑥𝑛) √𝑛 2 √𝑛 √𝑛 2 √𝑛 2 󵄩 󵄩 +𝛽𝑛 󵄩𝑇𝑢𝑛 −𝑇𝑧𝑛󵄩 2 1 4 4 1 + ((1 − )𝑥𝑛 + 𝑥𝑛) 󵄩 󵄩 √𝑛 2 √𝑛 √𝑛 2 ≤(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) 󵄩𝑢𝑛 −𝑦𝑛󵄩 󵄩 󵄩 2 4 2 +𝛽 𝛿 󵄩𝑢 −𝑧 󵄩 =(1− − − )𝑥 𝑛 󵄩 𝑛 𝑛󵄩 √𝑛 𝑛 𝑛√𝑛 𝑛 󵄩 󵄩 +(𝛼𝑛 +𝛽𝑛)𝜑(󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩), . . (30) 𝑛 󵄩 󵄩 󵄩 󵄩 2 4 2 󵄩𝑢 −𝑦 󵄩 = 󵄩𝑢 −(1−𝑎 −𝑏)𝑧 −𝑎 𝑇𝑧 −𝑏𝑇𝑥 󵄩 = ∏ (1 − − − )𝑥 . 󵄩 𝑛 𝑛󵄩 󵄩 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛󵄩 √ 𝑖 √ 0 𝑖=25 𝑖 𝑖 𝑖 󵄩 󵄩 ≤(1−𝑎𝑛 −𝑏𝑛) 󵄩𝑢𝑛 −𝑧𝑛󵄩 (26) 󵄩 󵄩 +𝑎 󵄩𝑢 −𝑇𝑢 +𝑇𝑢 −𝑇𝑧 󵄩 Now, let us compare these results as follows: 𝑛 󵄩 𝑛 𝑛 𝑛 𝑛󵄩 󵄩 󵄩 󵄨 󵄨 󵄨 𝑛 󵄨 +𝑏 󵄩𝑢 −𝑇𝑢 +𝑇𝑢 −𝑇𝑥 󵄩 󵄨V −0󵄨 󵄨 4𝑖 − 20√𝑖+16 󵄨 𝑛 󵄩 𝑛 𝑛 𝑛 𝑛󵄩 󵄨 𝑛 󵄨 = 󵄨∏ (1 − )󵄨 󵄨 𝑠 −0󵄨 󵄨 √ √ 󵄨 󵄩 󵄩 󵄩 󵄩 󵄨 𝑛 󵄨 󵄨𝑖=25 𝑖 𝑖−2𝑖−4 𝑖−2 󵄨 ≤(1−𝑎𝑛 −𝑏𝑛) 󵄩𝑢𝑛 −𝑧𝑛󵄩 +𝑎𝑛 󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩 (27) 󵄩 󵄩 󵄨 𝑛 󵄨 󵄩 󵄩 󵄨 4(√𝑖−1)(√𝑖−4) 󵄨 +𝑎𝑛 󵄩𝑇𝑢𝑛 −𝑇𝑧𝑛󵄩 󵄨 󵄨 = 󵄨∏ (1 − ) 󵄨 . 󵄨𝑖=25 (√𝑖 − 4) (𝑖√ +2 𝑖+4)+14 󵄨 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 +𝑏𝑛 󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩 +𝑏𝑛 󵄩𝑇𝑢𝑛 −𝑇𝑥𝑛󵄩 Since 󵄩 󵄩 󵄩 󵄩 ≤(1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿) 󵄩𝑢𝑛 −𝑧𝑛󵄩 +𝑏𝑛𝛿 󵄩𝑢𝑛 −𝑥𝑛󵄩 𝑛 4(√𝑖−1)(√𝑖−4) 󵄩 󵄩 +(𝑎 +𝑏) 󵄩𝑢 −𝑇𝑢 󵄩 0≤ lim ∏ (1 − ) 𝑛 𝑛 󵄩 𝑛 𝑛󵄩 𝑛→∞ (√𝑖 − 4) (𝑖√ +2 𝑖+4)+14 𝑖=25 󵄩 󵄩 +(𝑎𝑛 +𝑏𝑛)𝜑(󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩), 𝑛 1 (31) ≤ ∏ (1 − ) (28) 𝑛→∞lim 𝑖=25 𝑖 󵄩 󵄩 󵄩 󵄩 󵄩𝑢𝑛 −𝑧𝑛󵄩 = 󵄩𝑢𝑛 −(1−𝑐𝑛)𝑥𝑛 −𝑐𝑛𝑇𝑥𝑛󵄩 24 󵄩 󵄩 = =0, ≤(1−𝑐) 󵄩𝑢 −𝑥 󵄩 𝑛→∞lim 𝑛 𝑛 󵄩 𝑛 𝑛󵄩 󵄩 󵄩 +𝑐 󵄩𝑢 −𝑇𝑢 +𝑇𝑢 −𝑇𝑥 󵄩 finally we get 𝑛 󵄩 𝑛 𝑛 𝑛 𝑛󵄩 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄨V −0󵄨 ≤(1−𝑐) 󵄩𝑢 −𝑥 󵄩 +𝑐 󵄩𝑢 −𝑇𝑢 󵄩 󵄨 𝑛 󵄨 =0. 𝑛 󵄩 𝑛 𝑛󵄩 𝑛 󵄩 𝑛 𝑛󵄩 𝑛→∞lim 󵄨 󵄨 (29) 󵄨 𝑠𝑛 −0󵄨 󵄩 󵄩 +𝑐𝑛 󵄩𝑇𝑢𝑛 −𝑇𝑥𝑛󵄩 Thus, from Definition,weconcludethattheiteration 1 󵄩 󵄩 󵄩 󵄩 method (11) is faster than the iteration method (7). ≤(1−𝑐𝑛 (1−𝛿)) 󵄩𝑢𝑛 −𝑥𝑛󵄩 +𝑐𝑛 󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩 󵄩 󵄩 Theorem 5. Let 𝐶 be a nonempty closed convex subset of an +𝑐𝑛𝜑(󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩). arbitrary Banach space 𝐸 and 𝑇:𝐶a →𝐶 mapping satisfying (32) 6 Abstract and Applied Analysis

Substituting (32)in(31), we get Also, from triangle inequality we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑥 −𝑥∗󵄩 ≤ 󵄩𝑢 −𝑥 󵄩 + 󵄩𝑥∗ −𝑢 󵄩 󵄩𝑢𝑛 −𝑦𝑛󵄩 ≤[(1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿)(1−𝑐𝑛 (1−𝛿)) +𝑏𝑛𝛿] 󵄩 𝑛 󵄩 󵄩 𝑛 𝑛󵄩 󵄩 𝑛󵄩 (37) 󵄩 󵄩 ∗ × 󵄩𝑢 −𝑥 󵄩 and this leads to 𝑥𝑛 →𝑥 as 𝑛→∞. 󵄩 𝑛 𝑛󵄩 ∗ (ii)⇒(i):Now, suppose that 𝑥𝑛 →𝑥 as 𝑛→∞.We ∗ +[(1 − 𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿)𝑛 𝑐 +(𝑎𝑛 +𝑏𝑛)] will show that 𝑢𝑛 →𝑥 as 𝑛→∞.Using(4), (11), and (13), 󵄩 󵄩 the following estimates can be obtained: × 󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩 󵄩 󵄩 󵄩 󵄩𝑥𝑛+1 −𝑢𝑛+1󵄩 = 󵄩(1 − 𝛼𝑛 −𝛽𝑛)𝑦𝑛 +𝛼𝑛𝑇𝑦𝑛 +𝛽𝑛𝑇𝑧𝑛 +[(1−𝑎 −𝑏 +𝑎 𝛿) 𝑐 +(𝑎 +𝑏)] 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 󵄩 −(1−𝛼 −𝛽 )𝑢 −𝛼 𝑇𝑢 −𝛽 𝑇𝑢 󵄩 󵄩 󵄩 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛󵄩 ×𝜑(󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩). 󵄩 󵄩 󵄩 󵄩 (33) ≤(1−𝛼𝑛 −𝛽𝑛) 󵄩𝑦𝑛 −𝑢𝑛󵄩 +𝛼𝑛 󵄩𝑇𝑦𝑛 −𝑇𝑢𝑛󵄩 󵄩 󵄩 +𝛽 󵄩𝑇𝑧 −𝑇𝑢 󵄩 By combining (30), (32), and (33) and using the assumption 𝑛 󵄩 𝑛 𝑛󵄩 𝛼𝑛 +𝛽𝑛 ≥𝐴,wehave 󵄩 󵄩 ≤(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) 󵄩𝑦𝑛 −𝑢𝑛󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑢𝑛+1 −𝑥𝑛+1󵄩 ≤ [1−𝐴(1−𝛿)] 󵄩𝑢𝑛 −𝑥𝑛󵄩 󵄩 󵄩 +𝛼𝑛𝜑(󵄩𝑦𝑛 −𝑇𝑦𝑛󵄩)

+[(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) 󵄩 󵄩 󵄩 󵄩 +𝛽𝑛𝛿 󵄩𝑧𝑛 −𝑢𝑛󵄩 +𝛽𝑛𝜑(󵄩𝑧𝑛 −𝑇𝑧𝑛󵄩), (38) ×[(1 − 𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿)𝑛 𝑐 +(𝑎𝑛 +𝑏𝑛)] 󵄩 󵄩 󵄩 󵄩 󵄩𝑦𝑛 −𝑢𝑛󵄩 ≤(1−𝑎𝑛 −𝑏𝑛) 󵄩𝑧𝑛 −𝑢𝑛󵄩 +𝛽𝑛𝛿𝑐𝑛] 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 +𝑎𝑛 󵄩𝑇𝑧𝑛 −𝑧𝑛󵄩 +𝑎𝑛 󵄩𝑧𝑛 −𝑢𝑛󵄩 × 󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩 󵄩 󵄩 󵄩 󵄩 +𝑏𝑛 󵄩𝑇𝑥𝑛 −𝑥𝑛󵄩 +𝑏𝑛 󵄩𝑥𝑛 −𝑢𝑛󵄩 (39) + [(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 =(1−𝑏𝑛) 󵄩𝑧𝑛 −𝑢𝑛󵄩 +𝑏𝑛 󵄩𝑥𝑛 −𝑢𝑛󵄩 ×[(1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿)𝑛 𝑐 +(𝑎𝑛 +𝑏𝑛)] 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 +𝑎𝑛 󵄩𝑇𝑧𝑛 −𝑧𝑛󵄩 +𝑏𝑛 󵄩𝑇𝑥𝑛 −𝑥𝑛󵄩 , +𝛽𝑛𝛿𝑐𝑛 +(𝛼𝑛 +𝛽𝑛)] 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑧𝑛 −𝑢𝑛󵄩 = 󵄩(1 − 𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛 −𝑢𝑛󵄩 ×𝜑(󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩). (34) 󵄩 󵄩 󵄩 󵄩 ≤(1−𝑐𝑛) 󵄩𝑥𝑛 −𝑢𝑛󵄩 +𝑐𝑛 󵄩𝑇𝑥𝑛 −𝑥𝑛󵄩 (40) 󵄩 󵄩 Denote that +𝑐𝑛 󵄩𝑥𝑛 −𝑢𝑛󵄩 󵄩 󵄩 𝑎 = 󵄩𝑢 −𝑥 󵄩 , 󵄩 󵄩 󵄩 󵄩 𝑛 󵄩 𝑛 𝑛󵄩 = 󵄩𝑥𝑛 −𝑢𝑛󵄩 +𝑐𝑛 󵄩𝑇𝑥𝑛 −𝑥𝑛󵄩 . 𝜌=[1−𝐴(1−𝛿)] ∈ (0, 1) , By substituting (40)in(39), we obtain 󵄩 󵄩 󵄩 󵄩 𝜉𝑛 =[(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) 󵄩 󵄩 󵄩 󵄩 󵄩𝑦𝑛 −𝑢𝑛󵄩 ≤ 󵄩𝑥𝑛 −𝑢𝑛󵄩 󵄩 󵄩 ×[(1 − 𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿)𝑛 𝑐 +(𝑎𝑛 +𝑏𝑛)] 󵄩 󵄩 +[(1−𝑏𝑛)𝑐𝑛 +𝑏𝑛] 󵄩𝑇𝑥𝑛 −𝑥𝑛󵄩 (41) 󵄩 󵄩 +𝛽𝑛𝛿𝑐𝑛] 󵄩 󵄩 +𝑎𝑛 󵄩𝑇𝑧𝑛 −𝑧𝑛󵄩 . 󵄩 󵄩 (35) × 󵄩𝑢 −𝑇𝑢 󵄩 󵄩 𝑛 𝑛󵄩 Again by substituting (40)and(41)in(38)andusingthe assumption 𝛼𝑛 +𝛽𝑛 ≥𝐴,wehave +[(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) 󵄩 󵄩 󵄩𝑥𝑛+1 −𝑢𝑛+1󵄩 ×[(1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿)𝑛 𝑐 +(𝑎𝑛 +𝑏𝑛)] 󵄩 󵄩 ≤ (1−𝐴(1−𝛿)) 󵄩𝑥𝑛 −𝑢𝑛󵄩 +𝛽𝑛𝛿𝑐𝑛 +(𝛼𝑛 +𝛽𝑛)] 󵄩 󵄩 +[(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) [(1𝑛 −𝑏 )𝑐𝑛 +𝑏𝑛]+𝛽𝑛𝑐𝑛𝛿] ×𝜑(󵄩𝑢𝑛 −𝑇𝑢𝑛󵄩). 󵄩 󵄩 × 󵄩𝑇𝑥 −𝑥 󵄩 ∗ ∗ ∗ 󵄩 𝑛 𝑛󵄩 Since lim𝑛→∞‖𝑢𝑛−𝑥 ‖=0and 𝑇𝑥 =𝑥 ∈𝐹𝑇 =⌀̸ ,itfollows ‖𝑢 −𝑇𝑢 ‖=0 󵄩 󵄩 from (13)thatlim𝑛→∞ 𝑛 𝑛 .HencebyLemma 2 we +(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿)𝑛 𝑎 󵄩𝑇𝑧𝑛 −𝑧𝑛󵄩 see that 󵄩 󵄩 󵄩 󵄩 +𝛼𝑛𝜑(󵄩𝑦𝑛 −𝑇𝑦𝑛󵄩)+𝛽𝑛𝜑(󵄩𝑧𝑛 −𝑇𝑧𝑛󵄩). 󵄩 󵄩 󵄩𝑢𝑛 −𝑥𝑛󵄩 󳨀→ 0 as 𝑛󳨀→∞. (36) (42) Abstract and Applied Analysis 7

Now define Corollary 6. Let 𝐶 beanonemptyclosedconvexsubsetofan 󵄩 󵄩 arbitrary Banach space 𝐸 and 𝑇:𝐶a →𝐶 mapping satisfying 𝑎𝑛 = 󵄩𝑢𝑛 −𝑥𝑛󵄩 , (13) with 𝐹𝑇 =⌀̸ . If the initial point is the same for all iterations 𝜌=[1−𝐴(1−𝛿)] ∈ (0, 1) , and 𝛼𝑛 +𝛽𝑛 ≥𝐴>0,forall𝑛∈N, then the following expressions are equivalent. 𝜉𝑛 =[(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿) [(1𝑛 −𝑏 )𝑐𝑛 +𝑏𝑛]+𝛽𝑛𝑐𝑛𝛿] 󵄩 󵄩 (43) × 󵄩𝑇𝑥𝑛 −𝑥𝑛󵄩 (1) The Picard iteration (1) converges to the fixed point 𝑥∗ 𝑇 󵄩 󵄩 of . +(1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿)𝑛 𝑎 󵄩𝑇𝑧𝑛 −𝑧𝑛󵄩 󵄩 󵄩 󵄩 󵄩 +𝛼𝑛𝜑(󵄩𝑦𝑛 −𝑇𝑦𝑛󵄩)+𝛽𝑛𝜑(󵄩𝑧𝑛 −𝑇𝑧𝑛󵄩). (2) The Krasnoselskij iteration [40]convergestothefixed ∗ ∗ ∗ ∗ point 𝑥 of 𝑇. Since 𝑥𝑛 →𝑥 as 𝑛→∞and 𝑇𝑥 =𝑥 ∈𝐹𝑇,itfollows from (13)that 󵄩 󵄩 (3) The Mann iteration (4) converges to the fixed point 󵄩𝑦 −𝑇𝑦 󵄩 ∗ 󵄩 𝑛 𝑛󵄩 𝑥 of 𝑇. 󵄩 ∗󵄩 󵄩 ∗ 󵄩 ≤ 󵄩𝑦𝑛 −𝑥 󵄩 + 󵄩𝑥 −𝑇𝑦𝑛󵄩 (4) The Ishikawa iteration (5) converges to the fixed point 󵄩 ∗󵄩 󵄩 ∗ 󵄩 󵄩 ∗ ∗󵄩 ∗ ≤ 󵄩𝑦𝑛 −𝑥 󵄩 +𝛿󵄩𝑥 −𝑦𝑛󵄩 +𝜑(󵄩𝑥 −𝑇𝑥 󵄩) 𝑥 of 𝑇. 󵄩 ∗󵄩 ≤ (1+𝛿) (1 − 𝑎𝑛 −𝑏𝑛) 󵄩𝑧𝑛 −𝑥 󵄩 (5) The Noor iteration (6) converges to the fixed point 󵄩 ∗󵄩 ∗ + (1+𝛿) 𝑎𝑛 󵄩𝑇𝑧𝑛 −𝑥 󵄩 𝑥 of 𝑇. 󵄩 󵄩 󵄩 󵄩 + (1+𝛿) 𝑏 󵄩𝑇𝑥 −𝑥∗󵄩 +𝜑(󵄩𝑥∗ −𝑇𝑥∗󵄩) 𝑛 󵄩 𝑛 󵄩 󵄩 󵄩 ∗ (6) The S-iteration (8) converges to the fixed point 𝑥 of 𝑇. 󵄩 ∗󵄩 ≤ (1+𝛿) (1 − 𝑎𝑛 −𝑏𝑛) 󵄩(1 − 𝑐𝑛)𝑥𝑛 +𝑐𝑛𝑇𝑥𝑛 −𝑥 󵄩 󵄩 󵄩 󵄩 󵄩 + (1+𝛿) 𝑎 𝛿 󵄩𝑧 −𝑥∗󵄩 + (1+𝛿) 𝑎 𝜑(󵄩𝑥∗ −𝑇𝑥∗󵄩) (7) The two-step Mann iteration (9) converges to the fixed 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 󵄩 ∗ point 𝑥 of 𝑇. 󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 + (1+𝛿) 𝑏𝑛𝛿 󵄩𝑥𝑛 −𝑥 󵄩 + (1+𝛿) 𝑏𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩) 󵄩 ∗ ∗󵄩 (8) The SP iteration (10) converges to the fixed point +𝜑(󵄩𝑥 −𝑇𝑥 󵄩) ∗ 𝑥 of 𝑇. ≤[(1+𝛿) (1 − 𝑎𝑛 −𝑏𝑛)(1−𝑐𝑛) (9) The multistep iteration [14] converges to the fixed point + (1+𝛿) (1 − 𝑎 −𝑏)𝑐 𝛿+(1+𝛿) 𝑎 𝛿(1−𝑐 ) ∗ 𝑛 𝑛 𝑛 𝑛 𝑛 𝑥 of 𝑇. 2 + (1+𝛿) 𝑎𝑛𝛿 𝑐𝑛 + (1+𝛿) 𝑏𝑛𝛿] (10) The new multistep iteration [41]convergestothefixed 󵄩 ∗󵄩 ∗ × 󵄩𝑥𝑛 −𝑥 󵄩 󳨀→ 0 as 𝑛󳨀→∞. point 𝑥 of 𝑇. (44) 𝜑 (11) The new iteration (11) converges to the fixed point Since the function is continuous, we get ∗ 󵄩 󵄩 󵄩 󵄩 𝑥 of 𝑇. 󵄩𝑦 −𝑇𝑦 󵄩 = 󵄩𝑥 −𝑇𝑥 󵄩 𝑛→∞lim 󵄩 𝑛 𝑛󵄩 𝑛→∞lim 󵄩 𝑛 𝑛󵄩 Theorem 7. (𝐸, ‖ ⋅ ‖) 𝑇: 󵄩 󵄩 Let be an arbitrary Banach space, = 󵄩𝑧 −𝑇𝑧 󵄩 𝐸→𝐸 𝐸 𝐹 =⌀̸ 𝑥∗ 𝑛→∞lim 󵄩 𝑛 𝑛󵄩 a self-map of satisfying (13) with 𝑇 ,and the (45) unique fixed point of 𝑇.For𝑥0 ∈𝐸,let(𝑥𝑛) be the new iteration 󵄩 󵄩 = lim 𝜑(󵄩𝑦𝑛 −𝑇𝑦𝑛󵄩) method defined by (11) with real sequences (𝑎𝑛), (𝑏𝑛), (𝑐𝑛), (𝛼𝑛), 𝑛→∞ (𝛽 )⊂[0,1] 0<𝐴≤𝛼 +𝛽 𝑛∈N 󵄩 󵄩 𝑛 satisfying 𝑛 𝑛,forall .Then = 𝜑(󵄩𝑧 −𝑇𝑧 󵄩)=0. 𝑇 𝑛→∞lim 󵄩 𝑛 𝑛󵄩 the new iteration method (11) is -stable. (𝑦 ) 𝐸 Thus Lemma 2 and (42)give‖𝑥𝑛 −𝑢𝑛‖→0as 𝑛→∞. Proof. Let 𝑛 be an arbitrary sequence in .Define Also, from triangle inequality we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑢 −𝑥∗󵄩 ≤ 󵄩𝑢 −𝑥 󵄩 + 󵄩𝑥 −𝑥∗󵄩 󳨀→ 0 󵄩 󵄩 󵄩 𝑛 󵄩 󵄩 𝑛 𝑛󵄩 󵄩 𝑛 󵄩 (46) 𝜀𝑛 = 󵄩𝑦𝑛+1 −(1−𝛼𝑛 −𝛽𝑛)𝑢𝑛 −𝛼𝑛𝑇𝑢𝑛 −𝛽𝑛𝑇V𝑛󵄩 , (47) ∗ and this yields 𝑢𝑛 →𝑥 as 𝑛→∞.

With regard to ([5], Corollary 2) and Theorem 5,wecan for all 𝑛∈N,where𝑢𝑛 =(1−𝑎𝑛 −𝑏𝑛)V𝑛 +𝑎𝑛𝑇V𝑛 +𝑏𝑛𝑇𝑦𝑛 and ∗ without hesitation give the following corollary. V𝑛 =(1−𝑐𝑛)𝑦𝑛 +𝑐𝑛𝑇𝑦𝑛.Supposethat𝑥𝑛 →𝑥 as 𝑛→∞ 8 Abstract and Applied Analysis

∗ and lim𝑛→∞ 𝜖𝑛 =0.Then,weprovethatlim𝑛→∞ 𝑦𝑛 =𝑥. Thus an application of Lemma 2 to (51) yields lim𝑛→∞ 𝑦𝑛 = ∗ From condition (13), we have the following estimates: 𝑥 . ∗ Conversely, assume that lim𝑛→∞ 𝑦𝑛 =𝑥.Wenow 󵄩 ∗󵄩 󵄩 󵄩 󵄩𝑦𝑛+1 −𝑥 󵄩 ≤ 󵄩𝑦𝑛+1 −(1−𝛼𝑛 −𝛽𝑛)𝑢𝑛 −𝛼𝑛𝑇𝑢𝑛 −𝛽𝑛𝑇V𝑛󵄩 show that lim𝑛→∞ 𝜀𝑛 =0. From condition (13)andtriangle 󵄩 ∗󵄩 inequality we have + 󵄩(1 − 𝛼𝑛 −𝛽𝑛)𝑢𝑛 +𝛼𝑛𝑇𝑢𝑛 +𝛽𝑛𝑇V𝑛 −𝑥 󵄩 󵄩 󵄩 𝜀 = 󵄩𝑦 −(1−𝛼 −𝛽 )𝑢 −𝛼 𝑇𝑢 −𝛽 𝑇V 󵄩 󵄩 ∗󵄩 𝑛 󵄩 𝑛+1 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛󵄩 ≤𝜀𝑛 + 󵄩(1 − 𝛼𝑛 −𝛽𝑛)𝑢𝑛 +𝛼𝑛𝑇𝑢𝑛 +𝛽𝑛𝑇V𝑛 −𝑥 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩𝑦 −𝑥∗󵄩+󵄩𝑥∗ −(1 − 𝛼 −𝛽 )𝑢 −𝛼 𝑇𝑢 −𝛽 𝑇V 󵄩 󵄩 ∗󵄩 󵄩 𝑛+1 󵄩 󵄩 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛󵄩 ≤𝜀𝑛 +(1−𝛼𝑛 −𝛽𝑛) 󵄩𝑢𝑛 −𝑥 󵄩 󵄩 ∗󵄩 󵄩 ∗󵄩 󵄩 ∗󵄩 󵄩 ∗󵄩 ≤ 󵄩𝑦𝑛+1 −𝑥 󵄩 +(1−𝛼𝑛 −𝛽𝑛) 󵄩𝑢𝑛 −𝑥 󵄩 +𝛼𝑛𝛿 󵄩𝑢𝑛 −𝑥 󵄩 +𝛽𝑛𝛿 󵄩V𝑛 −𝑥 󵄩 󵄩 󵄩 󵄩 󵄩 +𝛼 󵄩𝑇𝑢 −𝑥∗󵄩 +𝛽 󵄩𝑇V −𝑥∗󵄩 󵄩 ∗ ∗󵄩 󵄩 ∗ ∗󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 +𝛼𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩)+𝛽𝑛𝜑(󵄩𝑥 −𝑇𝑥 󵄩) 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩𝑦 −𝑥∗󵄩 +[1−𝛼 −𝛽 +𝛼 𝛿] 󵄩𝑢 −𝑥∗󵄩 󵄩 ∗󵄩 󵄩 𝑛+1 󵄩 𝑛 𝑛 𝑛 󵄩 𝑛 󵄩 =𝜀𝑛 +[1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿] 󵄩𝑢𝑛 −𝑥 󵄩 󵄩 󵄩 +𝛽 𝛿 󵄩V −𝑥∗󵄩 󵄩 ∗󵄩 𝑛 󵄩 𝑛 󵄩 +𝛽𝑛𝛿 󵄩V𝑛 −𝑥 󵄩 , 󵄩 ∗󵄩 󵄩 ∗󵄩 (48) ≤ 󵄩𝑦𝑛+1 −𝑥 󵄩 +(1−(𝛼𝑛 +𝛽𝑛) (1−𝛿)) 󵄩𝑦𝑛 −𝑥 󵄩 . 󵄩 ∗󵄩 󵄩 ∗󵄩 (52) 󵄩𝑢𝑛 −𝑥 󵄩 ≤ 󵄩(1 − 𝑎𝑛 −𝑏𝑛) V𝑛 +𝑎𝑛𝑇V𝑛 +𝑏𝑛𝑇𝑦𝑛 −𝑥 󵄩

󵄩 ∗󵄩 󵄩 ∗󵄩 Since 𝛿∈[0,1)and 𝛼𝑛 +𝛽𝑛 ∈ [0, 1],forall𝑛∈N, ≤(1−𝑎𝑛 −𝑏𝑛) 󵄩V𝑛 −𝑥 󵄩 +𝑎𝑛 󵄩𝑇V𝑛 −𝑥 󵄩 󵄩 ∗󵄩 0<1−(𝛼 +𝛽 ) (1−𝛿) <1. +𝑏𝑛 󵄩𝑇𝑦𝑛 −𝑥 󵄩 𝑛 𝑛 (53) 󵄩 ∗󵄩 ≤[1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿] 󵄩V𝑛 −𝑥 󵄩 By taking the limit as 𝑛→∞of both sides of (52)and 󵄩 󵄩 ∗ 󵄩 󵄩 󵄩 󵄩 using the assumption lim𝑛→∞‖𝑦𝑛 −𝑥‖=0,wehave +𝑎 𝜑(󵄩𝑥∗ −𝑇𝑥∗󵄩)+𝑏 󵄩𝑇𝑦 −𝑥∗󵄩 𝑛 󵄩 󵄩 𝑛 󵄩 𝑛 󵄩 lim𝑛→∞ 𝜀𝑛 =0. 󵄩 ∗󵄩 ≤[1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿] 󵄩V𝑛 −𝑥 󵄩 Conflict of Interests 󵄩 ∗󵄩 󵄩 ∗ ∗󵄩 +𝑏𝑛𝛿 󵄩𝑦𝑛 −𝑥 󵄩 +(𝑏𝑛 +𝑎𝑛)𝜑(󵄩𝑥 −𝑇𝑥 󵄩) The authors declare that there is no conflict of interests 󵄩 ∗󵄩 󵄩 ∗󵄩 =[1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿] 󵄩V𝑛 −𝑥 󵄩 +𝑏𝑛𝛿 󵄩𝑦𝑛 −𝑥 󵄩 , regarding the publication of this paper. (49) 󵄩 ∗󵄩 󵄩 ∗󵄩 References 󵄩V𝑛 −𝑥 󵄩 ≤ 󵄩(1 − 𝑐𝑛)𝑦𝑛 +𝑐𝑛𝑇𝑦𝑛 −𝑥 󵄩 󵄩 󵄩 󵄩 󵄩 ≤(1−𝑐) 󵄩𝑦 −𝑥∗󵄩 +𝑐 󵄩𝑇𝑦 −𝑥∗󵄩 [1] R. P. Agarwal, O. O’Regan, and D. R. Sahu, “Iterative con- 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 󵄩 (50) structionoffixedpointsofnearlyasymptoticallynonexpansive 󵄩 󵄩 ≤[1−𝑐 (1−𝛿)] 󵄩𝑦 −𝑥∗󵄩 . mappings,” Journal of Nonlinear and Convex Analysis,vol.8,no. 𝑛 󵄩 𝑛 󵄩 1,pp.6179–6189,2007. Substituting (49)and(50)in(48)andusingtheassumption [2] V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,” Acta Mathematica 𝛼𝑛 +𝛽𝑛 ≥𝐴>0,forall𝑛∈N,weobtain Universitatis Comenianae,vol.73,no.1,pp.119–126,2004. 󵄩 ∗󵄩 [3] R. Chugh and V. Kumar, “Strong convergence of SP iterative 󵄩𝑦𝑛+1 −𝑥 󵄩 scheme for Quasi-contractive operators,” International Journal of Computer Applications, vol. 31, no. 5, pp. 21–27, 2011. ≤𝜀𝑛 +[[1−𝛼𝑛 −𝛽𝑛 +𝛼𝑛𝛿] [4] R. Chugh, V. Kumar, and S. Kumar, “Strong convergence of ×[[1−𝑎𝑛 −𝑏𝑛 +𝑎𝑛𝛿] [1𝑛 −𝑐 (1−𝛿)]+𝑏𝑛𝛿] a new three step iterative scheme in Banach space,” American Journal of Computational Mathematics,vol.2,pp.345–357,2012. +𝛽𝑛𝛿[1−𝑐𝑛 (1−𝛿)]] [5] F. Gursoy,¨ V. Karakaya, and B. E. Rhoades, “The equivalence 󵄩 ∗󵄩 among new multistep iteration, S-iteration and some other × 󵄩𝑦𝑛 −𝑥 󵄩 iterative schemes,” http://arxiv.org/pdf/1211.5701.pdf. ≤𝜀 +([1 − 𝛼 −𝛽 +𝛼 𝛿] [1−(𝑎 +𝑏) (1−𝛿)]+𝛽 𝛿) [6] S. Ishikawa, “Fixed points by a new iteration method,” Proceed- 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 ings of the American Mathematical Society,vol.44,pp.147–150, 󵄩 ∗󵄩 1974. × 󵄩𝑦𝑛 −𝑥 󵄩 [7] W.R. Mann, “Mean value methods in iterations,” Proceedings of 󵄩 ∗󵄩 ≤𝜀𝑛 +(1−(𝛼𝑛 +𝛽𝑛) (1−𝛿)) 󵄩𝑦𝑛 −𝑥 󵄩 the American Mathematical Society,vol.4,pp.506–510,1953. 󵄩 ∗󵄩 [8] M. A. Noor, “New approximation schemes for general vari- ≤𝜀𝑛 + (1−𝐴(1−𝛿)) 󵄩𝑦𝑛 −𝑥 󵄩 . ational inequalities,” Journal of Mathematical Analysis and (51) Applications,vol.251,no.1,pp.217–229,2000. Abstract and Applied Analysis 9

[9] M. Olantiwo, “Some stability and strong convergence results for [26]A.M.HarderandT.L.Hicks,“Stabilityresultsforfixedpoint the Jungck-Ishikawa iteration process,” Creative Mathematics iteration procedures,” Mathematica Japonica,vol.33,pp.693– and Informatics,vol.17,pp.33–42,2008. 706, 1988. [10] W. Phuengrattana and S. Suantai, “On the rate of convergence [27]A.M.HarderandT.L.Hicks,“Astableiterationprocedure of Mann, Ishikawa, Noor and SP-iterations for continuous for nonexpansive mappings,” Mathematica Japonica,vol.33,pp. functionsonanarbitraryinterval,”Journal of Computational 687–692, 1988. and Applied Mathematics,vol.235,no.9,pp.3006–3014,2011. [28] B. E. Rhoades, “Fixed point theorems and stability results for [11] S. Suantai, “Weak and strong convergence criteria of Noor fixed point iteration procedures,” Indian Journal of Pure and iterations for asymptotically nonexpansive mappings,” Journal Applied Mathematics,vol.21,no.1,pp.1–9,1990. of Mathematical Analysis and Applications,vol.311,no.2,pp. [29] B. E. Rhoades, “Fixed point theorems and stability results for 506–517, 2005. fixed point iteration procedures. II,” Indian Journal of Pure and [12] S. Thianwan, “Common fixed points of new iterations for two Applied Mathematics,vol.24,no.11,pp.691–703,1993. asymptotically nonexpansive nonself-mappings in a Banach [30] M. O. Osilike, “Stability results for fixed point iteration proce- space,” Journal of Computational and Applied Mathematics,vol. dures,” Journal of Nigerian Mathematical Society, vol. 26, no. 10, 224, no. 2, pp. 688–695, 2009. pp. 937–945, 1995. [13] B. E. Rhoades and S. M. Soltuz, “On the equivalence of [31] M. O. Osilike and A. Udomene, “Short proofs of stability results Mann and Ishikawa iteration methods,” International Journal for fixed point iteration procedures for a class of contractive- of Mathematics and Mathematical Sciences,vol.2003,no.7,pp. type mappings,” Indian Journal of Pure and Applied Mathemat- 451–459, 2003. ics,vol.30,no.12,pp.1229–1234,1999. [14] B. E. Rhoades and S. M. Soltuz, “The equivalence between [32] M. Ostrowski, “The round-off stability of iterations,” Zeitschrift Mann-Ishikawa iterations and multistep iteration,” Nonlinear fur¨ Angewandte Mathematik und Mechanik,vol.47,no.2,pp. Analysis,Theory,MethodsandApplications,vol.58,no.1-2,pp. 77–81, 1967. 219–228, 2004. [33] V. Berinde, Iterative Approximation of Fixed Points,Springer, [15] V. Berinde, “Picard iteration converges faster than Mann iter- Berlin, Germany, 2007. ation for a class of quasi-contractive operators,” Fixed Point Theory and Applications, vol. 2004, no. 2, pp. 97–105, 2004. [34] S. L. Singh and B. Prasad, “Some coincidence theorems and [16] N. Hussain, A. Rafiq, B. Damjanovic,´ and R. Lazovic,´ “On rate stability of iterative procedures,” Computers and Mathematics of convergence of various iterative schemes,” Fixed Point Theory with Applications,vol.55,no.11,pp.2512–2520,2008. and Applications, vol. 2011, p. 45, 2011. [35] E. Picard, “Memoire sur la theorie des equations aux derivees [17]Y.QingandB.E.Rhoades,“Commentsontherateofcon- partielles et la methode des approximations successives,” Jour- vergence between mann and ishikawa iterations applied to nal de Mathematiques´ Pures et Appliquees´ ,vol.6,pp.145–210, zamfirescu operators,” Fixed Point Theory and Applications,vol. 1890. 2008, Article ID 387504, 3 pages, 2008. [36] C. O. Imoru and M. O. Olatinwo, “On the stability of Picard and [18] B. E. Rhoades and S. M. Soltuz, “The equivalence of mann Mann iteration processes,” Carpathian Journal of Mathematics, iteration and ishikawa iteration for non-lipschitzian operators,” vol.19,no.2,pp.155–160,2003. International Journal of Mathematics and Mathematical Sci- [37] T. Zamfirescu, “Fix point theorems in metric spaces,” Archiv der ences, vol. 2003, no. 42, pp. 2645–2651, 2003. Mathematik, vol. 23, no. 1, pp. 292–298, 1972. [19] B. E. Rhoades and S. M. Soltuz, “The equivalence between the [38] R. Kannan, “Some results on fixed points,” Bulletin of Calcutta convergences of Ishikawa and Mann iterations for an asymptot- Mathematical Society,vol.10,pp.71–76,1968. ically pseudocontractive map,” JournalofMathematicalAnalysis [39] S. K. Chatterjea, “Fixed point theorems,” Comptes Rendus de and Applications,vol.283,no.2,pp.681–688,2003. l’Academie Bulgare des Sciences, vol. 25, pp. 727–730, 1972. [20] B. E. Rhoades and S. M. Soltuz, “The equivalence between the [40] H. Schaefer, “Uber die methode sukzessiver approximationen,” Mann and Ishikawa iterations dealing with generalized contrac- Jahresbericht-Deutsche Mathematiker-Vereinigung,vol.59,pp. tions,” International Journal of Mathematics and Mathematical 131–140, 1957. Sciences,vol.2006,ArticleID54653,5pages,2006. [41] F. Gursoy,¨ V. Karakaya, and B. E. Rhoades, “Data dependence [21] B. E. Rhoades and S. M. Soltuz, “The equivalence between the results of new multistep and S-iterative schemes for contractive- convergences of Ishikawa and Mann iterations for an asymp- like operators,” Fixed Point Theory and Applications,vol.2013,p. totically nonexpansive in the intermediate sense and strongly 76, 2013. successively pseudocontractive maps,” Journal of Mathematical Analysis and Applications,vol.289,no.1,pp.266–278,2004. [22] S. M. Soltuz, “The equivalence of Picard, Mann, and Ishikawa iterations dealing with quasi- contractive operators,” Mathemat- ical Communications,vol.10,pp.81–89,2005. [23] A. Rafiq, “On the equivalence of Mann and Ishikawa iteration methodes with errors,” Mathematical Communications,vol.11, pp.143–152,2006. [24] S. M. Soltuz, “The equivalence between Krasnoselkij, Mann, Ishikawa, Noor and multistep iteration,” Mathematical Commu- nications,vol.12,pp.53–61,2007. [25] X. Zhiqun, “Remarks of equivalence among Picard, Mann, and Ishikawa Iterations in normed spaces,” Fixed Point Theory and Applications,vol.2007,ArticleID61434,5pages,2007. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 540108, 10 pages http://dx.doi.org/10.1155/2013/540108

Research Article Strong Convergence of Iterative Algorithm for a New System of Generalized 𝐻(⋅, ⋅)-Cocoercive −𝜂 Operator Inclusions in Banach Spaces

Saud M. Alsulami,1 Eskandar Naraghirad,2 and Nawab Hussain1

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Yasouj University, Yasouj 75918, Iran

Correspondence should be addressed to Saud M. Alsulami; [email protected]

Received 10 September 2013; Revised 6 November 2013; Accepted 20 November 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 Saud M. Alsulami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce and study a new system of generalized 𝐻(⋅, ⋅)−𝜂-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with 𝐻(⋅, ⋅)−𝜂-cocoerciveoperators,wesuggestandanalyzeanewgeneralizedalgorithmofnonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

1. Introduction Veryrecently,Ahmadetal.[23]introducedanew𝐻(⋅, ⋅)− 𝜂-cocoercive operator and its resolvent operator in the setting The resolvent operator technique is a powerful tool to of Banach spaces. The authors proposed concrete examples study the approximation solvability of nonlinear variational in support of 𝐻(⋅, ⋅) −𝜂-cocoercive operators and they inequalities and variational inclusions, which have been also proved the Lipschitz continuity of resolvent operator applied widely to optimization and control, mechanics and associated with 𝐻(⋅, ⋅) −𝜂-cocoercive operator. Motivated physics, economics and transportation equilibrium, and and inspired by the research works mentioned above, in this engineering sciences, see, for example, [1–4] and the refer- paper, we introduce and study a new system of 𝐻(⋅, ⋅) − ences therein. 𝜂-cocoercive mapping inclusions in Banach spaces. Using In a series of papers [5–8], the authors investigated the resolvent operator associated with 𝐻(⋅, ⋅) −𝜂-cocoercive (𝐴, 𝜂)-accretive and 𝐻(⋅, ⋅)-accretive operators for solving mapping, we suggest and analyze a new general algorithm variational inclusions in Banach spaces. Convergence and and establish the existence and uniqueness of solutions for stabilityofiterativealgorithmsforthesystemsof(𝐴, 𝜂)- this system of 𝐻(⋅, ⋅) −𝜂-cocoercive mappings. accretive operators have been studied in [9, 10]. The notion of (𝐻, 𝜙)-monotone −𝜂 operators has been introduced and 2. Preliminaries investigated by the authors in [11]. Generalized mixed vari- ational inclusions involving (𝐻(⋅, ⋅), -monotone𝜂) operators Throughout this paper, we denote the set of positive integers 𝐻((⋅, ⋅), 𝜂) by N.Let𝑋 be a Banach space with the norm ‖⋅‖and the dual have been discussed in [12]. Some results on - ∗ ∗ ∗ space 𝑋 .Forany𝑥∈𝑋, we denote the value of 𝑥 ∈𝑋 at accretive operators and application for solving set-valued ∗ variational inclusions in Banach spaces have been proved in 𝑥 by ⟨𝑥, 𝑥 ⟩.When{𝑥𝑛} is a sequence in 𝑋,wedenotethe [7]. Some other related articles on the variational inclusion strong convergence of {𝑥𝑛} to 𝑥∈𝑋by 𝑥𝑛 →𝑥as 𝑛→∞. 𝑋 problems can be found in [13–22]. We denote by 2 the family of all nonempty subsets of 𝑋.Let 2 Abstract and Applied Analysis

𝐶𝐵(𝑋) be the family of all nonempty, closed, and bounded (v) Lipschitz continuous, if there exists a constant 𝜆𝐴 >0 subsets of 𝑋.TheHausdorff¨ metric on 𝐶𝐵(𝑋) [24] is defined such that by 󵄩 󵄩 󵄩 󵄩 󵄩𝐴𝑥 − 𝐴𝑦󵄩 ≤𝜆𝐴 󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦∈𝑋, (8) 𝐷 (𝐴,) 𝐵 = max {sup 𝑑 (𝑥,) 𝐵 , sup 𝑑 (𝐴, 𝑦)} , 𝑥∈𝐴 𝑦∈𝐵 (1) (vi) 𝛼-expansive, if there exists a constant 𝛼>0such that 𝐴, 𝐵 ∈ 𝐶𝐵 (𝑋) , 󵄩 󵄩 󵄩 󵄩 󵄩𝐴𝑥−𝐴𝑦󵄩 ≥𝛼󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦∈𝑋, (9) where 𝑑(𝑥, 𝐵) = inf𝑏∈𝐵‖𝑥−𝑏‖and 𝑑(𝐴, 𝑦) = inf𝑎∈𝐴‖𝑎 − 𝑦‖. (vii) 𝜂 is said to be Lipschitz continuous, if there exists a Definition 1 (see [25]). A continuous and strictly increasing constant 𝜏>0such that function 𝜙 : [0, +∞) → [0, ∞) such that 𝜙(0) = 0 and 󵄩 󵄩 󵄩 󵄩 lim𝑡→∞𝜙(𝑡) =∞ is called a gauge function. 󵄩𝜂(𝑥,𝑦)󵄩 ≤𝜏󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦∈𝑋. (10) Definition 2 (see [25]). Let 𝑋 be a Banach space. Given a ∗ Definition 5. Let 𝑋 be a Banach space. Let 𝐴, 𝐵 : 𝑋 →𝑋, 𝐻: 𝜙 𝐽 :𝑋 →𝑋 2 gauge function ,themapping 𝜙 corresponding 𝑋×𝑋, →𝑋 𝜂 : 𝑋×𝑋 →𝑋be four single-valued mappings ∗ to 𝜙 defined by 𝐽:𝑋𝑋 →2 󵄩 󵄩 and be the normalized duality mapping. Then, 𝐽 (𝑥) ={𝑥∗ ∈𝑋∗ :⟨𝑥,𝑥∗⟩=‖𝑥‖ 󵄩𝑥∗󵄩 , 𝜙 󵄩 󵄩 𝐻(𝐴, ⋅) 𝜂 𝐴 (2) (i) is said to be -cocoercive with respect to ,if 󵄩 ∗󵄩 𝜇>0 󵄩𝑥 󵄩 =𝜙(‖𝑥‖)}, ∀𝑥∈𝑋, there exists a constant such that is called the duality mapping with gauge function 𝜙. ⟨𝐻 (𝐴𝑥, 𝑢) − 𝐻 (𝐴𝑦, 𝑢) , 𝑗 (𝜂 (𝑥,𝑦))⟩ In particular, if 𝜙(𝑡),thedualitymap =𝑡 𝐽=𝐽𝜙 is called 󵄩 󵄩2 the normalized duality mapping. ≥𝜇󵄩𝐴𝑥−𝐴𝑦󵄩 , ∀𝑥,𝑦,𝑢∈𝑋, (11)

Lemma 3 (see [26]). Let 𝑋 be a real Banach space and 𝐽: 𝑗 (𝜂 (𝑥, 𝑦)) ∈ 𝐽 (𝜂 (𝑥,𝑦)), 𝑋∗ 𝑋→2 be the normalized duality mapping. Then, for any 𝑥, 𝑦 ∈𝑋, (ii) 𝐻(⋅, 𝐵) is said to be 𝜂-relaxed cocoercive with respect 𝐵 𝛾>0 󵄩 󵄩2 2 to , if there exists a constant such that 󵄩𝑥+𝑦󵄩 ≤ ‖𝑥‖ +2⟨𝑦, 𝑗 (𝑥+𝑦)⟩ , (3) ⟨𝐻 (𝑢, 𝐵𝑥) − 𝐻 (𝑢, 𝐵𝑦) , 𝑗 (𝜂 (𝑥,𝑦))⟩ for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦). 󵄩 󵄩2 ≥(−𝛾)󵄩𝐵𝑥 − 𝐵𝑦󵄩 , ∀𝑥,𝑦,𝑢∈𝑋, Definition 4. Let 𝑋 be a Banach space. Let 𝐴:𝑋and →𝑋 󵄩 󵄩 (12) 𝑋∗ 𝜂:𝑋×𝑋 →𝑋be two mappings and 𝐽:𝑋 →2 be the 𝑗 (𝜂 (𝑥, 𝑦)) ∈ 𝐽 (𝜂 (𝑥,𝑦)), normalized duality mapping. Then, 𝐴 is called

(i) 𝜂-cocoercive, if there exists a constant 𝜇1 >0such (iii) 𝐻(𝐴, ⋅) is said to be 𝑟1-Lipschitz continuous with that respect to 𝐴, if there exists a constant 𝑟1 >0such that 󵄩 󵄩2 ⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝜂 (𝑥, 𝑦))⟩≥𝜇 󵄩𝐴𝑥 − 𝐴𝑦󵄩 ,∀𝑥,𝑦∈𝑋, 󵄩 󵄩 󵄩 󵄩 1󵄩 󵄩 󵄩𝐻 (𝐴𝑥, 𝑢) −𝐻(𝐴𝑦,𝑢)󵄩 ≤𝑟 󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦,𝑢∈𝑋, (4) 󵄩 󵄩 1 󵄩 󵄩 (13) 𝑗(𝜂(𝑥,𝑦))∈𝐽(𝜂(𝑥,𝑦)), (iv) 𝐻(⋅, 𝐵) is said to be 𝑟2-Lipschitz continuous with (ii) 𝜂-accretive, if respect to 𝐵, if there exists a constant 𝑟2 >0such that ⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝜂 (𝑥, 𝑦))⟩ ≥ 0,∀𝑥,𝑦∈𝑋, 󵄩 󵄩 󵄩 󵄩 󵄩𝐻 (𝑢, 𝐵𝑥) −𝐻(𝑢, 𝐵𝑦)󵄩 ≤𝑟2 󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦,𝑢∈𝑋.(14) (5) 𝑗 (𝜂 (𝑥, 𝑦)) ∈ 𝐽 (𝜂 (𝑥,𝑦)), Definition 6. Let 𝑋 be a Banach space. A set-valued mapping 𝑋 (iii) 𝜂-strongly accretive, if there exists a constant 𝛽1 >0 𝑀:𝑋 →2 is said to be 𝜂-cocoercive, if there exists a such that constant 𝜇2 >0such that 󵄩 󵄩2 ⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝜂 (𝑥,𝑦))⟩≥𝛽 󵄩𝑥−𝑦󵄩 ,∀𝑥,𝑦∈𝑋, 2 1󵄩 󵄩 ⟨𝑢 − V, 𝑗 (𝜂 (𝑥, 𝑦))⟩ ≥𝜇 ‖𝑢−V‖ ,∀𝑥,𝑦∈𝑋, (6) 2 𝑗 (𝜂 (𝑥, 𝑦)) ∈ 𝐽 (𝜂 (𝑥,𝑦)), 𝑢∈𝑀(𝑥) , V ∈ 𝑀 (𝑦) , 𝑗 (𝜂 (𝑥, 𝑦)) ∈ 𝐽 (𝜂(𝑥,𝑦)).

(iv) 𝜂-relaxed cocoercive, if there exists a constant 𝛾1 >0 (15) such that Definition 7. Let 𝑋 be a Banach space. A mapping 𝑇:𝑋 → 󵄩 󵄩2 ⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝜂 (𝑥,1 𝑦))⟩≥(−𝛾 ) 󵄩𝐴𝑥−𝐴𝑦󵄩 ,∀𝑥,𝑦∈𝑋, 𝐶𝐵(𝑋) is said to be D-Lipschitz continuous, if there exists a constant 𝜆𝑇 >0such that 𝑗(𝜂(𝑥,𝑦))∈𝐽(𝜂(𝑥,𝑦)), 󵄩 󵄩 (7) D (𝑇𝑥, 𝑇𝑦) ≤𝜆𝑇 󵄩𝑥−𝑦󵄩 ,𝑥,𝑦∈𝑋. (16) Abstract and Applied Analysis 3

Definition 8. Let 𝑋 be a Banach space. Let 𝑇, 𝑄 : 𝑋→ Example 11. Let 𝑋 = 𝐶[0, 1],thespaceofallrealvalued 𝐶𝐵(𝑋) be the mappings. A mapping 𝑁:𝑋×𝑋→is 𝑋 continuous functions defined on closed interval [0, 1] with said to be the norm 󵄩 󵄩 󵄨 󵄨 󵄩𝑓󵄩 = max 󵄨𝑓 (𝑡)󵄨 . (i) Lipschitz continuous in the first argument with 𝑡∈[0,1] (23) respect to 𝑇, if there exists a constant 𝑡1 >0such that 𝐴, 𝐵 : 𝑋 →𝑋 󵄩 󵄩 󵄩 󵄩 Let be defined by 󵄩𝑁(𝑤1,⋅)−𝑁(𝑤2,⋅)󵄩 ≤𝑡1 󵄩𝑤1 −𝑤2󵄩 ,∀𝑢1,𝑢2 ∈𝑋, 2 2 𝐴(𝑓)=sin (𝑓) , 𝐵 (𝑔) = cos (𝑔) , ∀𝑓,𝑔 ∈𝑋. (24) 𝑤1 ∈𝑇(𝑢1), 𝑤2 ∈𝑇(𝑢2), (17) Let 𝐻(𝐴, 𝐵) : 𝑋 ×𝑋→ be defined by

(ii) Lipschitz continuous in the second argument with 𝐻 (𝐴 (𝑓) , 𝐵 (𝑔)) = 𝐴 (𝑓) + 𝐵 (𝑔) ,∀𝑓,𝑔∈𝑋. (25) 𝑄 𝑡 >0 respect to , if there exists a constant 2 such that 2 2 Suppose that 𝑀(𝑓) =𝑓 ,where𝑓 (𝑡) = 𝑓(𝑡)𝑓(𝑡) for all 𝑡∈ 󵄩 󵄩 󵄩 󵄩 [0, 1] 𝜆=1 󵄩𝑁(⋅,V1)−𝑁(⋅,V2)󵄩 ≤𝑡2 󵄩V1 − V2󵄩 ,∀𝑢1,𝑢2 ∈𝑋, .Then,for ,weconcludethat (18) 󵄩 󵄩 󵄩 󵄩 󵄩(𝐻 (𝐴,) 𝐵 +𝑀) (𝑓)󵄩 = 󵄩𝐴(𝑓)+𝐵(𝑓)+𝑓2󵄩 V1 ∈𝑄(𝑢1), V2 ∈𝑄(𝑢2), 󵄩 󵄩 󵄩 󵄩 󵄨 2 2 2 󵄨 = max 󵄨sin (𝑓 (𝑡))+cos (𝑓 (𝑡))+𝑓 (𝑡)󵄨 (iii) 𝜂-relaxed Lipschitz in the first argument with respect 𝑡∈[0,1] 󵄨 󵄨 (26) to 𝑇, if there exists a constant 𝜏1 >0such that =1+𝑓2 (𝑡) >0. ⟨𝑁 1(𝑤 ,⋅)−𝑁(𝑤2,⋅),𝑗(𝜂(𝑢1,𝑢2))⟩ Thisprovesthat0 ∉ (𝐻(𝐴, 𝐵)+𝑀)(𝑋) and 𝑀 is not 𝐻(𝐴, 𝐵)− 󵄩 󵄩2 ≤(−𝜏1) 󵄩𝑢1 −𝑢2󵄩 ,∀𝑢1,𝑢2 ∈𝑋, 𝜂-cocoercive with respect to 𝐴 and 𝐵. (19) 𝑤1 ∈𝑇(𝑢1), 𝑤2 ∈𝑇(𝑢2), Proposition 12 (see [23]). Let 𝐻(𝐴, 𝐵) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0and 𝜂-relaxed cocoercive with 𝑗(𝜂(𝑢 ,𝑢 )) ∈ 𝐽 (𝜂 (𝑢 ,𝑢 )) , 1 2 1 2 respect to 𝐵 with constant 𝛾>0, 𝐴 be 𝛼-expansive and 𝐵 be 𝑋 𝛽-Lipschitz continuous 𝜇>𝛾and 𝛼>𝛽.Let𝑀:𝑋 →2 be (iv) 𝜂-relaxed Lipschitz in the second argument with 𝐻(𝐴, 𝐵) −𝜂-cocoercive operator. Suppose that respect to 𝑄, if there exists a constant 𝜏2 >0such that ⟨𝑥 − 𝑦, 𝑗 (𝜂 (𝑢, V))⟩ ≥ 0, ∀ (V, 𝑦) ∈ 𝐺𝑟𝑎𝑝ℎ (𝑀) , ⟨𝑁 (⋅, V )−𝑁(⋅,V ),𝑗(𝜂(𝑢 ,𝑢 ))⟩ (27) 1 2 1 2 𝑗(𝜂(𝑢, V))∈𝐽(𝜂(𝑢, V)). 󵄩 󵄩2 ≤(−𝜏2) 󵄩𝑢1 −𝑢2󵄩 ,∀𝑢1,𝑢2 ∈𝑋, V1 ∈𝑄(𝑢1), (20) Then, 𝑥∈𝑀𝑢,where𝐺𝑟𝑎𝑝ℎ(𝑀) = {(𝑢, 𝑥) ∈ .𝑋×𝑋 :𝑥∈𝑀𝑢} V2 ∈𝑄(𝑢2), 𝑗(𝜂(𝑢1,𝑢2)) ∈ 𝐽 (𝜂1 (𝑢 ,𝑢2)) . Theorem 13 (see [23]). Let 𝐻(𝐴, 𝐵) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0and 𝜂-relaxed cocoercive 𝑋 𝐴, 𝐵 : 𝑋 →𝑋 Definition 9. Let be a Banach space. Let , with respect to 𝐵 with constant 𝛾>0, 𝐴 be 𝛼-expansive and 𝐻:𝑋×𝑋→ 𝑋𝜂:𝑋×𝑋→ 𝑋 , be four single-valued 𝐵 be 𝛽-Lipschitz continuous, 𝜇>𝛾and 𝛼>𝛽.Let𝑀 be an 𝑀:𝑋𝑋 →2 𝑀 mappings. Let be a set-valued mapping. 𝐻(⋅, ⋅) −𝜂-cocoercive operator with respect to 𝐴 and 𝐵.Then, 𝐻(⋅, ⋅) −𝜂 𝐴 −1 is said to be -cocoercive operator with respect to for each 𝜆>0,theoperator(𝐻(𝐴, 𝐵)+𝜆𝑀) is single-valued. and 𝐵,if𝑀 is 𝜂-cocoercive and (𝐻(𝐴, 𝐵) + 𝜆𝑀)(𝑋) =𝑋,for every 𝜆>0. Definition 14. Let 𝑋 be a Banach space. Let 𝐻(𝐴, 𝐵) be 𝜂- cocoercive with respect to 𝐴 with constant 𝜇>0and 𝜂- 𝑋=R × R 𝐴, 𝐵 : 𝑋 →𝑋 Example 10. Let and be defined by relaxed cocoercive with respect to 𝐵 with constant 𝛾>0, 𝐴 𝛼 𝐵 𝛽 𝜂 𝛽 𝐴(𝑥 ,𝑥 )=(2𝑥 −𝑥 ,𝑥 −𝑥 ), be -expansive be -Lipschitz continuous and be - 1 2 1 2 1 2 Lipschitz continuous, 𝜇>𝛾,and𝛼>𝛽.Let𝑀 be a 𝐻(⋅, ⋅) −𝜂-cocoercive operator with respect to 𝐴 and 𝐵.Then, 𝐵(𝑦1,𝑦2)=(−2𝑦2,𝑦1 −𝑦2), ∀(𝑥1,𝑥2),(𝑦1,𝑦2)∈𝑋. 𝑅𝐻(⋅,⋅)−𝜂 :𝑋 → 𝑋 (21) the resolvent 𝜆,𝑀 is defined by

𝐻(⋅,⋅)−𝜂 −1 Assume now that 𝐻(𝐴, 𝐵), 𝜂:𝑋×𝑋are →𝑋 defined by 𝑅𝜆,𝑀 (𝑢) = (𝐻 (𝐴,) 𝐵 +𝜆𝑀) (𝑢) ,∀𝑢∈𝑋. (28)

𝐻 (𝐴𝑥,) 𝐵𝑦 =𝐴𝑥+𝐵𝑦,( 𝜂 𝑥,) 𝑦 =𝑥−𝑦, ∀𝑥,𝑦∈𝑋. Theorem 15 (see [23]). Let 𝑋 be a Banach space. Let 𝐻(𝐴, 𝐵) (22) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0and 𝜂- relaxed cocoercive with respect to 𝐵 with constant 𝛾>0, 𝐴 be Let 𝑀=𝐼,where𝐼 is the identity mapping. Then, 𝑀 is 𝛼-expansive 𝐵 be 𝛽-Lipschitz continuous, and 𝜂 be 𝜌-Lipschitz 𝐻(⋅, ⋅) −𝜂-cocoercive with respect to 𝐴 and 𝐵. continuous; 𝜇>𝛾and 𝛼>𝛽.Let𝑀 be 𝐻(⋅, ⋅) −𝜂-cocoercive 4 Abstract and Applied Analysis

operator with respect to 𝐴 and 𝐵.Then,theresolventoperator (𝑖 = 1, 2), define the mappings 𝑆1 :𝑋1 ×𝑋2 →𝑋1 and 𝐻(⋅,⋅)−𝜂 2 2 𝑆 :𝑋 ×𝑋 →𝑋 𝑅𝜆,𝑀 :𝑋 →is 𝑋 𝜌/(𝜇𝛼 −𝛾𝛽 )-Lipschitz continuous, that 2 1 2 2 by is, 𝑆1 (𝑥, 𝑦) 󵄩 𝐻(⋅,⋅)−𝜂 𝐻(⋅,⋅)−𝜂 󵄩 󵄩 󵄩 𝐻 (⋅,⋅)−𝜂 󵄩𝑅𝜆,𝑀 (𝑢) −𝑅𝜆,𝑀 (V)󵄩 1 1 󵄩 󵄩 = ⋃ ⋃ 𝑅 [𝐻1 (𝐴1𝑥,1 𝐵 𝑥) − 𝜆1𝐹 (𝑤, V)], 𝜆1,𝑀(⋅,𝑥) 𝜌 (29) 𝑤∈𝑇(𝑥) V∈𝑄(𝑦) ≤ ‖𝑢−V‖ ,∀𝑢,V ∈𝑋. 𝜇𝛼2 −𝛾𝛽2 𝑆2 (𝑥, 𝑦)

𝐻 (⋅,⋅)−𝜂 = ⋃ ⋃ 𝑅 2 2 [𝐻 (𝐴 𝑦, 𝐵 𝑦) − 𝜆 𝐺 (𝑤, V)]. 3. Strong Convergence Theorem 𝜆 ,𝑁(⋅,𝑦) 2 2 2 2 𝑤∈𝑇(𝑥) V∈𝑄(𝑦) 2 In this section, using the resolvent operator technique asso- (32) ciated with 𝐻(⋅, ⋅) −𝜂-cocoercive operators, we propose a new generalized algorithm of nonlinear set-valued varia- For any given (𝑥0,𝑦0)∈𝑋1 ×𝑋2, 𝑤0 ∈𝑇(𝑥0), V0 ∈𝑄(𝑦0),let tional inclusions and establish strong convergence of iterative 𝐻1(⋅,⋅)−𝜂1 sequences produced by the method. 𝑧0 =𝑅 𝜆1,𝑀(⋅,𝑥0) For 𝑖=1,2,let𝑋𝑖 be real Banach spaces with the norm ‖⋅‖𝑖.Let𝐴𝑖,𝐵𝑖 :𝑋𝑖 →𝑋𝑖, 𝐻𝑖 :𝑋𝑖 ×𝑋𝑖 →𝑋𝑖, 𝜂𝑖 :𝑋𝑖 × ×[𝐻1 (𝐴1𝑥0,𝐵1𝑥0)−𝜆1𝐹(𝑤0, V0)] ∈ 𝑆1 (𝑥0,𝑦0), 𝑋𝑖 →𝑋𝑖, 𝐹:𝑋1 ×𝑋2 →𝑋1,and𝐺:𝑋1 ×𝑋2 →𝑋2 be 𝐻2(⋅,⋅)−𝜂2 single-valued mappings, and 𝑇:𝑋1 →𝐶𝐵(𝑋1), 𝑄:𝑋2 → 𝑢0 =𝑅 𝜆2,𝑁(⋅,𝑦0) 𝑋1 𝐶𝐵(𝑋2) be set-valued mappings. Let 𝑀:𝑋1 ×𝑋1 →2 , 𝑋2 𝑁:𝑋2 ×𝑋2 →2 be 𝐻1(⋅, ⋅) − 𝜂1-cocoercive and 𝐻2(⋅, ⋅) − ×[𝐻2 (𝐴2𝑦0,𝐵2𝑦0)−𝜆2𝐺(𝑤0, V0)] ∈ 𝑆2 (𝑥0,𝑦0). 𝜂2-cocoercive operators with respect to (𝐴1,𝐵1) and (𝐴2,𝐵2), (33) respectively. We consider the following problem. Find (𝑥, 𝑦)1 ∈𝑋 ×𝑋2, 𝑤∈𝑇(𝑥),andV ∈ 𝑄(𝑦) such that Since 𝑤0 ∈𝑇(𝑥0)⊂𝐶𝐵(𝑋1) and V0 ∈𝑄(𝑦0)⊂𝐶𝐵(𝑋2),in view of Nadler’s theorem [24], there exist 𝑤1 ∈𝑇(𝑥1) and 0∈𝑀(𝑥, 𝑥) +𝐹(𝑤, V) , V1 ∈𝑄(𝑦1) such that (30) 󵄩 󵄩 0 ∈ 𝑁 (𝑦, 𝑦) +𝐺 (𝑤, V) . 󵄩𝑤1 −𝑤0󵄩1 ≤ (1+1) 𝐷(𝑇(𝑥1),𝑇(𝑥0)) , 󵄩 󵄩 (34) 󵄩V − V 󵄩 ≤ (1+1) 𝐷(𝑄(𝑦 ),𝑄(𝑦 )) . We call problem (30)asystemofgeneralized𝐻(⋅, ⋅) −𝜂- 󵄩 1 0󵄩2 1 0 cocoercive operator inclusions. By induction, we define iterative sequences {𝑥𝑛}, {𝑦𝑛}, {𝑤𝑛}, Under the assumptions mentioned above, we have the and {V𝑛} as follows: following key and simple lemma.

𝐻1(⋅,⋅)−𝜂1 𝑥𝑛+1 =𝑅 Lemma 16. (𝑥, 𝑦)1 ∈𝑋 ×𝑋2, 𝑤∈𝑇(𝑥), V ∈ 𝑄(𝑦) is a solution 𝜆1,𝑀(⋅,𝑥𝑛) of problem (30) if and only if ×[𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] ∈ 𝑆1 (𝑥𝑛,𝑦𝑛), 𝐻 (⋅,⋅)−𝜂 𝑥=𝑅 1 [𝐻 (𝐴 𝑥, 𝐵 𝑥) − 𝜆 𝐹 (𝑤, V)], 𝐻 (⋅,⋅)−𝜂 𝜆 ,𝑀(⋅,𝑥) 1 1 1 1 𝑦 =𝑅 2 2 1 𝑛+1 𝜆 ,𝑁(⋅,𝑦 ) (31) 2 𝑛 𝐻2(⋅,⋅)−𝜂 𝑦=𝑅 [𝐻2 (𝐴2𝑦,2 𝐵 𝑦) − 𝜆2𝐺 (𝑤, V)], 𝜆2,𝑁(⋅,𝑦) ×[𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝜆2𝐺(𝑤𝑛, V𝑛)] ∈ 𝑆2 (𝑥𝑛,𝑦𝑛), 𝑤 ∈𝑇(𝑥), 𝐻1(⋅,⋅)−𝜂 −1 𝐻2(⋅,⋅)−𝜂 𝑛 𝑛 𝑅 =(𝐻1(𝐴1𝑥,1 𝐵 𝑥) + 𝜆1𝑀(⋅, 𝑥)) 𝑅 = where 𝜆1,𝑀(⋅,𝑥) , 𝜆1,𝑁(⋅,𝑦) −1 󵄩 󵄩 1 (𝐻2(𝐴2𝑦,2 𝐵 𝑦) + 𝜆2𝑁(⋅, 𝑦)) ,and𝜆1,𝜆2 >0are constants. 󵄩𝑤 −𝑤 󵄩 ≤(1+ )𝐷(𝑇(𝑥 ),𝑇(𝑥 )) , 󵄩 𝑛+1 𝑛󵄩1 𝑛+1 𝑛+1 𝑛 Proof . Thisisaneasyanddirectconsequenceof V ∈𝑄(𝑦), Definition 14. 𝑛 𝑛 󵄩 󵄩 1 𝑖=1,2 𝑋 󵄩V − V 󵄩 ≤(1+ ) 𝐷 (𝑄 (𝑦 ),𝑄(𝑦 )) , Algorithm 17. For ,let 𝑖 be real Banach spaces with 󵄩 𝑛+1 𝑛󵄩2 𝑛+1 𝑛+1 𝑛 ‖⋅‖ 𝐴 ,𝐵 :𝑋 →𝑋 𝐻 :𝑋×𝑋 →𝑋 the norm 𝑖.Let 𝑖 𝑖 𝑖 𝑖, 𝑖 𝑖 𝑖 𝑖, (35) 𝜂𝑖 :𝑋𝑖×𝑋𝑖 →𝑋𝑖, 𝐹:𝑋1×𝑋2 →𝑋1,and𝐺:𝑋1×𝑋2 →𝑋2 𝑇:𝑋 →𝐶𝐵(𝑋) 𝑄: be single-valued mappings, and 1 1 , where 𝑛 = 0, 1, 2, . .,and𝜆1,𝜆2 >0are constants. 𝑋2 →𝐶𝐵(𝑋2) be set-valued mappings. Let 𝑀:𝑋1 ×𝑋1 → 𝑋1 𝑋2 →2 , 𝑁:𝑋2 ×𝑋2 →2 be such that, for each fixed Theorem 18. For 𝑖=1,2,let𝑋𝑖 be real Banach spaces with 𝑥∈𝑋1, 𝑦∈𝑋2, 𝑀(⋅, 𝑥) and 𝑁(⋅, 𝑦) are 𝐻1(⋅, ⋅)−𝜂1-cocoercive the norm ‖⋅‖𝑖.Let𝐴𝑖,𝐵𝑖 :𝑋𝑖 →𝑋𝑖, 𝐻𝑖 :𝑋𝑖 ×𝑋𝑖 →𝑋𝑖, and 𝐻2(⋅, ⋅)−𝜂2-cocoercive operators with respect to (𝐴1,𝐵1) 𝜂𝑖 :𝑋𝑖×𝑋𝑖 →𝑋𝑖, 𝐹:𝑋1×𝑋2 →𝑋1, and 𝐺:𝑋1×𝑋2 →𝑋2 and (𝐴2,𝐵2),respectively.Foranygivenconstants𝜆𝑖 >0 be single-valued mappings, and 𝑇:𝑋1 →𝐶𝐵(𝑋1), 𝑄:𝑋2 → Abstract and Applied Analysis 5

𝑋1 𝐶𝐵(𝑋2) be set-valued mappings. Let 𝑀:𝑋1 ×𝑋1 →2 𝜇𝑖 >𝛾𝑖,𝛼𝑖 >𝛽𝑖,𝑖=1,2, 𝑋2 and 𝑁:𝑋2 ×𝑋2 →2 be such that, for each fixed 𝑥∈ 𝜌1 𝜌2 𝑋1,𝑦∈𝑋2, 𝑀(⋅, 𝑥) and 𝑁(⋅, 𝑦) are 𝐻1(⋅, ⋅)−𝜂1-cocoercive and 𝜃 = ,𝜃= , 1 𝜇 𝛼2 −𝛾𝛽2 2 𝜇 𝛼2 −𝛾𝛽2 𝐻2(⋅, ⋅) − 𝜂2-cocoercive operators with respect to (𝐴1,𝐵1) and 1 1 1 1 2 2 2 2 (𝐴2,𝐵2), respectively. Suppose that the following conditions are 𝜎 +𝜃 𝑝 +𝜃 𝑠 +𝜆 𝜃 𝑙 𝜆 <1, satisfied. 1 1 0 1 1 2 2 1 𝑇

𝜎2 +𝜃2𝑞0 +𝜃2𝑠2 +𝜆1𝜃1𝑡2𝜆𝑄 <1. (i) 𝐻𝑖(𝐴𝑖,𝐵𝑖) is 𝜂𝑖-cocoercive with respect to 𝐴𝑖 with constant 𝜇𝑖 and 𝜂𝑖-relaxed cocoercive with respect to 𝐵𝑖 (38) with constant 𝛾𝑖, 𝑖=1,2. Then, the iterative sequences {𝑥𝑛}, {𝑦𝑛}, {𝑤𝑛},and{V𝑛} gen- 𝐴 𝛼 𝐵 𝛽 𝑖= erated by Algorithm 17 converge strongly to 𝑥, 𝑦, 𝑤,andV, (ii) 𝑖 is 𝑖-expansive and 𝑖 is 𝑖-Lipschitz continuous, (𝑥, 𝑦, 𝑤, V) 1, 2. respectively, and is a solution of problem (30). 𝐻 (⋅,⋅)−𝜂 𝐻 (𝐴 ,𝐵) 𝑟 𝐴 1 1 (iii) 𝑖 𝑖 𝑖 is 𝑖-Lipschitz continuous with respect to 𝑖 Proof. In view of Theorem 13,theresolventoperator𝑅𝜆 ,𝑀 𝑠 𝐵 𝑖=1,2 1 and 𝑖-Lipschitz continuous with respect to 𝑖, . is 𝜃1-Lipschitz continuous. This, together with Algorithm 17 and (36), implies that (iv) 𝑇 is 𝐷-Lipschitz continuous with constant 𝜆𝑇 and 𝑄 is 󵄩 󵄩 𝐷-Lipschitz continuous with constant 𝜆𝑄. 󵄩𝑥𝑛+1 −𝑥𝑛󵄩1 𝐹 𝑡 𝑇 󵄩 (v) is 1-Lipschitz continuous with respect to in the first 󵄩 𝐻1(⋅,⋅)−𝜂1 = 󵄩𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] argument and 𝑡2-Lipschitz continuous with respect to 𝑄 󵄩 𝜆1,𝑀(⋅,𝑥𝑛) in the second argument. 𝐻 (⋅,⋅)−𝜂 −𝑅 1 1 𝜆 ,𝑀(⋅,𝑥 ) (vi) 𝐺 is 𝑙1-Lipschitz continuous with respect to 𝑇 in the first 1 𝑛−1 argument and 𝑙2-Lipschitz continuous with respect to 𝑄 󵄩 󵄩 in the second argument. ×[𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛−1)−𝜆1𝐹(𝑤𝑛−1, V𝑛−1)] 󵄩 󵄩1 𝜂 𝜌 𝑖=1,2 (vii) 𝑖 is 𝑖-Lipschitz continuous, . 󵄩 󵄩 𝐻1(⋅,⋅)−𝜂1 ≤ 󵄩𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] 󵄩 𝜆1,𝑀(⋅,𝑥𝑛) (viii) 𝐹 is 𝜂1-relaxed Lipschitz continuous with respect to 𝑇 in 󵄩 thefirstargumentand𝜂1-relaxed Lipschitz continuous 𝐻 (⋅,⋅)−𝜂 󵄩 1 1 󵄩 𝑄 −𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)]󵄩 with respect to inthesecondargumentwithconstants 𝜆1,𝑀(⋅,𝑥𝑛−1) 󵄩1 𝜏1 and 𝜏2,respectively. 󵄩 󵄩 𝐻1(⋅,⋅)−𝜂1 + 󵄩𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] (ix) 𝐺 is 𝜂2-relaxed Lipschitz continuous with respect to 𝑇 in 󵄩 𝜆1,𝑀(⋅,𝑥𝑛−1) thefirstargumentand𝜂2-relaxed Lipschitz continuous 𝐻1(⋅,⋅)−𝜂1 with respect to 𝑄 inthesecondargumentwithconstants −𝑅 𝜆1,𝑀(⋅,𝑥𝑛−1) 𝜖1 and 𝜖2, respectively. Furthermore, assume that there 󵄩 exist constants 𝜎1,𝜎2 >0such that 󵄩 ×[𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛−1)−𝜆1𝐹(𝑤𝑛−1, V𝑛−1)] 󵄩 󵄩1 󵄩 󵄩 󵄩 𝐻1(⋅,⋅)−𝜂1 𝐻1(⋅,⋅)−𝜂1 󵄩 󵄩 󵄩 󵄩𝑅 (𝑥) −𝑅 (𝑥)󵄩 ≤𝜎󵄩𝑥 −𝑥 󵄩 󵄩 𝜆1,𝑀(⋅,𝑥1) 𝜆1,𝑀(⋅,𝑥2) 󵄩 1󵄩 𝑛 𝑛−1󵄩1 1 (36) 󵄩 󵄩 ≤𝜎󵄩𝑥 −𝑥 󵄩 ,∀𝑥,𝑥,𝑥 ∈𝑋, 󵄩 1󵄩 1 2󵄩1 1 2 1 +𝜃1 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛) −𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛−1) 󵄩 󵄩 󵄩 𝐻 (⋅,⋅)−𝜂 𝐻 (⋅,⋅)−𝜂 󵄩 󵄩 󵄩 2 2 2 2 󵄩 󵄩 󵄩𝑅 (𝑦) − 𝑅 (𝑦)󵄩 −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛−1))󵄩1 󵄩 𝜆2,𝑁(⋅,𝑦1) 𝜆2,𝑁(⋅,𝑦2) 󵄩 2 (37) 󵄩 󵄩 󵄩 󵄩 ≤𝜎󵄩𝑥 −𝑥 󵄩 ≤𝜎2󵄩𝑦1 −𝑦2󵄩2,∀𝑦,𝑦1,𝑦2 ∈𝑋2, 1󵄩 𝑛 𝑛−1󵄩1 󵄩 +𝜃1 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛−1) and 𝜆1,𝜆2 >0are constants satisfying the following conditions: 󵄩 −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛))󵄩1 √ 2 𝑝0 = 𝑟1 +2𝜆1𝑡1𝜆𝑇 [𝑟1 +𝜆1𝑡1𝜆𝑇 +𝜌1]−2𝜆1𝜏1, 󵄩 󵄩 +𝜆1𝜃1󵄩𝐹(𝑤𝑛−1, V𝑛)−𝐹(𝑤𝑛−1, V𝑛−1)󵄩1 𝑞 = √𝑟2 +2𝜆 𝑙 𝜆 [𝑟 +𝜆 𝑙 𝜆 +𝜌]−2𝜆 𝜖 , 󵄩 󵄩 0 2 2 2 𝑄 2 2 2 𝑄 2 2 2 ≤𝜎1󵄩𝑥𝑛 −𝑥𝑛−1󵄩1 𝑟2 +2𝜆 𝑡 𝜆 [𝑟 +𝜆 𝑡 𝜆 +𝜌]>2𝜆 𝜏 , 󵄩 1 1 1 𝑇 1 1 1 𝑇 1 1 1 +𝜃1 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛) 2 󵄩 𝑟2 +2𝜆2𝑙2𝜆𝑄 [𝑟2 +𝜆2𝑙2𝜆𝑄 +𝜌2]>2𝜆2𝜖2, −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛))󵄩1 6 Abstract and Applied Analysis

󵄩 󵄩 +𝜃1󵄩𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛−1)󵄩1 𝑗[𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛) 󵄩 󵄩 +𝜆1𝜃1󵄩𝐹(𝑤𝑛−1, V𝑛)−𝐹(𝑤𝑛−1, V𝑛−1)󵄩1. −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛))] ⟩ (39) 󵄩 󵄩2 = 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛)󵄩1 𝐹 𝑡 𝑇 Since is 1-Lipschitz continuous with respect to in the −2𝜆1 ⟨𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛), first argument and 𝑡2-Lipschitz continuous in the second argument, 𝑇 is 𝜆𝑇-Lipschitz continuous, and 𝑄 is 𝜆𝑄- 𝑗[𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛) Lipschitz continuous, by Algorithm 17,weget −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛))] 󵄩 󵄩 󵄩𝐹(𝑤𝑛, V𝑛)−𝐹(𝑤𝑛−1, V𝑛)󵄩 󵄩 󵄩1 +𝑗 1(𝜂 (𝑥𝑛,𝑥𝑛−1)) ⟩ 󵄩 󵄩 ≤𝑡1󵄩𝑤𝑛 −𝑤𝑛−1󵄩 󵄩 󵄩1 +2𝜆1 ⟨𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛),𝑗(𝜂1 (𝑥𝑛,𝑥𝑛−1))⟩ 1 (40) 󵄩 󵄩2 ≤𝑡 (1 + )𝐷(𝑇(𝑥 ),𝑇(𝑥 )) ≤ 󵄩𝐻 (𝐴 𝑥 ,𝐵 𝑥 )−𝐻 (𝐴 𝑥 ,𝐵 𝑥 )󵄩 1 𝑛 𝑛 𝑛−1 󵄩 1 1 𝑛 1 𝑛 1 1 𝑛−1 1 𝑛 󵄩1 󵄩 󵄩 +2𝜆 󵄩𝐹(𝑤 , V )−𝐹(𝑤 , V )󵄩 1 󵄩 󵄩 1󵄩 𝑛 𝑛 𝑛−1 𝑛 󵄩1 ≤𝑡1𝜆𝑇 (1 + ) 󵄩𝑥𝑛 −𝑥𝑛−1󵄩 , 𝑛 1 󵄩 󵄩 ×[󵄩𝐻 (𝐴 𝑥 ,𝐵 𝑥 )−𝐻 (𝐴 𝑥 ,𝐵 𝑥 )󵄩 󵄩 󵄩 󵄩 1 1 𝑛 1 𝑛 1 1 𝑛−1 1 𝑛 󵄩1 󵄩𝐹(𝑤 , V )−𝐹(𝑤 , V )󵄩 󵄩 𝑛−1 𝑛 𝑛−1 𝑛−1 󵄩1 󵄩 󵄩 +𝜆 󵄩𝐹(𝑤 , V )−𝐹(𝑤 , V )󵄩 󵄩 󵄩 1󵄩 𝑛 𝑛 𝑛−1 𝑛 󵄩1 ≤𝑡󵄩V − V 󵄩 2󵄩 𝑛 𝑛−1󵄩2 󵄩 󵄩 +󵄩𝜂1 (𝑥𝑛,𝑥𝑛−1)󵄩1] 1 (41) ≤𝑡2 (1 + ) 𝐷 (𝑄𝑛 (𝑦 ),𝑄(𝑦𝑛−1)) 𝑛 +2𝜆1 ⟨𝐹 (𝑤𝑛, V𝑛) −𝐹(𝑤𝑛−1, V𝑛) ,𝑗(𝜂1 (𝑥𝑛,𝑥𝑛−1))⟩

1 󵄩 󵄩 󵄩 󵄩2 1 󵄩 󵄩 ≤𝑡𝜆 (1 + ) 󵄩𝑦 −𝑦 󵄩 . ≤𝑟2󵄩𝑥 −𝑥 󵄩 +2𝜆 𝑡 𝜆 (1 + ) 󵄩𝑥 −𝑥 󵄩 2 𝑄 𝑛 󵄩 𝑛 𝑛−1󵄩2 1 󵄩 𝑛 𝑛−1󵄩1 1 1 𝑇 𝑛 󵄩 𝑛 𝑛−1󵄩1 󵄩 󵄩 1 𝐻 (⋅, ⋅) 𝑟 𝐴 ×[𝑟󵄩𝑥 −𝑥 󵄩 +𝜆 𝑡 𝜆 (1 + ) As 1 is 1-Lipschitz continuous with respect to 1,we 1󵄩 𝑛 𝑛−1󵄩1 1 1 𝑇 𝑛 obtain 󵄩 󵄩 󵄩 󵄩 ×󵄩𝑥 −𝑥 󵄩 +𝜌󵄩𝑥 −𝑥 󵄩 ] 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛 𝑛−1󵄩1 1󵄩 𝑛 𝑛−1󵄩1 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛) −𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛)󵄩1 ≤𝑟1󵄩𝑥𝑛 −𝑥𝑛−1󵄩1. (42) 󵄩 󵄩2 −2𝜆1𝜏1󵄩𝑥𝑛 −𝑥𝑛−1󵄩1 2 1 Since 𝜂1 is 𝜌1-Lipschitz continuous, we conclude that =[𝑟 +2𝜆 𝑡 𝜆 (1 + ) 1 1 1 𝑇 𝑛 󵄩 󵄩 󵄩 󵄩 1 󵄩 󵄩2 󵄩𝜂 (𝑥 ,𝑥 )󵄩 ≤𝜌󵄩𝑥 −𝑥 󵄩 . ×[𝑟 +𝜆 𝑡 𝜆 (1+ )+𝜌 ]−2𝜆 𝜏 ]󵄩𝑥 −𝑥 󵄩 . 󵄩 1 𝑛 𝑛−1 󵄩1 1󵄩 𝑛 𝑛−1󵄩1 (43) 1 1 1 𝑇 𝑛 1 1 1 󵄩 𝑛 𝑛−1󵄩1 (45) Since 𝐻1(⋅, ⋅) is 𝜂1-relaxed Lipschitz continuous with respect to 𝑇 and 𝜂2-relaxed Lipschitz continuous with respect This implies that to 𝑄 in the first and second arguments with constants 𝜏1 and 󵄩 𝜏2,respectively,wehave 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛) 󵄩 −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛))󵄩1 ⟨𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛),𝑗(𝜂1 (𝑥𝑛,𝑥𝑛−1))⟩

󵄩 󵄩2 (44) 1 1 ≤−𝜏󵄩𝑥 −𝑥 󵄩 . ≤ √𝑟2 +2𝜆 𝑡 𝜆 (1+ )[𝑟 +𝜆 𝑡 𝜆 (1+ )+𝜌 ]−2𝜆 𝜏 1󵄩 𝑛 𝑛−1󵄩1 1 1 1 𝑇 𝑛 1 1 1 𝑇 𝑛 1 1 1 󵄩 󵄩 × 󵄩𝑥 −𝑥 󵄩 Employing Lemma 3 and taking into account (39)–(44), we 󵄩 𝑛 𝑛−1󵄩1 obtain 󵄩 󵄩 =𝑝𝑛󵄩𝑥𝑛 −𝑥𝑛−1󵄩1, 󵄩 (46) 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥𝑛−1,𝐵1𝑥𝑛) 󵄩2 where −𝜆1 (𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛))󵄩1 𝑝𝑛 󵄩 󵄩2 ≤ 󵄩𝐻 (𝐴 𝑥 ,𝐵 𝑥 )−𝐻 (𝐴 𝑥 ,𝐵 𝑥 )󵄩 1 1 󵄩 1 1 𝑛 1 𝑛 1 1 𝑛−1 1 𝑛 󵄩1 = √𝑟2 +2𝜆 𝑡 𝜆 (1 + )[[𝑟 +𝜆 𝑡 𝜆 (1 + )+𝜌 ]−2𝜆 𝜏 ]. 1 1 1 𝑇 𝑛 1 1 1 𝑇 𝑛 1 1 1 −2𝜆1 ⟨𝐹 𝑛(𝑤 , V𝑛)−𝐹(𝑤𝑛−1, V𝑛), (47) Abstract and Applied Analysis 7

󵄩 󵄩 Using 𝑠1-Lipschitz continuity of 𝐻1(⋅, ⋅) with respect to 𝐵1, ≤𝜎2󵄩𝑦𝑛 −𝑦𝑛−1󵄩2 we deduce that 󵄩 +𝜃2 󵄩𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛) 󵄩 󵄩 󵄩𝐻 (𝐴 𝑥 ,𝐵 𝑥 )−𝐻 (𝐴 𝑥 ,𝐵 𝑥 )󵄩 󵄩 󵄩 1 1 𝑛−1 1 𝑛 1 1 𝑛−1 1 𝑛−1 󵄩1 −𝜆2 (𝐺 𝑛(𝑤 , V𝑛)−𝐺(𝑤𝑛, V𝑛−1))󵄩2 󵄩 󵄩 (48) ≤𝑠󵄩𝑥 −𝑥 󵄩 . 󵄩 󵄩 1󵄩 𝑛 𝑛−1󵄩1 +𝜃2󵄩𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛)−𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛−1)󵄩2 󵄩 󵄩 +𝜆2𝜃2󵄩𝐺(𝑤𝑛, V𝑛−1)−𝐺(𝑤𝑛−1, V𝑛−1)󵄩 In view of (41), (46), (48), (39)becomes 2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤𝜎2󵄩𝑦𝑛 −𝑦𝑛−1󵄩 +𝜃2𝑞𝑛󵄩𝑦𝑛 −𝑦𝑛−1󵄩 +𝜃2𝑠2󵄩𝑦𝑛 −𝑦𝑛−1󵄩 󵄩 󵄩 2 2 2 󵄩𝑥 −𝑥 󵄩 󵄩 𝑛+1 𝑛󵄩1 1 󵄩 󵄩 +𝜆 𝜃 𝑡 𝜆 (1 + ) 󵄩𝑥 −𝑥 󵄩 󵄩 󵄩 󵄩 󵄩 2 2 1 𝑇 𝑛 󵄩 𝑛 𝑛−1󵄩1 ≤𝜎1󵄩𝑥𝑛 −𝑥𝑛−1󵄩 +𝜃1𝑝𝑛󵄩𝑥𝑛 −𝑥𝑛−1󵄩 1 1 󵄩 󵄩 =(𝜎 +𝜃 𝑞 +𝜃 𝑠 ) 󵄩𝑦 −𝑦 󵄩 󵄩 󵄩 2 2 𝑛 2 2 󵄩 𝑛 𝑛−1󵄩2 +𝜃1𝑠1󵄩𝑥𝑛 −𝑥𝑛−1󵄩1 1 󵄩 󵄩 1 󵄩 󵄩 +𝜆 𝜃 𝑙 𝜆 (1 + ) 󵄩𝑥 −𝑥 󵄩 , +𝜆 𝜃 𝑡 𝜆 (1 + ) 󵄩𝑦 −𝑦 󵄩 (49) 2 2 1 𝑇 𝑛 󵄩 𝑛 𝑛−1󵄩1 1 1 2 𝑄 𝑛 󵄩 𝑛 𝑛−1󵄩2 (50) 󵄩 󵄩 =(𝜎1 +𝜃1𝑝𝑛 +𝜃1𝑠1) 󵄩𝑥𝑛 −𝑥𝑛−1󵄩1 where 1 󵄩 󵄩 +𝜆1𝜃1𝑡2𝜆𝑄 (1 + ) 󵄩𝑦𝑛 −𝑦𝑛−1󵄩 . 𝑛 2 𝑞𝑛 1 1 = √𝑟2 +2𝜆 𝑙 𝜆 (1 + )[𝑟 +𝜆 𝑙 𝜆 (1 + )+𝜌 ]−2𝜆 𝜖 . 2 2 2 𝑄 𝑛 2 2 2 𝑄 𝑛 2 2 2 Similarly, we have (51)

󵄩 󵄩 In view of (49)and(50), we obtain 󵄩𝑦𝑛+1 −𝑦𝑛󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝐻 (⋅,⋅)−𝜂 󵄩𝑥 −𝑥 󵄩 + 󵄩𝑦 −𝑦 󵄩 = 󵄩𝑅 2 2 [𝐻 (𝐴 𝑦 ,𝐵 𝑦 )−𝜆 𝐺(𝑤 , V )] 󵄩 𝑛+1 𝑛󵄩1 󵄩 𝑛+1 𝑛󵄩2 󵄩 𝜆 ,𝑁(⋅,𝑦 ) 2 2 𝑛 2 𝑛 2 𝑛 𝑛 󵄩 2 𝑛 󵄩 󵄩 󵄩 󵄩 ≤𝑐𝑛󵄩𝑥𝑛 −𝑥𝑛−1󵄩1 +𝑑𝑛󵄩𝑦𝑛 −𝑦𝑛−1󵄩2 (52) 𝐻2(⋅,⋅)−𝜂2 −𝑅 [𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛−1) 󵄩 󵄩 󵄩 󵄩 𝜆2,𝑁(⋅,𝑦𝑛−1) 󵄩 󵄩 󵄩 󵄩 ≤𝑘𝑛 (󵄩𝑥𝑛 −𝑥𝑛−1󵄩1 + 󵄩𝑦𝑛 −𝑦𝑛−1󵄩2), 󵄩 󵄩 −𝜆𝐺(𝑤 , V )]󵄩 𝑐 =𝜎 +𝜃𝑝 +𝜃𝑠 +𝜆 𝜃 𝑙 𝜆 (1 + 1/𝑛) 𝑑 =𝜎 + 2 𝑛−1 𝑛−1 󵄩 where 𝑛 1 1 𝑛 1 1 2 2 1 𝑇 , 𝑛 2 2 𝜃 𝑞 +𝜃 𝑠 +𝜆 𝜃 𝑡 𝜆 (1 + 1/𝑛) 𝑘 = {𝑐 ,𝑑 } 󵄩 2 𝑛 2 2 1 1 2 𝑄 ,and 𝑛 max 𝑛 𝑛 . 󵄩 𝐻 (⋅,⋅)−𝜂 󵄩 2 2 Letting 𝑛→∞,weobtain𝑘𝑛 →𝑘,where ≤ 󵄩𝑅 [𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝜆2𝐺(𝑤𝑛, V𝑛)] 󵄩 𝜆2,𝑁(⋅,𝑦𝑛) 󵄩 𝑘= {𝜎 +𝜃 𝑝 +𝜃 𝑠 +𝜆 𝜃 𝑙 𝜆 , 𝐻 (⋅,⋅)−𝜂 󵄩 max 1 1 0 1 1 2 2 1 𝑇 2 2 󵄩 −𝑅 [𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝜆2𝐺(𝑤𝑛, V𝑛)]󵄩 (53) 𝜆2,𝑁(⋅,𝑦𝑛−1) 󵄩2 𝜎2 +𝜃2𝑞0 +𝜃2𝑠2 +𝜆1𝜃1𝑡2𝜆𝑄}. 󵄩 󵄩 𝐻2(⋅,⋅)−𝜂2 + 󵄩𝑅𝜆 ,𝑁(⋅,𝑦 ) [𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝜆2𝐺(𝑤𝑛, V𝑛)] 󵄩 2 𝑛−1 Next, we define the norm ‖⋅‖on 𝑋1 ×𝑋2 by

𝐻2(⋅,⋅)−𝜂2 󵄩 󵄩 󵄩 󵄩 −𝑅 [𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛−1) 󵄩(𝑥,) 𝑦 󵄩 = ‖𝑥‖ + 󵄩𝑦󵄩 , (𝑥,) 𝑦 ∈𝑋 ×𝑋 . 𝜆2,𝑁(⋅,𝑦𝑛−1) 󵄩 󵄩 1 󵄩 󵄩2 1 2 (54) 󵄩 󵄩 (𝑋 ×𝑋 ,‖⋅‖) −𝜆𝐺(𝑤 , V )]󵄩 One can easily check that 1 2 is a Banach space. 2 𝑛−1 𝑛−1 󵄩 2 Define 𝑎𝑛+1 =(𝑥𝑛+1,𝑦𝑛+1).Then,wehave 󵄩 󵄩 ≤𝜎2󵄩𝑦𝑛 −𝑦𝑛−1󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑎𝑛+1 −𝑎𝑛󵄩 = 󵄩𝑥𝑛+1 −𝑥𝑛󵄩1 + 󵄩𝑦𝑛+1 −𝑦𝑛󵄩2. (55) 󵄩 +𝜃2 󵄩𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛−1) In view of (38), we conclude that 0<𝑘<1. This implies 󵄩 𝑛 ∈ N 𝑘 ∈ (0, 1) 𝑘 ≤𝑘 −𝜆2 (𝐺 𝑛(𝑤 , V𝑛)−𝐺(𝑤𝑛−1, V𝑛−1))󵄩2 that there exist 0 and 0 such that 𝑛 0 for all 𝑛≥𝑛 󵄩 󵄩 0. It follows from (52)and(54)that ≤𝜎󵄩𝑦 −𝑦 󵄩 2󵄩 𝑛 𝑛−1󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩𝑎 −𝑎 󵄩 ≤𝑘 󵄩𝑎 −𝑎 󵄩 ,∀𝑛≥𝑛. 󵄩 󵄩 𝑛+1 𝑛󵄩 0 󵄩 𝑛 𝑛−1󵄩 0 (56) +𝜃2 󵄩𝐻2 (𝐴2𝑦𝑛,𝐵2𝑦𝑛)−𝐻2 (𝐴2𝑦𝑛−1,𝐵2𝑦𝑛−1) 󵄩 In view of (56), we obtain −𝜆2 (𝐺 𝑛(𝑤 , V𝑛)−𝐺(𝑤𝑛, V𝑛−1))󵄩2 󵄩 󵄩 𝑛−𝑛 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 0 󵄩 󵄩 󵄩 󵄩 󵄩𝑎𝑛+1 −𝑎𝑛󵄩 ≤𝑘0 󵄩𝑎𝑛 +1 −𝑎𝑛 󵄩 ,∀𝑛≥𝑛0. (57) +𝜆2𝜃2󵄩𝐺(𝑤𝑛, V𝑛−1)−𝐺(𝑤𝑛−1, V𝑛−1)󵄩2 󵄩 0 0 󵄩 8 Abstract and Applied Analysis 󵄩 󵄩 𝐻1(⋅,⋅)−𝜂1 This implies that for any 𝑚≥𝑛≥𝑛0, + 󵄩𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] 󵄩 𝜆1,𝑀(⋅,𝑥)

𝐻 (⋅,⋅)−𝜂 󵄩 󵄩 󵄩 󵄩 󵄩 1 1 󵄩 󵄩𝑥 −𝑥 󵄩 ≤ 󵄩𝑎 −𝑎 󵄩 −𝑅 [𝐻1 (𝐴1𝑥,1 𝐵 𝑥) − 𝜆1𝐹 (𝑤, V)]󵄩 󵄩 𝑚 𝑛󵄩1 󵄩 𝑚 𝑛󵄩 𝜆1,𝑀(⋅,𝑥) 󵄩1 𝑚−1 𝑚−1 󵄩 󵄩 󵄩 󵄩 𝑖−𝑛 󵄩 󵄩 (58) ≤𝜎󵄩𝑥 −𝑥󵄩 ≤ ∑ 󵄩𝑎 −𝑎󵄩 ≤ ∑ 𝑘 0 󵄩𝑎 −𝑎 󵄩 . 1󵄩 𝑛 󵄩1 󵄩 𝑖+1 𝑖󵄩 0 󵄩 𝑛0+1 𝑛0 󵄩 𝑖=𝑛 𝑖=𝑛 󵄩 +𝜃1 󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛) 󵄩 −[𝐻 (𝐴 𝑥, 𝐵 𝑥) − 𝜆 𝐹 (𝑤, V)]󵄩 Since 0<𝑘0 <1,itfollowsfrom(58)that‖𝑥𝑚 −𝑥𝑛‖1 →0 1 1 1 1 󵄩1 𝑛→∞ {𝑥 } and .Thisprovesthat 𝑛 is a Cauchy sequence in 󵄩 󵄩 ≤𝜎1󵄩𝑥𝑛 −𝑥󵄩 𝑋1.Similarly,weconcludethat{𝑦𝑛} is a Cauchy sequence in 󵄩 󵄩1 𝑋2.Thus,thereexist𝑥∈𝑋1 and 𝑦∈𝑋2 such that 𝑥𝑛 →𝑥 󵄩 󵄩 +𝜃1 [󵄩𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝐻1 (𝐴1𝑥,1 𝐵 𝑥)󵄩1 and 𝑦𝑛 →𝑦as 𝑛→∞. 󵄩 󵄩 𝑤𝑛 → 𝑤 ∈ 𝑇(𝑥) V𝑛 → V ∈ 𝑄(𝑦) 󵄩 󵄩 Next, we prove that and . +𝜆1󵄩𝐹(𝑤𝑛, V𝑛)−𝐹(𝑤, V)󵄩1] In view of Lipschitz continuity of 𝑇 and 𝑄 and Algorithm 17, 󵄩 󵄩 we obtain ≤𝜎1󵄩𝑥𝑛 −𝑥󵄩1 󵄩 󵄩 +𝜃 [󵄩𝐻 (𝐴 𝑥 ,𝐵 𝑥 )−𝐻 (𝐴 𝑥, 𝐵 𝑥 )󵄩 󵄩 󵄩 1 󵄩 󵄩 1 󵄩 1 1 𝑛 1 𝑛 1 1 1 𝑛 󵄩1 󵄩𝑤 −𝑤 󵄩 ≤(1+ )𝜆 󵄩𝑥 −𝑥 󵄩 , 󵄩 𝑛 𝑛−1󵄩1 𝑇󵄩 𝑛 𝑛−1󵄩1 󵄩 󵄩 𝑛 +󵄩𝐻 (𝐴 𝑥, 𝐵 𝑥 )−𝐻 (𝐴 𝑥, 𝐵 𝑥)󵄩 ] (59) 󵄩 1 1 1 𝑛 1 1 1 󵄩1 󵄩 󵄩 1 󵄩 󵄩 󵄩 󵄩 󵄩V𝑛 − V𝑛−1󵄩 ≤(1+ )𝜆𝑄󵄩𝑦𝑛 −𝑦𝑛−1󵄩 . +𝜆 𝜃 [󵄩𝐹(𝑤 , V )−𝐹(𝑤,V )󵄩 󵄩 󵄩2 𝑛 󵄩 󵄩2 1 1 󵄩 𝑛 𝑛 𝑛 󵄩1 󵄩 󵄩 +󵄩𝐹(𝑤,V𝑛)−𝐹(𝑤, V)󵄩1] From (59), we deduce that {𝑤𝑛}, {V𝑛} are Cauchy sequences 𝑋 𝑋 𝑤∈𝑇(𝑥) 󵄩 󵄩 󵄩 󵄩 in 1 and 2, respectively. Thus, there exist and ≤𝜎1󵄩𝑥𝑛 −𝑥󵄩1 +𝜃1 [𝑟1 +𝑠1] 󵄩𝑥𝑛 −𝑥󵄩1 V ∈ 𝑄(𝑦) 𝑤 →𝑤 V → V 𝑛→∞ such that 𝑛 and 𝑛 as .Since 󵄩 󵄩 󵄩 󵄩 +𝜆 𝜃 [𝑡 󵄩𝑤 −𝑤󵄩 +𝑡 󵄩V − V󵄩 ] 𝑇 is 𝐷-Lipschitz continuous with constant 𝜆𝑇,itisobvious 1 1 1󵄩 𝑛 󵄩1 2󵄩 𝑛 󵄩2 that 󵄩 󵄩 =(𝜎1 +𝜃1𝑟1 +𝜃1𝑠1) 󵄩𝑥𝑛 −𝑥󵄩1 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑑 (𝑤, 𝑇 (𝑥)) ≤ 󵄩𝑤−𝑤𝑛󵄩 +𝑑(𝑤𝑛,𝑇(𝑥)) 󵄩 󵄩 󵄩 󵄩 1 +𝜆1𝜃1 [𝑡1󵄩𝑤𝑛 −𝑤󵄩1 +𝑡2󵄩V𝑛 − V󵄩2]. 󵄩 󵄩 ≤ 󵄩𝑤−𝑤𝑛−1󵄩1 + 𝐷 (𝑇𝑛 (𝑥 ),𝑇(𝑥)) (62) 󵄩 󵄩 (60) 󵄩 󵄩 Since 𝑥𝑛 →𝑥, 𝑤𝑛 →𝑤,andV𝑛 → V as 𝑛→∞,itfollows ≤ 󵄩𝑤−𝑤𝑛󵄩1 from (62)that +𝜆 ‖𝑥 −𝑥‖ 󳨀→ 0 (𝑛󳨀→∞) . 󵄩 󵄩 𝑇 𝑛 1 󵄩𝑥 −𝑥 󵄩 =0, 𝑛→∞lim 󵄩 𝑛 0󵄩1 (63) 𝑇(𝑥) 𝑤∈𝑇(𝑥) By the closedness of ,weconcludethat . and hence 𝑥0 =𝑥. V ∈ 𝑄(𝑦) Similarly, we have . A similar argument shows that 𝑦0 =𝑦. Therefore, Assume now that 𝐻 (⋅,⋅)−𝜂 𝑥=𝑥 =𝑅 1 [𝐻 (𝐴 𝑥, 𝐵 𝑥) − 𝜆 𝐹 𝑤, V ], 0 𝜆 ,𝑀(⋅,𝑥) 1 1 1 1 ( ) 𝐻 (⋅,⋅)−𝜂 1 𝑥 =𝑅 1 [𝐻 (𝐴 𝑥, 𝐵 𝑥) − 𝜆 𝐹 (𝑤, V)], 0 𝜆 ,𝑀(⋅,𝑥) 1 1 1 1 𝐻 (⋅,⋅)−𝜂 (64) 1 𝑦=𝑦 =𝑅 2 [𝐻 (𝐴 𝑦, 𝐵 𝑦) − 𝜆 𝐺 (𝑤, V)]. 0 𝜆 ,𝑁(⋅,𝑦) 2 2 2 2 𝐻 (⋅,⋅)−𝜂 (61) 2 𝑦 =𝑅 2 [𝐻 (𝐴 𝑦, 𝐵 𝑦) − 𝜆 𝐺 (𝑤, V)]. 0 𝜆 ,𝑁(⋅,𝑦) 2 2 2 2 2 In view of Lemma 16,weconcludethat(𝑥, 𝑦, 𝑤, V) is a solution of problem (30), which completes the proof. Then, we have At the end of this paper, we include the following simple 󵄩 󵄩 example in support of Theorem 18. 󵄩𝑥𝑛+1 −𝑥0󵄩1 󵄩 𝑋=R2 󵄩 𝐻 (⋅,⋅)−𝜂 Example 19. Let with the usual inner product. We 󵄩 1 1 2 2 ≤ 󵄩𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] define two mappings 𝐴, 𝐵 : R → R by 󵄩 𝜆1,𝑀(⋅,𝑥𝑛)

𝐻 (⋅,⋅)−𝜂 󵄩 1 1 1 1 󵄩 𝐴 (𝑥) := ( 𝑥 −𝑥 ,𝑥 + 𝑥 ), −𝑅 [𝐻1 (𝐴1𝑥,1 𝐵 𝑥) − 𝜆1𝐹 (𝑤, V)]󵄩 1 2 1 2 𝜆1,𝑀(⋅,𝑥) 󵄩1 4 4 󵄩 󵄩 𝐻 (⋅,⋅)−𝜂 1 1 1 1 󵄩 1 1 (65) ≤ 󵄩𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)] 𝐵 (𝑥) := (− 𝑥1 + 𝑥2,− 𝑥1 − 𝑥2), 󵄩 𝜆1,𝑀(⋅.,𝑥𝑛) 4 4 4 4

𝐻 (⋅,⋅) −𝜂 󵄩 2 1 1 󵄩 ∀𝑥=(𝑥,𝑥 )∈R . −𝑅 [𝐻1 (𝐴1𝑥𝑛,𝐵1𝑥𝑛)−𝜆1𝐹(𝑤𝑛, V𝑛)]󵄩 1 2 𝜆1,𝑀(⋅,𝑥) 󵄩1 Abstract and Applied Analysis 9

2 2 2 Let a mapping 𝐻:R × R → R be defined by References

2 𝐻 (𝐴𝑥, 𝐵𝑦) := 𝐴𝑥 +𝐵𝑦,∀𝑥,𝑦∈ R . (66) [1] S. Adly, “Perturbed algorithms and sensitivity analysis for a general class of variational inclusions,” Journal of Mathematical Analysis and Applications,vol.201,no.2,pp.609–630,1996. By similar arguments, as in Example 4.1 of [27], we can prove the following. [2] A. Alotaibi, V. Kumar, and N. Hussain, “Convergence compar- ison and stability of Jungck-Kirk type algorithms for common (1) 𝐻(𝐴, 𝐵) is 4/17-cocoercive with respect to 𝐴 and 1- fixed point problems,” Fixed Point Theory and Applications,vol. 2013, article 173, 2013. relaxed cocoercive with respect to 𝐵. [3] L. C. Ceng, N. Husain, A. Latif, and J. C. Yao, “Strong conver- √ (2) 𝐴 is 17/𝑛-expansive, for 𝑛=4,5. gence for solving general system of variational inequalities and fixed point problems in Banach spaces,” Journal of Inequalities (3) 𝐵 is 1/√𝑛-expansive, for 𝑛=1,2. and Applications,vol.2013,article334,2013. √ (4) 𝐻(𝐴, 𝐵) is 17/𝑛-Lipschitz continuous with constant [4] H. R. Feng and X. P. Ding, “A new system of generalized √17/𝑛 with respect to 𝐴 and 𝐵,for𝑛 = 1, 2, . . . , 15,. 16 nonlinear quasi-variational-like inclusions with 𝐴-monotone 𝑓, 𝑔 : R2 → R2 operators in Banach spaces,” Journal of Computational and (5) Let be defined by Applied Mathematics,vol.225,no.2,pp.365–373,2009. 8 8 [5] H.-Y. Lan, Y. J. Cho, and R. U. Verma, “Nonlinear relaxed coco- 𝑓 (𝑥) := (8𝑥 − 𝑥 , 𝑥 +8𝑥 ), (𝐴, 𝜂) 1 5 2 5 1 2 ercive variational inclusions involving -accretive map- pings in Banach spaces,” Computers and Mathematics with 17 5 5 17 (67) Applications, vol. 51, no. 9-10, pp. 1529–1538, 2006. 𝑔 (𝑥) := (− 𝑥1 + 𝑥2, 𝑥1 + 𝑥2), 8 8 8 8 [6] Y.-P. Fang and N.-J. Huang, “𝐻-accretive operators and resol- 2 vent operator technique for solving variational inclusions in ∀𝑥=(𝑥1,𝑥2)∈R . Banach spaces,” Applied Mathematics Letters,vol.17,no.6,pp. 647–653, 2004. 2 2 2 𝐻((., .), 𝜂) (6) Now, we define a mapping 𝑀:R × R → R by [7]Z.B.WangandX.P.Ding,“ -accretive operators with an application for solving set-valued variational inclusions in 2 Banach spaces,” Computers and Mathematics with Applications, 𝑀 (𝑓𝑥, 𝑔𝑦) := 𝑓𝑥 − 𝑔𝑦,∀𝑥,𝑦∈ R . (68) vol.59,no.4,pp.1559–1567,2010. [8] Y.-Z. Zou and N.-J. Huang, “𝐻(., .)-accretive operator with an 2 2 Let 𝑅, 𝑆, 𝑇: R → R betheidentitymappings.Itisobvious application for solving variational inclusions in Banach spaces,” Applied Mathematics and Computation,vol.204,no.2,pp.809– that these mappings are 𝐷-Lipschitz continuous. 816, 2008. 2 2 (7) Assume that 𝐹, 𝐺 : R → R are defined by [9] M.-M. Jin, “Convergence and stability of iterative algorithm for anewsystemof(𝐴, 𝜂)-accretive mapping inclusions in Banach 1 1 1 1 spaces,” Computers and Mathematics with Applications,vol.56, 𝐹 (𝑥) := (− 𝑥 − 𝑥 , 𝑥 − 𝑥 ), 4 1 8 2 8 1 4 2 no.9,pp.2305–2311,2008. [10] H.-Y. Lan, “Stability of iterative processes with errors for a 1 1 1 1 (69) system of nonlinear (𝐴, 𝜂)-accretive variational inclusions in 𝐺 (𝑥) := (− 𝑥1 + 𝑥2,− 𝑥1 − 𝑥2), 8 5 5 8 Banach spaces,” Computers and Mathematics with Applications, vol.56,no.1,pp.290–303,2008. ∀𝑥 = (𝑥 ,𝑥 )∈R2. 1 2 [11] X.-P. Luo and N.-J. Huang, “(𝐻, 𝜙) − 𝜂−monotone operators in Banach spaces with an application to variational inclusions,” Itcouldeasilybeseenthatalltheaspectsofthehypothesesof Applied Mathematics and Computation,vol.216,no.4,pp.1131– Theorem 18 are satisfied, so we have the desired conclusion. 1139, 2010. [12] Z. Xu and Z. Wang, “AGeneralized mixed variational inclusion involving (𝐻(., .), 𝜂)-monotone operators in Banach spaces,” Conflict of Interests Journal of Mathematics Research,vol.2,no.3,pp.47–56,2010. The authors declare that there is no conflict of interests [13] R. U. Verma, “Approximation solvability of a class of nonlinear set-valued variational inclusions involving (𝐴, 𝜂)-monotone regarding the publication of this paper. mappings,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 969–975, 2008. Acknowledgments [14] H.-Y. Lan, “Nonlinear parametric multi-valued variational inclusion systems involving (𝐴,)-accretive 𝜂 mappings in This paper was funded by King Abdulaziz University, under Banach spaces,” Nonlinear Analysis: Theory, Methods & Grant no. 23-130-1433-HiCi. The authors, therefore, acknowl- Applications,vol.69,no.5-6,pp.1757–1767,2008. edge the technical and financial support of KAU. The authors [15] H.-Y. Lan, “(𝐴, 𝜂)-accretive mappings and set-valued varia- would like to thank the referees for their sincere evaluation tional inclusions with relaxed cocoercive mappings in Banach and constructive comments which improved the paper con- spaces,” Applied Mathematics Letters,vol.20,no.5,pp.571–577, siderably. 2007. 10 Abstract and Applied Analysis

[16] C.-L. Su, “Finger shape expansion and recognition by wavelet transform,” Applied Mathematics and Computation,vol.190,no. 2, pp. 1583–1592, 2007. [17] H.-Y. Lan, Y.-S. Cui, and Y. Fu, “New approximation-solvability of general nonlinear operator inclusion couples involving (𝐴, 𝜂, 𝑚)-resolvent operators and relaxed cocoercive type oper- ators,” Communications in Nonlinear Science and Numerical Simulation,vol.17,no.4,pp.1844–1851,2012. [18] N. Hussain and A. Rafiq, “On modified implicit Mann iteration method involving strictly hemicontractive mappings in smooth Banach spaces,” Journal of Computational Analysis and Applica- tions, vol. 15, no. 5, pp. 892–902, 2013. [19] N. Hussain, A. Rafiq, and L. B. Ciric, “Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces,” Fixed Point Theory and Applica- tions,vol.2012,article160,2012. [20] S. H. Khan, A. Rafiq, and N. Hussain, “A three-step iterative scheme for solving nonlinear 𝜙-strongly accretive operator equations in Banach spaces,” Fixed Point Theory and Applica- tions,vol.2012,article149,2012. [21] L.-C. Ceng, A. R. Khan, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for strongly positive and monotone operators,” Fixed Point Theory,vol.10,no.1,pp.35–71,2009. [22] R. Ahmad and M. Mursaleen, “System of generalized 𝐻- resolvent equations and the corresponding system of general- ized variational inclusions,” Hacettepe Journal of Mathematics and Statistics,vol.41,no.1,pp.33–45,2012. [23] R. Ahmad, M. Dilshad, M.-M. Wong, and J.-C. Yao, “Gener- alized set-valued variational-like inclusions involving 𝐻(., .)-𝜂- cocoercive operator in Banach spaces,” Journal of Inequalities and Applications,vol.2012,article149,2012. [24] S. B. Nadler, “Multivalued contraction mappings,” Pacific Jour- nal of Mathematics, vol. 30, no. 3, pp. 457–488, 1969. [25] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,vol.62ofMathematics and Its Appli- cations, Kluwer Academic Publishers, Dordrecht, The Nether- lands, 1990. [26] W. V. Petryshyn, “A characterization of strict convexity of Banach spaces and other uses of duality mappings,” Journal of Functional Analysis,vol.6,no.2,pp.282–291,1970. [27] R. Ahmad, M. Akram, and J. C. Yao, “Generalized monotone mapping with an application for solving a variational inclusion problem,” Journal of Optimization Theroy and Applications,vol. 157,no.2,pp.324–346,2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 974317, 8 pages http://dx.doi.org/10.1155/2013/974317

Research Article Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms

Seyit Temir1,2 and Adem Kiliçman3

1 Department of Mathematics, Arts and Science Faculty, Harran University, 63200 S¸anliurfa, Turkey 2 Graduate School of Natural and Applied Sciences, Adıyaman University, 02040 Adıyaman, Turkey 3 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Adem Kilic¸man; [email protected]

Received 6 October 2013; Accepted 25 November 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 S. Temir and A. Kilic¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let 𝑋 be a real Banach space and 𝐾 a nonempty closed convex subset of 𝑋.Let𝑇𝑖 : 𝐾 → 𝐾(𝑖= 1,2,...,𝑚)be 𝑚 asymptotically ∞ 𝑚 nonexpansive mappings with sequence {𝑘𝑛}⊂[1,∞), ∑𝑛=1(𝑘𝑛 −1)<∞,andF = ⋂𝑖=1 𝐹(𝑇𝑖) =0̸ ,where𝐹 is the set of fixed points ∞ ∞ ∞ of 𝑇𝑖.Supposethat{𝑎𝑖𝑛}𝑛=1, {𝑏𝑖𝑛}𝑛=1, 𝑖 = 1, 2, . . . ,𝑚 are appropriate sequences in [0, 1] and {𝑢𝑖𝑛}𝑛=1, 𝑖 = 1, 2, . . . ,𝑚 are bounded ∞ sequences in 𝐾 such that ∑𝑛=1 𝑏𝑖𝑛 <∞for 𝑖 = 1, 2, . . ..Wegive ,𝑚 {𝑥𝑛} defined by 𝑥1 ∈𝐾,𝑥𝑛+1 =(1−𝑎1𝑛 −𝑏1𝑛)𝑦𝑛+𝑚−2 + 𝑛 𝑛 𝑛 𝑎1𝑛𝑇1 𝑦𝑛+𝑚−2 +𝑏1𝑛𝑢1𝑛,𝑦𝑛+𝑚−2 =(1−𝑎2𝑛 −𝑏2𝑛)𝑦𝑛+𝑚−3 +𝑎2𝑛𝑇2 𝑦𝑛+𝑚−3 +𝑏2𝑛𝑢2𝑛,...,𝑦𝑛+2 =(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛)𝑦𝑛+1 +𝑎(𝑚−2)𝑛𝑇𝑚−2𝑦𝑛+1 + 𝑛 𝑛 𝑏(𝑚−2)𝑛𝑢(𝑚−2)𝑛,𝑦𝑛+1 =(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛)𝑦𝑛 +𝑎(𝑚−1)𝑛𝑇𝑚−1𝑦𝑛 +𝑏(𝑚−1)𝑛𝑢(𝑚−1)𝑛,𝑦𝑛 =(1−𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑥𝑛 +𝑎𝑚𝑛𝑇𝑚𝑥𝑛 +𝑏𝑚𝑛𝑢𝑚𝑛,𝑚≥2,𝑛≥1. The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite familyof asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.

1. Introduction L-Lipschitzian if there exists a positive constant 𝐿 such that 󵄩 𝑛 𝑛 󵄩 󵄩 󵄩 Let 𝐾 be a nonempty subset of a real Banach space 𝑋 and let 󵄩𝑇 𝑥−𝑇 𝑦󵄩 ≤𝐿󵄩𝑥−𝑦󵄩 (3) 𝑇:𝐾be →𝐾 a mapping. Let 𝐹(𝑇)={𝑥∈𝐾:𝑇𝑥=𝑥}be the set of fixed points of 𝑇. for all 𝑥, 𝑦 ∈𝐾 and 𝑛≥1. Amapping𝑇:𝐾is →𝐾 called nonexpansive if It is easy to see that if 𝑇 is asymptotically nonexpansive, then it is uniformly 𝐿-Lipschitzian with the uniform Lipschitz 𝐿= {𝑘 :𝑛≥1} 󵄩 󵄩 󵄩 󵄩 constant sup 𝑛 . 󵄩𝑇𝑥 − 𝑇𝑦󵄩 ≤ 󵄩𝑥−𝑦󵄩 (1) The class of asymptotically nonexpansive mappings which is an important generalization of the class nonex- pansive maps was introduced by Goebel and Kirk [1]. They for all 𝑥, 𝑦.Similarly, ∈𝐾 𝑇 is called asymptotically proved that every asymptotically nonexpansive self-mapping nonexpansive if there exists a sequence {𝑘𝑛}⊂[1,∞)with of a nonempty closed convex bounded subset of a uniformly lim𝑛→∞𝑘𝑛 =1such that convex Banach space has a fixed point. The main tool for approximation of fixed points of 󵄩 𝑛 𝑛 󵄩 󵄩 󵄩 󵄩𝑇 𝑥−𝑇 𝑦󵄩 ≤𝑘𝑛 󵄩𝑥−𝑦󵄩 (2) generalizations of nonexpansive mappings remains itera- tive technique. Iterative techniques for nonexpansive self- mappings in Banach spaces including Mann type (one-step), for all 𝑥, 𝑦 ∈𝐾 and 𝑛≥1.Themapping𝑇 is called uniformly Ishikawa type (two-step), and three-step iteration processes 2 Abstract and Applied Analysis have been studied extensively by various authors; see, for . . example, ([2–8]). Recently, Chidume and Ali [9] defined (4)andcon- 𝑦𝑛+2 =(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛)𝑦𝑛+1 structed the sequence for the approximation of common 𝑛 fixed points of finite families of asymptotically nonexpansive +𝑎(𝑚−2)𝑛𝑇𝑚−2𝑦𝑛+1 +𝑏(𝑚−2)𝑛𝑢(𝑚−2)𝑛, mappings. Yıldırım and Ozdemir¨ [10]introducedaniteration scheme for approximating common fixed points of a finite 𝑦𝑛+1 =(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛)𝑦𝑛 family of asymptotically quasi-nonexpansive self-mappings +𝑎 𝑇𝑛 𝑦 +𝑏 𝑢 , and proved some strong and weak convergence theorems for (𝑚−1)𝑛 𝑚−1 𝑛 (𝑚−1)𝑛 (𝑚−1)𝑛 such mappings in uniformly convex Banach spaces. Quan et 𝑦 =(1−𝑎 −𝑏 )𝑥 +𝑎 𝑇𝑛 𝑥 al. [11] studied sufficient and necessary conditions for finite 𝑛 𝑚𝑛 𝑚𝑛 𝑛 𝑚𝑛 𝑚 𝑛 step iterative schemes with mean errors for a finite family +𝑏𝑚𝑛𝑢𝑚𝑛, 𝑚≥2,𝑛≥1. of asymptotically quasi-nonexpansive mappings in Banach (5) spaces to converge to a common fixed point. Peng [12]proved the convergence of finite step iterative schemes with mean errors for asymptotically nonexpansive mappings in Banach 2. Preliminaries spaces. More recently Kızıltunc¸andTemir[13]introduced 𝑋 𝐾 and studied a new iteration process for a finite family of Let be a real Banach space, a nonempty closed convex 𝑋 𝐹(𝑇) 𝑇 nonself asymptotically nonexpansive mappings with errors in subset of ,and the set of fixed points of .ABanach 𝑋 Banach spaces. space is said to be uniformly convex if the modulus of 𝑋 In [9], the authors introduced an iterative process for convexity of a finite family of asymptotically nonexpansive mappings as 󵄩 󵄩 󵄩𝑥+𝑦󵄩 󵄩 󵄩 󵄩 󵄩 follows: 𝛿 (𝜀) = {1−󵄩 󵄩 : ‖𝑥‖ = 󵄩𝑦󵄩 =1,󵄩𝑥−𝑦󵄩 =𝜀} >0 inf 2 󵄩 󵄩 󵄩 󵄩 𝑥 ∈𝐾, 1 (6) 𝑛 𝑥𝑛+1 =(1−𝑎1𝑛)𝑥𝑛 +𝑎1𝑛𝑇1 𝑦𝑛+𝑚−2, for all 0<𝜀≤2(i.e., 𝛿 : (0, 2] → [0, 1]).Recallthata 𝑛 𝑋 𝑦𝑛+𝑚−2 =(1−𝑎2𝑛)𝑥𝑛 +𝑎2𝑛𝑇2 𝑦𝑛+𝑚−3, Banach space is said to satisfy Opial’s condition if, for each {𝑥 } 𝑋 𝑥 ⇀𝑥 (4) sequence 𝑛 in ,thecondition 𝑛 implies that . . 󵄩 󵄩 󵄩 󵄩 . 󵄩𝑥 −𝑥󵄩 < 󵄩𝑥 −𝑦󵄩 lim𝑛→∞ inf 󵄩 𝑛 󵄩 lim𝑛→∞ inf 󵄩 𝑛 󵄩 (7) 𝑛 𝑦𝑛+1 =(1−𝑎(𝑚−1)𝑛)𝑥𝑛 +𝑎(𝑚−1)𝑛𝑇𝑚−1𝑦𝑛, 𝑛 for all 𝑦∈𝑋with 𝑦 =𝑥̸ .Itiswellknownthatall𝑙𝑟 spaces for 𝑦𝑛 =(1−𝑎𝑚𝑛)𝑥𝑛 +𝑎𝑚𝑛𝑇 𝑥𝑛, if 𝑚 ≥ 2, 𝑛 ≥1, 𝑚 1<𝑟<∞have this property. However, the 𝐿𝑟 spaces do not have unless 𝑟=2. 𝑇 ,𝑇 ,...,𝑇 :𝐾 → 𝐾 𝑚 where 1 2 𝑚 are asymptotically Amapping𝑇:𝐾is →𝐾 said to be semicompact if, for {𝑎 }⊂[0,1] 𝑖=1,...,𝑚 nonexpansive mappings and 𝑖𝑛 for . any bounded sequence {𝑥𝑛} in 𝐾 such that ‖𝑥𝑛 −𝑇𝑥𝑛‖→0 Inspiredandmotivatedbythesefacts,itisourpurposein 𝑛→∞ {𝑥 } {𝑥 } as , there exists a subsequence say 𝑛𝑗 of 𝑛 such this paper to construct an iteration scheme for approximat- {𝑥 } 𝑥∗ 𝐾 𝑇 that 𝑛𝑗 converges strongly to some in . is said to be ing common fixed points of finite family of asymptotically {𝑥 } nonexpansive mappings and study weak and some strong completely continuous if for every bounded sequence 𝑛 in 𝐾, there exists a subsequence say {𝑥𝑛 } of {𝑥𝑛} such that the convergence theorems for such mappings in real Banach 𝑗 {𝑇𝑥 } spaces. sequence 𝑛𝑗 converges strongly to some element of the Let 𝑋 be a real Banach space and 𝐾 anonemptyclosed range of 𝑇. convex subset of 𝑋.Let𝑇𝑖 : 𝐾 → 𝐾 (𝑖 = 1,2,...,𝑚) The following lemmas were given in14 [ , 15], respectively, be 𝑚 asymptotically nonexpansive mappings with sequence and we need them to prove our main results. ∞ 𝑚 {𝑘𝑛}⊂[1,∞), ∑𝑛=1(𝑘𝑛 −1)< ∞,andF =⋂𝑖=1 𝐹(𝑇𝑖) =0̸ . ∞ ∞ Lemma 1. {𝑠 } {𝑡 } {𝜎 } Suppose that {𝑎𝑖𝑛}𝑛=1, {𝑏𝑖𝑛}𝑛=1, 𝑖 = 1,2,...,𝑚are appropriate Let 𝑛 , 𝑛 ,and 𝑛 be sequences of nonnegative ∞ 𝑛≥1 sequences in [0, 1] and {𝑢𝑖𝑛}𝑛=1, 𝑖 = 1,2,...,𝑚are bounded real numbers satisfying the following conditions: for all , ∞ 𝑠 ≤(1+𝜎)𝑠 +𝑡 ∑∞ 𝜎 <∞ ∑∞ 𝑡 <∞ sequences in 𝐾 such that ∑𝑛=1 𝑏𝑖𝑛 <∞for 𝑖=1,2,...,𝑚.Let 𝑛+1 𝑛 𝑛 𝑛,where 𝑛=1 𝑛 and 𝑛=1 𝑛 . {𝑥𝑛} be defined by Then 𝑠 𝑥1 ∈𝐾, (i) lim𝑛→∞ 𝑛 exists;

𝑛 (ii) in particular, if {𝑠𝑛} has a subsequence {𝑠𝑛 } converging 𝑥𝑛+1 =(1−𝑎1𝑛 −𝑏1𝑛)𝑦𝑛+𝑚−2 +𝑎1𝑛𝑇1 𝑦𝑛+𝑚−2 +𝑏1𝑛𝑢1𝑛, 𝑗 to 0, then lim𝑛→∞𝑠𝑛 =0. 𝑦𝑛+𝑚−2 =(1−𝑎2𝑛 −𝑏2𝑛)𝑦𝑛+𝑚−3 Lemma 2. Let 𝑝>1and 𝐶>0be two fixed numbers. Then a 𝑛 +𝑎2𝑛𝑇2 𝑦𝑛+𝑚−3 +𝑏2𝑛𝑢2𝑛, Banach space 𝑋 is uniformly convex if and only if there exists a Abstract and Applied Analysis 3 continuous, strictly increasing, convex function 𝑔:[0,∞)→ For each 𝑛≥1,using(5), we have [0, ∞) with 𝑔(0) = 0 such that 󵄩 󵄩 󵄩𝑦𝑛 −𝑝󵄩 󵄩 󵄩𝑝 𝑝 󵄩 󵄩𝑝 󵄩𝜆𝑥 + (1 − 𝜆)𝑦󵄩 ≤𝜆‖𝑥‖ + (1−𝜆) 󵄩𝑦󵄩 󵄩 𝑛 󵄩 = 󵄩((1 − 𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑥𝑛 +𝑎𝑚𝑛𝑇𝑚𝑥𝑛 +𝑏𝑚𝑛𝑢𝑚𝑛)−𝑝󵄩 󵄩 󵄩 (8) −𝑤 𝜆 𝑔(󵄩𝑥−𝑦󵄩) 𝑝 ( ) 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛 󵄩 ≤(1−𝑎𝑚𝑛 −𝑏𝑚𝑛) 󵄩𝑥𝑛 −𝑝󵄩 +𝑎𝑚𝑛 󵄩𝑇𝑚𝑥𝑛 −𝑝󵄩 for all 𝑥, 𝑦𝐶 ∈𝐵 := {𝑥 ∈ 𝑋 : ‖𝑥‖ ≤𝐶},and𝜆∈[0,1],where 󵄩 󵄩 𝑝 𝑝 +𝑏𝑚𝑛 󵄩𝑢𝑚𝑛 −𝑝󵄩 𝑤𝑝(𝜆) = 𝜆(1 −𝜆) +𝜆 (1 − 𝜆). 󵄩 󵄩 󵄩 󵄩 ≤(1−𝑎𝑚𝑛 −𝑏𝑚𝑛) 󵄩𝑥𝑛 −𝑝󵄩 +𝑎𝑚𝑛𝑘𝑛 󵄩𝑥𝑛 −𝑝󵄩 The following lemmas were proved in [3]. 󵄩 󵄩 +𝑏𝑚𝑛 󵄩𝑢𝑚𝑛 −𝑝󵄩 Lemma 3. Let 𝑋 be a uniformly convex Banach space and 󵄩 󵄩 𝐵𝐶 :={𝑥∈𝑋:‖𝑥‖≤𝐶},𝐶>0. Then there exists a ≤𝑘𝑛 󵄩𝑥𝑛 −𝑝󵄩 +𝑏𝑚𝑛𝑀, continuous, strictly increasing, convex function 𝑔:[0,∞)→ 󵄩 󵄩 󵄩𝑦 −𝑝󵄩 [0, ∞) with 𝑔(0) = 0 such that 󵄩 𝑛+1 󵄩 󵄩 󵄩 𝑛 󵄩 󵄩2 2 󵄩 󵄩2 = 󵄩((1 − 𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛)𝑦𝑛 +𝑎(𝑚−1)𝑛𝑇 𝑦𝑛 󵄩𝜆𝑥 + 𝜇𝑦 + ]𝑧󵄩 ≤𝜆‖𝑥‖ +𝜇󵄩𝑦󵄩 󵄩 (𝑚−1) (9) 󵄩 2 󵄩 󵄩 +𝑏 𝑢 )−𝑝󵄩 + ]‖𝑧‖ −(𝜆𝜇)𝑔(󵄩𝑥−𝑦󵄩) (𝑚−1)𝑛 (𝑚−1)𝑛 󵄩 󵄩 󵄩 ≤(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛) 󵄩𝑦𝑛 −𝑝󵄩 +𝑎(𝑚−1)𝑛 for all 𝑥, 𝑦,𝐶 𝑧∈𝐵 and 𝜆, 𝜇, ] ∈ [0, 1] with 𝜆+𝜇+] =1. 󵄩 𝑛 󵄩 󵄩 󵄩 × 󵄩𝑇 𝑦 −𝑝󵄩 +𝑏 󵄩𝑢 −𝑝󵄩 Lemma 4. Let 𝑋 be a uniformly convex Banach space, 𝐾 a 󵄩 (𝑚−1) 𝑛 󵄩 (𝑚−1)𝑛 󵄩 (𝑚−1)𝑛 󵄩 nonempty closed convex subset of 𝑋,and𝑇:𝐾an →𝐾 󵄩 󵄩 ≤(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛) 󵄩𝑦𝑛 −𝑝󵄩 +𝑎(𝑚−1)𝑛𝑘𝑛 asymptotically nonexpansive mapping. Then 𝐼−𝑇(𝐼 is identity ∗ 󵄩 󵄩 󵄩 󵄩 mapping) is demiclosed at zero; that is, if 𝑥𝑛 →𝑥 weakly and 󵄩 󵄩 󵄩 󵄩 ∗ × 󵄩𝑦𝑛 −𝑝󵄩 +𝑏(𝑚−1)𝑛 󵄩𝑢(𝑚−1)𝑛 −𝑝󵄩 𝑥𝑛 −𝑇𝑥𝑛 →0strongly, then 𝑥 ∈ 𝐹(𝑇),where𝐹(𝑇) is the set 𝑇 󵄩 󵄩 of fixed points of . ≤𝑘𝑛 󵄩𝑦𝑛 −𝑝󵄩 +𝑏(𝑚−1)𝑛𝑀 2 󵄩 󵄩 Definition 5. Afamily{𝑇𝑖 : 𝑖 ∈ {1,...,𝑚}}of asymptotically ≤𝑘 󵄩𝑥 −𝑝󵄩 +𝑏 𝑀𝑘 +𝑏 𝑀, 𝑚 𝑛 󵄩 𝑛 󵄩 𝑚𝑛 𝑛 (𝑚−1)𝑛 nonexpansive mappings on 𝐾 with F = ⋂ 𝐹(𝑇𝑖) =0̸ is said 𝑖=1 󵄩 󵄩 to satisfy condition (A) on 𝐾 if there exists a nondecreasing 󵄩𝑦𝑛+2 −𝑝󵄩 function 𝑓 : [0, ∞) → [0, ∞) with 𝑓(0) = 0, 𝑓(𝑟) >0,forall 󵄩 = 󵄩((1 − 𝑎 −𝑏 )𝑦 +𝑎 𝑇𝑛 𝑦 𝑟 ∈ (0, ∞) such that max1≤𝑖≤𝑚‖𝑥−𝑇𝑖𝑥‖≥ 𝑓(𝑑(𝑥, 𝐹)) for all 󵄩 (𝑚−2)𝑛 (𝑚−2)𝑛 𝑛+1 (𝑚−2)𝑛 (𝑚−2) 𝑛+1 𝑥∈𝐾. 󵄩 󵄩 +𝑏(𝑚−2)𝑛𝑢(𝑚−2)𝑛)−𝑝󵄩 󵄩 󵄩 3. Main Results ≤(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛) 󵄩𝑦𝑛+1 −𝑝󵄩 󵄩 𝑛 󵄩 In this section, we prove weak and strong convergence of +𝑎 󵄩𝑇 𝑦 −𝑝󵄩 the iterative sequence generated by iterative scheme (5)toa (𝑚−2)𝑛 󵄩 (𝑚−2) 𝑛+1 󵄩 common element of the sets of fixed points of a finite family 󵄩 󵄩 +𝑏(𝑚−2)𝑛 󵄩𝑢(𝑚−2)𝑛 −𝑝󵄩 of asymptotically nonexpansive mappings in a real Banach 󵄩 󵄩 space. ≤(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛) 󵄩𝑦𝑛+1 −𝑝󵄩 Lemma 6. 𝑋 𝐾 󵄩 󵄩 Let be a real Banach space and anonempty +𝑎(𝑚−2)𝑛𝑘𝑛 󵄩𝑦𝑛+1 −𝑝󵄩 𝑋 𝑇 :𝐾 → 𝐾 (𝑖=1,2,...,𝑚) closed convex subset of .Let 𝑖 be 󵄩 󵄩 +𝑏 󵄩𝑢 −𝑝󵄩 𝑚 asymptotically nonexpansive mappings with sequence {𝑘𝑛}⊂ (𝑚−2)𝑛 󵄩 (𝑚−2)𝑛 󵄩 ∞ 𝑚 [1, ∞), ∑𝑛=1(𝑘𝑛 −1)<∞,andF =⋂𝑖=1 𝐹(𝑇𝑖) =0̸ .Suppose 󵄩 󵄩 ∞ ∞ 󵄩 󵄩 ≤𝑘𝑛 󵄩𝑦𝑛+1 −𝑝󵄩 +𝑏(𝑚−2)𝑛𝑀 that {𝑎𝑖𝑛}𝑛=1, {𝑏𝑖𝑛}𝑛=1, 𝑖=1,2,...,𝑚are appropriate sequences 󵄩 󵄩 ∞ in [0, 1] and {𝑢𝑖𝑛}𝑛=1, 𝑖=1,2,...,𝑚are bounded sequences in 3 󵄩 󵄩 2 ∞ 󵄩 󵄩 ≤𝑘𝑛 󵄩𝑥𝑛 −𝑝󵄩 +𝑏𝑚𝑛𝑀𝑘𝑛 +𝑘𝑛𝑏(𝑚−1)𝑛𝑀+𝑏(𝑚−2)𝑛𝑀, 𝐾 such that ∑𝑛=1 𝑏𝑖𝑛 <∞for 𝑖=1,2,...,𝑚.Let{𝑥𝑛} be given {𝑥 } ‖𝑥 −𝑝‖ by (5).Then 𝑛 is bounded and lim𝑛→∞ 𝑛 exists for . 𝑝∈F. . ∞ 󵄩 󵄩 Proof. For any given 𝑝∈F,since{𝑢𝑖𝑛}𝑛=1, 𝑖=1,2,...,𝑚are 󵄩𝑦𝑛+𝑚−2 −𝑝󵄩 𝐾 bounded sequences in ,let 󵄩 𝑛 󵄩 = 󵄩((1 − 𝑎2𝑛 −𝑏2𝑛)𝑦𝑛+𝑚−3 +𝑎2𝑛𝑇2 𝑦𝑛+𝑚−3 +𝑏2𝑛𝑢2𝑛)−𝑝󵄩 󵄩 󵄩 𝑀= sup 󵄩𝑢𝑖𝑛 −𝑝󵄩 . 󵄩 󵄩 𝑛≥1, 𝑖=1,2,...,𝑚 (10) ≤(1−𝑎2𝑛 −𝑏2𝑛) 󵄩𝑦𝑛+𝑚−3 −𝑝󵄩 +𝑎2𝑛 4 Abstract and Applied Analysis

󵄩 𝑛 󵄩 󵄩 󵄩 × 󵄩𝑇2 𝑦𝑛+𝑚−3 −𝑝󵄩 +𝑏2𝑛 󵄩𝑢2𝑛 −𝑝󵄩 for 𝑖=1,...,𝑚.Then 󵄩 󵄩 󵄩 󵄩 ≤(1−𝑎 −𝑏 ) 󵄩𝑦 −𝑝󵄩 +𝑎 𝑘 󵄩𝑇 𝑥 −𝑥 󵄩 =0, (𝑖=1,2,...,𝑚) . 2𝑛 2𝑛 󵄩 𝑛+𝑚−3 󵄩 2𝑛 𝑛 𝑛→∞lim 󵄩 𝑖 𝑛 𝑛󵄩 (16) 󵄩 󵄩 󵄩 󵄩 × 󵄩𝑦 −𝑝󵄩 +𝑏 󵄩𝑢 −𝑝󵄩 󵄩 𝑛+𝑚−3 󵄩 2𝑛 󵄩 2𝑛 󵄩 Proof. Let 𝑝∈F.ThenbyLemma6,lim𝑛→∞‖𝑥𝑛 −𝑝‖ {𝑢 }∞ 𝑖 = 1,2,...,𝑚 󵄩 󵄩 exists. Since 𝑖𝑛 𝑛=1, are bounded sequences ≤𝑘𝑛 󵄩𝑦𝑛+𝑚−3 −𝑝󵄩 +𝑏2𝑛𝑀 󵄩 󵄩 in 𝐾,let𝑀=sup𝑛≥1, 𝑖=1,2,...,𝑚‖𝑢𝑖𝑛 −𝑝‖;moreover,itfollows (𝑚−1) 󵄩 󵄩 (𝑚−2) (𝑚−3) that {𝑦𝑛+𝑚−𝑖 −𝑝}is also bounded for each 𝑖 ∈ {2,3,...,𝑚}, ≤𝑘𝑛 󵄩𝑥𝑛 −𝑝󵄩 +𝑘𝑛 𝑏𝑚𝑛𝑀+𝑘𝑛 𝑏(𝑚−1)𝑛𝑀 and hence {(𝑢(𝑚−𝑖+1)𝑛 −𝑦𝑛+𝑖−1)} is also bounded for 𝑖∈ {1,2,...,𝑚}.Byusing(5), we obtain +⋅⋅⋅+𝑘𝑛𝑏3𝑛𝑀+𝑏2𝑛𝑀, 󵄩 󵄩2 󵄩 󵄩 󵄩𝑦 −𝑝󵄩 󵄩𝑥𝑛+1 −𝑝󵄩 󵄩 𝑛 󵄩 󵄩 𝑛 󵄩 󵄩 𝑛 󵄩2 = 󵄩((1 − 𝑎1𝑛 −𝑏1𝑛)𝑦𝑛+𝑚−2 +𝑎1𝑛𝑇1 𝑦𝑛+𝑚−2 +𝑏1𝑛𝑢1𝑛)−𝑝󵄩 = 󵄩((1 − 𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑥𝑛 +𝑎𝑚𝑛𝑇𝑚𝑥𝑛 +𝑏𝑚𝑛𝑢𝑚𝑛)−𝑝󵄩 󵄩 󵄩 ≤(1−𝑎 −𝑏 ) 󵄩𝑦 −𝑝󵄩 +𝑎 󵄩 󵄩2 󵄩 𝑛 󵄩2 1𝑛 1𝑛 󵄩 𝑛+𝑚−2 󵄩 1𝑛 ≤(1−𝑎𝑚𝑛 −𝑏𝑚𝑛) 󵄩𝑥𝑛 −𝑝󵄩 +𝑎𝑚𝑛󵄩𝑇𝑚𝑥𝑛 −𝑝󵄩 󵄩 𝑛 󵄩 󵄩 󵄩 × 󵄩𝑇1 𝑦𝑛+𝑚−2 −𝑝󵄩 +𝑏1𝑛 󵄩𝑢1𝑛 −𝑝󵄩 󵄩 󵄩2 +𝑏𝑚𝑛󵄩𝑢𝑚𝑛 −𝑝󵄩 󵄩 󵄩 ≤(1−𝑎1𝑛 −𝑏1𝑛) 󵄩𝑦𝑛+𝑚−2 −𝑝󵄩 +𝑎1𝑛𝑘𝑛 󵄩 𝑛 󵄩 −(1−𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩) 󵄩 󵄩 󵄩 󵄩 × 󵄩𝑦 −𝑝󵄩 +𝑏 󵄩𝑢 −𝑝󵄩 󵄩 𝑛+𝑚−2 󵄩 1𝑛 󵄩 1𝑛 󵄩 󵄩 󵄩2 2 ≤(1−𝑎𝑚𝑛 −𝑏𝑚𝑛) 󵄩𝑥𝑛 −𝑝󵄩 +𝑎𝑚𝑛𝑘𝑛 󵄩 󵄩 ≤𝑘𝑛 󵄩𝑦𝑛+𝑚−2 −𝑝󵄩 +𝑏1𝑛𝑀 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩 × 󵄩𝑥𝑛 −𝑝󵄩 +𝑏𝑚𝑛󵄩𝑢𝑛𝑚 −𝑝󵄩 ≤𝑘𝑚 󵄩𝑥 −𝑝󵄩 +𝑘(𝑚−1)𝑏 𝑀+𝑘(𝑚−2)𝑏 𝑀 𝑛 󵄩 𝑛 󵄩 𝑛 𝑚𝑛 𝑛 (𝑚−1)𝑛 󵄩 󵄩 −(1−𝑎 −𝑏 )𝑎 𝑔(󵄩𝑇𝑛 𝑥 −𝑥 󵄩) 2 𝑚𝑛 𝑚𝑛 𝑚𝑛 󵄩 𝑚 𝑛 𝑛󵄩 +⋅⋅⋅+𝑘𝑛𝑏3𝑛𝑀+𝑘𝑛𝑏2𝑛𝑀+𝑏1𝑛𝑀. 2󵄩 󵄩2 2 (11) ≤𝑘𝑛󵄩𝑥𝑛 −𝑝󵄩 +𝑏𝑚𝑛𝑀 −(1−𝑎𝑚𝑛 −𝑏𝑚𝑛) 󵄩 𝑛 󵄩 Then we have ×𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩), 󵄩 󵄩 󵄩 󵄩 𝑚 󵄩𝑥 −𝑝󵄩 ≤𝑘𝑚 󵄩𝑥 −𝑝󵄩 +𝑀∑𝑘(𝑖−1)𝑏 , 󵄩 󵄩2 󵄩 𝑛+1 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 𝑖𝑛 (12) 󵄩𝑦𝑛+1 −𝑝󵄩 𝑖=1 󵄩 = 󵄩((1 − 𝑎 −𝑏 )𝑦 +𝑎 which leads to 󵄩 (𝑚−1)𝑛 (𝑚−1)𝑛 𝑛 (𝑚−1)𝑛 󵄩 󵄩 𝑚 󵄩 󵄩 󵄩2 󵄩𝑥 −𝑝󵄩 ≤ (1+(𝑘 −1)) 󵄩𝑥 −𝑝󵄩 +𝜑 ,𝑛≥1, ×𝑇𝑛 𝑦 +𝑏 𝑢 )−𝑝󵄩 󵄩 𝑛+1 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 (13) (𝑚−1) 𝑛 (𝑚−1)𝑛 (𝑚−1)𝑛 󵄩 󵄩 󵄩2 where ≤(1−𝑎 −𝑏 ) 󵄩𝑦 −𝑝󵄩 𝑚 (𝑚−1)𝑛 (𝑚−1)𝑛 󵄩 𝑛 󵄩 (𝑖−1) 𝜑𝑛 =𝑀∑𝑘 𝑏𝑖𝑛. 󵄩 󵄩2 󵄩 󵄩2 𝑛 (14) +𝑎 󵄩𝑇𝑛 𝑦 −𝑝󵄩 +𝑏 󵄩𝑢 −𝑝󵄩 𝑖=1 (𝑚−1)𝑛󵄩 (𝑚−1) 𝑛 󵄩 (𝑚−1)𝑛󵄩 (𝑚−1)𝑛 󵄩 𝑚 𝑚−1 󵄩 󵄩 Since 𝑡 −1≤𝑚𝑡 (𝑡 − 1) for all 𝑡≥1, the only assumption −(1−𝑎 −𝑏 )𝑎 𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩) ∞ (𝑚−1)𝑛 (𝑚−1)𝑛 (𝑚−1)𝑛 󵄩 (𝑚−1) 𝑛 𝑛󵄩 ∑𝑛=1(𝑘𝑛−1) < ∞ is enough for the boundedness for {𝑘𝑛},then 𝑘 ⊂[1,𝐷] 𝑛≥1 𝐷 𝑘𝑚 −1≤ 󵄩 󵄩2 𝑛 ,forall ,andforsome .Hence 𝑛 ≤(1−𝑎 −𝑏 ) 󵄩𝑦 −𝑝󵄩 𝑚−1 ∞ 𝑚 (𝑚−1)𝑛 (𝑚−1)𝑛 󵄩 𝑛 󵄩 𝑚𝐷 (𝑘𝑛−1) holds for all 𝑛≥1. Therefore ∑𝑛=1(𝑘𝑛 −1) < ∞ ∞ 2󵄩 󵄩2 2 and also ∑𝑛=1 𝜑𝑛 <∞.Equation(13)andLemma1 guarantee 󵄩 󵄩 +𝑎(𝑚−1)𝑛𝑘𝑛󵄩𝑦𝑛 −𝑝󵄩 +𝑏(𝑚−1)𝑛𝑀 that the sequence {𝑥𝑛} is bounded and lim𝑛→∞‖𝑥𝑛 −𝑝‖exists. 󵄩 󵄩 −(1−𝑎 −𝑏 )𝑎 𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩) (𝑚−1)𝑛 (𝑚−1)𝑛 (𝑚−1)𝑛 󵄩 (𝑚−1) 𝑛 𝑛󵄩 Theorem 7. 𝑋 Let be a real uniformly convex Banach space 2󵄩 󵄩2 2 ≤𝑘𝑛󵄩𝑦𝑛 −𝑝󵄩 +𝑏(𝑚−1)𝑛𝑀 −(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛) and 𝐾 anonemptyclosedconvexsubsetof𝑋.Let𝑇𝑖 : 𝐾 → 𝐾 (𝑖 = 1,2,...,𝑚)be 𝑚 asymptotically nonexpansive 󵄩 𝑛 󵄩 ∞ 󵄩 󵄩 ×𝑎(𝑚−1)𝑛𝑔(󵄩𝑇(𝑚−1)𝑦𝑛 −𝑦𝑛󵄩) mappings with sequence {𝑘𝑛}⊂[1,∞), ∑𝑛=1(𝑘𝑛 −1)< ∞ 󵄩 󵄩 F =⋂𝑚 𝐹(𝑇 ) =0̸ {𝑎 }∞ {𝑏 }∞ 𝑖= and 𝑖=1 𝑖 .Supposethat 𝑖𝑛 𝑛=1, 𝑖𝑛 𝑛=1, 󵄩 󵄩2 ∞ ≤𝑘2 (𝑘2󵄩𝑥 −𝑝󵄩 +𝑏 𝑀2 −(1−𝑎 −𝑏 ) 1,2,...,𝑚are appropriate sequences in [0, 1] and {𝑢𝑖𝑛}𝑛=1, 𝑖= 𝑛 𝑛󵄩 𝑛 󵄩 𝑚𝑛 𝑚𝑛 𝑚𝑛 ∞ 1,2,...,𝑚are bounded sequences in 𝐾 such that ∑ 𝑏𝑖𝑛 <∞ 𝑛=1 󵄩 𝑛 󵄩 2 for 𝑖=1,2,...,𝑚.Let{𝑥𝑛} be given by (5).Supposethat ×𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩))+𝑏(𝑚−1)𝑛𝑀 󵄩 󵄩 0

󵄩 󵄩2 󵄩 󵄩2 󵄩𝑦𝑛+1 −𝑝󵄩 󵄩𝑥𝑛+1 −𝑝󵄩 4󵄩 󵄩2 2 2 2 𝑚−1 ≤𝑘𝑛󵄩𝑥𝑛 −𝑝󵄩 +𝑘𝑛𝑏𝑚𝑛𝑀 +𝑏(𝑚−1)𝑛𝑀 2𝑚󵄩 󵄩2 2 2𝑖 ≤𝑘 󵄩𝑥 −𝑝󵄩 +𝑀 ∑ 𝑘 𝑏 󵄩 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 (𝑖+1)𝑛 󵄩 𝑛 󵄩 𝑖=0 −(1−𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩) 󵄩 󵄩 𝑚−2 󵄩 𝑛 󵄩 (18) −(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛)𝑎(𝑚−1)𝑛𝑔(󵄩𝑇(𝑚−1)𝑦𝑛 −𝑦𝑛󵄩), 󵄩 󵄩 − ∑ ((1 − 𝑎(𝑖+1)𝑛 −𝑏(𝑖+1)𝑛)𝑎(𝑖+1)𝑛 󵄩 󵄩2 𝑖=0 󵄩𝑦𝑛+2 −𝑝󵄩 󵄩 𝑛 󵄩 ×𝑔(󵄩𝑇𝑖+1𝑦𝑛+𝑚−𝑖−2 −𝑦𝑛+𝑚−𝑖−2󵄩)) 󵄩 𝑛 = 󵄩((1 − 𝑎 −𝑏 )𝑦 +𝑎 𝑇 𝑦 󵄩 (𝑚−2)𝑛 (𝑚−2)𝑛 𝑛+1 (𝑚−2)𝑛 (𝑚−2) 𝑛+1 󵄩 𝑛 󵄩 −(1−𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩). 󵄩2 󵄩 +𝑏(𝑚−2)𝑛𝑢(𝑚−2)𝑛)−𝑝󵄩 0≤𝜃2 −1≤2𝜃(𝜃−1) 𝜃≥1 󵄩 󵄩2 Note that for all , the assumption ≤(1−𝑎 −𝑏 ) 󵄩𝑦 −𝑝󵄩 ∞ ∞ 2 (𝑚−2)𝑛 (𝑚−2)𝑛 󵄩 𝑛+1 󵄩 ∑𝑛=1(𝑘𝑛 −1) < ∞implies that ∑𝑛=1(𝑘𝑛 −1) < ∞.Since{𝑘𝑛} is 𝐷>0 𝑘 ∈ [1, 𝐷], 𝑛 ≥1 󵄩 󵄩2 bounded, there exists such that 𝑛 .Then +𝑎 󵄩𝑇𝑛 𝑦 −𝑝󵄩 𝑘2𝑚 − 1 ≤ 2𝑚𝐷2𝑚−1(𝑘 −1) 𝑛≥1 (𝑚−2)𝑛󵄩 (𝑚−2) 𝑛+1 󵄩 𝑛 𝑛 holds for all . Therefore, the ∑∞ (𝑘 −1) < ∞ ∑∞ (𝑘2𝑚−1) < ∞ 󵄩 󵄩2 assumption 𝑛=1 𝑛 implies that 𝑛=1 𝑛 . +𝑏(𝑚−2)𝑛󵄩𝑢(𝑚−2)𝑛 −𝑝󵄩 Then

−(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛)𝑎(𝑚−2)𝑛 󵄩 󵄩2 (𝑘2𝑚 −1)󵄩𝑥 −𝑝󵄩 ≤ 2𝑚𝐷2𝑚−1𝐶2 (𝑘 −1). 󵄩 󵄩 𝑛 󵄩 𝑛 󵄩 𝑛 (19) ×𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩) 󵄩 (𝑚−2) 𝑛+1 𝑛+1󵄩 󵄩 󵄩2 It follows from (18)and(19)that ≤(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛) 󵄩𝑦𝑛+1 −𝑝󵄩 2󵄩 󵄩2 2 󵄩 𝑛 󵄩 󵄩 󵄩 (1 − 𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑎𝑚𝑛𝑔(󵄩𝑇 𝑥𝑛 −𝑥𝑛󵄩) +𝑎(𝑚−2)𝑛𝑘𝑛󵄩𝑦𝑛+1 −𝑝󵄩 +𝑏(𝑚−2)𝑛𝑀 󵄩 𝑚 󵄩 𝑚−1 −(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛)𝑎(𝑚−2)𝑛 + ∑ ((1 − 𝑎(𝑖+1)𝑛 −𝑏(𝑖+1)𝑛)𝑎(𝑖+1)𝑛 󵄩 𝑛 󵄩 𝑖=0 ×𝑔(󵄩𝑇 𝑦𝑛+1 −𝑦𝑛+1󵄩) 󵄩 (𝑚−2) 󵄩 󵄩 󵄩 ×𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩)) 2󵄩 󵄩2 2 󵄩 𝑖+1 𝑛+𝑚−𝑖−2 𝑛+𝑚−𝑖−2󵄩 (20) ≤𝑘𝑛󵄩𝑦𝑛+1 −𝑝󵄩 +𝑏(𝑚−2)𝑛𝑀 󵄩 󵄩2 󵄩 󵄩2 2𝑚−1 2 ≤ 󵄩𝑥𝑛 −𝑝󵄩 − 󵄩𝑥𝑛+1 −𝑝󵄩 + 2𝑚𝐷 𝐶 (𝑘𝑛 −1) −(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛)𝑎(𝑚−2)𝑛 󵄩 󵄩 𝑚−1 ×𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩) 2 2𝑖 󵄩 (𝑚−2) 𝑛+1 𝑛+1󵄩 +𝑀 ∑ 𝑘𝑛 𝑏(𝑖+1)𝑛. 𝑖=0 2 4󵄩 󵄩2 2 2 2 ≤𝑘𝑛 (𝑘𝑛󵄩𝑥𝑛 −𝑝󵄩 +𝑘𝑛𝑏𝑚𝑛𝑀 +𝑏(𝑚−1)𝑛𝑀 󵄩 𝑛 󵄩 We first obtain that −(1−𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩) 󵄩 𝑛 󵄩 (1 − 𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑎𝑚𝑛𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩) −(1−𝑎(𝑚−1)𝑛 −𝑏(𝑚−1)𝑛)𝑎(𝑚−1)𝑛 󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩2 2𝑚−1 2 ×𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩))+𝑏 𝑀2 ≤ 󵄩𝑥𝑛 −𝑝󵄩 − 󵄩𝑥𝑛+1 −𝑝󵄩 + 2𝑚𝐷 𝐶 (𝑘𝑛 −1) 󵄩 (𝑚−1) 𝑛 𝑛󵄩 (𝑚−2)𝑛 (21) 𝑚−1 2 2𝑖 −(1−𝑎(𝑚−2)𝑛 −𝑏(𝑚−2)𝑛)𝑎(𝑚−2)𝑛 +𝑀 ∑ 𝑘𝑛 𝑏(𝑖+1)𝑛. 󵄩 󵄩 𝑖=0 ×𝑔(󵄩𝑇𝑛 𝑦 −𝑦 󵄩) 󵄩 (𝑚−2) 𝑛+1 𝑛+1󵄩 6󵄩 󵄩2 4 2 2 2 2 Now if 0

It follows from (22)thatforℓ≥𝑛0, It follows from (5)that 󵄩 󵄩 ℓ 󵄩𝑦 −𝑥 󵄩 󵄩 𝑛 󵄩 󵄩 𝑛 𝑛󵄩 ∑ 𝑔(󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩) 󵄩 𝑛 󵄩 𝑛=𝑛0 󵄩 󵄩 = 󵄩(1 − 𝑎𝑚𝑛 −𝑏𝑚𝑛)𝑥𝑛 +𝑎𝑚𝑛𝑇𝑚𝑥𝑛 +𝑏𝑚𝑛𝑢𝑚𝑛 −𝑥𝑛󵄩 (28) ℓ 󵄩 𝑛 󵄩 󵄩 󵄩 1 󵄩 󵄩2 󵄩 󵄩2 ≤𝑎 󵄩𝑇 𝑥 −𝑥 󵄩 +𝑏 󵄩𝑢 −𝑥 󵄩 . ≤ ( ∑ (󵄩𝑥 −𝑝󵄩 − 󵄩𝑥 −𝑝󵄩 ) 𝑚𝑛 󵄩 𝑚 𝑛 𝑛󵄩 𝑚𝑛 󵄩 𝑚𝑛 𝑛󵄩 𝜂(1−𝜂󸀠) 󵄩 𝑛 󵄩 󵄩 𝑛+1 󵄩 𝑛=𝑛0 (23) Equations (24)and(28) imply that ℓ + 2𝑚𝐷2𝑚−1𝐶2 ∑ (𝑘 −1) 󵄩 󵄩 𝑛 󵄩𝑦 −𝑥 󵄩 =0. 𝑛=𝑛0 𝑛→∞lim 󵄩 𝑛 𝑛󵄩 (29)

ℓ 𝑚−1 2 2𝑖 It follows from (5)that +𝑀 ∑ ∑ 𝑘𝑛 𝑏(𝑖+1)𝑛). 𝑛=𝑛 𝑖=0 0 󵄩 󵄩 󵄩𝑥𝑛+1 −𝑦𝑛+𝑚−2󵄩 ∞ 𝑛 Then ∑ 𝑔(‖𝑇 𝑥𝑛 −𝑥𝑛‖)<∞, and therefore 𝑛=𝑛0 𝑚 󵄩 𝑛 𝑛 = 󵄩(1 − 𝑎1𝑛 −𝑏1𝑛)𝑦𝑛+𝑚−2 +𝑎1𝑛𝑇1 𝑦𝑛+𝑚−2 lim𝑛→∞𝑔(‖𝑇𝑚𝑥𝑛 −𝑥𝑛‖)=0,andbypropertyof𝑔,we 𝑛 󵄩 (30) have lim𝑛→∞‖𝑇𝑚𝑥𝑛 −𝑥𝑛‖=0. By a similar method, together 󵄩 +𝑏1𝑛𝑢1𝑛 −𝑦𝑛+𝑚−2󵄩 with (20)andbypropertyof𝑔,wehave 󵄩 󵄩 󵄩 󵄩 ≤𝑎 󵄩𝑇𝑛𝑦 −𝑦 󵄩 +𝑏 󵄩𝑢 −𝑦 󵄩 . 󵄩 󵄩 1𝑛 󵄩 1 𝑛+𝑚−2 𝑛+𝑚−2󵄩 1𝑛 󵄩 1𝑛 𝑛+𝑚−2󵄩 󵄩𝑇𝑛 𝑥 −𝑥 󵄩 𝑛→∞lim 󵄩 𝑚 𝑛 𝑛󵄩 󵄩 𝑛 󵄩 Thus, (24)and(30)guaranteethat = lim 󵄩𝑇𝑚−1𝑦𝑛 −𝑦𝑛󵄩 𝑛→∞ 󵄩 󵄩 󵄩 󵄩 󵄩𝑥 −𝑦 󵄩 =0. = 󵄩𝑇𝑛 𝑦 −𝑦 󵄩 𝑛→∞lim 󵄩 𝑛+1 𝑛+𝑚−2󵄩 (31) 𝑛→∞lim 󵄩 𝑚−2 𝑛+1 𝑛+1󵄩 . Continuing in this fashion, for each 𝑖=2,...,𝑚we get, . (24) 󵄩 󵄩 󵄩 𝑛 󵄩 󵄩𝑥 −𝑦 󵄩 =0, = 󵄩𝑇 𝑦 −𝑦 󵄩 𝑛→∞lim 󵄩 𝑛+1 𝑛+𝑖−2󵄩 (32) 𝑛→∞lim 󵄩 𝑖 𝑛+𝑚−𝑖−1 𝑛+𝑚−𝑖−1󵄩 󵄩 󵄩 󵄩𝑥 −𝑥 󵄩 . 󵄩 𝑛+1 𝑛󵄩 . 󵄩 󵄩 = 󵄩𝑥 −𝑦 +𝑦 −⋅⋅⋅+𝑦 −𝑦 +𝑦 −𝑥 󵄩 󵄩 󵄩 󵄩 𝑛+1 𝑛+𝑚−2 𝑛+𝑚−2 𝑛+1 𝑛 𝑛 𝑛󵄩 = 󵄩𝑇𝑛𝑦 −𝑦 󵄩 =0 𝑛→∞lim 󵄩 1 𝑛+𝑚−2 𝑛+𝑚−2󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩𝑥𝑛+1 −𝑦𝑛+𝑚−2󵄩 + 󵄩𝑦𝑛+𝑚−2 −𝑦𝑛+𝑚−3󵄩 2≤𝑖<𝑚 󵄩 󵄩 󵄩 󵄩 for .Thus,weconcludethat +⋅⋅⋅+󵄩𝑦𝑛+1 −𝑦𝑛󵄩 + 󵄩𝑦𝑛 −𝑥𝑛󵄩 . 󵄩 𝑛 󵄩 (33) lim sup 󵄩𝑇𝑖−1𝑦𝑛+𝑚−𝑖 −𝑦𝑛+𝑚−𝑖󵄩 =0, 𝑛→∞ (25) Taking the limit on both sides inequality from (33), we have for 2≤𝑖≤𝑚.From(5)andfor𝑖=1,2,...,𝑚 󵄩 󵄩 󵄩𝑥 −𝑥 󵄩 =0. 󵄩 󵄩 𝑛→∞lim 󵄩 𝑛+1 𝑛󵄩 (34) 󵄩𝑦𝑛+𝑖 −𝑦𝑛+𝑖−1󵄩 󵄩 = 󵄩(1 − 𝑎(𝑚−𝑖)𝑛 −𝑏(𝑚−𝑖)𝑛)𝑦𝑛+𝑖−1 Since 𝑇𝑚 is an asymptotically nonexpansive mapping with 𝑘𝑛, 𝑛 󵄩 we have +𝑎(𝑚−𝑖)𝑛𝑇(𝑚−𝑖)𝑦𝑛+𝑖−1 +𝑏(𝑚−𝑖)𝑛 𝑢(𝑚−𝑖)𝑛 −𝑦𝑛+𝑖−1󵄩 󵄩 𝑛 󵄩 󵄩 󵄩𝑥𝑛+1 −𝑇𝑚𝑥𝑛+1󵄩 = 󵄩𝑎 (𝑇𝑛 𝑦 −𝑦 ) 󵄩 (𝑚−𝑖)𝑛 (𝑚−𝑖) 𝑛+𝑖−1 𝑛+𝑖−1 (26) 󵄩 󵄩 = 󵄩𝑥 −𝑥 +𝑥 −𝑇𝑛 𝑥 +𝑇𝑛 𝑥 −𝑇𝑛 𝑥 󵄩 󵄩 󵄩 𝑛+1 𝑛 𝑛 𝑚 𝑛 𝑚 𝑛 𝑚 𝑛+1󵄩 󵄩 +𝑏(𝑚−𝑖)𝑛 (𝑢(𝑚−𝑖)𝑛 −𝑦𝑛+𝑖−1) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 (35) 󵄩 ≤ 󵄩𝑥 −𝑥 󵄩 + 󵄩𝑇𝑛 𝑥 −𝑇𝑛 𝑥 󵄩 + 󵄩𝑇𝑛 𝑥 −𝑥 󵄩 󵄩 󵄩 󵄩 𝑛+1 𝑛󵄩 󵄩 𝑚 𝑛+1 𝑚 𝑛󵄩 󵄩 𝑚 𝑛 𝑛󵄩 ≤𝑎 󵄩𝑇𝑛 𝑦 −𝑦 󵄩 (𝑚−𝑖)𝑛 󵄩 (𝑚−𝑖) 𝑛+𝑖−1 𝑛+𝑖−1󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛 󵄩 ≤ 󵄩𝑥𝑛+1 −𝑥𝑛󵄩 +𝑘𝑛 󵄩𝑥𝑛+1 −𝑥𝑛󵄩 + 󵄩𝑇𝑚𝑥𝑛 −𝑥𝑛󵄩 . 󵄩 󵄩 +𝑏(𝑚−𝑖)𝑛 󵄩𝑢(𝑚−𝑖)𝑛 −𝑦𝑛+𝑖−1󵄩 . Taking the limit on both sides inequality (35), and by using This together with25 ( ) implies that for each 𝑖=1,2,...,𝑚−2 (24), we get 󵄩 󵄩 󵄩 󵄩 󵄩𝑦 −𝑦 󵄩 =0. 󵄩𝑥 −𝑇𝑛 𝑥 󵄩 =0. 𝑛→∞lim 󵄩 𝑛+𝑖 𝑛+𝑖−1󵄩 (27) 𝑛→∞lim 󵄩 𝑛+1 𝑚 𝑛+1󵄩 (36) Abstract and Applied Analysis 7

𝑚 ∞ ∞ Since 𝑇𝑚−1 is an asymptotically nonexpansive mapping with and F = ⋂𝑖=1 𝐹(𝑇𝑖) =0̸ .Supposethat{𝑎𝑖𝑛}𝑛=1, {𝑏𝑖𝑛}𝑛=1, 𝑖= ∞ 𝑘𝑛,wehave 1,2,...,𝑚are appropriate sequences in [0, 1] and {𝑢𝑖𝑛}𝑛=1, 𝑖= 1,2,...,𝑚 𝐾 ∑∞ 𝑏 <∞ 󵄩 󵄩 are bounded sequences in such that 𝑛=1 𝑖𝑛 󵄩𝑥 −𝑇𝑛 𝑥 󵄩 󵄩 𝑛+1 𝑚−1 𝑛+1󵄩 for 𝑖=1,2,...,𝑚.Supposethat 󵄩 𝑛 𝑛 𝑛 󵄩 = 󵄩𝑥𝑛+1 −𝑦𝑛 +𝑦𝑛 −𝑇𝑚−1𝑦𝑛 +𝑇𝑚−1𝑦𝑛 −𝑇𝑚−1𝑥𝑛+1󵄩 󵄩 󵄩 0

Theorem 8. Let 𝑋 be a real uniformly convex Banach space we have that lim𝑛→∞‖𝑥𝑛 −𝑝‖exists. It follows from (47)that 𝑑(𝑥 , F) and 𝐾 anonemptyclosedconvexsubsetof𝑋.Let𝑇𝑖 : lim𝑛→∞ 𝑛 exists. From condition (A) 𝐾 → 𝐾 (𝑖 = 1,2,...,𝑚)be 𝑚 asymptotically nonexpansive 󵄩 󵄩 {𝑘 }⊂[1,∞)∑∞ (𝑘 −1)< ∞ 0≤𝑓(𝑑(𝑥, F)) ≤ 󵄩𝑥 −𝑇 𝑥 󵄩 , mappings with sequence 𝑛 , 𝑛=1 𝑛 𝑛 󵄩 𝑛 𝑖0 𝑛󵄩 (48) 8 Abstract and Applied Analysis

‖𝑥 −𝑇𝑥 ‖ ‖𝑥 −𝑇𝑥 ‖ where 𝑛 𝑖0 𝑛 is max1≤𝑖≤𝑚 𝑛 𝑖 𝑛 .FromThe- [3]Y.J.Cho,H.Zhou,andG.Guo,“Weakandstrongconvergence ‖𝑥 −𝑇𝑥 ‖=0 theorems for three-step iterations with errors for asymptotically orem 7 lim𝑛→∞ 𝑛 𝑖0 𝑛 .Itthenfollows (48)thatlim𝑛→∞𝑓(𝑑(𝑥𝑛, F)) = 0.Bypropertyof𝑓, nonexpansive mappings,” Computers and Mathematics with lim𝑛→∞𝑑(𝑥𝑛, F)=0.Italsofollowsfrom(47)that Applications,vol.47,no.4-5,pp.707–717,2004. lim𝑛→∞‖𝑥𝑛 −𝑝‖=0. Therefore lim𝑛→∞𝑥𝑛 =𝑝∈F. [4] M. A. Noor, “New approximation schemes for general vari- ational inequalities,” Journal of Mathematical Analysis and Now, we prove the weak convergence of iteration (5) Applications,vol.251,no.1,pp.217–229,2000. for a family of asymptotically nonexpansive mappings in a [5]M.O.OsilikeandS.C.Aniagbosor,“Weakandstrongconver- uniformly convex Banach space. gence theorems for fixed points of asymptotically nonexpansive mappings,” Mathematical and Computer Modelling,vol.32,no. Theorem 10. Let 𝑋 be a uniformly convex Banach space 10, pp. 1181–1191, 2000. satisfying Opial’s condition, and let 𝐾 be a nonempty closed [6] S. Plubtieng, R. Wangkeeree, and R. Punpaeng, “On the conver- convex subset of 𝑋.Let𝑇𝑖 : 𝐾 → 𝐾 (𝑖 = 1,2,...,𝑚) gence of modified Noor iterations with errors for asymptotically be 𝑚 asymptotically nonexpansive mappings with sequence nonexpansive mappings,” Journal of Mathematical Analysis and ∞ ∞ ∞ {𝑘𝑛}, and let the sequences {𝑎𝑖𝑛}𝑛=1, {𝑏𝑖𝑛}𝑛=1,and{𝑢𝑖𝑛}𝑛=1, 𝑖= Applications,vol.322,no.2,pp.1018–1029,2006. 1,2,...,𝑚be the same as in Theorem 7. Then the sequence {𝑥𝑛} [7] J. Schu, “Weak and strong convergence to fixed points of asymp- defined by (5) convergesweaklytoacommonfixedpointof totically nonexpansive mappings,” Bulletin of the Australian {𝑇𝑖 :𝑖=1,...,𝑚}. Mathematical Society,vol.43,no.1,pp.153–159,1991. [8] B. Xu and M. A. Noor, “Fixed-point iterations for asymptotically Proof. It follows from Lemma 6 that lim𝑛→∞‖𝑥𝑛 −𝑝‖exists. nonexpansive mappings in Banach spaces,” Journal of Mathe- Therefore, {𝑥𝑛 −𝑝}is a bounded sequence in 𝑋.Thenby matical Analysis and Applications,vol.267,no.2,pp.444–453, the reflexivity of 𝑋 and the boundedness of {𝑥𝑛}, there exists 2002. {𝑥 } {𝑥 } 𝑥 ⇀𝑝 [9] C. E. Chidume and B. Ali, “Approximation of common fixed asubsequence 𝑛𝑗 of 𝑛 such that 𝑛𝑗 weakly. By points for finite families of nonself asymptotically nonexpansive Theorem 7,lim𝑛→∞‖𝑥𝑛 −𝑇𝑖𝑥𝑛‖=0,and𝐼−𝑇𝑖 is demiclosed mappings in Banach spaces,” Journal of Mathematical Analysis at 0 for 𝑖 = 1,2,...,𝑚.Soweobtain𝑇𝑖𝑝=𝑝for 𝑖= 1,2,...,𝑚 {𝑥 } 𝑝 and Applications,vol.326,no.2,pp.960–973,2007. .Finally,weprovethat 𝑛 converges to .Suppose ¨ 𝑝, 𝑞 ∈ 𝑤({𝑥 }) 𝑤({𝑥 }) [10] I. Yıldırım and M. Ozdemir, “Approximating common fixed 𝑛 ,where 𝑛 denotes the weak limit set of points of asymptotically quasi-nonexpansive mappings by a {𝑥𝑛} {𝑥𝑛 } {𝑥𝑚 } {𝑥𝑛} .Let 𝑗 and 𝑗 be two subsequences of which new iterative process,” Arabian Journal for Science and Engineer- converge weakly to 𝑝 and 𝑞, respectively. Opial’s condition ing,vol.36,no.3,pp.393–403,2011. ensures that 𝜔(𝑥𝑛) is a singleton set. It follows that 𝑝=𝑞.Thus [11] J. Quan, S.-S. Chang, and X. J. Long, “Approximation common {𝑥𝑛} convergesweaklytoanelementofF.Thiscompletesthe fixed point of asymptotically quasi-nonexpansive-type map- proof. pings by the finite steps iterative sequences,” Fixed Point Theory and Applications,vol.2006,ArticleID70830,8pages,2006. Conflict of Interests [12] J. W. Peng, “On the convergence of finite steps iterative sequences with errors asymptotically nonexpansive mappings,” The authors declare that there is no conflict of interests IAENG International Journal of Applied Mathematics,vol.37,no. regarding the publication of this paper. 2, 5 pages, 2007. [13] H. Kızıltunc¸ and S. Temir, “Convergence theorems by a new iteration process for a finite family of nonself asymptotically Acknowledgment nonexpansive mappings with errors in Banach spaces,” Comput- ers and Mathematics with Applications,vol.61,no.9,pp.2480– The authors are grateful for the very useful comments 2489, 2011. regarding detailed remarks which improved the presentation [14] K. K. Tan and H. K. Xu, “Approximating fixed points of andthecontentsofthepaper.Thefirstauthoracknowledged nonexpansive mappings by the Ishikawa iteration process,” that this paper was partially supported by Turkish Scientific Journal of Mathematical Analysis and Applications,vol.178,no. and Research Council (Tubitak)¨ Program 2224. This joint 2,pp.301–308,1993. work was done when the first author visited University Putra [15] H.-K. Xu, “Inequalities in Banach spaces with applications,” Malaysia as a visiting scientist during 5th February–15th Nonlinear Analysis: Theory, Methods & Applications,vol.16,no. February, 2013. Thus, he is very grateful to the administration 12, pp. 1127–1138, 1991. of INSPEM for providing him local hospitalities.

References

[1] K. Goebel and W. A. Kirk, “A fixed point theorem for asymp- totically nonexpansive mappings,” Proceedings of the American Mathmatical Society,vol.35,pp.171–174,1972. [2] S.C.Chang,Y.J.Cho,andH.Zhou,“Demi-closedprincipleand weak convergence problems for asymptotically nonexpansive mappings,” Journal of the Korean Mathematical Society,vol.38, no. 6, pp. 1245–1260, 2001. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 264520, 5 pages http://dx.doi.org/10.1155/2013/264520

Research Article 𝜆-Statistical Convergence in Paranormed Space

Mohammed A. Alghamdi1 and Mohammad Mursaleen2

1 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Correspondence should be addressed to Mohammad Mursaleen; [email protected]

Received 24 October 2013; Accepted 9 December 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 M. A. Alghamdi and M. Mursaleen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The concept of 𝜆-statistical convergence for sequences of real numbers was introduced in Mursaleen (2000). In this paper, we prove decomposition theorem for 𝜆-statistical convergence. We also define and study 𝜆-statistical convergence, 𝜆-statistically Cauchy, and strongly 𝜆𝑝-summability in Paranormed Space.

1. Introduction (𝑃4) if (𝛼𝑛) is a sequence of scalars with 𝛼𝑛 →𝛼0 (𝑛 → ∞) and 𝑥𝑛, 𝑎∈𝑋with 𝑥𝑛 → 𝑎 (𝑛 →∞) in the The notion of statistical convergence was first introduced by sense that 𝑔(𝑥𝑛 −𝑎)→ 0(𝑛→,then ∞) 𝛼𝑛𝑥𝑛 → Fast [1]. In the recent years, statistical summability became 𝛼0𝑎(𝑛→,inthesensethat ∞) 𝑔(𝛼𝑛𝑥𝑛 −𝛼0𝑎) → one of the most active areas of research in summability 0(𝑛→∞). theory, which was further generalized as lacunary statistical convergence [2], 𝜆-statical convergence [3], statistical 𝐴- A paranorm 𝑔 for which 𝑔(𝑥) =0 implies that 𝑥=𝜃is summability [4], and statistical 𝜎-convergence [5]. Maddox called a total paranorm on 𝑋,andthepair(𝑋, 𝑔) is called a [6] studied this notion in locally convex Hausdorff topolog- total Paranormed Space. ical spaces and Kolk [7] defined and studied this notion in Banach spaces while C¸akalli [8] extended it to topological 𝜆 Hausdorff groups. The concept of statistical convergence is 2. -Statistical Convergence studied in probabilistic normed space and in intuitionistic Let 𝜆=(𝜆𝑛) be a nondecreasing sequence of positive num- fuzzy normed spaces in [9, 10]. Recently, the statistical bers tending to ∞ such that convergence has been studied in Paranormed Space and locally solid Riesz spaces in [11, 12], respectively. Therefore, 𝜆𝑛+1 ≤𝜆𝑛 +1, 𝜆1 =0. (1) one can choose either some different setup to study these concepts or generalizing the existing concepts through dif- The generalized de la Vallee-Poussin´ mean is defined by ferent means. In this paper, we will study the concept of 𝜆- statistical convergence, 𝜆-statistical Cauchy, and strongly 𝜆𝑝- 1 𝑡 (𝑥) =: ∑ 𝑥 , summability in Paranormed Space. 𝑛 𝑗 (2) 𝜆𝑛 𝑗∈𝐼 A paranorm is a function 𝑔:𝑋 → R definedonalinear 𝑛 space 𝑋 such that for all 𝑥, 𝑦, 𝑧∈𝑋 where 𝐼𝑛 =[𝑛−𝜆𝑛 +1,𝑛]. (𝑃1) 𝑔(𝑥) =0 if 𝑥=𝜃, Asequence𝑥=(𝑥𝑗) is said to be (𝑉,𝜆)-summable to a number 𝐿 if (𝑃2) 𝑔(−𝑥) =, 𝑔(𝑥)

(𝑃3) 𝑔(𝑥 + 𝑦) ≤ 𝑔(𝑥) +𝑔(𝑦), 𝑡𝑛 (𝑥) 󳨀→ 𝐿 as 𝑛󳨀→∞. (3) 2 Abstract and Applied Analysis

Let 𝐾 be a subset of the set of natural numbers N.Then, Clearly, 𝑥=𝑦+𝑧and 𝑦 and 𝑧 arebounded,if𝑥 is bounded. the 𝜆-density of 𝐾 is defined as Also, we observe that for 𝑘>𝑁𝑗,wehave 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 1 󵄨 󵄨 󵄨𝑦 −𝐿󵄨 <𝜖 󵄨𝑦 −𝐿󵄨 = 󵄨𝑥 −𝐿󵄨 <𝜖 𝛿 (𝐾) = 󵄨{𝑛 − 𝜆 +1≤𝑗≤𝑛:𝑗∈𝐾}󵄨 . 󵄨 𝑘 󵄨 since 󵄨 𝑘 󵄨 󵄨 𝑘 󵄨 𝜆 lim𝑛 𝜆 󵄨 𝑛 󵄨 (4) 𝑛 󵄨 󵄨 −1 if 󵄨𝑥𝑘 −𝐿󵄨 <𝑗 , (10) The number sequence 𝑥=(𝑥𝑗) is said to be 𝜆-statistically 󵄨 󵄨 󵄨 󵄨 󵄨𝑦 −𝐿󵄨 = |𝐿−𝐿| =0 󵄨𝑥 −𝐿󵄨 >𝑗−1. convergent to the number 𝐿 (c.f. [3, 13, 14]) if 𝛿𝜆(𝐾(𝜖)); =0 󵄨 𝑘 󵄨 if 󵄨 𝑘 󵄨 that is, if for each 𝜖>0, Hence, lim𝑘𝑦𝑘 =𝐿,since𝜖 was arbitrary. 1 󵄨 󵄨 󵄨 󵄨 Next we observe that 󵄨{𝑘∈ 𝐼 : 󵄨𝑥 −𝐿󵄨 ≥𝜖}󵄨 =0. lim𝑛 󵄨 𝑛 󵄨 𝑘 󵄨 󵄨 (5) 𝜆𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨{𝑘∈ 𝐼 𝑛 :𝑧𝑘 =0}̸󵄨 ≥ 󵄨{𝑘∈ 𝐼 𝑛 : 󵄨𝑧𝑘󵄨 ≥𝜖}󵄨 (11) 𝑥 =𝐿 In this case we write st𝜆-lim𝑘 𝑘 andwedenotetheset for any natural number 𝑛 and 𝜖>0.Hence,lim𝑛(1/𝜆𝑛)|{𝑘∈ 𝜆 𝑆 𝜆 = of all -statistically convergent sequences by 𝜆.Incase 𝑛 𝐼𝑛 :𝑧𝑘 =0}|=0̸ ;thatis,𝑧 is 𝜆-statistically null. 𝑛 𝜆 𝜆 −1 , -density reduces to the natural density and -statistical Wenowshowthatif𝛿>0and 𝑗∈𝑁such that 𝑗 < convergence reduces to statistical convergence. This notion 𝛿,then|{𝑘∈ 𝐼 𝑛 :𝑧𝑘 =0}|̸ < 𝛿 for all 𝑛>𝑁𝑗.Recallfrom for double sequences has been studied in [15]. 𝑁 <𝑘≤𝑁 𝑧 =0̸ 𝑥=(𝑥) 𝜆 the construction that if 𝑗 𝑗+1,then 𝑘 only if Asequence 𝑘 is said to be strongly 𝑝- |𝑥 −𝐿|>𝑗−1 𝑁 <𝑘≤𝑁 summable (0<𝑝<∞)to the limit 𝐿 [14]if 𝑘 .Itfollowsthatif ℓ ℓ+1,then 󵄨 󵄨 {𝑘∈ 𝐼 :𝑧 =0}⊆{𝑘∈𝐼̸ : 󵄨𝑥 −𝐿󵄨 >ℓ−1}. 1 󵄨 󵄨𝑝 𝑛 𝑘 𝑛 󵄨 𝑘 󵄨 (12) ∑ 󵄨𝑥 −𝐿󵄨 =0, lim𝑛 󵄨 𝑘 󵄨 (6) 𝜆𝑛 𝑘∈𝐼𝑛 Consequently,

𝑥 →𝐿[𝑉] 𝐿 1 󵄨 󵄨 1 󵄨 󵄨 󵄨 −1 󵄨 andwewriteitas 𝑘 𝜆 𝑝.Inthiscase is called the 󵄨{𝑘∈ 𝐼 :𝑧 =0}̸󵄨 ≤ 󵄨{𝑘∈ 𝐼 : 󵄨𝑥 −𝐿󵄨 >ℓ }󵄨 [𝑉 ] 𝑥 𝜆 󵄨 𝑛 𝑘 󵄨 𝜆 󵄨 𝑛 󵄨 𝑘 󵄨 󵄨 𝜆 𝑝-limit of . 𝑛 𝑛 (13) The following relation was established in [14]. <ℓ−1 <𝑗−1 <𝛿,

Theorem 1. If 0<𝑝<∞ and a sequence 𝑥=(𝑥𝑘) is str- if 𝑁ℓ <𝑛≤𝑁ℓ+1 and ℓ>𝑗.Thatis, ongly 𝜆𝑝-summable to 𝐿,thenitis𝜆-statistically convergent to 𝐿. If a bounded sequence is 𝜆-statistically convergent to 𝐿,then 1 󵄨 󵄨 𝜆 𝐿 󵄨{𝑘∈ 𝐼 :𝑧 =0}̸󵄨 =0. it is strongly 𝑝-summable to . lim𝑛 𝜆 󵄨 𝑛 𝑘 󵄨 (14) The following theorem is 𝜆-statistical version of Connor’s 𝑛 Decomposition Theorem16 [ ]. This completes the proof of the theorem.

Theorem 2. If 𝑥=(𝑥𝑘) is strongly 𝜆𝑝-summable or statistica- lly 𝜆-convergent to 𝐿, then there is a convergent sequence 𝑦 and 3. Application to Fourier Series a 𝜆-statistically null sequence 𝑧 such that 𝑦 is convergent to Let 𝑓:T → C be a Lebesgue integrable function on the 𝐿, 𝑥 = 𝑦 +𝑧 and 1 torus T := [−𝜋, 𝜋);thatis,𝑓∈𝐿(T). The Fourier series of 𝑓 is defined by 1 󵄨 󵄨 lim 󵄨{𝑘∈ 𝐼 𝑛 :𝑧𝑘 =0}̸󵄨 =0. (7) 𝑛 𝜆 𝑖𝑗𝑥 𝑛 𝑓 (𝑥) ∼ ∑𝑓̂ (𝑗) 𝑒 ,𝑥∈T, 𝑗∈Z (15) Moreover, if 𝑥 is bounded, then 𝑦 and 𝑧 both are bounded. ̂ where the Fourier coefficients 𝑓(𝑗) are defined by Proof. By Theorem 1,itfollowsthat𝑥 is 𝜆-statistically con- 𝐿 𝑥 𝜆 𝐿 𝑁 =0 vergent to if is strongly 𝑝-summable to .Set 0 ̂ 1 −𝑖𝑗𝑡 𝑓(𝑗) := ∫ 𝑓 (𝑡) 𝑒 𝑑𝑡, 𝑗∈ Z. (16) and choose a strictly increasing sequence of positive integers 2𝜋 T 𝑁1 <𝑁2 <𝑁3 <⋅⋅⋅such that The symmetric partial sums of the series in(15) are defined 1 󵄨 󵄨 󵄨 󵄨 󵄨 −1 󵄨 −1 by 󵄨{𝑘∈ 𝐼 𝑛 : 󵄨𝑥𝑘 −𝐿󵄨 ≥𝑗 }󵄨 <𝑗 (8) 𝜆𝑛 𝑠 (𝑓; 𝑥) := ∑ 𝑓̂ (𝑗)𝑖𝑗𝑥 𝑒 ,𝑥∈T,𝑘∈N. 𝑘 (17) for 𝑛>𝑁𝑗.Define𝑦 and 𝑧 as follows. |𝑗|≤𝑘 If 𝑁0 <𝑘<𝑁1 set 𝑧𝑘 =0and 𝑦𝑘 =𝑥𝑘.Let𝑗≥1and 𝑁𝑗 <𝑘≤𝑁𝑗+1.Nowweset The conjugate series to the Fourier series in(15) is defined by [17,Vol.I,pp.49] 󵄨 󵄨 −1 𝑥 ,𝑧 =0, 󵄨𝑥 −𝐿󵄨 <𝑗 ; 𝑖𝑗𝑥 𝑦 ={ 𝑘 𝑘 if 󵄨 𝑘 󵄨 ∑ (−𝑖 𝑗) 𝑓̂ (𝑗) 𝑒 . 𝑘 󵄨 󵄨 −1 (9) sgn (18) 𝐿,𝑘 𝑧 =𝑥𝑘 −𝐿, if 󵄨𝑥𝑘 −𝐿󵄨 ≥𝑗 . 𝑗∈Z Abstract and Applied Analysis 3

Clearly, it follows from (15)and(18)that We define the following. ̂ 𝑖𝑗𝑥 ̂ 𝑖𝑗𝑥 ∑𝑓 (𝑗) 𝑒 +𝑖∑ (−𝑖 sgn 𝑗) 𝑓 (𝑗) 𝑒 Definition 5. Asequence𝑥=(𝑥𝑘) is said to be 𝜆-statistically 𝑗∈Z 𝑗∈Z convergent to the number 𝜉 in (𝑋, 𝑔) if, for each 𝜖>0, (19) ∞ 1 󵄨 󵄨 =1+2∑𝑓̂ (𝑗)𝑖𝑗𝑥 𝑒 , 󵄨{𝑘∈ 𝐼 :𝑔(𝑥 −𝜉)≥𝜖}󵄨 =0. lim𝑛 𝜆 󵄨 𝑛 𝑘 󵄨 (23) 𝑗=1 𝑛 (𝑔) 𝑥=𝜉 and the power series In this case we write st𝜆 -lim . ∞ Definition 6. Asequence𝑥=(𝑥𝑘) is said to be 𝜆-statistically 1+2∑𝑓(𝑗)𝑒̂ 𝑖𝑗𝑥 , 𝑧:=𝑟𝑒𝑖𝑥,0≤𝑟<1, where (20) Cauchy sequence in (𝑋, 𝑔) if for every 𝜖>0there exists a 𝑗=1 number 𝑁=𝑁(𝜖)such that |𝑧| < 1 is analytic on the open unit disk , due to the fact that 1 󵄨 󵄨 lim 󵄨{𝑗 ∈𝑛 𝐼 :𝑔(𝑥𝑗 −𝑥𝑁)≥𝜖}󵄨 =0. 󵄨 󵄨 1 𝑛 𝜆 󵄨 󵄨 (24) 󵄨 ̂ 󵄨 󵄨 󵄨 𝑛 󵄨𝑓 (𝑗)󵄨 ≤ ∫ 󵄨𝑓 (𝑡)󵄨 𝑑𝑡, 𝑗∈ Z. (21) 2𝜋 𝜋 Definition 7. Asequence𝑥=(𝑥𝑘) is said to be strongly 𝜆𝑝- ̂ 1 The conjugate function 𝑓 of a function 𝑓∈𝐿(T) is defined summable (0 < 𝑝 < ∞) to the limit 𝜉 in (𝑋, 𝑔) if by 1 ∑ (𝑔 (𝑥 −𝜉))𝑝 =0, 1 𝑓 (𝑥+𝑡) lim𝑛 𝜆 𝑘 (25) ̂ 𝑛 𝑘∈𝐼 𝑓 (𝑥) := −lim ∫ 𝑑𝑡 𝑛 𝜀↓0 𝜋 𝜀≤|𝑡|≤𝜋 2 tan (𝑡/2) 𝜋 (22) and we write it as 𝑥𝑘 → 𝜉[𝑉𝜆,𝑔]𝑝.Inthiscase𝜉 is called the 1 𝑓 (𝑥−𝑡) −𝑓(𝑥+𝑡) [𝑉 ,𝑔] 𝑥 = lim ∫ 𝑑𝑡 𝜆 𝑝-limit of . 𝜀↓0 𝜋 𝜀 2 tan (𝑡/2) Now we define another type of convergence in Para- 𝑓(𝑥)̂ in the “principal value” sense and that exists at almost normed Space. every 𝑥∈T. 𝜆 The following is -statistical version of [18](c.f.[19, Definition 8. Asequence(𝑥𝑘) in a Paranormed Space (𝑋, 𝑔) ∗ Theorem 2.1 (ii)]). is said to st𝜆(𝑔)-convergent to 𝜉∈𝑋if there exists an index 𝐾={𝑘 <𝑘 <⋅⋅⋅<𝑘 <⋅⋅⋅}⊆N 𝑛 = 1, 2, . . 1 set 1 2 𝑛 , ,with Theorem 3. If 𝑓∈𝐿(T),thenforany𝑝>0its Fourier 𝛿 (𝐾) = 1 𝑔(𝑥 −𝜉)→0(𝑛→∞) 𝜆 such that 𝑘𝑛 .Inthiscase, series is strongly 𝜆𝑝-summable to 𝑓(𝑥) at almost every 𝑥∈T. ∗ we write 𝜉=st𝜆(𝑔)-lim𝑥. Furthermore, its conjugate series (18) is strongly 𝜆𝑝-summable ̂ for any 𝑝>0to the conjugate function 𝑓(𝑥) defined in (22) at First we prove the following results on 𝜆-statistical con- almost every 𝑥∈T. vzergence in (𝑋, 𝑔).

From Theorems 1 and 3, we easily get the following useful Theorem 9. If 𝑔-lim 𝑥=𝜉,thenst𝜆(𝑔)-lim 𝑥=𝜉but converse result. neednotbetrueingeneral.

1 Theorem 4. If 𝑓∈𝐿(T),thenitsFourierseriesis𝜆-stati- Proof. Let 𝑔-lim 𝑥=𝜉. Then, for every 𝜀>0,thereisa stically convergent to 𝑓(𝑥) at almost every 𝑥∈T.Furthermore, positive integer 𝑁 such that its conjugate series (18) is 𝜆-statistically convergent to the conj- ̂ 𝑔(𝑥 −𝜉)<𝜀 ugate function 𝑓(𝑥) defined in (22) at almost every 𝑥∈T. 𝑛 (26)

for all 𝑛≥𝑁.Sincetheset𝐴(𝜖) := {𝑘∈ N :𝑔(𝑥𝑘 −𝜉)≥𝜀}is 4. 𝜆-Statistical Convergence in finite, 𝛿𝜆(𝐴(𝜖)).Hence,st =0 𝜆(𝑔)-lim 𝑥=𝜉. Paranormed Space The following example shows that the converse need not be true. Recently, statistical convergence, statistical Cauchy, and stro- 1/(𝑘+1) ngly Cesaro` summability have been studied in Paranormed Example 10. Let 𝑋 = ℓ(1/𝑘) := {𝑥𝑘 =(𝑥 ):∑𝑘 |𝑥𝑘| < 1/(𝑘+1) Space by Alotaibi and Alroqi [11]. ∞} with the paranorm 𝑔(𝑥)∑ =( 𝑘 |𝑥𝑘| ).Defineasequ- 𝜆 In this paper, we define and study the notion of -sum- ence 𝑥=(𝑥𝑘) by mable, 𝜆-statistical convergence, 𝜆-statistical Cauchy, and 𝜆 strongly 𝑝-summability in Paranormed Space. 𝑘, if 𝑛−[𝜆𝑛]+1≤𝑘≤𝑛,𝑛∈N; (𝑋, 𝑔) 𝑥𝑘 := { (27) Let be a Paranormed Space. 0, otherwise, Asequence𝑥=(𝑥𝑘) is said to be convergent to the number 𝜉 in (𝑋, 𝑔) if, for every 𝜀>0, there exists a positive and write integer 𝑘0 such that 𝑔(𝑥𝑘 −𝜉)<𝜀whenever 𝑘≥𝑘0.Inthis case, we write 𝑔-lim 𝑥=𝜉,and𝜉 is called the 𝑔-limit of 𝑥. 𝐾 (𝜀) := {𝑘≤ 𝑛 : 𝑔 (𝑥 𝑘)≥𝜀}, 0<𝜀<1. (28) 4 Abstract and Applied Analysis

𝑛∈𝑀 (𝑥 ) 𝑔 We see that Nowwehavetoshowthat,for 𝑟, 𝑘𝑛 is -con- 𝜉 (𝑥 ) 𝑔 1/(𝑘+1) vergent to .Oncontrarysupposethat 𝑘𝑛 is not -con- 𝑘 , if 𝑛−[𝜆𝑛]+1≤𝑘≤𝑛,𝑛∈N; 𝜉 𝜀>0 𝑔(𝑥 −𝜉)≥𝜀 𝑔(𝑥𝑘):={ (29) vergent to . Therefore, there is such that 𝑘𝑛 0, otherwise, 𝑀 := {𝑛 ∈ N :𝑔(𝑥 −𝜉)<𝜀} for infinitely many terms. Let 𝜀 𝑘𝑛 and 𝜀>1/𝑟, 𝑟∈N. and hence Then 1, 𝑛−[𝜆 ]+1≤𝑘≤𝑛,𝑛∈N; if 𝑛 𝛿𝜆 (𝑀𝜀)=0, (34) lim𝑔(𝑥𝑘):={ (30) 𝑘 0, otherwise. and by (32), 𝑀𝑟 ⊂𝑀𝜀.Hence𝛿𝜆(𝑀𝑟)=0,whichcontradicts (𝑥 ) 𝑔 𝜉 𝑥 Therefore 𝑔-lim 𝑥 does not exist. On the other hand (33)andwegetthat 𝑘𝑛 is -convergent to .Hence, is ∗ 𝛿𝜆(𝐾(𝜀));thatis,st =0 𝜆(𝑔)-lim 𝑥=0. st𝜆(𝑔)-convergent to 𝜉. ∗ This completes the proof of the theorem. Conversely, suppose that 𝑥 is st𝜆(𝑔)-convergent to 𝜉.Then there exists a set 𝐾={𝑘1 <𝑘2 <𝑘3 < ⋅⋅⋅ < 𝑘𝑛 < ⋅⋅⋅}with 𝜆 We can easily prove the following results on -statistical 𝛿𝜆(𝐾) = 1 such that 𝑔-lim𝑛→∞𝑥𝑘 =𝜉. Therefore, there is (𝑋, 𝑔) 𝑛 convergence in similar to those of [11]. a positive integer 𝑁 such that 𝑔(𝑥𝑛 −𝜉)<𝜀for 𝑛≥𝑁.Put 󸀠 𝐾𝜀 := {𝑛 ∈ N :𝑔(𝑥𝑛−𝜉) ≥ 𝜀}and 𝐾 := {𝑘𝑁+1,𝑘𝑁+2,...}.Then Theorem 11. If a sequence 𝑥=(𝑥𝑘) is 𝜆-statistically conver- 󸀠 󸀠 𝛿𝜆(𝐾 )=1and 𝐾𝜀 ⊆ N −𝐾 which implies that 𝛿𝜆(𝐾𝜀)=0. gent in (𝑋, 𝑔),then𝑠𝑡𝜆(𝑔)-limit is unique. Hence 𝑥=(𝑥𝑘) is 𝜆-statistically convergent to 𝜉;thatisst𝜆(𝑔)- lim 𝑥=𝜉. Theorem 12. Let 𝑠𝑡𝜆(𝑔)-lim 𝑥=𝜉1 and 𝑠𝑡𝜆(𝑔)- lim 𝑦=𝜉2. This completes the proof of the theorem. Then,

(i) 𝑠𝑡𝜆(𝑔)-lim (𝑥 ± 𝑦)1 =𝜉 ±𝜉2, Conflict of Interests (ii) 𝑠𝑡𝜆(𝑔)-lim 𝛼𝑥=𝛼𝜉1,𝛼∈R. The authors declare that there is no conflict of interests Theorem 13. Let (𝑋, 𝑔) be a complete Paranormed Space. regarding the publication of this paper. Then a sequence 𝑥=(𝑥𝑘) of points in (𝑋, 𝑔) is 𝜆-statistically convergent if and only if it is 𝜆-statistically Cauchy. Acknowledgment

Theorem 14. (a) If 0<𝑝<∞and 𝑥𝑘 → 𝜉[𝑉𝜆,𝑔]𝑝,then ThisworkwasfundedbytheDeanshipofScientificResearch 𝑥=(𝑥𝑘) is 𝜆-statistically convergent to 𝜉 in (𝑋, 𝑔). (DSR), King Abdulaziz University, Jeddah, under Grant no. (b) If 𝑥=(𝑥𝑘) is bounded and 𝜆-statistically convergent to (130-073-D1434). The authors, therefore, acknowledge with 𝜉 in (𝑋, 𝑔),then𝑥𝑘 → 𝜉[𝑉𝜆,𝑔]𝑝. thanks DSR technical and financial support.

Theorem 15. Let (𝑋, 𝑔) be a complete Paranormed Space. References Then a sequence 𝑥=(𝑥𝑘) of points in (𝑋, 𝑔) is 𝜆-statistically convergent if and only if it is 𝜆-statistically Cauchy. [1] H. Fast, “Sur la convergence statistique,” Colloquium Mathe- maticum,vol.2,pp.241–244,1951. Note that the proof of Theorem 2.4 [11]isincorrectand [2] J. A. Fridy and C. Orhan, “Lacunary statistical convergence,” the correct proof is given in the following theorem which is Pacific Journal of Mathematics,vol.160,no.1,pp.43–51,1993. generalization of Theorem 2.4 [11].Anotherformofthisresult [3] Mursaleen, “𝜆-statistical convergence,” Mathematica Slovaca, is given in [20] for ideal convergence. vol. 50, no. 1, pp. 111–115, 2000. [4]O.H.H.EdelyandM.Mursaleen,“Onstatistical𝐴-summa- Theorem 16. A sequence 𝑥=(𝑥𝑘) in (𝑋, 𝑔) is 𝜆-statistically ∗ bility,” Mathematical and Computer Modelling,vol.49,no.3-4, convergent to 𝜉 if and only if it is 𝑠𝑡𝜆(𝑔)-convergent to 𝜉. pp. 672–680, 2009. [5] M. Mursaleen and O. H. H. Edely, “On the invariant mean and 𝑥=(𝑥) 𝜆 Proof. Suppose that 𝑘 is -statistically convergent to statistical convergence,” Applied Mathematics Letters,vol.22,no. 𝜉;thatis,𝑠𝑡𝜆(𝑔)-lim 𝑥=𝜉.Now,writefor𝑟 = 1, 2, .. 11,pp.1700–1704,2009. 1 [6] I. J. Maddox, “Statistical convergence in a locally convex space,” 𝐾𝑟 := {𝑛 ∈ N :𝑔(𝑥𝑘 −𝜉)≥ }, Mathematical Proceedings of the Cambridge Philosophical Soci- 𝑛 𝑟 ety, vol. 104, no. 1, pp. 141–145, 1988. (31) 1 [7] E. Kolk, “The statistical convergence in Banach spaces,” Tartu 𝑀𝑟 := {𝑛 ∈ N :𝑔(𝑥𝑘 −𝜉)< } (𝑟 = 1, 2, .) . . ¨ 𝑛 𝑟 Ulikooli Toimetised,no.928,pp.41–52,1991. [8] H. C¸ akalli, “On statistical convergence in topological groups,” Then 𝛿𝜆(𝐾𝑟)=0, Pure and Applied Mathematika Sciences,vol.43,no.1-2,pp.27– 31, 1996. 𝑀1 ⊃𝑀2 ⊃⋅⋅⋅⊃𝑀𝑖 ⊃𝑀𝑖+1 ⊃⋅⋅⋅ , (32) [9] S. Karakus, “Statistical convergence on probabilistic normed spaces,” Mathematical Communications,vol.12,no.1,pp.11–23, 𝛿𝜆 (𝑀𝑟) = 1, 𝑟 = 1, 2, . . . (33) 2007. Abstract and Applied Analysis 5

[10] S. Karakus, K. Demirci, and O. Duman, “Statistical convergence on intuitionistic fuzzy normed spaces,” Chaos, Solitons and Fra- ctals,vol.35,no.4,pp.763–769,2008. [11] A. Alotaibi and A. M. Alroqi, “Statistical convergence in a paranormed space,” Journal of Inequalities and Applications,vol. 2012,article39,2012. [12] H. Albayrak and S. Pehlivan, “Statistical convergence and stati- stical continuity on locally solid Riesz spaces,” Topology and Its Applications,vol.159,no.7,pp.1887–1893,2012. [13] S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence through de la Vallee-Poussinmeaninlocallysolid´ Riesz spaces,” Advances in Difference Equations,vol.2013,article 66, 2013. [14] M. Mursaleen and A. Alotaibi, “Statistical summability and approximation by de la Vallee-Poussin´ mean,” Applied Math- ematics Letters, vol. 24, no. 3, pp. 320–324, 2011, Erratum in Applied Mathematics Letters, vol. 25, p. 665, 2012. [15] F. Moricz,´ “Statistical convergence of multiple sequences,” Arc- hiv der Mathematik,vol.81,no.1,pp.82–89,2003. [16] J. S. Connor, “The statistical and strong 𝑝-Cesaro` convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47–63, 1988. [17] A. Zygmund, Trigonometric Series, Cambridge University Press, New York, NY, USA, 1959. [18] A. Zygmund, “On the convergence and summability of power series on the circle of convergence. II,” Proceedings of the London Mathematical Society,vol.47,pp.326–350,1942. [19] F. Moricz,´ “Statistical convergence of sequences and series of complex numbers with applications in Fourier analysis and summability,” Analysis Mathematica,vol.39,no.4,pp.271–285, 2013. [20] M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca,vol.62,no. 1,pp.49–62,2012. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 539240, 6 pages http://dx.doi.org/10.1155/2013/539240

Research Article On Modified Mellin Transform of Generalized Functions

S. K. Q. Al-Omari1 and Adem KJlJçman2

1 Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan 2 Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Adem Kılıc¸man; [email protected]

Received 10 September 2013; Accepted 14 November 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 S. K. Q. Al-Omari and A. Kılıc¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the modified Mellin transform on certain function space of generalized functions. We first obtain the convolution theorem for the classical and distributional modified Mellin transform. Then we describe the domain and range spaces where the extended modified transform is well defined. Consistency,lution, convo analyticity, continuity, and sufficient theorems for the proposed transform have been established. An inversion formula is also obtained and many properties are given.

1. Introduction From [1]ithasbeennotedthat 𝜇 The Mellin transform of a suitably restricted function over 𝜇(𝑓⋎𝑔)(𝑦)=𝜇𝑓 (𝑦)𝑔 𝜇 (𝑦) . (3) R+((0, ∞)) wasdefinedonsomestripinthecomplexplane [1], where many of the transform properties are obtained Utilizing the Mellin-type convolution product the following by applying change of variables to various properties of the theorem is essential for our next investigations. Laplace transformation. The Mellin transform is extended to 1 distributions in [1]andtoBoehmiansin[2]. Theorem 1 (convolution theorem). Let L be the Lebesgue 1 By combining Fourier and Mellin transforms, the space of integrable functions and 𝑓, 𝑔 ∈ L ;then obtained transform is called Fourier-Mellin transform which 𝑚 𝑚 has many applications in digital signals, image processing, 𝜇 (𝑓 ⋎ 𝑔) (𝑦)𝑓 =𝜇 (𝑦)𝑔 𝜇 (𝑦) , (4) and ship target recognition by sonar system and radar signals 𝑚 as well. where 𝜇𝑓 and 𝜇𝑔 are the Mellin-type and Mellin transforms of The modified Mellin transform of a suitably restricted 𝑓 and 𝑔,respectively. function 𝑓 over R+ was introduced by [3] Proof. By the definition of the Mellin and modified Mellin 𝑚 𝑚 𝑦−1 𝜇𝑓 (𝑦) =: 𝜇 𝑓(𝑦)=:∫ 𝑓 (𝑥) 𝑦𝑥 𝑑𝑥 (1) transforms we have R+ 𝜇𝑚 (𝑓 ⋎ 𝑔) (𝑦) = ∫ (𝑓 ⋎ 𝑔) (𝑥) 𝑦𝑥𝑦−1 𝑑𝑥 as a scale-invariant transform. Then, as earlier, combining the R+ modified Mellin transform with the Fourier transform gives 𝑥 the Fourier-modified Mellin transform3 [ , Equation 16]. = ∫ (∫ 𝑓 (𝜏) 𝑔( )𝜏−1 𝑑𝜏) 𝑦𝑥𝑦−1 𝑑𝑥 𝜏 The Mellin-type convolution product of two functions 𝑓 R+ R+ 𝑔 and is given by 𝑥 = ∫ 𝑦𝑓 (𝜏) 𝜏−1 (∫ 𝑔( )𝑥𝑦−1𝑑𝑥) 𝑑𝜏. 𝑥 −1 R R 𝜏 (𝑓⋎𝑔) (𝑥) = ∫ 𝑓 (𝜏) 𝑔 ( ) 𝜏 𝑑𝜏. + + 𝜏 (2) R+ (5) 2 Abstract and Applied Analysis

Employing Fubinis theorem, then the substitution 𝑧=𝑥/𝜏 Therefore, denote by 𝜗(R+) the space of test functions of together with simple computations establishes that bounded support over R+; then the convolution product of 𝑓∈𝜇𝑎,𝑏́ and 𝑔∈𝜗(R+) canbegivenas 𝜇𝑚 (𝑓 ⋎ 𝑔) (𝑦)𝑚 =𝜇 (𝑦) 𝜇 (𝑦) . 𝑓 𝑔 (6) 1 𝑥 (𝑓 ⋎ 𝑔) (𝑥) =⟨𝑓(𝑡) , 𝑔( )⟩ , (12) Hence the theorem is proved. 𝑡 𝑡 where 𝑥∈R+. 2. Modified Mellin Transform of Distribution 3. Boehmians Let 𝜇𝑎,𝑏 be the space of smooth functions 𝜑 over R+ such that [1] Let G agroupandS asubgroupofG.Weassumethattoeach 󵄨 󵄨 pair of elements 𝑓∈G and 𝜔∈S is assigned the product 󵄨 𝑑𝑘 󵄨 𝛾 (𝜑) = 󵄨𝜉 (𝑥) 𝑥𝑘−1 𝜑 (𝑥)󵄨 𝑓∗𝑔such that 𝑘 sup 󵄨 𝑎,𝑏 𝑘 󵄨 (7) 𝑥∈R 󵄨 𝑑𝑥 󵄨 + (1) if 𝜔, 𝜓 ∈ S,then𝜔∗𝜓∈S and 𝜔∗𝜓=𝜓∗𝜔; is finite, 𝑘=0,1,2,...,where (2) if 𝑓∈G and 𝜔, 𝜓 ∈ S,then(𝑓∗𝜔)∗𝜓 = 𝑓∗(𝜔⋆𝜓); 𝑓, 𝑔 ∈ G,𝜔∈S 𝜆∈R −𝑎 (3) if ,and ,then 𝑥 , for 0<𝑥≤1, 𝜉𝑎,𝑏 (𝑥) ≜{ −𝑏 (8) (𝑓 + 𝑔) ∗ 𝜔 = 𝑓 ∗ 𝜔 + 𝑔 ∗ 𝜔, 𝜆 (𝑓 ∗ 𝜔) =(𝜆𝑓)∗𝜔. 𝑥 , for 1<𝑥<∞, (13) and 𝑎, 𝑏 ∈ R+. Let Δ be a family of sequences from S such that Then 𝜇𝑎,𝑏 is linear space under addition and multi- 𝜇 plication by complex numbers. 𝑎,𝑏 canalsobegenerated (1) if 𝑓, 𝑔 ∈ G,(𝛿𝑛)∈Δ,and𝑓∗𝛿𝑛 = 𝑔∗𝛿𝑛 (𝑛=1,2,...), (𝛾 )∞ by the multinorms 𝑘 0 which turns to be a countably then 𝑓=𝑔,forall𝑛; multinormed space. (2) if (𝜔𝑛), (𝛿𝑛)∈Δ,then(𝜔𝑛 ∗𝜓𝑛)∈Δ. Denote by 𝜇𝑎,𝑏́ the complete strong dual space of 𝜇𝑎,𝑏;then it is assigned the weak topology. Let 𝜉(R+) be the space of Elements of Δ will be called delta sequences. smooth functions over R+;thenforany𝑎, 𝑏 ∈ R,𝜇𝑎,𝑏 is dense Consider the class A of pair of sequences defined by in 𝜉(R+) and the topology assigned to 𝜇𝑎,𝑏 is stronger than A ={((𝑓),(𝜔 )) : (𝑓 )⊆GN,(𝜔 )∈Δ}, that induced on 𝜇𝑎,𝑏 by 𝜉(R+) andisidentifiedwithasubspace 𝑛 𝑛 𝑛 𝑛 (14) of 𝜇𝑎,𝑏́ . for each 𝑛∈N. The straightforward conclusion is that the kernel function 𝑦−1 𝑚 An element ((𝑓𝑛), (𝜔𝑛)) ∈ A is called a quotient of (𝑦𝑥 ) of 𝜇𝑓 is a member of 𝜇𝑎,𝑏 for 𝑎≤Re 𝑦≤𝑏. sequences, denoted by [𝑓𝜐/𝜔𝑛],if𝑓𝑛 ∗𝜔𝑚 =𝑓𝑚 ∗𝜔𝑛,for 𝑓∈𝜇́ This usually leads to the following definition: let 𝑎,𝑏; all 𝑛, 𝑚 ∈ N. 𝜇̂𝑚 𝑓 then the distributional transform 𝑓 of is defined as Two quotients of sequences 𝑓𝑛/𝜔𝑛 and 𝑔𝑛/𝜓𝑛 are said to be equivalent, 𝑓𝑛/𝜔𝑛 ∼𝑔𝑛/𝜓𝑛 ,if𝑓𝑛 ∗𝜓𝑚 =𝑔𝑚 ∗𝜔𝑛,forall ̂𝑚 𝑦−1 𝑛, 𝑚 ∈ N 𝜇𝑓 (𝑦) ≜ ⟨𝑓 (𝑥) ,𝑦𝑥 ⟩, 𝑦∈Ω𝑓, (9) . The relation ∼ is an equivalent relation on A.The 𝑓 /𝜔 [𝑓 /𝜔 ] where 𝑓∈𝜇𝑎,𝑏́ and Ω𝑓 ={𝑦∈C :𝑎

𝛿-Convergence. AsequenceofBoehmians(𝛽𝑛) in 𝛽1 is said to respectively. Therefore, by the convolution theorem and (19) 𝛿 we get that be 𝛿 convergent to a Boehmian 𝛽 in 𝛽1, denoted by 𝛽𝑛 →𝛽󳨀 , (𝜔 ) if there exists a delta sequence 𝑛 such that (𝑔 ⋏ 𝜓) (𝑦) = 𝑔 (𝑦) 𝜓 (𝛽 ∗𝜔 ) , (𝛽∗𝜔 ) ∈ G,∀𝑘,𝑛∈N, 𝑛 𝑛 𝑛 ̂𝑚 =(𝜇𝑓 𝜇𝜑)(𝑦) (20)

(𝛽𝑛 ∗𝜔𝑘)󳨀→(𝛽∗𝜔𝑘) as 𝑛󳨀→∞,in G, (17) =𝜇𝑚 (𝑓 ⋎ 𝜑) (𝑦) . 𝑘∈N. for every 𝑚 Since 𝑓⋎𝜑∈𝜇𝑎,𝑏́ ,itfollows𝑔⋏𝜓∈ℓ . The following is equivalent for the statement of 𝛿- Hencewehaveprovedthetheorem. 𝛿 𝑒 𝑒 convergence: 𝛽𝑛 →󳨀 𝛽 (𝑛 →∞) in 𝛽1 if and only if there Theorem 4. Let 𝜓1,𝜓2 ∈𝜗;then𝜓1 ⋏𝜓2 ∈𝜗. 𝑓𝑛,𝑘,𝑓𝑘 ∈ G (𝜔𝑘)∈Δ 𝛽𝑛 =[𝑓𝑛,𝑘/𝜔𝑘] is and such that , 𝑒 𝛽=[𝑓𝑘/𝜔𝑘] and for each 𝑘∈N, 𝑓𝑛,𝑘 →𝑓𝑘 as 𝑛→∞ Proof. By definition of 𝜗 we can find 𝜑1,𝜑2 ∈𝜗such that G 𝜓 =𝜇 𝜓 =𝜇 in . 1 𝜑1 and 2 𝜑2 . AsequenceofBoehmians(𝛽𝑛) in 𝛽1 is said to be Δ Therefore, by [1], Δ convergent to a Boehmian 𝛽 in 𝛽1, denoted by 𝛽𝑛 󳨀→𝛽,if 𝜓1 ⋏𝜓2 =𝜓1 (𝑦)2 𝜓 (𝑦) there exists a (𝜔𝑛)∈Δsuch that (𝛽𝑛 −𝛽)∗𝜔𝑛 ∈𝛽1,forall 𝑛∈N,and(𝛽𝑛 −𝛽)∗𝜔𝑛 →0as 𝑛→∞in 𝛽1.See[2, 5–15]. =𝜇 (𝑦) 𝜇 (𝑦) 𝜑1 𝜑2 (21) =𝜇(𝜑 ⋎𝜑 )(𝑦). 4. Modified Mellin Transform of Boehmian 1 2 𝑒 In this section we discuss the modified Mellin transform on But since 𝜑1 ⋎𝜑2 ∈𝜗,weget𝜓1 ⋏𝜓2 ∈𝜗.Thuswehavethe spaces of Boehmians. Consider the group 𝜇𝑎,𝑏́ (R+) and 𝜗(R+) theorem. as a subgroup of 𝜇𝑎,𝑏́ (R+).Let⋎ be the operation between Theorem 5. 𝑔 ,𝑔 ∈ℓ𝑚 𝜓∈𝜗𝑒 (𝑔 +𝑔 )⋏𝜓= 𝜇𝑎,𝑏́ (R+) and 𝜗(R+) and Δ the set of delta sequences given by Let 1 2 and ;then 1 2 𝑔 ⋏𝜓+𝑔 ⋏𝜓 (𝛼𝑔)⋏𝜓=𝑔⋏(𝛼𝜓)=𝛼(𝑔 ⋏𝜓) [2] 1 2 and 1 .

(1) ∫ 𝜑𝑛(𝑡)𝑑𝑡,forall =1 𝑛∈N; Proof. Is straightforward. R+ Theorem 6. 𝑔 →𝑔 ℓ𝑚,𝜓∈𝜗𝑒 𝑔 ⋏𝜓 → 𝑔⋏𝜓 ∫ |𝜑 (𝑡)|𝑑𝑡 ≤ m 𝑛∈N m >0 Let 𝑛 in ;then 𝑛 (2) R 𝑛 ,forall ,forsome ; 𝑚 + as 𝑛→∞in ℓ .

(3) supp 𝜑𝑛 ⊂(𝑎𝑛,𝑏𝑛),forall𝑛∈N for some 0<𝑎𝑛 < Proof. Caneasilybechecked. 𝑏𝑛 <∞with 𝑎𝑛 →1, 𝑏𝑛 →1as 𝑛→∞. 𝑚 𝑒 Theorem 7. Let 𝑔∈ℓ and (𝜓𝑛)∈Δ ;then𝑔⋏𝜓𝑛 →𝑔as 𝛽 𝜇́ ,𝜗(R ) R+ Let 1 be the Boehmian space obtained from 𝑎,𝑏 + and 𝑛→∞ ↷ . 𝑚 Δ;then𝛽1 will serve as the domain space of 𝜇 . Our next objective is to construct a range space, say 𝛽2, Proof. By (19)wehave ↷ 𝑚 for 𝜇 . ̂𝑚 (𝑔⋏𝜓𝑛)(𝑦)=(𝜇𝑓 𝜇𝜑 )(𝑦), (22) Let 𝑛 ̂𝑚 𝑒 𝑒 𝑓∈𝜇𝑎,𝑏́ (𝜑𝑛)∈Δ 𝜇 =𝑔 𝜇𝜑 = Δ ={𝜇 :(𝜑)∈Δ}, 𝜗 ={𝜇 :𝜑∈𝜗}, where and are such that 𝑓 and 𝑛 R+ 𝜑𝑛 𝑛 𝜑 𝜓𝑛,forall𝑛∈N. (18) Since 𝜇𝜑 (𝑦)→1as 𝑛→∞on compact subsets of R+, 𝑚 ̂𝑚 ́ 𝑛 ℓ ={𝜇𝑓 :𝑓∈𝜇𝑎,𝑏}. ̂𝑚 (22)impliesthat(𝑔 ⋏𝑛 𝜓 )(𝑦) → 𝜇𝑓 (𝑦) = 𝑔(𝑦),forall𝑦,as 𝑛→∞ 𝑚 𝑒 . Hence we obtain the theorem. For 𝑔∈ℓ ,𝜓∈𝜗 , define Theorem 8. (𝜓 ), (𝜃 )∈Δ𝑒 𝜓 ⋏𝜃 ∈Δ𝑒 Let 𝑛 𝑛 R+ ;then 𝑛 𝑛 R+ . (𝑔⋏𝜓)(𝑦) =𝑔(𝑦) 𝜓 (𝑦) . (19) Let 𝜓𝑛 =𝜇𝛼 ,𝜃𝑛 =𝜇𝜎 ; then taking into account the fact (𝜓 ⋏𝜃)(𝑦)𝑛 = (𝜇 𝑛 ⋏𝜇 )(𝑦) = 𝜇(𝛼 ⋎𝜎)(𝑦) We have the following theorem. that 𝑛 𝑛 𝛼𝑛 𝜎𝑛 𝑛 𝑛 ,since 𝛼𝑛 ⋎𝜎𝑛 ∈Δ, this theorem follows. 𝑚 𝑒 𝑚 Theorem 3. Let 𝑔∈ℓ and 𝜓∈𝜗,then𝑔⋏𝜓∈ℓ . The Boehmian space 𝛽2 is therefore constructed. In addition, scalar multiplication, differentiation, and 𝑚 𝑒 Proof. Let 𝑔 and 𝜓 belong to ℓ and 𝜗 ;respectively.Then the operation ⋏ in 𝛽2 are defined similar to that of usual 𝑚 there are 𝑓∈𝜇𝑎,𝑏́ and 𝜑∈𝜗such that 𝑔=𝜇𝑓 and 𝜓=𝜇𝜑, Boehmian spaces. 4 Abstract and Applied Analysis

𝑚 ̂𝑚 ̂𝑚 ́ ́ Each 𝑔∈ℓ can be identified by a member of 𝛽2 given as The fact that 𝜇𝑓 is injective, 𝜇𝑓 : 𝜇𝑎,𝑏 → 𝜇𝑎,𝑏,implies that 𝑓𝑛 ⋏𝜑𝑚 =𝑓𝑚 ⋏𝜑𝑛,𝑚,𝑛∈N. 𝑔⋏𝜓𝑛 𝑔󳨀→[ ] as 𝑛󳨀→∞, (23) Thus 𝑓𝑛/𝜑𝑛 is quotient of sequences in 𝛽1.Hence, 𝜓𝑛 [𝑓𝑛/𝜑𝑛]∈𝛽1 and 𝑒 where 𝜓𝑛 ∈ΔR . + ↷ 𝑓 𝑔 𝜇𝑚 ([ 𝑛 ]) = [ 𝑛 ]. ↷ (28) 𝑚 𝜑 𝜓 Definition 9. The extended modified Mellin transform 𝜇 : 𝑛 𝑛 𝛽 →𝛽 1 2 is defined by Hence the theorem is proved. ↷ ̂𝑚 𝜇𝑓 𝑓 𝑚 [ 𝑛 ] 𝑛 Theorem 13 (generalized convolution theorem). Let 𝛽= 𝜇 (𝛽) = ,∀𝛽=[]∈𝛽1. (24) 𝜇𝜑 𝜑𝑛 [𝑓𝑛/𝜑𝑛]∈𝛽1 and 𝛾=[𝜅𝑛/𝜙𝑛]∈𝛽1;then [ 𝑛 ] ↷ ↷ ↷ Theorem 10. The extended modified Mellin transform is well 𝑚 𝑚 𝑓𝑛 𝑚 𝜅𝑛 𝜇 (𝛽⋎𝛾) = 𝜇 ([ ]) ⋏ 𝜇 ([ ]) . (29) defined. 𝜑𝑛 𝜙𝑛

Proof. The proof of this theorem is straightforward. See [11– Proof. Assume that the requirements of the theorem satisfy 13]. for some 𝛽 and 𝛾∈𝛽1;thenusingDefinition 9 and the operation ⋏ yields Theorem 11 (consistency theorem). The extended modified ↷ 𝜇𝑚 ↷ ↷ 𝑓 ⋎𝜅 𝜇𝑚 (𝑓 ⋎𝜅 ) Mellin transform is consistent with the distributional 𝜇𝑚 (𝛽 ⋎ 𝛾) = 𝜇𝑚 ([ 𝑛 𝑛 ]) = [ 𝑛 𝑛 ] ̂𝑚 ̂𝑚 ́ ́ 𝜑 ⋎𝜙 𝜇(𝜑 ⋎𝜙 ) 𝜇𝑓 (𝜇𝑓 : 𝜇𝑎,𝑏 → 𝜇𝑎,𝑏). 𝑛 𝑛 𝑛 𝑛 𝜇̂𝑚 ⋏𝜇𝑚 𝜇̂𝑚 𝜇𝑚 (30) 𝑓∈𝜇́ 𝛽 𝛽 𝑓 𝜅𝑛 𝑓 𝜅 Proof. For every 𝑎,𝑏,let be its representative in 1;then = [ 𝑛 ] = [ 𝑛 ] ⋏[ 𝑛 ]. 𝛽=[(𝑓⋎𝜑𝑛)/𝜑𝑛],where𝜑𝑛 ∈Δ,forall𝑛.Thenitisclearthat 𝜇𝜑 ⋏𝜇𝜙 𝜇𝜑 𝜇𝜙 [ 𝑛 𝑛 ] [ 𝑛 ] 𝑛 𝜑𝑛 is independent of the representative, for all 𝑛. Therefore Therefore ↷ ↷ 𝑚 𝑚 𝑓⋎𝜑𝑛 𝜇 (𝛽) = 𝜇 ([ ]) ↷ ↷ ↷ 𝑚 𝑚 𝑓𝑛 𝑚 𝜅𝑛 𝜑𝑛 𝜇 (𝛽 ⋎ 𝛾) = 𝜇 ([ ]) ⋏ 𝜇 ([ ]) . 𝜑 𝜙 (31) ̂𝑚 𝑛 𝑛 𝜇 𝜇𝜑 . .= [ 𝑓 𝑛 ] i e This completes the proof. 𝜇𝜑 (25) [ 𝑛 ] ↷ ̂𝑚 𝑚 (𝜇 )𝜇𝜑 Theorem 14. The extended modified Mellin transform 𝜇 : [ 𝑓 𝑛 ] i.e.= 𝛽1 →𝛽2 is bijective. 𝜇𝜑 [ 𝑛 ] ↷ ↷ 𝑚 𝑚 ̂𝑚 𝜇 [𝑓 /𝜑 ]=𝜇 [𝜅 /𝜙 ] which is the representative of 𝜇 in 𝜇𝑎,𝑏́ . Proof. Assume 𝑛 𝑛 𝑛 𝑛 ;thenitfollows 𝑓 ̂𝑚 from the concept of quotients of sequences that 𝜇 ⋏𝜇𝜙 = Hencewehavetheproof. 𝑓𝑛 𝑚 𝜇̂𝑚 ⋏𝜇 𝜇𝑚(𝑓 ⋎𝜙 )=𝜇𝑚(𝜅 ⋎𝜑 ) 𝜅𝑚 𝜑𝑛 . Therefore, 𝑛 𝑚 𝑚 𝑛 .Theproperty Theorem 12 (necessity theorem). Let [𝑔𝑛/𝜓𝑛]∈𝛽2; then the 𝑚 that 𝜇 is one to one implies 𝑓𝑛 ⋎𝜙𝑚 =𝜅𝑚 ⋎𝜑𝑛. Therefore, necessary and sufficient condition that [𝑔𝑛/𝜓𝑛]is to be in the ↷ 𝜇𝑚 𝑔 𝜇̂𝑚 𝑛∈N 𝑓𝑛 𝜅𝑛 range of is that 𝑛 belongs to range of 𝑓 for every . [ ] = [ ] . (32) 𝜑𝑛 𝜙𝑛 ↷ 𝑚 [𝑔 /𝜓 ] 𝜇 ; 𝑔 ↷ Proof. Let 𝑛 𝑛 be in the range of then of course 𝑛 𝑚 ̂𝑚 ̂𝑚 Next to establish that 𝜇 is onto, let [𝜇𝑓 /𝜇𝜑 ](∈ 𝛽2) be arbitr- belongs to the range of (𝜇𝑓 ),forall𝑛∈N. 𝑛 𝑛 𝑚 𝑚 𝑚 𝜇̂ ⋏𝜇 = 𝜇̂ ⋏𝜇 𝑚, 𝑛 ∈ N 𝑔 𝜇̂ ary; then 𝑓 𝜑 𝑓 𝜑 for every .Hence To establish the converse, let 𝑛 be in the range of 𝑓 ,for 𝑛 𝑚 𝑚 𝑛 𝑓 ,𝑓 ∈ 𝜇́ 𝜇𝑚(𝑓 ⋎𝜑 )=𝜇𝑚(𝑓 ⋎𝜑 ) ́ ̂𝑚 𝑛 𝑚 𝑎,𝑏 are such that 𝑛 𝑚 𝑚 𝑛 ,forall all 𝑛∈N. Then there is 𝑓𝑛 ∈ 𝜇𝑎,𝑏 such that 𝜇𝑓 =𝑔𝑛,𝑛∈N. 𝑛 𝑚, 𝑛 ∈ N. Since [𝑔𝑛/𝜓𝑛]∈𝛽2, Hence, the Boehmian [𝑓𝑛/𝜑𝑛] belongs to 𝛽1 and satisfies 𝑔𝑛 ⋎𝜓𝑚 =𝑔𝑚 ⋎𝜓𝑛, (26) ↷ 𝜇̂𝑚 𝑚 𝑓𝑛 𝑓𝑛 for all 𝑚, 𝑛 ∈ N. Therefore, 𝜇 [ ]=[ ] . (33) 𝜑𝑛 𝜇𝜑 𝑚 𝑚 [ 𝑛 ] 𝜇 (𝑓𝑛 ⋏𝜑𝑛) =𝜇 (𝑓𝑚 ⋏𝜑𝑛) , ∀𝑚, 𝑛 ∈ N, (27) where 𝑓𝑛 ∈ 𝜇𝑎,𝑏́ and 𝜑𝑛 ∈Δ. This completes the proof of the theorem. Abstract and Applied Analysis 5

↷ ↷ 𝑚 𝑚 𝑚 −1 ̂ ̂ ̂ 𝑚 𝑚 The continuity of 𝜇𝑓 implies 𝜇𝑓 → 𝜇𝑓 as 𝑛→∞. Now we introduce (𝜇 ) as the inverse transform of 𝜇 , 𝑛,𝑘 𝑛,𝑘 𝑘 where Thus, −1 𝑚 𝑚 𝑚 𝑚 𝜇̂ ̂𝑚 ↷ −1 𝜇̂ (𝜇̂) (𝜇̂) 𝑓 𝜇𝑓 𝑓 [ 𝑓 𝑓 ] [ 𝑛,𝑘 ] [ 𝑘 ] (𝜇𝑚) ([ 𝑛 ])= 𝑛 𝑛 , 󳨀→ (42) [ −1 ] (34) 𝜇 𝜇 𝜇𝜑 𝜑𝑘 𝜑𝑘 𝑛 (𝜇𝜑 ) (𝜇𝜑 ) [ ] [ ] [ ] [ 𝑛 𝑛 ] ↷ [𝑓 /𝜑 ]∈𝛽 𝑚 for every 𝑛 𝑛 1. as 𝑛→∞in 𝛽2.Thisprovescontinuityof𝜇 . 𝛿 ̂𝑚 𝑒 Next, let 𝑔𝑛 →𝑔∈𝛽󳨀 2 as 𝑛→∞;thenwehave𝑔𝑛 = Theorem 15. Let [𝜇 /𝜇𝜑 ]∈𝛽2 and 𝜙∈𝜗,then 𝑓𝑛 𝑛 ̂𝑚 ̂𝑚 ̂𝑚 ̂𝑚 𝑚 [𝜇 /𝜇𝜑 ] 𝑔=[𝜇 /𝜇𝜑 ] 𝜇 , 𝜇 ∈ℓ 𝑓𝑛,𝑘 𝑘 and 𝑓𝑘 𝑘 for some 𝑓𝑛,𝑘 𝑓𝑘 ,where ↷ −1 𝜇̂𝑚 𝑓 𝑓 𝜇̂𝑚 → 𝜇̂𝑚 𝑛→∞ 𝑚 [ 𝑛 ] 𝑛 𝑓 𝑓 as .Hence (𝜇 ) ( ⋏𝜙)=[ ]⋎𝜙, (35) 𝑛,𝑘 𝑘 𝜇𝜑 𝜑𝑛 [ 𝑛 ] −1 −1 (𝜇̂𝑚 ) (𝜇̂𝑚 ) 󳨀→ (𝜇̂𝑚 ) (𝜇̂𝑚 ) (43) 𝑓𝑛, 𝑓𝑛,𝑘 𝑓𝑘 𝑓𝑘 ↷ ̂𝑚 k 𝑓 𝜇𝑓 𝑚 𝑛 [ 𝑛 ] 𝜇 ([ ]⋎𝜙)= ⋏𝜙. (36) as 𝑛→∞in 𝛽1.Thatis, 𝜑𝑛 𝜇𝜑 [ 𝑛 ] −1 −1 (𝜇̂𝑚 ) (𝜇̂𝑚 ) (𝜇̂𝑚 ) (𝜇̂𝑚 ) [ 𝑓 𝑓 ] [ 𝑓 𝑓 ] Proof. We prove (35) and omit the proof of (36)duetoits [ 𝑛,𝑘 𝑛,𝑘 ] = [ 𝑛,𝑘 𝑛,𝑘 ] ̂𝑚 𝑚 −1 similarity. Given [𝜇 /𝜇𝜑 ]∈𝛽2 and 𝜓∈𝜗such that 𝜙=𝜇 𝜑𝑘 𝑓𝑛 𝑛 𝜓 (𝜇𝜑 ) (𝜇𝜑 ) [ ] [ 𝑘 𝑘 ] then employing (34)yields −1 (44) 𝑚 𝑚 𝑚 𝑚 (𝜇̂) (𝜇̂) ↷ −1 𝜇̂ ↷ −1 𝜇̂ ⋏𝜙 𝑓 𝑓 𝑓 𝑓 [ 𝑘 𝑘 ] 𝑚 [ 𝑛 ] 𝑚 [ 𝑛 ] 󳨀→ [ ] (𝜇 ) ( ⋏𝜙)=(𝜇 ) ( ) 𝜑 𝜇𝜑 𝜇𝜑 𝑘 [ 𝑛 ] [ 𝑛 ] [ ] −1 (37) 𝑛→∞ (𝜇𝑚 ) (𝜇̂𝑚 ⋏𝜇𝑚) as . [ 𝑓𝑛 𝑓 𝜓 ] = [ 𝑛 ] . Hence −1 −1 −1 (𝜇 ) (𝜇 ) 𝑚 𝑚 𝑚 𝑚 𝜑𝑛 𝜑𝑛 (𝜇̂) (𝜇̂) (𝜇̂) (𝜇̂) [ ] [ 𝑓 𝑓 ] [ 𝑓 𝑓 ] 𝑛,𝑘 𝑛,𝑘 󳨀→ 𝑘 𝑘 [ ] [ −1 ] (45) Using (19)gives 𝜑𝑘 (𝜇𝜑 ) (𝜇𝜑 ) [ ] [ 𝑘 𝑘 ] 𝑚 −1 𝑚 𝑚 ↷ −1 𝜇̂𝑚 (𝜇 ) (𝜇̂𝜇 ) 𝑚 𝑓 [ 𝑓𝑛 𝑓𝑛 𝜓 ] as 𝑛→∞. (𝜇 ) ([ 𝑛 ] ⋏𝜙)=[ ] . 𝜇 −1 (38) That is, 𝜑𝑛 (𝜇 ) (𝜇 ) [ ] 𝜑𝑛 𝜑𝑛 [ ] ↷ −1 ↷ −1 𝑚 𝑚 Hence the convolution theorem gives (𝜇 ) 𝑔𝑛 󳨀→ (𝜇 ) 𝑔 (46) −1 ↷ −1 ̂𝑚 𝑚 𝑚 𝜇𝑓 (𝜇𝑓 ) (𝜇𝑓 (𝑓𝑛 ⋎𝜙)) 𝑛→∞ (𝜇𝑚) ([ 𝑛 ] ⋏𝜙)=[ 𝑛 𝑛 ] . as .Thiscompletestheproof. −1 𝜇𝜑 [ 𝑛 ] [ (𝜇𝜑 ) (𝜇𝜑 ) ] ↷ ↷ 𝑛 𝑛 Theorem 17. 𝜇𝑚 :𝛽 →𝛽 (𝜇𝑚 )−1 :𝛽 →𝛽 (39) 1 2 and 2 1 are continuous with respect to Δ-convergence. Thus Δ ↷ −1 𝜇̂𝑚 𝛽 󳨀→𝛽 𝛽 𝑛→∞ 𝑓 ∈ 𝜇́ 𝑚 𝑓 𝑓𝑛 Proof. Let 𝑛 in 1 as .Then,wefind 𝑛 𝑎,𝑏 (𝜇 ) ([ 𝑛 ] ⋏𝜙)=[ ]⋎𝜙. (40) 𝜇 𝜑 and (𝜑𝑘)∈Δsuch that (𝛽𝑛 −𝛽)⋏𝜑𝑘 =[(𝑓𝑛 ⋏𝜑𝑘)/𝜑𝑘 ] and 𝜑𝑛 𝑛 [ ] 𝑓𝑛 →0as 𝑛→∞. Therefore Proof of the second part is similar. ↷ 𝑚 𝑚 𝜇 (𝑓𝑛 ⋎𝜑𝑘) This completes the proof of the theorem. 𝜇 ((𝛽 −𝛽)⋎𝜑 )=[ ]. 𝑛 𝑘 𝜇 (47) 𝜑𝑘 ↷ ↷ 𝑚 𝑚 −1 ↷ Theorem 16. 𝜇 :𝛽1 →𝛽2 and (𝜇 ) :𝛽2 →𝛽1 are 𝑚 ̂𝑚 ̂𝑚 Hence, 𝜇 ((𝛽𝑛 −𝛽)⋎𝜑𝑘)=[(𝜇 ⋏𝜇𝜑 )/𝜇𝜑 ]=𝜇 →0as 𝛿 𝑓𝑛 𝑘 𝑘 𝑓𝑛 continuous with respect to -convergence. 𝑚 𝑛→∞in ℓ . 𝛿 Therefore Proof. Let 𝛽𝑛 →𝛽󳨀 in 𝛽1 as 𝑛→∞;thenweestablishthat ↷ ↷ ↷ 𝑚 𝑚 𝜇𝑚 ((𝛽 −𝛽)⋎𝜑 ) 𝜇 𝛽𝑛 → 𝜇 𝛽 as 𝑛→∞.Let𝑓𝑛,𝑘 and 𝑓𝑘 be in 𝜇𝑎,𝑏́ such 𝑛 𝑛 that ↷ ↷ (48) =(𝜇𝑚 𝛽 − 𝜇𝑚 𝛽) ⋏ 𝜇 󳨀→ 𝑛󳨀→∞. 𝑓𝑛,𝑘 𝑓𝑘 𝑛 𝜑𝑘 as 𝛽𝑛 =[ ], 𝛽=[ ] (41) 𝜑𝑘 𝜑𝑘 ↷ Δ ↷ 𝑚 󳨀→𝜇𝑚 𝛽 𝑛→∞ and 𝑓𝑛,𝑘 →𝑓𝑘 as 𝑛→∞for every 𝑘∈N. Hence, 𝜇 𝛽𝑛 as . 6 Abstract and Applied Analysis

Proof of the second part is analogous. Detailed proof is as [5]S.K.Q.Al-Omari,“Distributionalandtempereddistributional follows. diffraction Fresnel transforms and their extension to Boehmian Δ spaces,” Italian Journal of Pure and Applied Mathematics,no.30, Finally, let 𝑔𝑛 󳨀→𝑔in 𝛽2 as 𝑛→∞;thenwecanfind ̂𝑚 𝑚 ̂𝑚 pp.179–194,2013. 𝜇𝑓 ∈ℓ such that (𝑔𝑛 −𝑔)⋏𝜇𝜑 =[(𝜇𝑓 ⋏𝜇𝜑 )/𝜇𝜑 ] and 𝑘 𝑘 𝑘 𝑘 𝑘 [6]P.K.Banerji,S.K.Al-Omari,andL.Debnath,“Tempereddis- ̂𝑚 𝑒 𝜇 →0 𝑛→∞ (𝜇𝜑 )∈Δ 𝑓𝑘 as for some 𝑘 R+ . tributional Fourier sine (cosine) transform,” Integral Transforms Next, we have and Special Functions,vol.17,no.11,pp.759–768,2006. [7] P. Mikusinski,´ “Fourier transform for integrable Boehmians,” 𝑚 −1 𝑚 ↷ −1 (𝜇 ) (𝜇̂ ⋏𝜇 ) The Rocky Mountain Journal of Mathematics,vol.17,no.3,pp. [ 𝑓𝑘 𝑓 𝜑𝑘 ] (𝜇𝑚) ((𝑔 −𝑔)⋏𝜇 )=[ 𝑘 ] . 577–582, 1987. 𝑛 𝜑𝑘 −1 (49) (𝜇𝜑 ) (𝜇𝜑 ) [8] P. Mikusinski,´ “Tempered Boehmians and ultradistributions,” [ 𝑘 𝑘 ] Proceedings of the American Mathematical Society,vol.123,no. Thus, by (34)weget 3,pp.813–817,1995. [9] P.Mikusinski,´ “Convergence of Boehmians,” Japanese Journal of −1 ↷ 𝑓 ⋎𝜑 Mathematics,vol.9,no.1,pp.159–179,1983. (𝜇𝑚) ((𝑔 −𝑔)⋏𝜇 )=[ 𝑛 𝑘 ]=𝑓 󳨀→ 0 (50) 𝑛 𝜑𝑘 𝑛 [10]S.K.Q.Al-Omari,D.Loonker,P.K.Banerji,andS.L.Kalla, 𝜑𝑘 “Fourier sine (cosine) transform for ultradistributions and their as 𝑛→∞in 𝜇𝑎,𝑏́ . extensions to tempered and ultraBoehmian spaces,” Integral Therefore Transforms and Special Functions,vol.19,no.5-6,pp.453–462, 2008. ↷ −1 𝑚 [11] S. K. Q. Al-Omari, “Hartley transforms on a certain space of (𝜇 ) ((𝑔𝑛 −𝑔)⋏𝜇𝜑 ) 𝑘 generalized functions,” Georgian Mathematical Journal,vol.20, no.3,pp.415–426,2013. −1 −1 (51) ↷ ↷ [12] S. K. Q. Al-Omari and A. Kılıc¸man, “Note on Boehmians for =((𝜇𝑚) 𝑔 −(𝜇𝑚) 𝑔) ⋎ 𝜑 󳨀→ 0 𝑛 𝑘 class of optical Fresnel wavelet transforms,” Journal of Function Spaces and Applications,vol.2012,ArticleID405368,14pages, as 𝑛→∞. 2012. Thus, we have [13]S.K.Q.Al-OmariandA.Kılıc¸man, “On generalized hartley- Hilbert and Fourier-Hilbert transforms,” Advances in Difference −1 −1 ↷ Δ ↷ Equations,vol.2012,article232,12pages,2012. (𝜇𝑚) 𝑔 󳨀→(𝜇𝑚) 𝑔 (52) 𝑛 [14] T. K. Boehme, “The support of Mikusinski´ operators,” Transac- tions of the American Mathematical Society,vol.176,pp.319–334, as 𝑛→∞in 𝛽1. 1973. This completes the proof of the theorem. [15]S.K.Q.Al-OmariandA.Kılıc¸man, “On diffraction Fresnel tra- nsforms for Boehmians,” Abstract and Applied Analysis,vol. Conflict of Interests 2011, Article ID 712746, 11 pages, 2011. The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment The authors acknowledge that this research was partially supported by Universiti Putra Malaysia under the ERGS 1- 2013/5527179.

References

[1] A. H. Zemanian, Generalized Integral Transformations,Dover, New York, NY, USA, 1987. [2] S.K.Q.Al-Omari,“OnthedistributionalMellintransformation and its extension to Boehmian spaces,” International Journal of Contemporary Mathematical Sciences,vol.6,no.17–20,pp.801– 810, 2011. [3] J. Yang, T. K. Sarkar, and P. Antonik, “Applying the Fourier- modified Mellin transform to Doppler-distorted waveforms,” DigitalSignalProcessing,vol.17,no.6,pp.1030–1039,2007. [4] S. K. Q. Al-Omari, “A Mellin transform for a space of Lebesgue integrable Boehmians,” International Journal of Contemporary Mathematical Sciences,vol.6,no.29–32,pp.1597–1606,2011. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 827205, 9 pages http://dx.doi.org/10.1155/2013/827205

Research Article New Fixed Point Results with PPF Dependence in Banach Spaces Endowed with a Graph

N. Hussain,1 S. Khaleghizadeh,2 P. Salimi,3 and F. Akbar4

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran 3 Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran 4 Department of Mathematics, GDCW, Bosan Road, Multan, Pakistan

Correspondence should be addressed to P. Salimi; [email protected]

Received 30 October 2013; Accepted 21 November 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 N. Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the concept of an 𝛼𝑐-admissible non-self-mappings with respect to 𝜂𝑐 and establish the existence of PPF dependent fixed and coincidence point theorems for 𝛼𝑐𝜂𝑐-𝜓-contractive non-self-mappings in the Razumikhin class. As applications of our PPF dependent fixed point and coincidence point theorems, we derive some new fixed and coincidence point results for 𝜓-contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some PPF dependent fixed point results in the literature. Several interesting consequences of our theorems are also provided.

1. Introduction and Preliminaries consideration (see [9]). On the other hand, Samet et al. [22] introduced the concept of 𝛼-admissible self-mappings and In nonlinear functional analysis, one of the most significant proved fixed point results for 𝛼-admissible contractive map- research areas is fixed point theory. On the other hand, pings in complete metric spaces and provided application fixed point theory has an application in distinct branches of the obtained results to ordinary differential equations. of mathematics and also in different sciences, such as engi- More recently, Salimi et al. [24] modified the notions of 𝛼- neering, computer science, and economics. In 1922, Banach 𝜓-contractive and 𝛼-admissible mappings and established proved that every contraction in a complete metric space fixed point theorems to generalize the results in [22]. In this has a unique fixed point. This celebrated result have been paper, we introduce the concept of an 𝛼𝑐-admissible non- generalized and improved by many authors in the context self-mapping with respect to 𝜂𝑐 and establish the existence of different abstract spaces for various operators (see [1–31] of PPF dependent fixed and coincidence point theorems and references therein). In 1997, Bernfeld et al. [5]introduced for 𝛼𝑐𝜂𝑐-𝜓-contractive non-self-mappings in the Razumikhin the concept of fixed point for mappings that have different class. As applications of our PPF dependent fixed point domains and ranges, which is called PPF dependent fixed and coincidence point theorems, we derive some new fixed point or the fixed point with PPF dependence. Furthermore, 𝜓 they gave the notion of Banach type contraction for non-self- and coincidence point results for -contractions whenever mapping and also proved the existence of PPF dependent the range space is endowed with a graph or with a partial fixed point theorems in the Razumikhin class for Banach order. The obtained results generalize, extend, and modify type contractions (see [17]). The PPF dependent fixed point some PPF dependent fixed results in the literature. Several theorems are useful for proving the solutions of nonlinear interesting consequences of our theorems are also provided. functional differential and integral equations which may Throughout this paper, we assume that (𝐸, ‖ ⋅𝐸 ‖ ) is a depend upon the past history, present data, and future Banach space, 𝐼 denotes a closed interval [𝑎, 𝑏] in R,and 2 Abstract and Applied Analysis

𝐸0 = 𝐶(𝐼, 𝐸) denotes the set of all continuous 𝐸-valued Note that if we take 𝜂(𝑥, 𝑦), =1 then this definition reduces 𝐼 ‖⋅‖ 𝛼(𝑥, 𝑦) =1 functions on equipped with the supremum norm 𝐸0 to Definition.Also,ifwetake, 5 ,thenwesaythat defined by 𝑇 is an 𝜂-subadmissible mapping. 󵄩 󵄩 󵄩 󵄩 󵄩𝜙󵄩 = sup󵄩𝜙 (𝑡)󵄩 . 󵄩 󵄩𝐸0 󵄩 󵄩𝐸 (1) The following result is a proper generalization of the 𝑡∈𝐼 above-mentioned results. For a fixed element 𝑐∈𝐼, the Razumikhin or minimal Theorem 7 (see [24]). Let (𝑋, 𝑑) be a complete metric space class of functions in 𝐸0 is defined by and let 𝑇 be an 𝛼-admissible mapping. Assume that 󵄩 󵄩 󵄩 󵄩 R𝑐 ={𝜙∈𝐸0 : 󵄩𝜙󵄩 = 󵄩𝜙 (𝑐)󵄩 }. 󵄩 󵄩𝐸0 󵄩 󵄩𝐸 (2) 𝑥, 𝑦 ∈ 𝑋, 𝛼 (𝑥, 𝑦) ≥ 1 󳨐⇒ 𝑑 (𝑇𝑥, 𝑇𝑦) ≤𝜓(𝑀(𝑥,𝑦)), (6) Clearly, every constant function from 𝐼 to 𝐸 belongs to R𝑐. where 𝜓∈Ψand Definition 1. Let R𝑐 be the Razumikhin class, then 𝑑 (𝑥, 𝑇𝑥) +𝑑(𝑦,𝑇𝑦) (i) the class R𝑐 is algebraically closed with respect to 𝑀(𝑥,𝑦)=max {𝑑 (𝑥, 𝑦), , difference, if 𝜙−𝜉∈R𝑐 whenever 𝜙, 𝜉∈R𝑐; 2 R (7) (ii) the class 𝑐 is topologically closed if it is closed with 𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥) respect to the topology on 𝐸0 generated by the norm }. ‖⋅‖ 2 𝐸0 .

Definition 2 (see [5]). A mapping 𝜙∈𝐸0 is said to be a PPF Also, suppose that the following assertions hold: dependent fixed point or a fixed point with PPF dependence (i) there exists 𝑥0 ∈𝑋such that 𝛼(𝑥0,𝑇𝑥0)≥1, of mapping 𝑇:𝐸0 →𝐸if 𝑇𝜙 = 𝜙(𝑐) for some 𝑐∈𝐼. (ii) either 𝑇 is continuous or for any sequence {𝑥𝑛} in 𝑋 Definition 3 (see [17]). Let 𝑆:𝐸0 →𝐸0 and let 𝑇:𝐸0 →𝐸. with 𝛼(𝑥𝑛,𝑥𝑛+1)≥1for all 𝑛∈N ∪{0}and 𝑥𝑛 →𝑥 Apoint𝜙∈𝐸0 is said to be a PPF dependent coincidence as 𝑛→+∞,wehave𝛼(𝑥𝑛,𝑥)≥1for all 𝑛∈N ∪{0}. point or a coincidence point with PPF dependence of 𝑆 and 𝑇 𝑇 if 𝑇𝜙 = (𝑆𝜙)(𝑐) for some 𝑐∈𝐼. Then has a fixed point. 𝛼 𝜓 Definition 4 (see [5]). The mapping 𝑇:𝐸0 →𝐸is called a For more details on modified - -contractive mappings Banach type contraction if there exists 𝑘∈[0,1)such that and related fixed point results we refer the reader to [8, 13, 14, 󵄩 󵄩 󵄩 󵄩 25, 26]. 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤𝑘󵄩𝜙−𝜉󵄩 , 󵄩 󵄩𝐸 󵄩 󵄩𝐸0 (3) 2. PPF Dependent Fixed and Coincidence for all 𝜙, 𝜉∈𝐸0. Point Results Sametetal.[22] defined the notion of 𝛼-admissible First we define the notion of non-self 𝛼-admissible mapping mappings as follows. with respect to 𝜂 as follows. Definition 5. Let 𝑇 be a self-mapping on 𝑋 and let 𝛼:𝑋× Definition 8. Let 𝑐∈𝐼and let 𝑇:𝐸0 →𝐸, 𝛼, 𝜂:𝐸×𝐸 → 𝑋→[0,+∞)be a function. We say that 𝑇 is an 𝛼-admissible [0, ∞).Wesaythat𝑇 is an 𝛼𝑐-admissible non-self-mapping mapping if with respect to 𝜂𝑐 if for 𝜙, 𝜉∈𝐸0, 𝑥, 𝑦 ∈ 𝑋, 𝛼 (𝑥,) 𝑦 ≥1󳨐⇒𝛼(𝑇𝑥, 𝑇𝑦) ≥1. (4) 𝛼(𝜙(𝑐) ,𝜉(𝑐))≥𝜂(𝜙(𝑐) ,𝜉(𝑐)) In [22] the authors considered the family Ψ of nonde- (8) 󳨐⇒𝛼(𝑇𝜙,𝑇𝜉)≥𝜂(𝑇𝜙,𝑇𝜉). creasing functions 𝜓:[0,+∞)→[0,+∞)such that ∑+∞ 𝜓𝑛(𝑡) < +∞ 𝑡>0 𝜓𝑛 𝑛 𝑛=1 for each ,where is the th iterate 𝜂(𝜙(𝑐), 𝜉(𝑐)) =1 𝑇 𝛼 of 𝜓. Note that if we take ,thenwesay is an 𝑐- admissible non-self-mapping. Also, if we take 𝛼(𝜙(𝑐), 𝜉(𝑐)) = Salimi et al. [24] modified and generalized the notions 1 𝑇 𝜂 of 𝛼-𝜓-contractive mappings and 𝛼-admissible mappings as ,thenwesaythat is an 𝑐-subadmissible non-self- follows. mapping. 𝐸=R Definition 6 (see [24]). Let 𝑇 be a self-mapping on 𝑋 and 𝛼, Example 9. Let bearealBanachspacewithusualnorm 𝜂:𝑋×𝑋 → [0,+∞) 𝑇 and let 𝐼 = [0, 1].Define𝑇:𝐸0 →𝐸by 𝑇𝜙 = (1/2)𝜙(1) for be two functions. We say that is an 𝜙∈𝐸 𝛼 𝜂:𝐸×𝐸 → [0,+∞) 𝛼-admissible mapping with respect to 𝜂 if all 0 and , by 4 8 𝛼 (𝑥, 𝑦) ≥ 𝜂 (𝑥, 𝑦) 󳨐⇒ 𝛼 (𝑇𝑥, 𝑇𝑦) ≥𝜂(𝑇𝑥,𝑇𝑦), {𝑥 +𝑦 +1, if 𝑥≥𝑦 𝛼(𝑥,𝑦)= 1 (5) { , , (9) 𝑥, 𝑦 ∈ 𝑋. {3 otherwise Abstract and Applied Analysis 3

4 𝜂(𝑥, 𝑦) =𝑥 +1/2.Then,𝑇 is an 𝛼1-admissible mapping with By continuing this process, by induction, we can build a respect to 𝜂1. In fact, if 𝛼(𝜙(1), 𝜉(1)) ≥ 𝜂(𝜙(1), 𝜉(1)),then sequence {𝜙𝑛} in R𝑐 ⊆𝐸0 such that 𝜙(1) ≥ 𝜉(1) and so, (1/2)𝜙(1) ≥ (1/2)𝜉(1).Thatis,𝑇𝜙 ≥ 𝑇𝜉 𝛼(𝑇𝜙, 𝑇𝜉) ≥ 𝜂(𝑇𝜙, 𝑇𝜉) which implies that . 𝑇𝜙𝑛−1 =𝜙𝑛 (𝑐) ,∀𝑛∈N. (14) Denote with Ψ the family of nondecreasing functions 𝜓: ∞ 𝑛 Since R𝑐 isalgebraicallyclosedwithrespecttodifference, [0, +∞) → [0, +∞) such that ∑𝑛=1 𝜓 (𝑡) < +∞ for all 𝑡>0, 𝑛 where 𝜓 is the 𝑛th iterate of 𝜓. it follows that ThefollowingRemarkisobvious. 󵄩 󵄩 󵄩 󵄩 󵄩𝜙𝑛−1 −𝜙𝑛󵄩 = 󵄩𝜙𝑛−1 (𝑐) −𝜙𝑛 (𝑐)󵄩 ,∀𝑛∈N. 󵄩 󵄩𝐸0 󵄩 󵄩𝐸 (15) Remark 10. If 𝜓∈Ψ,then𝜓(𝑡) <𝑡 for all 𝑡>0. If there exists 𝑛0 ∈ N such that 𝜙𝑛 (𝑐) =𝑛 𝜙 +1(𝑐) = 𝑇𝜙𝑛 ,then 𝑇:𝐸 →𝐸𝛼 𝜂:𝐸×𝐸→ [0,∞) 0 0 0 Definition 11. Let 0 , , be 𝜙𝑛 is a PPF dependent fixed point of 𝑇 and we have nothing 𝑐∈𝐼 0 three mappings and .Then, toprove.Henceweassumethat𝜙𝑛−1 =𝜙̸ 𝑛 for all 𝑛∈N. Since 𝑇 is an 𝛼𝑐-admissible non-self-mapping with (i) 𝑇 is an 𝛼𝑐𝜂𝑐-𝜓-contractive non-self-mapping if respect to 𝜂𝑐 and 𝛼(𝜙(𝑐) ,𝜉(𝑐))≥𝜂(𝜙(𝑐) ,𝑇𝜙) (10) 𝛼(𝜙0 (𝑐) ,𝜙1 (𝑐))=𝛼(𝜙0 (𝑐) ,𝑇𝜙0) 󵄩 󵄩 󳨐⇒ 󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 ≤ 𝜓 (𝑀 (𝜙, 𝜉)), (16) ≥𝜂(𝜙0 (𝑐) ,𝑇𝜙0)=𝜂(𝜙0 (𝑐) ,𝜙1 (𝑐)),

(ii) 𝑇 is a modified 𝛼𝑐-𝜓-contractive non-self-mapping if so, 󵄩 󵄩 𝛼 (𝜙 (𝑐) ,𝜉(𝑐)) ≥1󳨐⇒󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 ≤𝜓(𝑀 (𝜙, 𝜉)) , (11) 𝛼(𝜙1 (𝑐) ,𝑇𝜙1)≥𝜂(𝜙1 (𝑐) ,𝑇𝜙1). (17) where 𝜓∈Ψand By continuing this process we get 𝑀(𝜙,𝜉) 󵄩 󵄩 󵄩 󵄩 𝛼(𝜙 (𝑐) ,𝜙 (𝑐))=𝛼(𝜙 (𝑐) ,𝑇𝜙 ) 󵄩𝜙 (𝑐) −𝑇𝜙󵄩 + 󵄩𝜉 (𝑐) −𝑇𝜉󵄩 𝑛−1 𝑛 𝑛−1 𝑛−1 󵄩 󵄩 󵄩 󵄩𝐸 󵄩 󵄩𝐸 (18) = max {󵄩𝜙−𝜉󵄩𝐸 , , 0 2 (12) ≥𝜂(𝜙𝑛−1 (𝑐) ,𝑇𝜙𝑛−1), 󵄩 󵄩 󵄩 󵄩 󵄩𝜙 (𝑐) −𝑇𝜉󵄩 + 󵄩𝜉 (𝑐) −𝑇𝜙󵄩 󵄩 󵄩𝐸 󵄩 󵄩𝐸 }. for all 𝑛∈N.Thenfrom(10)weget 2 󵄩 󵄩 󵄩 󵄩 󵄩𝜙𝑛 −𝜙𝑛+1󵄩 = 󵄩𝜙𝑛 (𝑐) −𝜙𝑛+1 (𝑐)󵄩 󵄩 󵄩𝐸0 󵄩 󵄩𝐸 The following theorem is our first main result in this (19) section. 󵄩 󵄩 = 󵄩𝑇𝜙𝑛−1 −𝑇𝜙𝑛󵄩𝐸 ≤𝜓(𝑀(𝜙𝑛−1,𝜙𝑛)) ,

Theorem 12. Let 𝑇:𝐸0 →𝐸, 𝛼, 𝜂:𝐸×𝐸→ [0,∞)be three mappings that satisfy the following assertions: where 𝑐∈𝐼 R (i) there exists such that 𝑐 is topologically closed 𝑀(𝜙𝑛−1,𝜙𝑛) and algebraically closed with respect to difference; 󵄩 󵄩 (ii) 𝑇 is an 𝛼𝑐-admissible non-self-mapping with respect to 󵄩 󵄩 = max { 󵄩𝜙𝑛−1 −𝜙𝑛󵄩 , 𝜂𝑐; 𝐸0 󵄩 󵄩 󵄩 󵄩 (iii) 𝑇 is an 𝛼𝑐𝜂𝑐-𝜓-contractive non-self-mapping; 󵄩𝜙 (𝑐) −𝑇𝜙 󵄩 + 󵄩𝜙 (𝑐) −𝑇𝜙 󵄩 󵄩 𝑛−1 𝑛−1󵄩𝐸 󵄩 𝑛 𝑛󵄩𝐸 , (iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ 2 𝛼(𝜙 (𝑐), 𝜙 (𝑐)) ≥ 𝜂(𝜙 (𝑐), 𝜙 (𝑐)) 𝑛∈ and 𝑛 𝑛+1 𝑛 𝑛+1 for all 󵄩 󵄩 󵄩 󵄩 N ∪{0} 𝛼(𝜙 (𝑐), 𝜙(𝑐)) ≥𝜂(𝜙 (𝑐), 𝑇𝜙 ) 𝑛∈ 󵄩𝜙𝑛−1 (𝑐) −𝑇𝜙𝑛󵄩 + 󵄩𝜙𝑛 (𝑐) −𝑇𝜙𝑛−1󵄩 ,then 𝑛 𝑛 𝑛 for all 󵄩 󵄩𝐸 󵄩 󵄩𝐸 } N ∪{0}; 2

(v) there exists 𝜙0 ∈ R𝑐 such that 𝛼(𝜙0(𝑐), 𝑇𝜙0)≥ 𝜂(𝜙 (𝑐), 𝑇𝜙 ) 󵄩 󵄩 0 0 . = {󵄩𝜙𝑛−1 −𝜙𝑛󵄩 , max 󵄩 󵄩𝐸0 Then, 𝑇 has a 𝑃𝑃𝐹 dependent fixed point. 󵄩 󵄩 󵄩 󵄩 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 + 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 󵄩 𝑛−1 𝑛 󵄩𝐸 󵄩 𝑛 𝑛+1 󵄩𝐸 , Proof. Let, 𝜙0 ∈ R𝑐.Since𝑇𝜙0 ∈𝐸, there exists 𝑥1 ∈𝐸such 2 that 𝑇𝜙0 =𝑥1.Choose𝜙1 ∈ R𝑐 such that, 󵄩 󵄩 󵄩 󵄩 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 + 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 󵄩 𝑛−1 𝑛+1 󵄩𝐸 󵄩 𝑛 𝑛 󵄩𝐸 } 𝑥1 =𝜙1 (𝑐) . (13) 2 4 Abstract and Applied Analysis

𝛼(𝜙 (𝑐), 𝜙(𝑐)) ≥𝜂(𝜙 (𝑐), 𝑇𝜙 ) 𝑛∈ 󵄩 󵄩 From (iv) we have 𝑛 𝑛 𝑛 for all = max {󵄩𝜙𝑛−1 −𝜙𝑛󵄩 , 𝐸0 N ∪{0}.By(10)wehave 󵄩 ∗ ∗ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑇𝜙 −𝜙 (𝑐)󵄩𝐸 󵄩𝜙𝑛−1 −𝜙𝑛󵄩 + 󵄩𝜙𝑛 −𝜙𝑛+1󵄩 󵄩 󵄩𝐸0 󵄩 󵄩𝐸0 , 󵄩 ∗ 󵄩 󵄩 ∗ 󵄩 2 ≤ 󵄩𝑇𝜙 −𝑇𝜙𝑛󵄩𝐸 + 󵄩𝑇𝜙𝑛 −𝜙 (𝑐)󵄩𝐸 󵄩 󵄩 󵄩 ∗ 󵄩 󵄩 ∗ 󵄩 󵄩𝜙𝑛−1 −𝜙𝑛+1󵄩𝐸 = 󵄩𝑇𝜙 −𝑇𝜙 󵄩 + 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 0 } 󵄩 𝑛󵄩𝐸 󵄩 𝑛+1 󵄩𝐸 (27) 2 ∗ 󵄩 ∗󵄩 ≤ 𝜓 (𝑀 (𝜙 ,𝜙𝑛)) + 󵄩𝜙𝑛+1 −𝜙 󵄩 󵄩 󵄩𝐸0 󵄩 󵄩 󵄩 󵄩 <𝑀(𝜙∗,𝜙 )+󵄩𝜙 −𝜙∗󵄩 , ≤ {󵄩𝜙𝑛−1 −𝜙𝑛󵄩 , 𝑛 󵄩 𝑛+1 󵄩𝐸 max 󵄩 󵄩𝐸0 0

󵄩 󵄩 󵄩 󵄩 where 󵄩𝜙𝑛−1 −𝜙𝑛󵄩𝐸 + 󵄩𝜙𝑛 −𝜙𝑛+1󵄩𝐸 ∗ 0 0 } 𝑀(𝜙 ,𝜙 ) 2 𝑛 󵄩 󵄩 󵄩 󵄩 󵄩 ∗ 󵄩 󵄩 󵄩 󵄩 󵄩 = max {󵄩𝜙 −𝜙𝑛󵄩 , ≤ {󵄩𝜙𝑛−1 −𝜙𝑛󵄩 , 󵄩𝜙𝑛 −𝜙𝑛+1󵄩 } 󵄩 󵄩𝐸0 max 󵄩 󵄩𝐸0 󵄩 󵄩𝐸0 󵄩 󵄩 󵄩 󵄩 (20) 󵄩𝜙∗ (𝑐) −𝑇𝜙∗󵄩 + 󵄩𝜙 (𝑐) −𝑇𝜙 󵄩 󵄩 󵄩𝐸 󵄩 𝑛 𝑛󵄩𝐸 , 2 which implies that 󵄩 󵄩 󵄩 󵄩 󵄩𝜙∗ (𝑐) −𝑇𝜙 󵄩 + 󵄩𝜙 (𝑐) −𝑇𝜙∗󵄩 󵄩 󵄩 󵄩 𝑛󵄩𝐸 󵄩 𝑛 󵄩𝐸 } 󵄩𝜙𝑛 −𝜙𝑛+1󵄩𝐸 0 2 (28) (21) 󵄩 󵄩 󵄩 󵄩 ≤𝜓(max {󵄩𝜙𝑛−1 −𝜙𝑛󵄩 , 󵄩𝜙𝑛 −𝜙𝑛+1󵄩 }) . 𝐸0 𝐸0 󵄩 ∗ 󵄩 = {󵄩𝜙 −𝜙𝑛󵄩 , max 󵄩 󵄩𝐸0

{‖𝜙𝑛−1 −𝜙𝑛‖ ,‖𝜙𝑛 −𝜙𝑛+1‖ }=‖𝜙𝑛 −𝜙𝑛+1‖𝐸 Now, if max 𝐸0 𝐸0 0 , 󵄩 󵄩 󵄩 󵄩 󵄩𝜙∗ (𝑐) −𝑇𝜙∗󵄩 + 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 then 󵄩 󵄩𝐸 󵄩 𝑛 𝑛+1 󵄩𝐸 , 2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝜙𝑛 −𝜙𝑛+1󵄩 ≤𝜓(󵄩𝜙𝑛 −𝜙𝑛+1󵄩 )<󵄩𝜙𝑛 −𝜙𝑛+1󵄩 (22) 󵄩 ∗ 󵄩 󵄩 ∗󵄩 󵄩 󵄩𝐸0 󵄩 󵄩𝐸0 󵄩 󵄩𝐸0 󵄩𝜙 (𝑐) −𝜙 (𝑐)󵄩 + 󵄩𝜙 (𝑐) −𝑇𝜙 󵄩 󵄩 𝑛+1 󵄩𝐸 󵄩 𝑛 󵄩𝐸 }. 2 which is a contradiction. Hence, Taking limit as 𝑛→∞in the above inequality we get 󵄩 󵄩 󵄩 󵄩 󵄩𝜙𝑛 −𝜙𝑛+1󵄩𝐸 ≤𝜓(󵄩𝜙𝑛−1 −𝜙𝑛󵄩𝐸 ), (23) 0 0 󵄩 󵄩 1󵄩 󵄩 󵄩𝑇𝜙∗ −𝜙∗ (𝑐)󵄩 ≤ 󵄩𝑇𝜙∗ −𝜙∗ (𝑐)󵄩 . 󵄩 󵄩𝐸 2󵄩 󵄩𝐸 (29) for all 𝑛∈N.So, ∗ ∗ ∗ ∗ Therefore, ‖𝑇𝜙 −𝜙 (𝑐)‖𝐸 =0.Thatis,𝑇𝜙 =𝜙(𝑐).This 󵄩 󵄩 𝑛 󵄩 󵄩 ∗ 󵄩𝜙𝑛 −𝜙𝑛+1󵄩 ≤𝜓 (󵄩𝜙0 −𝜙1󵄩 ) , implies that 𝜙 is a PPF dependent fixed point of 𝑇 in R𝑐. 󵄩 󵄩𝐸0 󵄩 󵄩𝐸0 (24) 𝜂(𝜙, 𝜉) =1 𝜙 𝜉∈𝐸 𝑛∈N If in Theorem 12 we take for all , 0,then for all . we deduce the following corollary. Fix 𝜖>0, then there exists 𝑁∈N such that Corollary 13. Let 𝑇:𝐸0 →𝐸and 𝛼:𝐸×𝐸 → [0,∞)be 𝑛 󵄩 󵄩 ∑ 𝜓 (󵄩𝜙0 −𝜙1󵄩 )<𝜖 ∀𝑛∈N. 𝐸0 (25) two mappings satisfy that the following assertions: 𝑛≥𝑁 (i) there exists 𝑐∈𝐼such that R𝑐 is topologically closed Let 𝑚, 𝑛∈N with 𝑚>𝑛≥𝑁.Bytriangularinequalityweget and algebraically closed with respect to difference; (ii) 𝑇 is an 𝛼𝑐-admissible non-self-mapping; 𝑚−1 󵄩 󵄩 󵄩 󵄩 (iii) 𝑇 is a modified 𝛼𝑐-𝜓-contractive non-self-mapping; 󵄩𝜙𝑛 −𝜙𝑚󵄩 ≤ ∑ 󵄩𝜙𝑘 −𝜙𝑘+1󵄩 󵄩 󵄩𝐸0 󵄩 󵄩𝐸0 𝑘=𝑛 (iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ (26) and 𝛼(𝜙𝑛(𝑐),𝑛+1 𝜙 (𝑐)) ≥ 1 for all 𝑛∈N ∪{0},then 𝑛 󵄩 󵄩 ≤ ∑ 𝜓 (󵄩𝜙0 −𝜙1󵄩 )<𝜖. 𝛼(𝜙 (𝑐), 𝜙(𝑐)) ≥1 𝑛∈N ∪{0} 󵄩 󵄩𝐸0 𝑛 for all ; 𝑛≥𝑁 (v) there exists 𝜙0 ∈ R𝑐 such that 𝛼(𝜙0(𝑐), 𝑇𝜙0)≥1. ‖𝜙 −𝜙 ‖ =0 {𝜙 } 𝑇 Consequently, lim𝑚,𝑛, → +∞ 𝑛 𝑚 𝐸0 .Hence 𝑛 is a Then, has a PPF dependent fixed point. Cauchy sequence in R𝑐 ⊆𝐸0.Bythecompletenessof𝐸0, {𝜙𝑛} ∗ ∗ converges to a point 𝜙 ∈𝐸0,thatis,𝜙𝑛 →𝜙,as𝑛→∞. We now introduce the notion of 𝛼𝑐-admissible mapping ∗ Since R𝑐 is topologically closed, we deduce that 𝜙 ∈ R𝑐. with respect to 𝜂𝑐 for the pair of maps (𝑆, 𝑇) as follows. Abstract and Applied Analysis 5

Definition 14. Let 𝑐∈𝐼, 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and Proof. As 𝑆:𝐸0 →𝐸0, so there exists 𝐹0 ⊆𝐸0 such that 𝛼 𝜂:𝐸×𝐸→ [0,∞) (𝑆, 𝑇) 𝑆(𝐹 )=𝑆(𝐸) 𝑆| 𝑇(𝐹 )⊆𝑇(𝐸)⊆ let , . We say that the pair is an 0 0 and 𝐹0 is one-to-one. Since 0 0 𝛼𝑐-admissible with respect to 𝜂𝑐,iffor𝜙, 𝜉∈𝐸0, 𝐸,wecandefinethemappingA :𝑆(𝐹0)→𝐸by A(𝑆𝜙) = 𝑇𝜙 𝜙∈𝐹 𝑆| A for all 0.Since 𝐹0 is one-to-one, then is well defined. 𝛼 ((𝑆𝜙) (𝑐) , (𝑆𝜉)(𝑐)) ≥ 𝜂 ((𝑆𝜙) (𝑐) , (𝑆𝜉)(𝑐)) Let (30) 󳨐⇒𝛼(𝑇𝜙,𝑇𝜉)≥𝜂(𝑇𝜙,𝑇𝜉). 𝛼 ((𝑆𝜙) (𝑐) , (𝑆𝜉)(𝑐)) ≥ 𝜂 ((𝑆𝜙) (𝑐) ,𝑇𝜙), then (34) Note that if we take 𝜂((𝑆𝜙)(𝑐), 𝑆(𝜉)(𝑐)),thenwesaythat =1 𝛼 ((𝑆𝜙) (𝑐) , (𝑆𝜉)(𝑐)) ≥ 𝜂 ((𝑆𝜙) (𝑐) , A (𝑆𝜙)) . the pair (𝑆, 𝑇) is an 𝛼𝑐-admissible mapping. Also, if we take 𝛼((𝑆𝜙)(𝑐), 𝑆(𝜉)(𝑐)) =1,thenwesaythatthepair(𝑆, 𝑇) is an Therefore, by (31)wehave 𝜂 𝑐-subadmissible mapping. 󵄩 󵄩 󵄩A (𝑆𝜙) − A (𝑆𝜉)󵄩𝐸 ≤𝜓(𝑁 (𝜙, 𝜉)) , (35) Now we introduce the notion of 𝛼𝑐𝜂𝑐-𝜓-contractiveness for the pair (𝑆, 𝑇) as follows. where 𝑁(𝜙,𝜉) Definition 15. Let 𝑐∈𝐼, 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝛼, 𝜂:𝐸×𝐸 → [0,∞).Then, 󵄩 󵄩 = {󵄩𝑆𝜙 −󵄩 𝑆𝜉 , max 󵄩 󵄩𝐸0 (i) we say that the pair (𝑆, 𝑇) is an 𝛼𝑐𝜂𝑐-𝜓-contractive if 󵄩 󵄩 󵄩 󵄩 𝛼 ((𝑆𝜙) (𝑐) ,𝜉(𝑐)) ≥ 𝜂 ((𝑆𝜙) (𝑐) ,𝑇𝜙) 󵄩(𝑆𝜙) (𝑐) − A (𝑆𝜙)󵄩 + 󵄩(𝑆𝜉)(𝑐) − A (𝑆𝜉)󵄩 󵄩 󵄩𝐸 󵄩 󵄩𝐸 , 󵄩 󵄩 (31) 2 󳨐⇒ 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤ 𝜓 (𝑁 (𝜙, 𝜉)); 󵄩 󵄩𝐸 󵄩 󵄩 󵄩 󵄩 󵄩(𝑆𝜙) (𝑐) − A (𝑆𝜉)󵄩 + 󵄩(𝑆𝜉)(𝑐) − A (𝑆𝜙)󵄩 󵄩 󵄩𝐸 󵄩 󵄩𝐸 }. (ii) we say that the pair (𝑆, 𝑇) is a modified 𝛼𝑐-𝜓- 2 contractive if (36) 󵄩 󵄩 𝛼 ((𝑆𝜙) (𝑐) ,𝜉(𝑐)) ≥1󳨐⇒󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 ≤𝜓(𝑁 (𝜙, 𝜉)) , This shows that A is an 𝛼𝑐𝜂𝑐-𝜓-contractive non-self- (32) mapping. Further, all other conditions of Theorem 12 hold true for A. Thus, there exists PPF dependent fixed point 𝜑∈ where 𝜓∈Ψand 𝑆(𝐹0) of A;thatis,A𝜑=𝜑(𝑐).Since𝜑∈𝑆(𝐹0), so there exists 𝜔∈𝐹0 such that 𝑆𝜔.Thus, =𝜑 𝑁(𝜙,𝜉) 𝑇𝜔 = A (𝑆𝜔) = A𝜑=𝜑(𝑐) = (𝑆𝜔)(𝑐) . (37) 󵄩 󵄩 = max {󵄩𝑆𝜙 −󵄩 𝑆𝜉 , 󵄩 󵄩𝐸0 That is, 𝜔 is a PPF dependent coincidence point of 𝑆 and 𝑇. 󵄩 󵄩 󵄩 󵄩 󵄩(𝑆𝜙) (𝑐) −𝑇(𝑆𝜙)󵄩 + 󵄩(𝑆𝜉)(𝑐) −𝑇(𝑆𝜉)󵄩 󵄩 󵄩𝐸 󵄩 󵄩𝐸 , 2 Corollary 17. Let 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸, 𝛼:𝐸×𝐸 → 󵄩 󵄩 󵄩 󵄩 󵄩(𝑆𝜙) (𝑐) −𝑇(𝑆𝜉)󵄩 + 󵄩(𝑆𝜙) (𝑐) −𝑇(𝑆𝜉)󵄩 [0, ∞) be three mappings satisfying the following assertions: 󵄩 󵄩𝐸 󵄩 󵄩𝐸 }. 2 (i) there exists 𝑐∈𝐼such that 𝑆(R𝑐) is topologically closed (33) and algebraically closed with respect to difference;

(ii) the pair (𝑆, 𝑇) is an 𝛼𝑐-admissible; Theorem 16. Let 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸, 𝛼, 𝜂:𝐸×𝐸 → [0, ∞) be four mappings satisfying the following assertions: (iii) the pair (𝑆, 𝑇) is a modified 𝛼𝑐-𝜓-contractive; 𝑐∈𝐼 𝑆(R ) (iv) if {𝑆𝜙𝑛} is a sequence in 𝐸0 such that 𝑆𝜙𝑛 →𝑆𝜙as (i) there exists such that 𝑐 is topologically closed 𝑛→∞ 𝛼((𝑆𝜙 )(𝑐), (𝑆𝜙 )(𝑐)) ≥ 1 𝑛∈ and algebraically closed with respect to difference; and 𝑛 𝑛+1 for all N ∪{0},then𝛼((𝑆𝜙𝑛)(𝑐), 𝑆𝜙(𝑐)) ≥1 for all 𝑛∈N ∪{0}; (ii) the pair (𝑆, 𝑇) is an 𝛼𝑐-admissible with respect to 𝜂𝑐; (v) there exists 𝑆𝜙0 ∈𝑆(R𝑐) such that 𝛼(𝑆𝜙0(𝑐), 𝑇𝜙0)≥1. (iii) the pair (𝑆, 𝑇) is an 𝛼𝑐𝜂𝑐-𝜓-contractive; Then, 𝑆 and 𝑇 have a PPF dependent coincidence point. (iv) if {𝑆𝜙𝑛} is a sequence in 𝐸0 such that 𝑆𝜙𝑛 → 𝑆𝜙 as 𝑛→∞and 𝛼((𝑆𝜙𝑛)(𝑐), (𝑆𝜙𝑛+1)(𝑐)) ≥ 𝜂((𝑆𝜙𝑛)(𝑐), (𝑆𝜙𝑛+1)(𝑐)) for all 𝑛∈N ∪{0},then 3. Some Results in Banach Spaces Endowed 𝛼((𝑆𝜙 )(𝑐), 𝑆𝜙(𝑐)) ≥ 𝜂((𝑆𝜙 )(𝑐), 𝑇𝜙 ) 𝑛∈N ∪ 𝑛 𝑛 𝑛 for all with a Graph {0}; Consistent with Jachymski [15], let (𝐸, 𝑑) be a metric space (v) there exists 𝑆𝜙0 ∈𝑆(R𝑐) such that 𝛼(𝑆𝜙0(𝑐), 𝑇𝜙0)≥ where 𝑑(𝑥,𝑦) = ‖𝑥−𝑦‖𝐸 for all 𝑥, 𝑦 ∈𝐸 and Δ denotes 𝜂(𝑆𝜙0(𝑐), 𝑇𝜙0). the diagonal of the Cartesian product of 𝑋×𝑋. Consider Then, 𝑆 and 𝑇 have a PPF dependent coincidence point. a directed graph 𝐺 such that the set 𝑉(𝐺) of its vertices 6 Abstract and Applied Analysis coincides with 𝑋,andtheset𝐸(𝐺) of its edges contains all Similarly as an application of Corollary 17,wecanprove loops; that is, 𝐸(𝐺) ⊇Δ. We assume that 𝐺 has no parallel the following Theorem. edges, so we can identify 𝐺 with the pair (𝑉(𝐺), 𝐸(𝐺)). Moreover, we may treat 𝐺 as a weighted graph (see [16,page Theorem 20. Let 𝑐∈𝐼, 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝐸 309]) by assigning to each edge the distance between its endowed with a graph 𝐺. Suppose that the following assertions vertices. If 𝑥 and 𝑦 are vertices in a graph 𝐺,thenapathin𝐺 hold true: 𝑥 𝑦 𝑁(𝑁∈N) {𝑥 }𝑁 𝑁+1 from to of length is a sequence 𝑖 𝑖=0 of (i) there exists 𝑐∈𝐼such that 𝑆(R𝑐) is topologically closed 𝑥 =𝑥 𝑥 =𝑦 (𝑥 ,𝑥)∈𝐸(𝐺) vertices such that 0 , 𝑁 and 𝑖−1 𝑖 for and algebraically closed with respect to difference; 𝑖=1,...,𝑁.Agraph𝐺 is connected if there is a path between (ii) if ((𝑆𝜙)(𝑐), (𝑆𝜉)(𝑐)) ∈𝐸(𝐺),then(𝑇𝜙, 𝑇𝜉) ∈𝐸(𝐺); any two vertices. 𝐺 is weakly connected if 𝐺̃is connected (see for more details [6, 11, 15]). (iii) assume that 󵄩 󵄩 ((𝑆𝜙) (𝑐) , (𝑆𝜉)(𝑐))∈𝐸(𝐺) 󳨐⇒ 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤ 𝜓 (𝑁 (𝜙, 𝜉)) Definition 18 (see [15]). Let (𝑋, 𝑑) be a metric space endowed 󵄩 󵄩𝐸 with a graph 𝐺. We say that a self-mapping 𝑇:𝑋 →𝑋 (42) is a Banach 𝐺-contraction or simply a 𝐺-contraction if 𝑇 for 𝜙, 𝜉∈𝐸0,where𝜓∈Ψ; preserves the edges of 𝐺;thatis, (iv) if {𝑆𝜙𝑛} is a sequence in 𝐸0 such that 𝑆𝜙𝑛 →𝑆𝜙as ∀𝑥,𝑦∈𝑋, (𝑥,𝑦)∈𝐸(𝐺) 󳨐⇒(𝑇𝑥,𝑇𝑦)∈𝐸(𝐺) (38) 𝑛→∞and ((𝑆𝜙𝑛)(𝑐), (𝑆𝜙𝑛+1)(𝑐)) ∈ 𝐸(𝐺) for all 𝑛∈ N∪{0} ((𝑆𝜙 )(𝑐), 𝑆𝜙(𝑐)) ∈ 𝐸(𝐺) 𝑛∈N∪{0} and 𝑇 decreases weights of the edges of 𝐺 in the following ,then 𝑛 for all ; way: (v) there exists 𝑆𝜙0 ∈𝑆(R𝑐) such that (𝑆𝜙0(𝑐), 𝑇𝜙0)∈ 𝐸(𝐺). ∃𝛼 ∈ (0, 1) such that ∀𝑥, 𝑦 ∈ 𝑋, (39) Then, 𝑆 and 𝑇 have a 𝑃𝑃𝐹 dependent coincidence point. (𝑥, 𝑦) ∈𝐸 (𝐺) 󳨐⇒ 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝛼𝑑 (𝑥,𝑦). The study of existence of fixed points in partially ordered sets has been initiated by Ran and Reurings [27]with Theorem 19. Let 𝑇:𝐸0 →𝐸and 𝐸 endowed with a graph ´ 𝐺. Suppose that the following assertions hold true: applications to matrix equations. Agarwal et al. [1, 2], Ciric´ et al. [7], and Hussain et al. [11, 12] presented some new 𝑐∈𝐼 R (i) there exists such that 𝑐 is topologically closed results for nonlinear contractions in partially ordered Banach and algebraically closed with respect to difference; and metric spaces with applications. Here as an application (ii) if (𝜙(𝑐), 𝜉(𝑐)) ∈𝐸(𝐺),then(𝑇𝜙, 𝑇𝜉) ∈𝐸(𝐺); of our results we deduce some new PPF dependent fixed (iii) assume that and coincidence point results whenever the range space is endowed with a partial order. 󵄩 󵄩 (𝜙 (𝑐) ,𝜉(𝑐))∈𝐸(𝐺) 󳨐⇒ 󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 ≤ 𝜓 (𝑀 (𝜙, 𝜉)), (40) Definition 21. Let 𝑐∈𝐼, 𝑇:𝐸0 →𝐸and 𝐸 endowed with 𝜙, 𝜉0 ∈𝐸 , 𝑤ℎ𝑒𝑟𝑒 𝜓∈Ψ; a partial order ⪯.Wesaythat𝑇 is a 𝑐-increasing non-self- mapping if for 𝜙, 𝜉∈𝐸0 with 𝜙(𝑐) ⪯ 𝜉(𝑐) we have 𝑇𝜙 ⪯ 𝑇𝜉. (iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ and (𝜙𝑛(𝑐),𝑛+1 𝜙 (𝑐)) ∈ 𝐸(𝐺) for all 𝑛∈N ∪{0},then Definition 22. Let 𝑐∈𝐼, 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝐸 (𝜙𝑛(𝑐), 𝜙(𝑐)) ∈ 𝐸(𝐺) for all 𝑛∈N ∪{0}; endowed with a partial order ⪯. We say that the pair (𝑆, 𝑇) is 𝑐 𝜙 𝜉∈𝐸 (𝑆𝜙)(𝑐) ⪯ (𝑆𝜉)(𝑐) (v) there exists 𝜙0 ∈ R𝑐 such that (𝜙0(𝑐), 𝑇𝜙0)∈𝐸(𝐺). -increasing if for , 0 with we have 𝑇𝜙 ⪯ 𝑇𝜉. Then, 𝑇 has a PPF dependent fixed point. Theorem 23. Let 𝑇:𝐸0 →𝐸and 𝐸 endowed with a partial 𝛼:𝐸×𝐸 → [0,+∞) Proof. Define by order ⪯. Suppose that the following assertions holds true:

{1, if (𝑥, 𝑦) ∈𝐸 (𝐺) (i) there exists 𝑐∈𝐼such that R𝑐 is topologically closed 𝛼(𝑥,𝑦)={1 (41) and algebraically closed with respect to difference; otherwise. {2 (ii) 𝑇 is a 𝑐-increasing non-self-mapping; 𝑇 𝛼 First, we prove that is an 𝑐-admissible non-self-mapping. (iii) Assume that Assume that 𝛼(𝜙(𝑐), 𝜉(𝑐)).Then,wehave ≥1 (𝜙(𝑐), 𝜉(𝑐)) ∈ 󵄩 󵄩 𝐸(𝐺).From(ii),wehave(𝑇𝜙, 𝑇𝜉) ∈𝐸(𝐺);thatis,𝛼(𝑇𝜙, 𝑇𝜉) ≥ 󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 ≤ 𝜓 (𝑀 (𝜙, 𝜉)) (43) 1.Thus𝑇 is an 𝛼𝑐-admissible non-self-mapping. From (v) 𝜙 𝜉∈𝐸 𝜙(𝑐) ⪯ 𝜉(𝑐) 𝜓∈Ψ there exists 𝜙0 ∈ R𝑐 such that 𝛼(𝜙0(𝑐), 𝑇𝜙0)≥1.Let, holds for all , 0 with where ; {𝜙𝑛} be a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ (iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ and (𝜙𝑛(𝑐),𝑛+1 𝜙 (𝑐)) ∈ 𝐸(𝐺) for all 𝑛∈N ∪{0}.Then, and 𝜙𝑛(𝑐) ⪯𝑛+1 𝜙 (𝑐) for all 𝑛∈N ∪{0},then𝜙𝑛(𝑐) ⪯ 𝛼(𝜙𝑛(𝑐),𝑛+1 𝜙 (𝑐)) ≥ 1 for all 𝑛∈N∪{0}.Thus,from(iv)weget, 𝜙(𝑐) for all 𝑛∈N ∪{0}; (𝜙𝑛(𝑐), 𝜙) ∈ 𝐸(𝐺) for all 𝑛∈N ∪{0}.Thatis,𝛼(𝜙𝑛(𝑐), 𝜙) ≥1 (v) there exists 𝜙0 ∈ R𝑐 such that 𝜙0(𝑐) ⪯ 𝑇𝜙0. for all 𝑛∈N ∪{0}. Therefore all conditions of Corollary 13 hold true and 𝑇 has a PPF dependent fixed point. Then, 𝑇 has a PPF dependent fixed point. Abstract and Applied Analysis 7

Proof. Define 𝛼:𝐸×𝐸 → [0,+∞)by Proof. Let 𝛼(𝜙(𝑐), 𝜉(𝑐)). ≥1 Hence, from (iii) we have 󵄩 󵄩 󵄩 󵄩 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤𝛼(𝜙(𝑐) ,𝜉(𝑐)) 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤ 𝜓 (𝑀 (𝜙, 𝜉)). {1, if 𝑥⪯𝑦 󵄩 󵄩𝐸 󵄩 󵄩𝐸 𝛼(𝑥,𝑦)= 1 (44) (47) { . {2 otherwise That is, all conditions of Corollary 13 are satisfied and 𝑇 has a 𝑇 𝛼 First, we prove that is an 𝑐-admissible non-self-mapping. PPF dependent fixed point. Assume that 𝛼(𝜙(𝑐), 𝜉(𝑐)).Then,wehave ≥1 𝜙(𝑐) ⪯ 𝜉(𝑐). Since 𝑇 is 𝑐-increasing, we get 𝑇𝜙 ⪯ 𝑇𝜉;thatis,𝛼(𝑇𝜙, 𝑇𝜉). ≥1 Similarly we can prove the following results. Thus 𝑇 is an 𝛼𝑐-admissible non-self-mapping. From (v) there Theorem 26. 𝑇:𝐸 →𝐸 𝛼:𝐸×𝐸→ [0,∞) exists 𝜙0 ∈ R𝑐 such that 𝜙0(𝑐) ⪯ 𝑇𝜙0.Thatis,𝛼(𝜙0(𝑐), 𝑇𝜙0)≥ Let 0 and be 1.Let{𝜙𝑛} be a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ two mappings that satisfy the following assertions: 𝛼(𝜙 (𝑐), 𝜙 (𝑐)) ≥ 1 𝑛∈N ∪{0} 𝜙 (𝑐) ⪯ and 𝑛 𝑛+1 for all .Then, 𝑛 𝑐∈𝐼 R 𝜙 (𝑐) 𝑛∈N ∪{0} 𝜙 (𝑐) ⪯ 𝜙(𝑐) (i) there exists such that 𝑐 is topologically closed 𝑛+1 for all .Thus,from(iv)weget 𝑛 and algebraically closed with respect to difference; for all 𝑛∈N∪{0}.Thatis,𝛼(𝜙𝑛(𝑐), 𝜙(𝑐)) ≥1 for all 𝑛∈N∪{0}. 𝑇 𝛼 Therefore all conditions of Corollary 13 hold true and 𝑇 has a (ii) is an 𝑐-admissible mapping; PPF dependent fixed point. (iii) assume that Similarly we can prove following Theorem. 󵄩 󵄩 𝛼(𝜙(𝑐),𝜉(𝑐)) (󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 +𝜖) ≤𝜓(𝑀 (𝜙, 𝜉)) +𝜖 (48) Theorem 24. Let 𝑐∈𝐼, 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝜙 𝜉∈𝐸 𝜖≥1 𝜓∈Ψ 𝐸 endowed with a partial order ⪯. Suppose that the following holds for all , 0,where and ; assertions hold true: (iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ 𝛼(𝜙 (𝑐), 𝜙 (𝑐)) ≥ 1 𝑛∈N ∪{0} (i) there exists 𝑐∈𝐼such that 𝑆(R𝑐) is topologically closed and 𝑛 𝑛+1 for all ,then 𝛼(𝜙 (𝑐), 𝜙(𝑐)) ≥1 𝑛∈N ∪{0} and algebraically closed with respect to difference; 𝑛 for all ; 𝜙 ∈ R 𝛼(𝜙 (𝑐), 𝑇𝜙 )≥1 (ii) the pair (𝑆, 𝑇) is a 𝑐-increasing mapping; (v) there exists 0 𝑐 such that 0 0 . (iii) assume that Then, 𝑇 has a PPF dependent fixed point. 󵄩 󵄩 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤𝜓(𝑁 (𝜙, 𝜉)) 󵄩 󵄩𝐸 (45) Theorem 27. Let 𝑇:𝐸0 →𝐸and 𝛼:𝐸×𝐸→ [0,∞)be two mappings that satisfy the following assertions: holds for all 𝜙, 𝜉∈𝐸0 with (𝑆𝜙)(𝑐) ⪯ (𝑆𝜉)(𝑐),where 𝜓∈Ψ; (i) there exists 𝑐∈𝐼such that R𝑐 is topologically closed and algebraically closed with respect to difference; (iv) if {𝑆𝜙𝑛} is a sequence in 𝐸0 such that 𝑆𝜙𝑛 →𝑆𝜙as 𝑛→∞and (𝑆𝜙𝑛)(𝑐) ⪯ (𝑆𝜙𝑛+1)(𝑐) for all 𝑛∈N ∪{0}, (ii) 𝑇 is an 𝛼𝑐-admissible mapping; (𝑆𝜙 )(𝑐) ⪯ 𝑆𝜙(𝑐) 𝑛∈N ∪{0} then 𝑛 for all ; (iii) assume that 𝑆𝜙 ∈𝑆(R ) 𝑆𝜙 (𝑐) ⪯ 𝑇𝜙 (v) there exists 0 𝑐 such that 0 0. ‖𝑇𝜙−𝑇𝜉‖ (𝛼 (𝜙 (𝑐) ,𝜉(𝑐)) −1+𝜖󸀠) 𝐸 ≤𝜖𝜓(𝑀(𝜙,𝜉)) (49) Then, 𝑆 and 𝑇 have a PPF dependent coincidence point. 󸀠 holds for all 𝜙, 𝜉∈𝐸0,where1<𝜖≤𝜖 and 𝜓∈Ψ; 4. Further Consequences (iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ 𝛼(𝜙 (𝑐), 𝜙 (𝑐)) ≥ 1 𝑛∈N ∪{0} 4.1. Consequences of Corollary 13 and 𝑛 𝑛+1 for all ,then 𝛼(𝜙𝑛(𝑐), 𝜙(𝑐)) ≥1 for all 𝑛∈N ∪{0}; Theorem 25. 𝑇:𝐸 →𝐸 𝛼:𝐸×𝐸→ [0,∞) Let 0 and be (v) there exists 𝜙0 ∈ R𝑐 such that 𝛼(𝜙0(𝑐), 𝑇𝜙0)≥1. two mappings that satisfy the following assertions: Then, 𝑇 has a 𝑃𝑃𝐹 dependent fixed point. (i) there exists 𝑐∈𝐼such that R𝑐 is topologically closed and algebraically closed with respect to difference; 4.2. Consequences of Corollary 17 (ii) 𝑇 is an 𝛼𝑐-admissible mapping;

(iii) assume that Theorem 28. Let 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝛼:𝐸×𝐸→ [0, ∞) 󵄩 󵄩 be three mappings that satisfy the following assertions: 𝛼 (𝜙 (𝑐) ,𝜉(𝑐)) 󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 ≤𝜓(𝑀 (𝜙, 𝜉)) (46) (i) there exists 𝑐∈𝐼such that 𝑆(R𝑐) is topologically closed holds for all 𝜙, 𝜉∈𝐸0,where𝜓∈Ψ; and algebraically closed with respect to difference;

(iv) if {𝜙𝑛} is a sequence in 𝐸0 such that 𝜙𝑛 →𝜙as 𝑛→∞ (ii) the pair (𝑆, 𝑇) is an 𝛼𝑐-admissible; 𝛼(𝜙 (𝑐), 𝜙 (𝑐)) ≥ 1 𝑛∈N ∪{0} and 𝑛 𝑛+1 for all ,then (iii) assume that 𝛼(𝜙𝑛(𝑐), 𝜙(𝑐)) ≥1 for all 𝑛∈N ∪{0}; 󵄩 󵄩 𝛼((𝑆𝜙)(𝑐) , (𝑆𝜉)(𝑐)) 󵄩𝑇𝜙 − 𝑇𝜉󵄩 ≤ 𝜓 (𝑁 (𝜙, 𝜉)) (v) there exists 𝜙0 ∈ R𝑐 such that 𝛼(𝜙0(𝑐), 𝑇𝜙0)≥1. 󵄩 󵄩𝐸 (50)

Then, 𝑇 has a PPF dependent fixed point. holds for all 𝜙, 𝜉∈𝐸0,where𝜓∈Ψ; 8 Abstract and Applied Analysis

(iv) if {𝑆𝜙𝑛} is a sequence in 𝐸0 such that 𝑆𝜙𝑛 →𝑆𝜙as References 𝑛→∞and 𝛼((𝑆𝜙𝑛)(𝑐), (𝑆𝜙𝑛+1)(𝑐)) ≥ 1 for all 𝑛∈ [1] R. P. Agarwal, M. A. El-Gebeily, and D. O’Regan, “Generalized N ∪{0},then𝛼((𝑆𝜙𝑛)(𝑐), 𝑆𝜙(𝑐)) ≥1 for all 𝑛∈N ∪{0}; contractions in partially ordered metric spaces,” Applicable (v) there exists 𝑆𝜙0 ∈𝑆(R𝑐) such that 𝛼(𝑆𝜙0(𝑐), 𝑇𝜙0)≥1. Analysis,vol.87,no.1,pp.1–8,2008. [2] R. P. Agarwal, N. Hussain, and M. A. Taoudi, “Fixed point the- Then, 𝑆 and 𝑇 have a PPF dependent coincidence point. orems in ordered Banach spaces and applications to nonlinear integral equations,” Abstract and Applied Analysis,vol.2012, Theorem 29. Let 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝛼:𝐸×𝐸→ Article ID 245872, 15 pages, 2012. [0, ∞) be three mappings that satisfy the following assertions: [3] M. A. Abbas and T. Nazir, “Common fixed point of a power graphic contraction pair in partial metric spaces endowed with 𝑐∈𝐼 𝑆(R ) (i) there exists such that 𝑐 is topologically closed graph,” Fixed Point Theory and Applications,vol.2013,p.20, and algebraically closed with respect to difference; 2013. [4] A. Gh. B. Ahmad, Z. Fadail, H. K. Nashine, Z. Kadelburg, and (ii) the pair (𝑆, 𝑇) is an 𝛼𝑐-admissible; S. Radenovic, “Some new common fixed point results through (iii) assume that generalized altering distances on partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, p. 120, 2012. 󵄩 󵄩 𝛼((𝑆𝜙)(𝑐),(𝑆𝜉)(𝑐)) (󵄩𝑇𝜙 − 𝑇𝜉󵄩𝐸 +𝜖) ≤𝜓(𝑁 (𝜙, 𝜉)) +𝜖 (51) [5] S. R. Bernfeld, V. Lakshmikatham, and Y. M. Reddy, “Fixed point theorems of operators with PPF dependence in Banach spaces,” Applicable Analysis,vol.6,no.4,pp.271–280,1977. holds for all 𝜙, 𝜉∈𝐸0,where𝜖≥1and 𝜓∈Ψ; [6] F. Bojor, “Fixed point theorems for Reich type contractions (iv) if {𝑆𝜙𝑛} is a sequence in 𝐸0 such that 𝑆𝜙𝑛 →𝑆𝜙as on metric spaces with a graph,” Nonlinear Analysis, Theory, 𝑛→∞and 𝛼((𝑆𝜙𝑛)(𝑐), (𝑆𝜙𝑛+1)(𝑐)) ≥ 1 for all 𝑛∈ Methods and Applications,vol.75,no.9,pp.3895–3901,2012. N ∪{0} 𝛼((𝑆𝜙 )(𝑐), 𝑆𝜙(𝑐)) ≥1 𝑛∈N ∪{0} ,then 𝑛 for all ; [7] L. Ciri´ c,´ M. Abbas, R. Saadati, and N. Hussain, “Common fixed points of almost generalized contractive mappings in ordered (v) there exists 𝑆𝜙0 ∈𝑆(R𝑐) such that 𝛼(𝑆𝜙0(𝑐), 𝑇𝜙0)≥1. metric spaces,” Applied Mathematics and Computation,vol.217, Then, 𝑆 and 𝑇 have a PPF dependent coincidence point. no.12,pp.5784–5789,2011. [8] L. B. Ciri´ c,´ S. M. A. Alsulami, P. Salimi, and P. Vetro, “PPF ∝ Theorem 30. Let 𝑆:𝐸0 →𝐸0, 𝑇:𝐸0 →𝐸and 𝛼:𝐸×𝐸→ dependent fixed point results for triangular 𝑐-admissible [0, ∞) be three mappings that satisfy the following assertions: mapping,” The Scientific World Journal.Inpress. [9] B. C. Dhage, “Some basic random fixed point theorems with (i) there exists 𝑐∈𝐼such that 𝑆(R𝑐) is topologically closed PPF dependence and functional random differential equations,” and algebraically closed with respect to difference; Differential Equations & Applications,vol.4,pp.181–195,2012. [10] M. Geraghty, “On contractive mappings,” Proceedings of the (𝑆, 𝑇) 𝛼 (ii) the pair is an 𝑐-admissible; American Mathematical Society,vol.40,pp.604–608,1973. (iii) assume that [11] N. Hussain, S. Al-Mezel, and P. Salimi, “Fixed points for 𝜓- graphic contractions with application to integral equations,” ‖𝑇𝜙−𝑇𝜉‖ Abstract and Applied Analysis,vol.2013,ArticleID575869,11 (𝛼 ((𝑆𝜙) (𝑐) , (𝑆𝜉)(𝑐))−1+𝜖󸀠) 𝐸 ≤𝜖𝜓(𝑁(𝜙,𝜉)) (52) pages, 2013. [12] N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnosel’skii and 󸀠 holds for all 𝜙, 𝜉∈𝐸0,where1<𝜖≤𝜖 and 𝜓∈Ψ; Ky Fan type fixed point theorems in ordered Banach spaces,” {𝑆𝜙 } 𝐸 𝑆𝜙 →𝑆𝜙 Journal of Nonlinear and Convex Analysis,vol.11,no.3,pp.475– (iv) if 𝑛 is a sequence in 0 such that 𝑛 as 489, 2010. 𝑛→∞and 𝛼((𝑆𝜙𝑛)(𝑐), (𝑆𝜙𝑛+1)(𝑐)) ≥ 1 for all 𝑛∈ N ∪{0} 𝛼((𝑆𝜙 )(𝑐), 𝑆𝜙(𝑐)) ≥1 𝑛∈N ∪{0} [13] M. A. Kutbi, N. Hussain, and P. Salimi, “Best proximity ,then 𝑛 for all ; point results for modified 𝛼-𝜓-proximal rational contractions,” (v) there exists 𝑆𝜙0 ∈𝑆(R𝑐) such that 𝛼(𝑆𝜙0(𝑐), 𝑇𝜙0)≥1. Abstract and Applied Analysis,vol.2013,ArticleID927457,14 pages, 2013. Then, 𝑆 and 𝑇 have a PPF dependent coincidence point. [14] N. Hussain, P. Salimi, and A. Latif, “Fixed point results for single and set-valued 𝛼-𝜂-𝜓-contractive mappings,” Fixed Point Theory and Applications, vol. 2013, p. 212, 2013. Conflict of Interests [15] J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American The authors declare that there is no conflict of interests Mathematical Society,vol.136,no.4,pp.1359–1373,2008. regarding the publication of this paper. [16] R. Johnsonbaugh, Discrete Mathematics, Prentice-Hall, New Jersey, NJ, USA, 1997. Acknowledgment [17] A. Kaewcharoen, “PPF dependent common fixed point theo- rems for mappings in Bnach spaces,” Journal of Inequalities and This article was funded by the Deanship of Scientific Research Applications,vol.2013,p.287,2013. (DSR), King Abdulaziz University, Jeddah. Therefore, the first [18] E. Karapinar, P. Kumam, and P. Salimi, “On 𝛼-𝜓-Meir-Keeler author acknowledges with thanks DSR, KAU for financial contractive mappings,” Fixed Point Theory and Applications,vol. support. 2013,p.94,2013. Abstract and Applied Analysis 9

[19] E. Karapinar and B. Samet, “Generalized (𝛼-𝜓) contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis,vol.2012,ArticleID793486,17 pages, 2012. [20] F. Khojasteh, E. Karapinar, and S. Radenovic, “𝜃-metric space: a generalization,” Mathematical Problems in Engineering,vol. 2013, Article ID 504609, 7 pages, 2013. [21] M. Mursaleen, S. Mohiuddine, and R. P. Agarwal, “Coupled fixed point theorems for 𝛼-𝜓-contractive type mappings in partially ordered metric spaces,” Fixed Point Theory and Appli- cations, vol. 2012, p. 228, 2012. [22] B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for 𝛼-𝜓-contractive type mappings,” Nonlinear Analysis, Theory, Methods and Applications,vol.75,no.4,pp.2154–2165,2012. [23] M. Samreen and T. Kamran, “Fixed point theorems for integral G-contraction,” Fixed Point Theory and Applications,vol.2013, p. 149, 2013. [24] P. Salimi, A. Latif, and N. Hussain, “Modified 𝛼-𝜓-contractive mappings with applications,” Fixed Point Theory and Applica- tions, vol. 2013, p. 151, 2013. [25] P. Salimi, C. Vetro, and P. Vetro, “Some new fixed point results in non-Archimedean fuzzy metric spaces,” Nonlinear Analysis: Modelling and Control, vol. 18, no. 3, pp. 344–358, 2013. [26] P. Salimi, C. Vetro, and P. Vetro, “Fixed point theorems for twisted (𝛼,𝛽)-𝜓-contractive type mappings and applications,” Filomat,vol.27,no.4,pp.605–615,2013. [27] A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. [28] N. Hussain, M. H. Shah, and M. A. Kutbi, “Coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a Q-function,” Fixed Point Theory and Applications, vol. 2011, Article ID 703938, 21 pages, 2011. [29] Y. J. Cho, M. H. Shah, and N. Hussain, “Coupled fixed points of weakly F-contractive mappings in topological spaces,” Applied Mathematics Letters,vol.24,no.7,pp.1185–1190,2011. [30] M. A. Kutbi, J. Ahmad, and A. Azam, “On fixed points of 𝛼-𝜓-contractive multi-valued mappings in cone metric spaces,” Abstract and Applied Analysis,vol.2013,ArticleID313782,6 pages, 2013. [31] R. P.Agarwal, P.Kumam, and W.Sintunavarat, “PPF dependent fixed point theorems for an 𝛼𝑐-admissible non-self mapping in the Razumikhin class,” Fixed Point Theory and Applications,vol. 2013, article 280, 2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 214123, 8 pages http://dx.doi.org/10.1155/2013/214123

Research Article Further Refinements of Jensen’s Type Inequalities for the Function Defined on the Rectangle

M. Adil Khan,1 G. Ali Khan,1 T. Ali,1 T. Batbold,2 and A. Kiliçman3

1 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan 2 Institute of Mathematics, National University of Mongolia, P.O. Box 46A/104, 14201 Ulaanbaatar, Mongolia 3 Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), 43400Serdang,Selangor,Malaysia

Correspondence should be addressed to A. Kilic¸man; [email protected]

Received 26 September 2013; Accepted 10 November 2013

Academic Editor: Abdullah Alotaibi

Copyright © 2013 M. Adil Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give refinement of Jensen’s type inequalities given by Bakula and Pecariˇ c´ (2006) for the co-ordinate convex function. Also we establish improvement of Jensen’s inequality for the convex function of two variables.

𝑛 1. Introduction a nonnegative m-tuple such that 𝑃𝑛 =∑𝑖=1 𝑝𝑖 >0and 𝑊𝑚 = 𝑚 ∑𝑖=1 𝑤𝑗 >0,then Jensen’s inequality for convex functions plays a crucial role in the theory of inequalities due to the fact that other 1 𝑛 1 𝑚 inequalities such as the arithmetic mean-geometric mean 𝑓( ∑𝑝 𝑥 , ∑𝑤 𝑦 ) 𝑃 𝑖 𝑖 𝑊 𝑗 𝑗 inequality, the Holder¨ and Minkowski inequalities, and the 𝑛 𝑖=1 𝑚 𝑗=1 Ky Fan inequality, can be obtained as particular cases of it. Therefore, it is worth studying it thoroughly and refining it 1 { 1 𝑛 1 𝑚 } ≤ ∑𝑝 𝑓(𝑥, 𝑦) + ∑𝑤 𝑓(𝑥, 𝑦 ) from different point of view. There are many refinements of { 𝑖 𝑖 𝑗 𝑗 } (1) 2 𝑃𝑛 𝑊𝑚 Jensen’s inequality; see, for example, [1–14] and the references { 𝑖=1 𝑗=1 } in them. 𝑛 𝑚 2 1 Afunction𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R, [𝑎, 𝑏] × [𝑐, 𝑑]⊂ R ≤ ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 ), 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 with 𝑎<𝑏and 𝑐<𝑑is called convex on the co-ordinates 𝑛 𝑚 𝑖=1𝑗=1 𝑓𝑦 : [𝑎, 𝑏] → R 𝑓𝑦(𝑡) = if the partial mappings defined as 𝑛 𝑚 𝑓(𝑡, 𝑦) and 𝑓𝑥 :[𝑐,𝑑]→ Rdefined as 𝑓𝑥(𝑠) = 𝑓(𝑥, 𝑠) are where 𝑥=(1/𝑃𝑛)∑𝑖=1 𝑝𝑖𝑥𝑖,and𝑦=(1/𝑊𝑚)∑𝑗=1 𝑤𝑗𝑦𝑗. convex for all 𝑥∈[𝑎,𝑏], 𝑦∈[𝑐,𝑑].Notethateveryconvex function 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R is co-ordinate convex, but Recently Dragomir has given new refinement for Jensen theconverseisnotgenerallytrue[8]. inequality in [9]. The purpose of this paper is to give The following theorem has been given in4 [ ]. related refinements of Jensen’s type inequalities (1)forthe co-ordinate convex function. We will also discuss some Theorem 1. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R be a convex function particular interesting cases. We establish improvement of on the co-ordinates on [𝑎, 𝑏]×[𝑐, 𝑑].Ifx is an n-tuple in [𝑎, 𝑏], Jensen’s inequality for the convex function defined on the y is m-tuple in [𝑐, 𝑑], p is a nonnegative n-tuple, and w is rectangles. For related improvements of Jensen’s inequality, 2 Abstract and Applied Analysis

see, for example, [1, 2, 9, 13, 14]. For further several related 𝐷𝑘 (𝑓, p, x) integral inequalities, see [15]. := 𝐷 (𝑓, p, x, {𝑘})

2. Main Results 𝑝 𝑃 −𝑝 ∑𝑛 𝑝 𝑥 −𝑝 𝑥 = 𝑘 𝑓(𝑥 , 𝑦) + 𝑛 𝑘 𝑓( 𝑖=1 𝑖 𝑖 𝑘 𝑘 , 𝑦) , 𝑃 𝑘 𝑃 𝑃 −𝑝 Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R be convex on the co-ordinate 𝑛 𝑛 𝑛 𝑘 on [𝑎, 𝑏] × [𝑐, 𝑑].If𝑥𝑖 ∈ [𝑎, 𝑏], 𝑦𝑗 ∈[𝑐,𝑑], 𝑝𝑖,𝑤𝑗 >0, 𝑖∈ 𝑛 𝐷𝑙 (𝑓, w, y) {1,2,...,𝑛}, 𝑗 ∈ {1,2,...,𝑚} with 𝑃𝑛 = ∑𝑖=1 𝑝𝑖,and𝑊𝑚 = 𝑚 ∑𝑗=1 𝑤𝑗,thenforanysubsets𝐼 ⊂ {1,2,...,𝑛} and 𝐽⊂ := 𝐷 (𝑓, w, y, {𝑙}) {1,2,...,𝑚},weassumethat𝐼 := {1,2,...,𝑛}\ 𝐼 and 𝐽:= 𝑚 𝑤 𝑊 −𝑤 ∑𝑗=1 𝑤𝑗𝑦𝑗 −𝑤𝑙𝑦𝑙 {1,2,...,𝑚}\.Define 𝐽 𝑃𝐼 = ∑𝑖∈𝐼 𝑝𝑖, 𝑃𝐼 = ∑𝑖∈𝐼 𝑝𝑖, 𝑊𝐽 = 𝑙 𝑚 𝑙 = 𝑓(𝑥,𝑙 𝑦 )+ 𝑓(𝑥, ). ∑𝑗∈𝐽 𝑤𝑗,and𝑊𝐽 = ∑𝑗∈𝐽 𝑤𝑗.Forthefunction𝑓 and the 𝑛-, 𝑊𝑚 𝑊𝑚 𝑊𝑚 −𝑤𝑙 𝑚 x =(𝑥,𝑥 ,...,𝑥 ) y =(𝑦,𝑦 ,...,𝑦 ) p =(𝑝, -tuples, 1 2 𝑛 , 1 2 𝑚 , 1 (3) 𝑝2,...,𝑝𝑛),andw =(𝑤1,𝑤2,...,𝑤𝑚),wedefinethefollowing functionals: The following refinement of (1)holds.

𝐷(𝑓,p, x,𝐼,𝑦𝑗) Theorem 2. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R be a co-ordinate convex function on [𝑎, 𝑏] × [𝑐, 𝑑].If𝑥𝑖 ∈ [𝑎, 𝑏], 𝑦𝑗 ∈[𝑐,𝑑], 𝑝𝑖, 𝑃 𝑃 𝑛 𝐼 1 𝐼 1 𝑤 >0 𝑖 ∈ {1,2,...,𝑛} 𝑗 ∈ {1,2,...,𝑚} 𝑃 = ∑ 𝑝 = 𝑓( ∑𝑝𝑖𝑥𝑖,𝑦𝑗)+ 𝑓( ∑𝑝𝑖𝑥𝑖,𝑦𝑗), 𝑗 , , with 𝑛 𝑖=1 𝑖, 𝑚 𝑃𝑛 𝑃𝐼 𝑖∈𝐼 𝑃𝑛 𝑃𝐼 𝑖∈𝐼 and 𝑊𝑚 = ∑𝑗=1 𝑤𝑗,thenforanysubsets𝐼 ⊂ {1,2,...,𝑛} and 𝐽⊂{1,2,...,𝑚},onehas 𝐷(𝑓,w, y,𝐽,𝑥𝑖) 𝑓(𝑥, 𝑦) 𝑊 1 𝑊 1 = 𝐽 𝑓(𝑥, ∑𝑤 𝑦 )+ 𝐽 𝑓(𝑥, ∑𝑤 𝑦 ), 1 𝑊 𝑖 𝑊 𝑗 𝑗 𝑊 𝑖 𝑊 𝑗 𝑗 ≤ [𝐷 (𝑓, w, y,𝐽)+𝐷(𝑓,p, x,𝐼)] 𝑚 𝐽 𝑗∈𝐽 𝑚 𝐽 𝑗∈𝐽 2

𝐷(𝑓,p, x,𝐼) 1 1 𝑛 1 𝑚 ≤ [ ∑𝑝 𝑓(𝑥, 𝑦) + ∑𝑤 𝑓(𝑥, 𝑦 )] 2 𝑃 𝑖 𝑖 𝑊 𝑗 𝑗 [ 𝑛 𝑖=1 𝑚 𝑗=1 ] 𝑃𝐼 1 𝑃𝐼 1 = 𝑓( ∑𝑝𝑖𝑥𝑖, 𝑦) + 𝑓( ∑𝑝𝑖𝑥𝑖, 𝑦) , 𝑃 𝑃 𝑃 𝑃 𝑛 𝑚 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 1 1 1 ≤ [ ∑𝑝 𝐷(𝑓,w, y,𝐽,𝑥)+ ∑𝑤 𝐷(𝑓,p, x,𝐼,𝑦 )] 2 𝑃 𝑖 𝑖 𝑊 𝑗 𝑗 𝐷(𝑓,w, y,𝐽) [ 𝑛 𝑖=1 𝑚 𝑗=1 ] 1 𝑛 𝑚 𝑊 1 𝑊 1 ≤ ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 ), 𝐽 𝐽 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 = 𝑓(𝑥, ∑𝑤𝑗𝑦𝑗)+ 𝑓(𝑥, ∑𝑤𝑗𝑦𝑗), 𝑛 𝑚 𝑖=1𝑗=1 𝑊𝑚 𝑊𝐽 𝑊𝑚 𝑊 𝑗∈𝐽 𝐽 𝑗∈𝐽 (4) (2) 𝑛 𝑚 𝑛 𝑚 where 𝑥=(1/𝑃𝑛)∑𝑖=1 𝑝𝑖𝑥𝑖,and 𝑦=(1/𝑊𝑚)∑𝑗=1 𝑤𝑗𝑦𝑗. where 𝑥=(1/𝑃𝑛) ∑𝑖=1 𝑝𝑖𝑥𝑖,and 𝑦=(1/𝑊𝑚) ∑𝑗=1 𝑤𝑗𝑦𝑗. 𝐼={𝑘}𝑘 ∈ {1,...,𝑛} It is worth to observe that for , , Proof. One-dimensional Jensen’s inequality gives us and 𝐽={𝑙}, 𝑙∈{1,...,𝑚},wehavethefunctionals 1 𝑚 𝐷𝑘 (𝑓, p, x,𝑦𝑗) 𝑓(𝑥𝑖, 𝑦) ≤ ∑𝑤𝑗𝑓(𝑥𝑖,𝑦𝑗), 𝑊𝑚 𝑗=1 := 𝐷 (𝑓, p, x, {𝑘} ,𝑦𝑗) (5) 1 𝑛 𝑓(𝑥, 𝑦 )≤ ∑𝑝 𝑓(𝑥,𝑦 ). ∑𝑛 𝑝 𝑥 −𝑝 𝑥 𝑗 𝑖 𝑖 𝑗 𝑝𝑘 𝑃𝑛 −𝑝𝑘 𝑖=1 𝑖 𝑖 𝑘 𝑘 𝑃𝑛 𝑖=1 = 𝑓(𝑥𝑘,𝑦𝑗)+ 𝑓( ,𝑦𝑗), 𝑃𝑛 𝑃𝑛 𝑃𝑛 −𝑝𝑘 As we have 𝐷𝑙 (𝑓, w, y,𝑥𝑖)

𝐷(𝑓,p, x,𝐼,𝑦𝑗) := 𝐷 (𝑓, w, y, {𝑙} ,𝑥𝑖) 𝑚 (6) 𝑃𝐼 1 𝑃𝐼 1 𝑤 𝑊 −𝑤 ∑𝑗=1 𝑤𝑗𝑦𝑗 −𝑤𝑙𝑦𝑙 = 𝑓( ∑𝑝 𝑥 ,𝑦 )+ 𝑓( ∑𝑝 𝑥 ,𝑦 ), = 𝑙 𝑓(𝑥,𝑦)+ 𝑚 𝑙 𝑓(𝑥, ), 𝑖 𝑖 𝑗 𝑖 𝑖 𝑗 𝑖 𝑙 𝑖 𝑃𝑛 𝑃𝐼 𝑖∈𝐼 𝑃𝑛 𝑃𝐼 𝑊𝑚 𝑊𝑚 𝑊𝑚 −𝑤𝑙 𝑖∈𝐼 Abstract and Applied Analysis 3

so by Jensen’s inequality, we have Multiplying (9)and(10), respectively, by 𝑤𝑗 and 𝑝𝑖 and sum- ming over 𝑖 and 𝑗,weobtain 𝐷(𝑓,p, x,𝐼,𝑦 ) 𝑗 1 𝑚 1 𝑚 ∑𝑤𝑗𝑓(𝑥,𝑗 𝑦 )≤ ∑𝑤𝑗𝐷(𝑓,p, x,𝐼,𝑦𝑗) 𝑊𝑚 𝑗=1 𝑊𝑚 𝑗=1 𝑃𝐼 1 𝑃𝐼 1 = 𝑓( ∑𝑝𝑖𝑥𝑖,𝑦𝑗)+ 𝑓( ∑𝑝𝑖𝑥𝑖,𝑦𝑗) (11) 𝑃 𝑃 𝑃 𝑃 𝑛 𝑚 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 1 ≤ ∑∑𝑝𝑖𝑤𝑗𝑓(𝑥𝑖,𝑦𝑗), 𝑃𝑛𝑊𝑚 𝑃𝐼 1 𝑃𝐼 1 𝑖=1𝑗=1 ≤ ∑𝑝𝑖𝑓(𝑥𝑖,𝑦𝑗)+ ∑𝑝𝑖𝑓(𝑥𝑖,𝑦𝑗) 𝑃 𝑃 𝑃 𝑃 𝑛 𝑛 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 1 1 ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) ≤ ∑𝑝𝑖𝐷(𝑓,w, y,𝐽,𝑥𝑖) 1 1 (7) 𝑃 𝑃 = ∑𝑝 𝑓(𝑥,𝑦 )+ ∑𝑝 𝑓(𝑥,𝑦 ) 𝑛 𝑖=1 𝑛 𝑖=1 𝑃 𝑖 𝑖 𝑗 𝑃 𝑖 𝑖 𝑗 (12) 𝑛 𝑖∈𝐼 𝑛 𝑖∈𝐼 1 𝑛 𝑚 ≤ ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 ). 1 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 = ∑ 𝑝 𝑓(𝑥,𝑦 ) 𝑛 𝑚 𝑖=1𝑗=1 𝑃 𝑖 𝑖 𝑗 𝑛 𝑖∈𝐼∪𝐼 Adding (11)and(12), we have 1 𝑛 󳨐⇒ 𝐷 (𝑓, p, x,𝐼,𝑦 )≤ ∑𝑝 𝑓(𝑥,𝑦 ). 𝑛 𝑚 𝑗 𝑃 𝑖 𝑖 𝑗 1 1 1 𝑛 𝑖=1 [ ∑𝑝 𝑓(𝑥, 𝑦) + ∑𝑤 𝑓(𝑥, 𝑦 )] 2 𝑃 𝑖 𝑖 𝑊 𝑗 𝑗 [ 𝑛 𝑖=1 𝑚 𝑗=1 ] As the function 𝑓 is convex on the first co-ordinate, so we have 1 1 𝑛 ≤ [ ∑𝑝 𝐷(𝑓,w, y,𝐽,𝑥) 2 𝑃 𝑖 𝑖 [ 𝑛 𝑖=1 𝐷(𝑓,p, x,𝐼,𝑦𝑗) (13) 1 𝑚 𝑃 1 𝑃 1 + ∑𝑤 𝐷(𝑓,p, x,𝐼,𝑦 )] = 𝐼 𝑓( ∑𝑝 𝑥 ,𝑦 )+ 𝐼 𝑓( ∑𝑝 𝑥 ,𝑦 ) 𝑊 𝑗 𝑗 𝑃 𝑃 𝑖 𝑖 𝑗 𝑃 𝑃 𝑖 𝑖 𝑗 𝑚 𝑗=1 ] 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 1 𝑛 𝑚 𝑃 1 𝑃 1 ≤ ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 ). ≥𝑓( 𝐼 ∑𝑝 𝑥 + 𝐼 ∑𝑝 𝑥 ,𝑦 ) 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 𝑃 𝑃 𝑖 𝑖 𝑃 𝑃 𝑖 𝑖 𝑗 𝑛 𝑚 𝑖=1𝑗=1 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 Again by one-dimensional Jensen’s inequality, we have 1 1 =𝑓( ∑𝑝 𝑥 + ∑𝑝 𝑥 ,𝑦 ) 𝑖 𝑖 𝑖 𝑖 𝑗 1 𝑛 𝑃𝑛 𝑖∈𝐼 𝑃𝑛 (8) 𝑖∈𝐼 𝑓(𝑥, 𝑦) ≤ ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) , 𝑃𝑛 𝑖=1 1 (14) =𝑓( ∑ 𝑝 𝑥 ,𝑦 ) 1 𝑚 𝑃 𝑖 𝑖 𝑗 𝑓(𝑥, 𝑦) ≤ ∑𝑤 𝑓(𝑥, 𝑦 ). 𝑛 𝑖∈𝐼∪𝐼 𝑗 𝑗 𝑊𝑚 𝑗=1 1 𝑛 =𝑓( ∑𝑝𝑖𝑥𝑖,𝑦𝑗) As we have the functional 𝑃𝑛 𝑖=1 𝑃𝐼 1 𝑃𝐼 1 1 𝑛 𝐷(𝑓,p, x,𝐼)= 𝑓( ∑𝑝 𝑥 , 𝑦) + 𝑓( ∑𝑝 𝑥 , 𝑦) , 󳨐⇒ 𝐷 (𝑓, p, x,𝐼,𝑦 )≥𝑓( ∑𝑝 𝑥 ,𝑦 ). 𝑃 𝑃 𝑖 𝑖 𝑃 𝑃 𝑖 𝑖 𝑗 𝑖 𝑖 𝑗 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 𝑃𝑛 𝑖=1 (15)

Now, from (7)and(8), we have so by Jensen’s inequality, we get

𝑛 𝑃 1 1 𝐷(𝑓,p, x,𝐼)= 𝐼 𝑓( ∑𝑝 𝑥 , 𝑦) 𝑓(𝑥,𝑗 𝑦 )≤𝐷(𝑓,p, x,𝐼,𝑦𝑗)≤ ∑𝑝𝑖𝑓(𝑥𝑖,𝑦𝑗). (9) 𝑖 𝑖 𝑃𝑛 𝑃𝐼 𝑃𝑛 𝑖=1 𝑖∈𝐼 𝑃 1 Similarly, we can write + 𝐼 𝑓( ∑𝑝 𝑥 , 𝑦) 𝑃 𝑃 𝑖 𝑖 𝑛 𝐼 𝑖∈𝐼 𝑚 1 𝑃 1 𝑃 1 𝑓(𝑥𝑖, 𝑦) ≤ 𝐷 (𝑓, w, y,𝐽,𝑥𝑖)≤ ∑𝑤𝑗𝑓(𝑥𝑖,𝑦𝑗). 𝐼 𝐼 (10) ≤ ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) + ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) 𝑊𝑚 𝑗=1 𝑃 𝑃 𝑃 𝑃 𝑛 𝐼 𝑖∈𝐼 𝑛 𝐼 𝑖∈𝐼 4 Abstract and Applied Analysis

1 1 = ∑𝑝 𝑓(𝑥, 𝑦) + ∑𝑝 𝑓(𝑥, 𝑦) Combining (13)and(20), we have 𝑃 𝑖 𝑖 𝑃 𝑖 𝑖 𝑛 𝑖∈𝐼 𝑛 𝑖∈𝐼

1 𝑓(𝑥, 𝑦) = ∑ 𝑝𝑖𝑓(𝑥𝑖, 𝑦) 𝑃𝑛 𝑖∈𝐼∪𝐼 1 ≤ [𝐷 (𝑓, w, y,𝐽)+𝐷(𝑓,p, x,𝐼)] 1 𝑛 2 󳨐⇒ 𝐷 (𝑓, p, x,𝐼)≤ ∑𝑝 𝑓(𝑥, 𝑦) , 𝑃 𝑖 𝑖 𝑛 𝑖=1 1 1 𝑛 1 𝑚 ≤ [ ∑𝑝 𝑓(𝑥, 𝑦) + ∑𝑤 𝑓(𝑥, 𝑦 )] (16) 2 𝑃 𝑖 𝑖 𝑊 𝑗 𝑗 [ 𝑛 𝑖=1 𝑚 𝑗=1 ] and as the function 𝑓 is convex on the first co-ordinate, so 1 1 𝑛 ≤ [ ∑𝑝 𝐷(𝑓,w, y,𝐽,𝑥) (21) we have 2 𝑃 𝑖 𝑖 [ 𝑛 𝑖=1 𝑃 1 𝐷(𝑓,p, x,𝐼)= 𝐼 𝑓( ∑𝑝 𝑥 , 𝑦) 1 𝑚 𝑃 𝑃 𝑖 𝑖 + ∑𝑤 𝐷(𝑓,p, x,𝐼,𝑦 )] 𝑛 𝐼 𝑖∈𝐼 𝑊 𝑗 𝑗 𝑚 𝑗=1 ]

𝑃𝐼 1 𝑛 𝑚 + 𝑓( ∑𝑝𝑖𝑥𝑖, 𝑦) 1 𝑃 𝑃 ≤ ∑∑𝑝𝑖𝑤𝑗𝑓(𝑥𝑖,𝑦𝑗). 𝑛 𝐼 𝑖∈𝐼 𝑃𝑛𝑊𝑚 𝑖=1𝑗=1 𝑃 1 𝑃 1 ≥𝑓( 𝐼 ∑𝑝 𝑥 + 𝐼 ∑𝑝 𝑥 , 𝑦) 𝑖 𝑖 𝑖 𝑖 The following cases from the above inequalities are of 𝑃𝑛 𝑃𝐼 𝑃𝑛 𝑃 𝑖∈𝐼 𝐼 𝑖∈𝐼 interest [6, 7]. 1 1 =𝑓( ∑𝑝 𝑥 + ∑𝑝 𝑥 , 𝑦) 𝑖 𝑖 𝑖 𝑖 Remark 3. We observe that the inequalities in (4)canbe 𝑃𝑛 𝑖∈𝐼 𝑃𝑛 𝑖∈𝐼 written equivalently as 1 1 𝑛 =𝑓( ∑ 𝑝 𝑥 , 𝑦) = 𝑓 ( ∑𝑝 𝑥 , 𝑦) 𝑃 𝑖 𝑖 𝑃 𝑖 𝑖 𝑛 𝑛 𝑖=1 1 𝑛 𝑚 𝑖∈𝐼∪𝐼 ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 ) 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 󳨐⇒ 𝐷 (𝑓, p, x,𝐼) ≥𝑓(𝑥, 𝑦) . 𝑛 𝑚 𝑖=1𝑗=1 (17) 𝑛 1 [ 1 ≥ max ∑𝑝𝑖𝐷(𝑓,w, y,𝐽,𝑥𝑖) 𝐼⊂{1,...,𝑛} 2 𝑃𝑛 𝑖=1 Now from (16)and(17), we have 𝐽⊂{1,...,𝑚} [

1 𝑛 𝑚 𝑓(𝑥, 𝑦) ≤ 𝐷 (𝑓, p, x,𝐼)≤ ∑𝑝 𝑓(𝑥, 𝑦) . 1 ] 𝑖 𝑖 (18) + ∑𝑤𝑗𝐷(𝑓,p, x,𝐼,𝑦𝑗) , 𝑃𝑛 𝑊 𝑖=1 𝑚 𝑗=1 ]

𝑛 𝑚 Similarly, we can prove that 1 1 ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) + ∑𝑤𝑗𝑓(𝑥,𝑗 𝑦 ) 𝑃𝑛 𝑖=1 𝑊𝑚 𝑗=1 1 𝑚 𝑓(𝑥, 𝑦) ≤ 𝐷 (𝑓, w, y,𝐽)≤ ∑𝑤 𝑓(𝑥, 𝑦 ). 𝑛 𝑊 𝑗 𝑗 (19) [ 1 𝑚 𝑗=1 ≤ min ∑𝑝𝑖𝐷(𝑓,w, y,𝐽,𝑥𝑖) 𝐼⊂{1,...,𝑛} 𝑃𝑛 𝐽⊂{1,...,𝑚} [ 𝑖=1 Adding (18)and(19), we get 1 𝑚 + ∑𝑤 𝐷(𝑓,p, x,𝐼,𝑦 )] , 𝑊 𝑗 𝑗 1 𝑚 𝑗=1 𝑓(𝑥, 𝑦) ≤ [𝐷 (𝑓, w, y,𝐽)+𝐷(𝑓,p, x,𝐼)] ] 2 1 𝑛 1 𝑚 ∑𝑝 𝑓(𝑥, 𝑦) + ∑𝑤 𝑓(𝑥, 𝑦 ) 𝑛 𝑚 𝑃 𝑖 𝑖 𝑊 𝑗 𝑗 1 [ 1 1 ] 𝑛 𝑖=1 𝑚 𝑗=1 ≤ ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) + ∑𝑤𝑗𝑓(𝑥,𝑗 𝑦 ) . 2 𝑃𝑛 𝑖=1 𝑊𝑚 𝑗=1 [ ] ≥ max [𝐷 (𝑓, w, y,𝐽)+𝐷(𝑓,p, x,𝐼)], 𝐼⊂{1,...,𝑛} (20) 𝐽⊂{1,...,𝑚} Abstract and Applied Analysis 5

𝑓(𝑥, 𝑦) min 𝐷𝑘 (𝑓, p, x,𝑦𝑗)≥ min 𝐷(𝑓,p, x;𝐼,𝑦𝑗), 𝑘∈{1,...,𝑛} 𝐼⊂{1,...,𝑛} 1 ≤ [𝐷 (𝑓, w, y,𝐽)+𝐷(𝑓,p, x,𝐼)]. min min 𝐷𝑙 (𝑓, w, y,𝑥𝑖)≥ min 𝐷(𝑓,w, y;𝐽,𝑥𝑖). 𝐼⊂{1,...,𝑛} 2 𝑙∈{1,...,𝑚} 𝐽⊂{1,...,𝑚} 𝐽⊂{1,...,𝑚} (24) (22)

These inequalities imply the following results: We discuss the following particular cases of the above 1 𝑛 𝑚 inequalities which is of interest [6]. ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 ) In the case when 𝑝𝑖 =1and 𝑤𝑗 =1for 𝑖 ∈ {1,...,𝑛} 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 𝑛 𝑚 𝑖=1𝑗=1 and 𝑗 ∈ {1,...,𝑚}, consider the natural numbers 𝑘, 𝑙 with 1≤𝑘≤𝑛−1and 1≤𝑙≤𝑚−1and define 𝑛 1 [ 1 ≥ max ∑𝑝𝑖𝐷𝑙 (𝑓, w, y,𝑥𝑖) 𝑘∈{1,...,𝑛} 2 𝑃𝑛 𝑖=1 𝑙∈{1,...,𝑚} [ 𝐷𝑘 (𝑓, x,𝑦𝑗) 1 𝑚 + ∑𝑤 𝐷 (𝑓, p, x,𝑦 )] , 𝑘 1 𝑘 𝑛−𝑘 1 𝑛 𝑊 𝑗 𝑘 𝑗 = 𝑓( ∑𝑥 ,𝑦 )+ 𝑓( ∑ 𝑥 ,𝑦 ), 𝑚 𝑗=1 𝑛 𝑘 𝑖 𝑗 𝑛 𝑛−𝑘 𝑖 𝑗 ] 𝑖=1 𝑖=𝑘+1 1 𝑛 1 𝑚 𝐷𝑙 (𝑓, y,𝑥𝑖) ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) + ∑𝑤𝑗𝑓(𝑥,𝑗 𝑦 ) 𝑃𝑛 𝑖=1 𝑊𝑚 𝑗=1 𝑙 1 𝑙 𝑚−𝑙 1 𝑚 = 𝑓(𝑥, ∑𝑦 )+ 𝑓(𝑥, ∑ 𝑦 ), 𝑛 𝑚 𝑖 𝑙 𝑗 𝑚 𝑖 𝑚−𝑙 𝑗 [ 1 𝑗=1 𝑗=𝑙+1 ≤ min ∑𝑝𝑖𝐷𝑙 (𝑓, w, y,𝑥𝑖) 𝑘∈{1,...,𝑛} 𝑃𝑛 𝑖=1 𝑙∈{1,...,𝑚} [ 𝑘 1 𝑘 𝑛−𝑘 1 𝑛 𝐷 (𝑓, x)= 𝑓( ∑𝑥 , 𝑦) + 𝑓( ∑ 𝑥 , 𝑦) , 𝑘 𝑛 𝑘 𝑖 𝑛 𝑛−𝑘 𝑖 𝑚 𝑖=1 𝑖=𝑘+1 1 ] + ∑𝑤𝑗𝐷𝑘 (𝑓, p, x,𝑦𝑗) , 𝑊 𝑙 𝑚 𝑚 𝑗=1 𝑙 1 𝑚−𝑙 1 ] 𝐷 (𝑓, y)= 𝑓(𝑥, ∑𝑦 )+ 𝑓(𝑥, ∑ 𝑦 ). 𝑙 𝑚 𝑙 𝑗 𝑚 𝑚−𝑙 𝑗 1 𝑛 1 𝑚 𝑗=1 𝑗=𝑙+1 ∑𝑝𝑖𝑓(𝑥𝑖, 𝑦) + ∑𝑤𝑗𝑓(𝑥,𝑗 𝑦 ) (25) 𝑃𝑛 𝑖=1 𝑊𝑚 𝑗=1

≥ max [𝐷𝑙 (𝑓, w, y)+𝐷𝑘 (𝑓, p, x)] , 𝑘∈{1,...,𝑛} 𝑙∈{1,...,𝑚} We can give the following result.

1 Corollary 4. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R be a co-ordinate 𝑓(𝑥, 𝑦) ≤ min [𝐷𝑙 (𝑓, w, y)+𝐷𝑘 (𝑓, p, x)] . 𝑘∈{1,...,𝑛} 2 convex function on [𝑎, 𝑏]×[𝑐, 𝑑].If 𝑥𝑖 ∈ [𝑎, 𝑏] and 𝑦𝑗 ∈[𝑐,𝑑], 𝑙∈{1,...,𝑚} then for any 𝑘∈{1,...,𝑛−1} and 𝑙∈{1,...,𝑚−1},onehas (23)

Moreover, from the above, we also have 1 𝑛 1 𝑚 𝑓( ∑𝑥 , ∑𝑦 ) 𝑛 𝑖 𝑚 𝑗 max 𝐷(𝑓,p, x;𝐼)≥ max 𝐷𝑘 (𝑓, p, x), 𝑖=1 𝑗=1 𝐼⊂{1,...,𝑛} 𝑘∈{1,...,𝑛} 1 ≤ [𝐷𝑙 (𝑓, y)+𝐷𝑘 (𝑓, x)] max 𝐷(𝑓,w, y;𝐽)≥ max 𝐷𝑙 (𝑓, w, y), 2 𝐽⊂{1,...,𝑚} 𝑙∈{1,...,𝑚} 𝑛 𝑚 1 [ 1 1 ] max 𝐷(𝑓,p, x;𝐼,𝑦𝑗)≥ max 𝐷𝑘 (𝑓, p, x, y𝑗), ≤ ∑𝑓(𝑥𝑖, 𝑦) + ∑𝑓(𝑥,𝑗 𝑦 ) 𝐼⊂{1,...,𝑛} 𝑘∈{1,...,𝑛} 2 𝑛 𝑚 (26) [ 𝑖=1 𝑗=1 ]

max 𝐷(𝑓,w, y;𝐽,𝑥𝑖)≥ max 𝐷𝑙 (𝑓, w, y,𝑥𝑖), 𝑛 𝑚 𝐽⊂{1,...,𝑚} 𝑙∈{1,...,𝑚} 1 [ 1 1 ] ≤ ∑𝐷𝑙 (𝑓, y,𝑥𝑖)+ ∑𝐷𝑘 (𝑓, x,𝑦𝑗) 2 𝑛 𝑖=1 𝑚 𝑗=1 min 𝐷𝑘 (𝑓, p, x)≥ min 𝐷(𝑓,p, x;𝐼), [ ] 𝑘∈{1,...,𝑛} 𝐼⊂{1,...,𝑛} 1 𝑛 𝑚 ≤ ∑∑𝑓(𝑥𝑖,𝑦𝑗). min 𝐷𝑙 (𝑓, w, y)≥ min 𝐷(𝑓,w, y;𝐽), 𝑚𝑛 𝑙∈{1,...,𝑚} 𝐽⊂{1,...,𝑚} 𝑖=1𝑗=1 6 Abstract and Applied Analysis

𝑝 −𝑞/𝑝 In particular, we have the bounds Proof. Using the functions 𝑓(𝑥) =𝑥 , 𝑝>1, 𝑥𝑖 →𝑥𝑖𝑦𝑖 , 𝑞 𝑛 𝑚 1 and 𝑝𝑖 →𝑦𝑖 in (28), we get (29). ∑∑𝑓(𝑥,𝑦 ) 𝑚𝑛 𝑖 𝑗 𝑖=1𝑗=1 Remark 7. As mentioned above from the inequalities in (29), 𝑛 we can write 1 [ 1 ≥ max ∑𝐷𝑙 (𝑓, y,𝑥𝑖) 𝑘∈{1,...,𝑛} 1/𝑝 1/𝑞 2 𝑛 𝑖=1 𝑛 𝑛 𝑙∈{1,...,𝑚} [ (∑𝑥𝑝) (∑𝑦𝑞) 𝑚 𝑖 𝑖 1 𝑖=1 𝑖=1 + ∑𝐷 (𝑓, x,𝑦 )] , 𝑚 𝑘 𝑗 𝑗=1 ] 𝑛 𝑝/𝑞 { −𝑝/𝑞 𝑝 [ 𝑞 𝑞 ≥ max (∑𝑦𝑖 ) (∑𝑦𝑖 ) (∑𝑥𝑖𝑦𝑖) 𝐼⊂{1,...,𝑛} { 1 𝑛 1 𝑚 [ 𝑖=1 { 𝑖∈𝐼 𝑖∈𝐼 ∑𝑓(𝑥𝑖, 𝑦) + ∑𝑓(𝑥,𝑗 𝑦 ) 𝑛 𝑚 −𝑝/𝑞 𝑝 1/𝑝 𝑖=1 𝑗=1 } 𝑞 ] +(∑𝑦𝑖) (∑𝑥𝑖𝑦𝑖) } , 1 𝑛 (27) [ 𝑖∈𝐼 𝑖∈𝐼 ≤ min ∑𝐷𝑙 (𝑓, y,𝑥𝑖) }] 𝑘∈{1,...,𝑛} 𝑛 𝑙∈{1,...,𝑚} [ 𝑖=1 𝑛 ∑𝑥 𝑦 1 𝑚 𝑖 𝑖 ] 𝑖=1 + ∑𝐷𝑘 (𝑓, x,𝑦𝑗) , 𝑚 𝑗=1 ] 𝑛 𝑝/𝑞 { −𝑝/𝑞 𝑝 1 𝑛 1 𝑚 [ 𝑞 𝑞 ≤ min (∑𝑦𝑖 ) (∑𝑦𝑖 ) (∑𝑥𝑖𝑦𝑖) ∑𝑓(𝑥, 𝑦) + ∑𝑓(𝑥, 𝑦 ) 𝐼⊂{1,...,𝑛} { 𝑖 𝑗 𝑖=1 𝑖∈𝐼 𝑖∈𝐼 𝑛 𝑖=1 𝑚 𝑗=1 [ {

≥ max [𝐷𝑙 (𝑓, y)+𝐷𝑘 (𝑓, x)] , −𝑝/𝑞 𝑝 1/𝑝 𝑘∈{1,...,𝑛} } 𝑙∈{1,...,𝑚} +(∑𝑦𝑞) (∑𝑥 𝑦) ] . 1 𝑖 𝑖 𝑖 } 𝑖∈𝐼 𝑖∈𝐼 𝑓(𝑥, 𝑦) ≤ min [𝐷𝑙 (𝑓, y)+𝐷𝑘 (𝑓, x)] . }] 𝑘∈{1,...,𝑛} 2 𝑙∈{1,...,𝑚} (30)

Remark 5. Note that if we substitute 𝑚=1, 𝑓(𝑥,1 𝑦 )→ The following improvement of Jensen’s inequality is valid. 𝑓(𝑥), 𝐷(𝑓, w, y,𝐽,𝑥𝑖)=𝑓(𝑥), 𝐷(𝑓, p, x,𝐼,𝑦𝑗)=𝐷(𝑓,p, x,𝐼), and 𝐷(𝑓, w, y,𝐽)= 𝑓(𝑥) in Theorem 2,wegetthefollowing Theorem 8. Let 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑]→ R be convex on the co- 2 result of Dragomir [9] for convex function defined on the ordinates of [𝑎, 𝑏]×[𝑐, 𝑑]⊆ R .If x is an n-tuple in [𝑎, 𝑏], y is 𝑛 interval and ∑𝑖=1 𝑝𝑖 =𝑃𝑛 >0, an m-tuple in [𝑐, 𝑑], p is a nonnegative n-tuple such that 𝑃𝑛 = 𝑛 ∑𝑖=1 𝑝𝑖 >0,andw is a nonnegative m-tuple such that 𝑊𝑚 = 1 𝑛 𝑃 1 𝑃 1 𝑚 𝐼 𝐼 ∑𝑗=1 𝑤𝑗 >0,then 𝑓( ∑𝑝𝑖𝑥𝑖)≤ 𝑓( ∑𝑝𝑖𝑥𝑖)+ 𝑓( ∑𝑝𝑖𝑥𝑖) 𝑃𝑛 𝑖=1 𝑃𝑛 𝑃𝐼 𝑖∈𝐼 𝑃𝑛 𝑃𝐼 𝑖∈𝐼 𝑛 𝑚 1 𝑛 1 ∑∑𝑝𝑖𝑤𝑗𝑓(𝑥𝑖,𝑦𝑗)−𝑓(𝑥, 𝑦) ≤ ∑𝑝𝑖𝑓(𝑥𝑖). 𝑃𝑛𝑊𝑚 𝑖=1𝑗=1 𝑃𝑛 𝑖=1 󵄨 (28) 󵄨 𝑛 𝑚 󵄨 1 󵄨 󵄨 ≥ 󵄨 ∑∑𝑝 𝑤 󵄨𝑓(𝑥,𝑦 )−𝑓(𝑥, 𝑦)󵄨 The following refinement of Holder¨ inequality holds. 󵄨𝑃 𝑊 𝑖 𝑗 󵄨 𝑖 𝑗 󵄨 󵄨 𝑛 𝑚 𝑖=1𝑗=1 Corollary 6. x =(𝑥,𝑥 ,...,𝑥 ) y =(𝑦,𝑦 ,...,𝑦 ) 󵄨 (31) Let 1 2 𝑛 and 1 2 𝑛 𝑛 𝑚 󵄨 1 󵄨𝜕𝑓+ (𝑥, 𝑦) be two positive n-tuples. Then for (1/𝑝) + (1/𝑞), =1 𝑝, 𝑞>1, − ∑∑𝑝 𝑤 󵄨 (𝑥 − 𝑥) 𝑃 𝑊 𝑖 𝑗 󵄨 𝜕𝑤 𝑖 one has 𝑛 𝑚 𝑖=1𝑗=1 󵄨 𝑛 󵄨 󵄨 ∑𝑥 𝑦 󵄨 󵄨 𝑖 𝑖 𝜕𝑓+ (𝑥, 𝑦) 󵄨 󵄨 𝑖=1 + (𝑦 − 𝑦) 󵄨 󵄨 , 𝜕𝑧 𝑗 󵄨 󵄨 𝑛 𝑝/𝑞 { −𝑝/𝑞 𝑝 󵄨 󵄨 [ 𝑞 𝑞 ≤ (∑𝑦𝑖 ) {(∑𝑦𝑖 ) (∑𝑥𝑖𝑦𝑖) 𝑥=(1/𝑃)∑𝑛 𝑝 𝑥 𝑦=(1/𝑊 )∑𝑚 𝑤 𝑦 [ 𝑖=1 { 𝑖∈𝐼 𝑖∈𝐼 where 𝑛 𝑖=1 𝑖 𝑖,and 𝑚 𝑗=1 𝑗 𝑗. −𝑝/𝑞 𝑝 1/𝑝 (29) } 𝑞 ] Proof. Since 𝑓 is convex on [𝑎, 𝑏] × [𝑐, 𝑑], therefore we have +(∑𝑦𝑖 ) (∑𝑥𝑖𝑦𝑖) } 𝑖∈𝐼 𝑖∈𝐼 }] 𝑓(𝑥,𝑦)−𝑓(𝑤,) 𝑧 𝑛 1/𝑝 𝑛 1/𝑞 𝑝 𝑞 ≤(∑𝑥 ) (∑𝑦 ) . 𝜕𝑓 (𝑤,) 𝑧 𝜕𝑓 (𝑤,) 𝑧 (32) 𝑖 𝑖 ≥ + (𝑥−𝑤) + + (𝑦 − 𝑧) . 𝑖=1 𝑖=1 𝜕𝑤 𝜕𝑧 Abstract and Applied Analysis 7

From the above inequality, we have One has 𝑛 𝑚 𝜕𝑓 (𝑥, 𝑦) ∑∑𝑝 𝑤 + (𝑥 − 𝑥) 𝑖 𝑗 𝜕𝑤 𝑖 𝜕𝑓 (𝑤,) 𝑧 𝜕𝑓 (𝑤,) 𝑧 𝑖=1𝑗=1 𝑓(𝑥,𝑦)−𝑓(𝑤,) 𝑧 − + (𝑥−𝑤) − + (𝑦 − 𝑧) 𝜕𝑤 𝜕𝑧 𝑚 𝑛 𝑛 𝜕𝑓+ (𝑥, 𝑦) 1 󵄨 𝜕𝑓 𝑤, 𝑧 = ∑𝑤 (∑𝑝 𝑥 −𝑃 ⋅ ∑𝑝 𝑥 )=0, 󵄨 + ( ) 𝑗 𝜕𝑤 𝑖 𝑖 𝑛 𝑃 𝑖 𝑖 (36) = 󵄨𝑓(𝑥,𝑦)−𝑓(𝑤,) 𝑧 − (𝑥−𝑤) 𝑗=1 𝑖=1 𝑛 𝑖=1 󵄨 𝜕𝑤 󵄨 𝑛 𝑚 𝜕𝑓 (𝑥, 𝑦) 𝜕𝑓+ (𝑤,) 𝑧 󵄨 ∑∑𝑝 𝑤 + (𝑦 − 𝑦) = 0. − (𝑦 − 𝑧)󵄨 𝑖 𝑗 𝜕𝑧 𝑗 𝜕𝑧 󵄨 𝑖=1𝑗=1 󵄨 󵄨 󵄨 󵄨 Therefore (35)becomes ≥ 󵄨 󵄨𝑓(𝑥,𝑦)−𝑓(𝑤,) 𝑧 󵄨 󵄨 𝑛 𝑚 𝑛 𝑚 ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 )−∑∑𝑝 𝑤 𝑓(𝑥, 𝑦) 󵄨 󵄨 󵄨 𝑖 𝑗 𝑖 𝑗 𝑖 𝑗 󵄨𝜕𝑓+ (𝑤,) 𝑧 𝜕𝑓+ (𝑤,) 𝑧 󵄨 󵄨 𝑖=1𝑗=1 𝑖=1𝑗=1 − 󵄨 (𝑥−𝑤) + (𝑦 − 𝑧)󵄨 󵄨 . 󵄨 𝜕𝑤 𝜕𝑧 󵄨 󵄨 󵄨 󵄨 𝑛 𝑚 (33) 󵄨 󵄨 󵄨 ≥ 󵄨∑∑𝑝 𝑤 󵄨𝑓(𝑥,𝑦 )−𝑓(𝑥, 𝑦)󵄨 󵄨 𝑖 𝑗 󵄨 𝑖 𝑗 󵄨 󵄨𝑖=1𝑗=1 𝑛 󵄨 (37) Let 𝑥→𝑥𝑖, 𝑦→𝑦𝑗, 𝑤→∑𝑖=1 𝑝𝑖𝑥𝑖/𝑃𝑛 := 𝑥,and𝑧→ 𝑛 𝑚 󵄨 𝑛 󵄨𝜕𝑓+ (𝑥, 𝑦) ∑ 𝑤 𝑦 /𝑊 := 𝑦 − ∑∑𝑝 𝑤 󵄨 (𝑥 − 𝑥) 𝑗=1 𝑗 𝑗 𝑚 ,then(33)becomes 𝑖 𝑗 󵄨 𝜕𝑤 𝑖 𝑖=1𝑗=1 󵄨 󵄨 󵄨 󵄨 󵄨 𝜕𝑓+ (𝑥, 𝑦) 󵄨 󵄨 𝑓(𝑥𝑖,𝑦𝑗)−𝑓(𝑥, 𝑦) + (𝑦 − 𝑦) 󵄨 󵄨 . 𝜕𝑧 𝑗 󵄨 󵄨 󵄨 󵄨 𝜕𝑓 (𝑥, 𝑦) 𝜕𝑓 (𝑥, 𝑦) − + (𝑥 − 𝑥) − + (𝑦 − 𝑦) 1/𝑃 𝑊 𝜕𝑤 𝑖 𝜕𝑧 𝑗 Multiplying both hand sides by 𝑛 𝑚,wehave 𝑛 𝑚 󵄨 1 󵄨 󵄨 󵄨 (34) ∑∑𝑝 𝑤 𝑓(𝑥,𝑦 )−𝑓(𝑥, 𝑦) ≥ 󵄨 󵄨𝑓(𝑥,𝑦 )−𝑓(𝑥, 𝑦)󵄨 𝑃 𝑊 𝑖 𝑗 𝑖 𝑗 󵄨 󵄨 𝑖 𝑗 󵄨 𝑛 𝑚 𝑖=1𝑗=1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑛 𝑚 󵄨𝜕𝑓 (𝑥, 𝑦) 𝜕𝑓 (𝑥, 𝑦) 󵄨 󵄨 󵄨 1 󵄨 󵄨 − 󵄨 + (𝑥 − 𝑥) + + (𝑦 − 𝑦)󵄨 󵄨 . ≥ 󵄨 ∑∑𝑝 𝑤 󵄨𝑓(𝑥,𝑦 )−𝑓(𝑥, 𝑦)󵄨 󵄨 𝑖 𝑗 󵄨 󵄨 󵄨𝑃 𝑊 𝑖 𝑗 󵄨 𝑖 𝑗 󵄨 󵄨 𝜕𝑤 𝜕𝑧 󵄨 󵄨 󵄨 𝑛 𝑚 𝑖=1𝑗=1 (38) 𝑛 𝑚 󵄨 1 󵄨𝜕𝑓+ (𝑥, 𝑦) − ∑∑𝑝𝑖𝑤𝑗 󵄨 (𝑥𝑖 − 𝑥) Multiplying (34)by𝑝𝑖 𝑤𝑗 𝑖 𝑗 󵄨 and and summing over and ,we 𝑃𝑛𝑊𝑚 𝑖=1𝑗=1 󵄨 𝜕𝑤 have 󵄨 󵄨 󵄨 𝜕𝑓+ (𝑥, 𝑦) 󵄨 󵄨 + (𝑦 − 𝑦)󵄨 󵄨 . 𝑛 𝑚 𝑛 𝑚 𝜕𝑧 𝑗 󵄨 󵄨 󵄨 󵄨 ∑∑𝑝𝑖𝑤𝑗𝑓(𝑥𝑖,𝑦𝑗)−∑∑𝑝𝑖𝑤𝑗𝑓(𝑥, 𝑦) 𝑖=1𝑗=1 𝑖=1𝑗=1 This completes the proof. 𝑛 𝑚 𝜕𝑓+ (𝑥, 𝑦) − ∑∑𝑝𝑖𝑤𝑗 (𝑥𝑖 − 𝑥) Conflict of Interests 𝑖=1𝑗=1 𝜕𝑤

𝑛 𝑚 The authors declare that they have no conflict of interests 𝜕𝑓+ (𝑥, 𝑦) regarding publication of this paper. − ∑∑𝑝𝑖𝑤𝑗 (𝑦𝑗 − 𝑦) 𝑖=1𝑗=1 𝜕𝑧 󵄨 (35) Acknowledgments 󵄨 𝑛 𝑚 󵄨 󵄨 󵄨 ≥ 󵄨∑∑𝑝 𝑤 󵄨𝑓(𝑥,𝑦 )−𝑓(𝑥, 𝑦)󵄨 󵄨 𝑖 𝑗 󵄨 𝑖 𝑗 󵄨 The authors are grateful to the referees for the useful com- 󵄨𝑖=1𝑗=1 ments regarding presentation in the early version of the paper. 󵄨 The last author also acknowledges that the present work was 𝑛 𝑚 󵄨𝜕𝑓 (𝑥, 𝑦) 󵄨 + partially supported by the University Putra Malaysia (UPM). −∑∑𝑝𝑖𝑤𝑗 󵄨 (𝑥𝑖 − 𝑥) 𝑖=1𝑗=1 󵄨 𝜕𝑤 󵄨 󵄨 References 𝜕𝑓 (𝑥, 𝑦) 󵄨 󵄨 + 󵄨 󵄨 + (𝑦𝑗 − 𝑦)󵄨 󵄨 . 𝜕𝑧 󵄨 󵄨 [1] M. Adil Khan, M. Anwar, J. Jakˇsetic,´ and J. Pecariˇ c,´ “On some 󵄨 improvements of the Jensen inequality with some applications,” 8 Abstract and Applied Analysis

Journal of Inequalities and Applications,vol.2009,ArticleID 323615, 15 pages, 2009. [2] M. Adil Khan, S. Khalid, and J. Pecariˇ c,´ “Improvement of Jensen’s inequality in terms of Gateauxˆ derivatives for convex functions in linear spaces with applications,” Kyungpook Math- ematical Journal,vol.52,no.4,pp.495–511,2012. [3]M.A.Khan,S.Khalid,andJ.Pecariˇ c,´ “Refinements of some majorization type inequalities,” Journal of Mathematical Inequalities,vol.7,no.1,pp.73–92,2013. [4] M. Klariciˇ cBakulaandJ.Pe´ cariˇ c,´ “On the Jensen’s inequality for convex functions on the co-ordinates in a rectangle from the plane,” Taiwanese Journal of Mathematics,vol.10,no.5,pp. 1271–1292, 2006. [5] L. Horvath,´ “A parameter-dependent refinement of the discrete Jensen’s inequality for convex and mid-convex functions,” Journal of Inequalities and Applications, vol. 2011, article 26, 2011. [6] S. S. Dragomir, “A refinement of Jensen’s inequality with applications for f-divergence measures,” Taiwanese Journal of Mathematics,vol.14,no.1,pp.153–164,2010. [7] S. S. Dragomir, “Some refinements of Jensen’s inequality,” Journal of Mathematical Analysis and Applications,vol.168,no. 2, pp. 518–522, 1992. [8] S. S. Dragomir, “On Hadamard’s inequality for the convex mappings defined on a ball in the space and applications,” Mathematical Inequalities & Applications,vol.3,no.2,pp.177– 187, 2000. [9]S.S.Dragomir,“AnewrefinementofJensen’sinequalityin linear spaces with applications,” Mathematical and Computer Modelling,vol.52,no.9-10,pp.1497–1505,2010. [10] J. Rooin, “Some refinements of discrete Jensen’s inequality and some of its applications,” Nonlinear Functional Analysis and Applications,vol.12,no.1,pp.107–118,2007. [11] L.-C. Wang, X.-F. Ma, and L.-H. Liu, “A note on some new refinements of Jensen’s inequality for convex functions,” Journal of Inequalities in Pure and Applied Mathematics,vol.10,no.2, article 48, 2009. [12] X. L. Tang and J. J. Wen, “Some developments of refinned Jensen’s inequality,” Journal of Southwest University for Nation- alities,vol.29,pp.20–26,2003. [13]S.HussainandJ.Pecariˇ c,´ “An improvement of Jensen’s inequal- ity with some applications,” Asian-European Journal of Mathe- matics,vol.2,no.1,pp.85–94,2009. [14] G. Zabandan and A. Kılıc¸man, “A new version of Jensen’s inequality and related results,” Journal of Inequalities and Appli- cations,vol.2012,article238,2012. [15] G. Zabandan and A. Kılıc¸man, “Several integral inequalities and an upper bound for the bidimensional Hermite-Hadamard inequality,” Journal of Inequalities and Applications,vol.2013, article 27, 2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 841585, 7 pages http://dx.doi.org/10.1155/2013/841585

Research Article On the Generalized Krätzel Transform and Its Extension to Bohemian Spaces

S. K. Q. Al-Omari1 and Adem KJlJçman2

1 Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan 2 Department of Mathematics and Institute of Mathematical Research, University of Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Adem Kılıc¸man; [email protected]

Received 4 September 2013; Accepted 12 November 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 S. K. Q. Al-Omari and A. Kılıc¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the Kratzel¨ transform on certain class of generalized functions. We propose operations that lead to the construction of desired spaces of generalized functions. The Kratzel¨ transform is extended and some of its properties are obtained.

1. Introduction where ((𝜌−1)/2)𝜌(1/2) In recent years, integral transforms of Bohemian have com- (𝜌) (2𝜋) 𝜆V (𝑥) = prised an active area of research. Several integral transforms 𝑇(V +1−(1/𝜌)) are extended to various spaces of Bohemian, especially, that (4) 𝑥 𝜌V ∞ permit a factorization property of Fourier convolution type. ×( ) ∫ (𝑡𝜌 −1)V−(1/𝜌)𝑒−𝑥𝑡 𝑑𝑡, On the other hand, several integral transforms that have 𝜌 1 not permitted a factorization property of Fourier convolution 𝜌∈N V >−1+(1/𝜌),𝑥>0 𝑙(𝜌) type are also extended to various spaces of Bohemian. In the for and Re .The V transform was sequence of these integral transforms, the Kratzel¨ transform extended to generalized functions in [3]andtodistributions 𝑙𝜌(𝑋) [1] in [4]. By ,wedenotethespaceofequivalenceclassesof measurable functions 𝑓:𝑋 → R such that (K𝜌𝑓) (𝑥) = ∫ ZV (𝑥𝑦) 𝑓 (𝑦) 𝑑𝑦, V 𝜌 (1) 𝜌 R 󵄨 󵄨 + ∫ 󵄨𝑓󵄨 𝑑𝜇 < ∞, (5) 𝑋 where where two measurable functions are equivalent when they are V V−1 −𝑡𝜌−𝑥𝑦/𝑡 𝜇 Z𝜌 (𝑥𝑦) = ∫ 𝑡 𝑒 𝑑𝑡, (2) equal to a.e. R 𝜌 𝜌 + For 𝑙 (𝑋),the𝑙 -norm is defined as 𝜌>0(∈N), V ∈ C,𝑥>0is extended to certain 1/𝜌 𝑟 󵄩 󵄩 󵄨 󵄨𝜌 space of ultra-Bohemian, denoted by S+(𝑙 ,𝛼,(𝛼𝑖), 𝑎) and 󵄩𝑓󵄩𝑙𝜌 =(∫ 󵄨𝑓󵄨 𝑑𝜇) . (6) 𝑟 𝑋 S+(𝑙 ,𝛼,{𝛼𝑖}, 𝑎), 1 ≤ 𝑟 ≤∞,respectively. Another form of the Kratzel¨ transform was initially The Banach space 𝑙𝜌,𝜇 of Lebesgue measurable functions is introduced in [2], and defined as a generalization of the defined by [3,page446] Laplace transform in [3]as 󵄩 󵄩 󵄩 −𝜇 󵄩 󵄩𝑓󵄩𝜌,𝜇 = 󵄩𝑥 𝑓󵄩𝜌 <∞. (7) (𝜌) (𝜌) (𝑙V 𝑓) (𝑥) = ∫ 𝜆V (𝑥𝑦) 𝑓 (𝑦) 𝑑𝑦, (3) R+ Due to [3, Proposition 2.1], we state the following theorem. 2 Abstract and Applied Analysis

󸀠 N Theorem 1. Let 1≤𝜌≤∞,𝜇, V ∈ C, 𝜌 > 0, (1/𝜌)+(1/𝜌 )= (3) a collection Δ⊂𝑌 such that 󸀠 (𝜌) 1 and Re 𝜇>−(1/𝜌)−min{0, 𝜌 Re V} and then 𝑙V is a 𝑙 𝑙 continuous linear mapping from 𝜌,𝜇 into 𝜌,2/(𝜌−𝜇−1). (i) for all (𝜐𝑛), (𝜎𝑛)∈Δ,wehave(𝜐𝑛 ∗𝜎𝑛)∈Δ, (ii) If 𝑥⊙𝜐𝑛 =𝑦⊙𝜐𝑛,then𝑥=𝑦where 𝑥, 𝑦 ∈ Note that we always assume that the hypothesis of 𝑋, (𝜐 )∈Δ 𝑛∈N Theorem 1 is satisfied. By ⋎ we denote the Mellin-type 𝑛 ,and . convolution product of first order defined by[5] Then the set Δ that satisfies (i) and (ii) is called the set of −1 −1 (𝑓 ⋎ 𝜑) (𝑦) = ∫ 𝑓(𝑦𝑡 )𝑡 𝜑 (𝑡) 𝑑𝑡. (8) all delta sequences. R+ Let 𝐴={(𝑥𝑛,𝜐𝑛):𝑥𝑛 ∈𝑋,(𝜐𝑛)∈Δ,𝑥𝑛 ⊙𝜐𝑚 =𝑥𝑚 ⊙ 𝜐𝑛, for all 𝑚, 𝑛 ∈ N}.Thenwesay(𝑥𝑛,𝜐𝑛)∼(𝑦𝑛,𝜎𝑛) if there D(R ) D By + , or simply , denote the Schwartz space of test are (𝑥𝑛,𝜐𝑛), (𝑦𝑛,𝜎𝑛)∈𝐴,suchthat𝑥𝑛 ⊙𝜎𝑚 =𝑦𝑚 ⊙𝜐𝑛,for R functions of compact support defined on +.Thenwehave all 𝑚, 𝑛 ∈ N.Therelation∼ is an equivalence relation in 𝐴. the following definition. The space of equivalence classes in 𝐴 is called the space of BH BH 𝑓, 𝑔 ∈𝑙 𝑓 𝑔 Bohemian and denoted by . Each element of is called Definition 2. Let 𝜌,𝜇.For and , we define the Bohemian. Then the convergence in BH is defined as follows. operation ⊗ given by

(1) (ℎ𝑛)∈BH is said to be 𝛿-convergent to ℎ∈BH, (𝑓⊗𝑔) (𝑥) = ∫ 𝑓 (𝑥𝑡) 𝑔 (𝑡) 𝑑𝑡. 𝛿 (9) ℎ →ℎ󳨀 𝑛→∞ ∃ R+ denoted by 𝑛 as ,if adeltasequence (𝜐𝑛) such that (ℎ𝑛 ⊙𝜐𝑛), (ℎ ⊙ 𝜐𝑛)∈𝑋,and(ℎ𝑛 ⊙𝜐𝑘)→ Then we easily see that the operations8 in( )and(9)are (ℎ ⊙ 𝜐𝑘)∈𝑋as 𝑛→∞,forall𝑘, 𝑛 ∈ N. very basic for the next construction of the desired Bohemian (2) (ℎ𝑛)∈BH is said to be Δ-convergent to ℎ∈BH, spaces. Δ denoted by ℎ𝑛 󳨀→ℎas 𝑛→∞,if∃ a (𝜐𝑛)∈Δ Theorem 3. 𝑓∈𝑙 𝜑∈D 𝑙(𝜌)(𝑓 ⋎ 𝜑)(𝑥) = Let 𝜌,𝜇 and ,then V such that (ℎ𝑛 −ℎ)⊙𝜐𝑛 ∈𝑋,forall𝑛∈N,and (𝜌) (ℎ −ℎ)⊙𝜐 →0∈𝑋 𝑛→∞ ((𝑙V 𝑓) ⊗ 𝜑)(𝑥). 𝑛 𝑛 as . 𝛿 Proof. Let 𝑥>0.For𝑓∈𝑙𝜌,𝜇 and 𝜑∈D we, by (3)and(8), The following theorem is equivalent to the statement of - get that convergence. 𝑙(𝜌) (𝑓 ⋎ 𝜑) (𝑥) 𝛿 V Theorem 4. ℎ𝑛 →ℎ∈󳨀 BH as 𝑛→∞ifandonlyifthereare 𝑓𝑛,𝑘,𝑓𝑘 ∈𝑋and 𝜐𝑘 ∈Δsuch that ℎ𝑛 =[𝑓𝑛,𝑘/𝜐𝑘], ℎ = [𝑓𝑘/𝜐𝑘] −1 −1 (𝜌) 𝑘∈N,𝑓 →𝑓∈𝑋 𝑛→∞ = ∫ (∫ 𝑓(𝑦𝑡 )𝑡 𝜑 (𝑡) 𝑑𝑡) 𝜆V (𝑥𝑦) 𝑑𝑦 and, for all 𝑛,𝑘 𝑘 as .See[6–17]for R R (10) + + more details.

−1 −1 = ∫ (∫ 𝑓(𝑦𝑡 )𝑡 𝜑 (𝑡) 𝑑𝑡) . BH (𝑙 ) BH (𝑙 ) R+ R+ 3. The Spaces 1 𝜌,𝜇 and 2 𝜌,2/(𝜌−𝜇−1)

Achangeofvariables𝑦=𝑡𝑧in (10)implies In this section we construct the space BH(𝑙𝜌,𝜇,(D,⋎),⋎,Δ) (or BH1(𝑙𝜌,𝜇)) and the space BH(𝑙𝜌,2/(𝜌−𝜇−1),(D,⋎),⊗,Δ) (𝜌) (𝜌) ( BH (𝑙 )) 𝑙V (𝑓 ⋎ 𝜑) (𝑥) = ∫ (∫ 𝑓 (𝑧) 𝜆V (𝑥𝑡𝑧) 𝑑𝑧) 𝜑 (𝑡) 𝑑𝑡 or 2 𝜌,2/(𝜌−𝜇−1) of Bohemian and give their properties. R+ R+ At the first step, we prove the following connecting (11) theorem. (𝜌) = ∫ (𝑙V 𝑓) (𝑥𝑡) 𝜑 (𝑡) 𝑑𝑡. R + Theorem 5. Let 𝑓∈𝑙𝜌,𝜇 and 𝜑, 𝜓 ∈ D;then,𝑓⊗(𝜑⋎𝜓)= (𝑓 ⊗ 𝜑) ⊗𝜓. Hence, by (9), we have

(𝜌) (𝜌) Proof. Let 𝑓∈𝑙𝜌,𝜇 and 𝜑, 𝜓 ∈ D; then, applying the Fubinitz 𝑙 (𝑓 ⋎ 𝜑) (𝑥) = ((𝑙 𝑓) ⊗ 𝜑) (𝑥) . (12) V V theorem yields

This completes the proof of theorem. (𝑓 ⊗ (𝜑 ⋎ 𝜓)) (𝑥)

2. General Bohemian = ∫ 𝑓 (𝑥𝑡) (𝜑 ⋎ 𝜓) (𝑡) 𝑑𝑡 R+ The structure necessary for the construction of Bohemian 𝑡 (13) consists of the following: = ∫ 𝑓 (𝑥𝑡) (∫ 𝜑( )𝜓(𝑦)𝑦−1 𝑑𝑦) 𝑑𝑡 R R 𝑦 (1) a group 𝑋 and a commutative semigroup (𝑌,∗); + + ⊙:𝑋×𝑌 →𝑋 𝑥⊙(𝜐 ∗𝜐 )= 𝑡 (2) a operation such that 1 2 = ∫ (∫ 𝑓 (𝑥𝑡) 𝜑( )𝑦−1 𝑑𝑦) 𝜓 (𝑦) 𝑑𝑦. (𝑥⊙𝜐 )⊙𝜐 𝑥∈𝑋 𝜐 ,𝜐 ,∈𝑌 𝑦 1 2,forall and 1 2 ; R+ R+ Abstract and Applied Analysis 3

The change of variables 𝑡=𝑦𝑧implies 𝑑𝑡 = 𝑦𝑑𝑧 and further Theorem 8. Let 𝑓∈𝑙𝜌,𝜇 and (𝜇𝑛)∈Δthen 𝑓⋎𝜇𝑛 →𝑓as 𝑛→∞. (𝑓 ⊗ (𝜑 ⋎ 𝜓)) (𝑥) = ∫ (∫ 𝑓(𝑥𝑦𝑧)𝜑(𝑧) 𝑑𝑧) 𝜓 (𝑦) 𝑑𝑦. D 𝑙 R+ R+ Proof. Since is a dense subspace of 𝜌 then it is a dense (14) subspace of 𝑙𝜌,𝜇.Hence,therecanbefound𝜓∈D such that 󵄩 󵄩 󵄩𝑓−𝜓󵄩 <𝜀 Therefore, 󵄩 󵄩𝜌,𝜇 (20)

(𝑓 ⊗ (𝜑 ⋎ 𝜓)) (𝑥) = ∫ (𝑓⊗𝜑)(𝑥𝑦)𝜓(𝑦)𝑑𝑦. (15) for 𝜀>0.Also,from(18), we have R+ 󵄩 󵄩𝜌 󵄩 󵄩𝜌 󵄩 󵄩 󵄩(𝑓 − 𝜓)𝑛 ⋎𝜇 󵄩 ≤ 󵄩𝑓󵄩 󵄩𝜇𝑛󵄩 1 . (21) Hence, (15)implies 𝜌,𝜇 𝜌,𝜇 𝑙 𝜇 −1 −1 (𝑓⊗(𝜑⋎𝜓)) (𝑥) = ((𝑓⊗𝜑) ⊗𝜓) (𝑥) . (16) Let 𝑔(𝑡) =𝑦 𝜓(𝑦𝑡 )𝑡 then; 𝑔(𝑡) ∈ D and hence uniformly continuous on R+. This completes the proof of the theorem. Therefore, there is 𝛿>0such that 󵄨 󵄨 󵄨 󵄨 Next, forthcoming theorems prove the existence of the 󵄨𝑔(𝑦)−𝑔(𝑥)󵄨 <𝜀 whenever 󵄨𝑦−𝑥󵄨 <𝛿. (22) space BH1(𝑙𝜌,𝜇). Thus, using (22), we get Theorem 6. Let 𝑓∈𝑙𝜌,𝜇 and 𝜑∈D and then 𝑓⋎𝜑∈𝑙𝜌,𝜇. 󵄩 󵄩𝜌 󵄩(𝜓 ×𝑛 𝜇 −𝜓)(𝑦)󵄩𝜌,𝜇 Proof. For each 𝑓∈𝑙𝜌,𝜇 and 𝜑∈D,wehave 󵄨 󵄨𝜌 󵄨 󵄨𝜌 󵄨 𝜇 󵄨 󵄨 󵄨 = ∫ 󵄨𝑦 (𝜓 ×𝑛 𝜇 −𝜓)(𝑦)󵄨 𝑑𝑦 󵄩 𝜇 󵄩𝜌 󵄨 𝜇 󵄨 󵄨 󵄨 󵄩𝑥 (𝑓 ⋎ 𝜑) (𝑥)󵄩 = ∫ 󵄨𝑥 ∫ 𝑓 (𝑥𝑡) 𝜑 (𝑡) 𝑑𝑡󵄨 𝑑𝑥. R+ 󵄩 󵄩𝜌,𝜇 󵄨 󵄨 (17) R 󵄨 R 󵄨 + 󵄨 + 󵄨 󵄨 󵄨 𝜇 −1 −1 ≤ ∫ ∫ 󵄨𝑦 (𝜓 (𝑦𝑡 )𝑡 𝜇𝑛 (𝑡) By Jensen’s theorem, we write R+ R+ 󵄩 󵄩𝜌 󵄨𝜌 󵄩𝑥𝜇 (𝑓 ⋎ 𝜑) (𝑥)󵄩 󵄨 󵄩 󵄩𝜌,𝜇 −𝜓 (𝑦)𝑛 )𝜇 (𝑡)󵄨 𝑑𝑡 𝑑𝑦 󵄨 󵄨𝜌 (23) 󵄨 󵄨 󵄨 𝜇 −1 −1 󵄨𝜌 󵄨 𝜇 󵄨 ≤ ∫ ∫ 󵄨𝑦 (𝜓 (𝑦𝑡 )𝑡 − 𝜓 (𝑦))󵄨 ≤ ∫ ∫ 󵄨𝑥 ∫ 𝑓 (𝑥𝑡) 𝜑 (𝑡) 𝑑𝑡󵄨 𝑑𝑥 󵄨 󵄨 󵄨 󵄨 R+ R+ R+ R+ 󵄨 R+ 󵄨 󵄨 󵄨 (18) × 󵄨𝜇 (𝑡)󵄨 𝑑𝑡 𝑑𝑦 󵄨 𝜇 󵄨𝜌 󵄨 󵄨 󵄨 𝑛 󵄨 ≤ ∫ ∫ 󵄨𝑥 𝑓 (𝑥𝑡)󵄨 󵄨𝜑 (𝑡)󵄨 𝑑𝑡 𝑑𝑥 R R + + 󵄨 󵄨𝜌 󵄨 󵄨 = ∫ ∫ 󵄨𝑔(𝑦)−𝑔(1)󵄨 󵄨𝜇𝑛 (𝑡)󵄨 𝑑𝑡 𝑑𝑦 󵄩 󵄩𝜌 󵄨 󵄨 R+ R+ ≤ 󵄩𝑓󵄩𝜌,𝜇 ∫ 󵄨𝜑 (𝑡)󵄨 𝑑𝑡, 𝐾 𝜌 󵄨 󵄨 = ∫ ∫ 𝜀 󵄨𝜇𝑛 (𝑡)󵄨 𝑑𝑡 𝑑𝑦. where 𝐾 = [𝑎, 𝑏], 𝑏 >𝑎0 is a compact subset containing R+ R+ the support of 𝜑.Hence,from(18), we get By (22), supp 𝜇𝑛(𝑡) → 0 as 𝑛→∞implies that there can 󵄩 𝜇 󵄩𝜌 󵄩 󵄩𝜌 𝑁∈N 𝜇 ⊆ [0, 𝛿] 𝑛≥𝑁 󵄩𝑥 (𝑓 ⋎ 𝜑) (𝑥)󵄩𝜌,𝜇 ≤ 󵄩𝑓󵄩𝜌,𝜇𝑀 (𝑏−𝑎) , (19) be found ,suchthatsupp 𝑛 ,forall . Further, the fact that the function 𝜓 is of compact support where 𝑀 is certain positive real number. This completes the thus this implies that supp 𝜓(𝑦)⊆𝐾=[𝑎,𝑏],where𝐾 is a proof of the theorem. compact subset of R+.Thus,from(23), we write Theorem 7. 𝑓∈𝑙 𝜑, 𝜓 ∈ D 𝑏 𝛿 Let 𝜌,𝜇 and and then 󵄩 󵄩𝜌 𝜌 󵄩𝜓⋎𝜇𝑛 −𝜓󵄩𝜌,𝜇 ≤𝜀 ∫ ∫ 𝑀𝑑𝑡 (i) 𝑓⋎(𝜑+𝜓)=𝑓⋎𝜑+𝑓⋎𝜓; 𝑎 0 (24) 𝜌 (ii) (𝛼𝑓) ⋎ 𝜑 = 𝛼(𝑓 ⋎𝜑); =𝜀 (𝑏−𝑎)(2𝛿) 𝑀. (𝑓 )∈𝑙 𝑓 →𝑓 𝑛→∞ 𝜑∈D (iii) let 𝑛 𝜌,𝜇 and 𝑛 as then for , Now, we have and we have 𝑓𝑛 ⋎𝜑 → 𝑓⋎𝜑as 𝑛→∞; 󵄩 󵄩 󵄩 󵄩 󵄩𝑓×𝜇 −𝑓󵄩 ≤ 󵄩(𝑓 − 𝜓) ⋎𝜇 󵄩 (iv) 𝑓 ⋎ (𝜑 ⋎ 𝜓) = (𝑓 ⋎𝜑)𝜓. 󵄩 𝑛 󵄩𝜌,𝜇 󵄩 𝑛󵄩𝜌,𝜇 (25) 󵄩 󵄩 󵄩 󵄩 Proof of Part (i), (ii), and (iii) follows from properties + 󵄩𝜓×𝜇𝑛 −𝜓󵄩𝜌,𝜇 + 󵄩(𝑓 − 𝜓)󵄩𝜌,𝜇. of integral operators ∫. Similarly, the proof of Part (iv) is straightforward from the properties of ⋎ proved in [5]. On using (20), (21)and(24)provethat Following theorem is straightforward, and detailed proof 󵄩 󵄩 󵄩𝑓×𝜇 −𝑓󵄩 󳨀→ 0 𝜖󳨀→0. is omitted. 󵄩 𝑛 󵄩𝜌,𝜇 as (26) 4 Abstract and Applied Analysis

Thus the theorem is proved. Then the Bohemian space Proof. Let [(𝑓𝑛)/(𝜇𝑛)], [(𝑔𝑛)/(𝜓𝑛)] ∈ BH1(𝑙𝜌,𝜇) be such that BH1(𝑙𝜌,𝜇) is therefore constructed. The operations such as [(𝑓𝑛)/(𝜇𝑛)] = [(𝑔𝑛)/(𝜓𝑛)],andthen𝑓𝑛 ⋎𝜓𝑚 =𝑔𝑚 ⋎𝜇𝑛 = (𝜌) (𝜌) sum and multiplication by a scalar of two Bohemian in 𝑔𝑛 ⋎𝜇𝑚.UsingTheorem 4 implies 𝑙V 𝑓𝑛 ⊗𝜓𝑚 =𝑙V 𝑔𝑛 ⊗𝜇𝑚,for BH (𝑙 ) 1 𝜌,𝜇 are defined in a natural way all 𝑛,. 𝑚 The idea of quotient of sequences in BH2(𝑙𝜌,2/(𝜌−𝜇−1)) (𝑓 ) (𝑔 ) (𝑓 ⋎𝜏 +𝑔 ⋎𝜇 ) implies that [ 𝑛 ]+[ 𝑛 ]=[ 𝑛 𝑛 𝑛 𝑛 ], (𝜇 ) (𝜏 ) (𝜇 ⋎𝜏) (𝜌) (𝜌) 𝑛 𝑛 𝑛 𝑛 𝑙V 𝑓𝑛 𝑙V 𝑔𝑛 (27) is equivalent to . (30) (𝑓 ) (𝑎𝑓 ) 𝜇𝑛 𝜓𝑛 𝛼[ 𝑛 ]=[ 𝑛 ], (𝜇𝑛) (𝜇𝑛) That is, where 𝛼 is a complex number. (𝜌) (𝜌) (𝑙V 𝑓𝑛) (𝑙V 𝑔𝑛) Similarly, the operation ⋎ and differentiation are defined [ ]=[ ]. (31) by (𝜇𝑛) (𝜓𝑛) (𝑓 ) (𝑔 ) (𝑓 ⋎𝑔 ) [ 𝑛 ]⋎[ 𝑛 ]=[ 𝑛 𝑛 ], Toprovepart(ii)ofthetheorem,if[(𝑓𝑛)/(𝜇𝑛)], [(𝑔𝑛)/(𝜓𝑛)] ∈ (𝜇𝑛) (𝜏𝑛) (𝜇𝑛 ⋎𝜏𝑛) BH1(𝑙𝜌,𝜇),then 𝛼 (28) 𝛼 (𝑓𝑛) (D 𝑓𝑛) (𝜌) (𝜌) D [ ]=[ ]. ̂ (𝑓 ) (𝑔 ) (𝑙V 𝑓𝑛 ⊗𝜓𝑛 +𝑙V 𝑔𝑛 ⊗𝜇𝑛) 𝑙(𝜌) ([ 𝑛 ]+[ 𝑛 ]) = [ ]. (𝜇𝑛) (𝜇𝑛) V (𝜇𝑛) (𝜓𝑛) (𝜇𝑛 ⊗𝜓𝑛) Now constructing the space BH2(𝑙𝜌,2/(𝜌−𝜇−1)) follows from (32) theoremswhichwereusedforconstructingthespace BH1(𝑙𝜌,𝜇). Therefore, the corresponding proofs of Theorems Hence 10 and 11 are omitted. ̂(𝜌) (𝑓𝑛) (𝑔𝑛) ̂(𝜌) (𝑓𝑛) ̂(𝜌) (𝑔𝑛) 𝑙V ([ ]+[ ]) = 𝑙V [ ]+𝑙V [ ]. Theorem 9. Let 𝑓∈𝑙𝜌,2/(𝜌−𝜇−1) and 𝜑∈D and then 𝑓⊗𝜑∈ (𝜇𝑛) (𝜓𝑛) (𝜇𝑛) (𝜓𝑛) 𝑙𝜌,2/(𝜌−𝜇−1). (33)

Theorem 10. Let 𝑓∈𝑙𝜌,2/(𝜌−𝜇−1),𝜑,𝜓∈D and then the Also, if 𝛼∈C,then following hold: (𝑙(𝜌)𝑓 ) (𝑙(𝜌) (𝛼𝑓 )) 𝑓⊗(𝜑+𝜓)=𝑓⊗𝜑+𝑓⊗𝜓 ̂(𝜌) (𝑓𝑛) V 𝑛 V 𝑛 (i) ; 𝛼𝑙V [ ]=𝛼[ ]=[ ]. (34) (𝜇 ) (𝜇 ) (𝜇 ) (ii) (𝛼𝑓) ⊗ 𝜑 = 𝛼(𝑓 ⊗𝜑); 𝑛 𝑛 𝑛 𝑓 →𝑓 𝑙 𝑛→∞ 𝑓 ⊗𝜑 → (iii) if 𝑛 in 𝜌,2/(𝜌−𝜇−1),as then 𝑛 Hence 𝑓⊗𝜑as 𝑛→∞; (𝜇 )∈Δ 𝑓⊗𝜇 →𝑓 𝑛→∞ ̂(𝜌) (𝑓𝑛) ̂(𝜌) (𝑓𝑛) (iv) if 𝑛 ,then 𝑛 as . 𝛼𝑙V [ ]=𝑙V (𝛼 [ ]) . (35) (𝜇𝑛) (𝜇𝑛) Theorem 11. Let 𝑓∈𝑙𝜌,2/(𝜌−𝜇−1) and 𝜑, 𝜓 ∈ D and then 𝑓⊗ (𝜑 ⋎ 𝜓) = (𝑓 ⊗ 𝜑)⊗𝜓. This completes the proof. (𝜌) ProofofthistheoremissimilartothatofTheorem 6. Definition 13. Let [(𝑙V 𝑓𝑛)/(𝜇𝑛)] ∈ BH2(𝑙𝜌,2/(𝜌−𝜇−1)).We Thus the space BH2(𝑙𝜌,2/(𝜌−𝜇−1)) can be regarded as Bohemian ̂(𝜌) define the inverse of 𝑙V transform of the Bohemian space. (𝜌) [(𝑙V 𝑓𝑛)/(𝜇𝑛)] as

−1 4. Krätzel Transform of Bohemian −1 (𝑙(𝜌)𝑓 ) (𝑙(𝜌)) (𝑙(𝜌)𝑓 ) (𝑓 ) ̂(𝜌) V 𝑛 [ V V 𝑛 ] 𝑛 𝑙V [ ]= =[ ], (36) By aid of Theorem 4,wehavetherighttodefinetheKratzel¨ (𝜇 ) (𝜇 ) (𝜇 ) 𝑛 [ 𝑛 ] 𝑛 transform of [(𝑓𝑛)/(𝜇𝑛)] ∈ BH1(𝑙𝜌,𝜇) as the Bohemian (𝜇 )∈Δ (𝑙(𝜌)𝑓 ) for each 𝑛 . ̂(𝜌) (𝑓𝑛) V 𝑛 𝑙V [ ]=[ ] (29) (𝜇 ) (𝜇 ) ̂(𝜌) 𝑛 𝑛 Theorem 14. 𝑙V : BH1(𝑙𝜌,𝜇)→BH2(𝑙𝜌,2/(𝜌−𝜇−1)) is an isomorphism. is embedded in the space BH2(𝑙𝜌,2/(𝜌−𝜇−1)).

̂(𝜌) ̂(𝜌) ̂(𝜌) Proof. Let 𝑙V [(𝑓𝑛)/(𝜇𝑛)] = 𝑙V [(𝑔𝑛)/(𝜓𝑛)] ∈ BH2(𝑙𝜌,2/(𝜌−𝜇−1)) Theorem 12. The mapping 𝑙V : BH1(𝑙𝜌,𝜇)→ 𝑙(𝜌)𝑓 ⊗𝜓 =𝑙(𝜌)𝑔 ⊗𝜇 BH2(𝑙𝜌,2/(𝜌−𝜇−1)) is and then by (29)weget V 𝑛 𝑚 V 𝑚 𝑛. Therefore, Theorem 4 implies (i) well defined, (𝜌) (𝜌) (ii) linear. 𝑙V (𝑓𝑛 ⋎𝜓𝑚)=𝑙V (𝑔𝑚 ⋎𝜇𝑛). (37) Abstract and Applied Analysis 5

Thus 𝑓𝑛 ⋎𝜓𝑚 =𝑔𝑚 ⋎𝜇𝑛,forall𝑚, 𝑛 ∈ N. The concept of such that 𝑓𝑛,𝑘 →𝑓𝑘 as 𝑛→∞for every 𝑘∈N.Hence, BH (𝑙 ) (𝜌) (𝜌) quotients of equivalent classes of 1 𝜌,𝜇 then gives 𝑙V 𝑓𝑛,𝑘 →𝑙V 𝑓𝑘 ∈𝑙𝜌,2/(𝜌−𝜇−1) as 𝑛→∞.Thus,

(𝑓𝑛) (𝑔𝑛) 𝑙(𝜌)𝑓 𝑙(𝜌)𝑓 [ ]=[ ]. (38) [ V 𝑛,𝑘 ] 󳨀→ [ V 𝑘 ] ∈ BH (𝑙 ) (𝜇𝑛) (𝜓𝑛) 2 𝜌,2/(𝜌−𝜇−1) (43) 𝜇𝑘 𝜇𝑘 ̂(𝜌) 𝑛→∞ This proves that 𝑙V is injective. as . (𝜌) (𝜌) To prove the Part (ii), Let 𝑔𝑛,𝑔∈BH2(𝑙𝜌,2/(𝜌−𝜇−1)) be such To show that 𝑙V is surjective, let [(𝑙V 𝑓𝑛)/(𝜇𝑛)] ∈ (𝜌) 𝛿 BH2(𝑙𝜌,2/(𝜌−𝜇−1)).Then(𝑙V 𝑓𝑛)/(𝜇𝑛) is a quotient of sequences that 𝑔𝑛 →𝑔󳨀 as 𝑛→∞.Then,onceagain,byTheorem 5, (𝜌) (𝜌) 𝑔 =[𝑙(𝜌)𝑓 /𝜇 ] 𝑔=[𝑙(𝜌)𝑓 /𝜇 ] 𝑓 ,𝑓 ∈ in BH2(𝑙𝜌,2/(𝜌−𝜇−1)). Hence, 𝑙V 𝑓𝑛 ⊗𝜇𝑚 =𝑙V 𝑓𝑚 ⊗𝜇𝑛,forall 𝑛 V 𝑛,𝑘 𝑘 and V 𝑘 𝑘 for some 𝑛,𝑘 𝑘 (𝜌) 𝑙 𝑙(𝜌)𝑓 →𝑙(𝜌)𝑓 𝑛→∞ [𝑓 /𝜇 ]→ 𝑚, 𝑛 ∈ N. Once again, Theorem 4 implies that 𝑙V (𝑓𝑛 ⋎𝜇𝑚)= 𝜌,𝜇 and V 𝑛,𝑘 V 𝑘 as .Hence 𝑛,𝑘 𝑘 (𝜌) [𝑓𝑘/𝜇𝑘] as 𝑛→∞. 𝑙V (𝑓𝑚 ⋎𝜇𝑛).Hence[(𝑓𝑛)/(𝜇𝑛)] ∈ BH1(𝑙𝜌,𝜇) satisfies Using (36), we get (𝑙(𝜌)𝑓 ) (𝜌) (𝑓𝑛) V 𝑛 −1 (𝜌) −1 (𝜌) 𝑙 [ ]=[ ]. ̂(𝜌) 𝑙V 𝑓𝑛,𝑘 ̂(𝜌) 𝑙 𝑓𝑘 V (39) 𝑙 [ ] 󳨀→ 𝑙 [ V ] 𝑛→∞. (𝜇𝑛) (𝜇𝑛) V V as (44) 𝜇𝑘 𝜇𝑘 This completes the proof of the theorem. −1 ̂(𝜌) ̂(𝜌) 𝑙V 𝑙V (𝜌) Now, we establish that and are continuous with Theorem 15. Let 𝜓∈D and [(𝑙V 𝑓𝑛)/(𝜇𝑛)] ∈ respect to Δ-convergence. BH2(𝑙𝜌,2/(𝜌−𝜇−1)) then one has Δ Let 𝛽𝑛 󳨀→𝛽in BH1(𝑙𝜌,𝜇) as 𝑛→∞. Then, there exist −1 𝑓 ∈𝑙 (𝜇 )∈Δ (𝛽 −𝛽)⋎𝜇 = [((𝑓 )⋎𝜇 )/𝜇 ] ̂(𝜌) (𝜌) 𝑛 𝜌,𝜇 and 𝑛 such that 𝑛 𝑛 𝑛 𝑘 𝑘 𝑙V ([(𝑙 𝑓 )/(𝜇 )] ⊗ 𝜓) = [(𝑓 )/(𝜇 )] ⋎ 𝜓 (i) V 𝑛 𝑛 𝑛 𝑛 ; and 𝑓𝑛 →0as 𝑛→∞. By applying (29)weget ̂(𝜌) (𝜌) 𝑙V ([(𝑓 )/(𝜇 )] ⋎ 𝜓) = [(𝑙 𝑓 )/(𝜇 )] ⊗ 𝜓 (𝜌) (ii) 𝑛 𝑛 V 𝑛 𝑛 . ̂(𝜌) 𝑙V ((𝑓𝑛)⋎𝜇𝑘) 𝑙V ((𝛽 −𝛽)⋎𝜇 )=[ ]. (45) 𝑛 𝑛 𝜇 Proof. By using (29), we have 𝑘 ̂ (𝜌) (𝜌) −1 (𝜌) (𝜌) (𝜌) −1 (𝑙 𝑓 ) (𝑙 ) ((𝑙 𝑓 )⊗𝜓) Hence, we have 𝑙V ((𝛽𝑛 −𝛽)⋎𝜇𝑛) = [((𝑙 𝑓𝑛)⊗𝜇𝑘)/𝜇𝑘]= ̂(𝜌) V 𝑛 [ V V 𝑛 ] V 𝑙V ([ ]⊗𝜓)= . (𝜌) 𝑙V 𝑓𝑛 →0as 𝑛→∞in 𝑙𝜌,2/(𝜌−𝜇−1). (𝜇𝑛) (𝜇𝑛) [ ] Therefore (40) ̂(𝜌) ̂(𝜌) ̂(𝜌) 𝑙V ((𝛽𝑛 −𝛽)⋎𝜇𝑛)=(𝑙V 𝛽𝑛 − 𝑙V 𝛽)⊗ 𝜇 𝑛 󳨀→ 0 Theorem 4 then gives (46) −1 (𝑙(𝜌)𝑓 ) as 𝑛󳨀→∞. ̂(𝜌) V 𝑛 (𝑓𝑛)⋎𝜓 (𝑓𝑛) 𝑙V ([ ]⊗𝜓)=[ ]=[ ]⋎𝜓. Δ (𝜇𝑛) (𝜇𝑛) (𝜇𝑛) ̂(𝜌) ̂(𝜌) Hence, 𝑙V 𝛽𝑛 󳨀→ 𝑙V 𝛽 as 𝑛→∞. (41) Δ Finally, let 𝑔𝑛 󳨀→𝑔∈BH2(𝑙𝜌,2/(𝜌−𝜇−1)) as 𝑛→∞and Hence the part (i) of the theorem is proved. The proof of part (𝜌) then we find 𝑙V 𝑓𝑘 ∈𝑙𝜌,2/(𝜌−𝜇−1) such that (𝑔𝑛 −𝑔)⊗𝜇𝑛 = (ii) is similar thus we omit the details. This completes the [(𝑙(𝜌)𝑓 ⊗𝜇)/𝜇 ] 𝑙(𝜌)𝑓 →0 𝑛→∞ (𝜇 )∈ proof of the theorem. V 𝑘 𝑘 𝑘 and V 𝑘 as for some 𝑛 Δ. Theorem 16. The mappings Now, using (36), we obtain −1 ̂(𝜌) −1 (𝑙(𝜌)) (𝑙(𝜌)𝑓 ⊗𝜇 ) (i) 𝑙V : BH1(𝑙𝜌,𝜇)→BH2(𝑙𝜌,2/(𝜌−𝜇−1)) are continuous ̂(𝜌) [ V V 𝑘 𝑘 ] 𝑙V ((𝑔 −𝑔)⊗𝜇 )= . (47) with respect to 𝛿 and Δ-convergence. 𝑛 𝑛 𝜇 [ 𝑘 ] ̂−1 𝑙(𝜌) : BH (𝑙 )→BH (𝑙 ) (ii) V 2 𝜌,2/(𝜌−𝜇−1) 1 𝜌,𝜇 are continuous Theorem 4 implies with respect to 𝛿 and Δ-convergence. ̂−1 (𝑓 )⋎𝜇 𝑙(𝜌) ((𝑔 −𝑔)⊗𝜇 )=[ 𝑛 𝑘 ]=𝑓 󳨀→ 0 ̂ ̂−1 V 𝑛 𝑛 𝑛 (𝜌) (𝜌) 𝜇𝑘 Proof. At first, let us show that 𝑙V and 𝑙V are continuous (48) with respect to 𝛿-convergence. 𝑛󳨀→∞ 𝑙 . 𝛿 as in 𝜌,𝜇 Let 𝛽𝑛 →𝛽󳨀 in BH1(𝑙𝜌,𝜇) as 𝑛→∞andthenweshow ̂(𝜌) ̂(𝜌) Thus that 𝑙V 𝛽𝑛 → 𝑙V 𝛽 as 𝑛→∞.ByvirtueofTheorem 5,we −1 ̂(𝜌) can find 𝑓𝑛,𝑘 and 𝑓𝑘 in 𝑙𝜌,𝜇 such that 𝑙V ((𝑔𝑛 −𝑔)⊗𝜇𝑛)

−1 −1 (49) 𝑓𝑛,𝑘 𝑓𝑘 ̂(𝜌) ̂(𝜌) 𝛽𝑛 =[ ], 𝛽=[ ] (42) =(𝑙V 𝑔𝑛 − 𝑙V 𝑔) ⋎𝑛 𝜇 󳨀→ 0 as 𝑛󳨀→∞. 𝜇𝑘 𝜇𝑘 6 Abstract and Applied Analysis

−1 −1 ̂(𝜌) Δ ̂(𝜌) Proof. Let the hypothesis of the theorem be satisfied for some From this, we find that 𝑙V 𝑔𝑛 󳨀→ 𝑙V 𝑔 as 𝑛→∞in [(𝑓𝑛)/(𝜑𝑛)] and [(𝜅𝑛)/(𝜇𝑛)]. Therefore, BH1(𝑙𝜌,𝜇). This completes the proof of the theorem. ̂(𝜌) (𝑓𝑛) (𝜅𝑛) ̂(𝜌) (𝑓𝑛 ⋎𝜅𝑛) 𝑙V ([ ]⋎[ ]) = 𝑙V ([ ]) ̂(𝜌) (𝜑𝑛) (𝜇𝑛) (𝜑𝑛 ⋎𝜇𝑛) Theorem 17. The 𝑙V transform is consistent with the classical (𝜌) (𝜌) transform 𝑙V . ((𝑙 𝑓 )⊗𝜅 ) =[ V 𝑛 𝑛 ] (𝜑 ⊗𝜇 ) (55) Proof. For every 𝑓∈𝑙𝜌,𝜇,let𝛽 be its representative in 𝑛 𝑛 BH1(𝑙𝜌,𝜇) and then 𝛽=[(𝑓⋎(𝜇𝑛))/(𝜇𝑛)] where (𝜇𝑛)∈Δ,for (𝑙(𝜌)𝑓 ) all 𝑛∈N.Since(𝜇𝑛) is independent from the representative, V 𝑛 (𝜅𝑛) 𝑛∈N =[ ]⊗[ ]. for all , therefore (𝜑𝑛) (𝜇𝑛)

(𝜌) This completes the proof. ̂(𝜌) ̂(𝜌) 𝑓⋎(𝜇𝑛) 𝑙V 𝑓⊗(𝜇𝑛) 𝑙V (𝛽) = 𝑙V ([ ]) = [ ], (50) (𝜇 ) (𝜇 ) 𝑛 𝑛 Conflict of Interests

(𝜌) The authors declare that there is no conflict of interests which is the representative of 𝑙 𝑓∈𝑙𝜌,2/(𝜌−𝜇−1).Thusthis V regarding the publication of this paper. completes the proof.

Theorem 18. Let [(𝑔𝑛)/(𝜓𝑛)] ∈ BH2(𝑙𝜌,2/(𝜌−𝜇−1)) and then a Acknowledgment necessary and sufficient condition for [(𝑔𝑛)/(𝜓𝑛)] to be in the ̂ 𝑙(𝜌) 𝑔 𝑙(𝜌) 𝑛∈N The authors gratefully acknowledge that this research was range of V is that 𝑛 belongs to range of V ,forall . partially supported by the University Putra Malaysia under

̂(𝜌) the ERGS Grant Scheme having project no. 5527068. Proof. Let [(𝑔𝑛)/(𝜓𝑛)] be in the range of 𝑙V and then it is clear (𝜌) that 𝑔𝑛 belongs to the range of 𝑙V ,forall𝑛∈N. (𝜌) References To establish the converse, let 𝑔𝑛 be in the range of 𝑙V ,for (𝜌) [1] S. K. Q. Al-Omari and A. Kılıc¸man, “Unified treatment of the all 𝑛∈N. Then there is 𝑓𝑛 ∈𝑙𝜌,𝜇 such that 𝑙V 𝑓𝑛 =𝑔𝑛,𝑛∈N. Kratzel¨ transformation for generalized functions,” Abstract and Since [(𝑔𝑛)/(𝜓𝑛)] ∈ BH2(𝑙𝜌,2/(𝜌−𝜇−1)), Applied Analysis,vol.2013,ArticleID750524,7pages,2013. [2] E. Kratzel,¨ “Eine Verallgemeinerung der Laplace- und Meijer- Transformation,” Wissenschaftliche Zeitschrift der Friedrich- 𝑔𝑛 ⊗𝜓𝑚 =𝑔𝑚 ⊗𝜓𝑛, (51) Schiller,vol.14,pp.369–381,1965. (𝜌) [3] D. I. Cruz-Baez´ and J. Rodr´ıguez, “The 𝐿] -transformation for all 𝑚, 𝑛 ∈ N, therefore, on McBride’s spaces of generalized functions,” Commentationes Mathematicae Universitatis Carolinae,vol.39,no.3,pp.445– 452, 1998. (𝜌) (𝜌) 𝑙V (𝑓𝑛 ⋎𝜑𝑛)=𝑙V (𝑓𝑚 ⋎𝜑𝑛) , ∀𝑚, 𝑛 ∈ N, (52) [4] G. L. N. Rao and L. Debnath, “Ageneralized Meijer transforma- tion,” International Journal of Mathematics and Mathematical Sciences, vol. 8, no. 2, pp. 359–365, 1985. where 𝑓𝑛 ∈𝑙𝜌,𝜇 and 𝜑𝑛 ∈Δ,forall𝑛∈N.Thenitfollows 𝑓 ⋎𝜑 =𝑓 ⋎𝜑 𝑚, 𝑛 ∈ N 𝑓 /𝜑 [5] A. H. Zemanian, Generalized Integral Transformations,Dover, that we get 𝑛 𝑚 𝑚 𝑛,forall .Thus 𝑛 𝑛 New York, NY, USA, 2nd edition, 1987. is a quotient of sequences in BH1(𝑙𝜌,𝜇). Therefore, we have [(𝑓 )/(𝜑 )] ∈ BH (𝑙 ) [6] V. Karunakaran and R. A. Chella Rajathi, “Gelfand transform 𝑛 𝑛 1 𝜌,𝜇 and for a Boehmian space of analytic functions,” Annales Polonici Mathematici,vol.101,no.1,pp.39–45,2011. ̂ (𝑓 ) (𝑔 ) [7]J.BeardsleyandP.Mikusinski,´ “Asheaf of Boehmians,” Annales 𝑙(𝜌) ([ 𝑛 ]) = [ 𝑛 ]. Polonici Mathematici,vol.107,no.3,pp.293–307,2013. V (𝜑 ) (𝜓 ) (53) 𝑛 𝑛 [8] D. Nemzer, “One-parameter groups of Boehmians,” Bulletin of the Korean Mathematical Society, vol. 44, no. 3, pp. 419–428, 2007. Hence the theorem is proved. [9] P. Mikusinski,´ “Fourier transform for integrable Boehmians,” The Rocky Mountain Journal of Mathematics,vol.17,no.3,pp. Theorem 19. If 𝛽=[(𝑓𝑛)/(𝜑𝑛)] ∈ BH1(𝑙𝜌,𝜇) and 𝛾= 577–582, 1987. [(𝜅𝑛)/(𝜇𝑛)] ∈ BH1(𝑙𝜌,𝜇), then one has [10] P. Mikusinski,´ “Tempered Boehmians and ultradistributions,” Proceedings of the American Mathematical Society,vol.123,no. (𝑙(𝜌)𝑓 ) 3,pp.813–817,1995. ̂(𝜌) (𝑓𝑛) (𝜅𝑛) V 𝑛 (𝜅𝑛) 𝑙V ([ ]⋎[ ]) = [ ]⊗[ ]. (54) [11] P.Mikusinski,´ “Convergence of Boehmians,” Japanese Journal of (𝜑𝑛) (𝜇𝑛) (𝜑𝑛) (𝜇𝑛) Mathematics,vol.9,no.1,pp.159–179,1983. Abstract and Applied Analysis 7

[12] S. K. Q. Al-Omari and A. Kılıc¸man, “On generalized Hartley- Hilbert and Fourier-Hilbert transforms,” Advances in Difference Equations,vol.2012,article232,12pages,2012. [13]T.K.Boehme,“ThesupportofMikusinski´ operators,” Transac- tions of the American Mathematical Society,vol.176,pp.319–334, 1973. [14] D. Nemzer, “Boehmians on the torus,” Bulletin of the Korean Mathematical Society,vol.43,no.4,pp.831–839,2006. [15] V. Karunakaran and C. Ganesan, “Fourier transform on inte- grable Boehmians,” Integral Transforms and Special Functions, vol. 20, no. 11-12, pp. 937–941, 2009. [16] S. K. Q. Al-Omari and A. Kılıc¸man, “Some remarks on the extended Hartley-Hilbert and Fourier-Hilbert transforms of Boehmians,” Abstract and Applied Analysis,vol.2013,ArticleID 348701,6pages,2013. [17] D. Nemzer, “A note on the convergence of a series in the space of Boehmians,” Bulletin of Pure and Applied Mathematics,vol.2, no. 1, pp. 63–69, 2008. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 947492, 3 pages http://dx.doi.org/10.1155/2013/947492

Research Article Applications of Hankel and Regular Matrices in Fourier Series

Abdullah Alotaibi1 and M. Mursaleen2 1 DepartmentofMathematics,KingAbdulAzizUniversity,P.O.Box80203,Jeddah21589,SaudiArabia 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Correspondence should be addressed to M. Mursaleen; [email protected]

Received 19 September 2013; Accepted 18 November 2013

Academic Editor: Adem Kılıc¸man

Copyright © 2013 A. Alotaibi and M. Mursaleen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.

∞ 1. Introduction and Preliminaries sequence 𝑦=(𝑦𝑛),where𝑦𝑛 =∑𝑘=0 ℎ𝑛+𝑘𝑥𝑘 provided that the series converges for each 𝑛 = 0, 1, 2, ..Anoperator . 𝑇 𝑋 𝑌 𝐴=(𝑎 )∞ Let and be two sequence spaces and let 𝑛𝑘 𝑛;𝑘=1 which transforms 𝑥 into 𝑦 as described is called the operator be an infinite matrix of real or complex numbers. We write induced by the Hankel matrix 𝐻.In[1]wecanfindthe 𝐴𝑥 = (𝐴 (𝑥)) 𝐴 (𝑥) = ∑ 𝑎 𝑥 𝑛 provided that 𝑛 𝑘 𝑛𝑘 𝑘 converges for applications of Hankel operators to approximation theory, 𝑛 𝑥=(𝑥) 𝐴 𝐿 each .Asequence 𝑘 is said to be -summable to if prediction theory, and linear system theory. Hankel matrices 𝐴 (𝑥) = 𝐿 𝑥=(𝑥)∈𝑋 𝐴𝑥 ∈𝑌 lim𝑛 𝑛 .If 𝑘 implies that ,then have a number of applications in various fields. 𝐴 𝑋 𝑌 we say that defines a matrix transformation from into Recently, Al-Homidan [2]provedthatHankelmatrices (𝑋, 𝑌) 𝑋 and by we denote the class of such matrices. If and are regular and obtained the sum of the conjugate Fourier 𝑌 𝑋 𝑌 are equipped with the limits -lim and -lim, respectively, series under certain conditions on the entries of Hankel 𝐴 ∈ (𝑋, 𝑌) 𝑌 𝐴 (𝑥) = 𝑋 𝑥 𝑥∈ 𝑋 and -lim𝑛 𝑛 -lim𝑘 𝑘 for all ,then matrix. Most recently, Alghamdi and Mursaleen [3]proved 𝐴 𝑋 𝑌 we say that is a regular map from into and in this case that Hankel matrices are strongly regular. Strongly regular 𝐴 ∈ (𝑋, 𝑌) 𝐴 ∈ (𝑐, 𝑐) we write reg.Thematrices reg are called matrices are those matrices which transform almost conver- 𝑐 regular, where denotes the space of all convergent sequences. gent sequences into convergent sequences leaving the limit The following are well-known Silverman-Toeplitz¨ condi- invariant [4]. 𝐴 tions for regularity of . Our aim here is to find necessary and sufficient conditions Lemma 1. 𝐴=(𝑎 )∞ for Hankel matrix as well as any arbitrary nonnegative regular 𝑛𝑘 𝑛,𝑘=1 is regular if and only if matrix to sum the derived Fourier series and conjugate (i) ‖𝐴‖ = sup𝑛 ∑𝑘 |𝑎𝑛𝑘|<∞, Fourier series.

(ii) lim𝑛𝑎𝑛𝑘 =0for each 𝑘, ∑ 𝑏 =1 (iii) lim𝑛 𝑘 𝑛𝑘 . 2. Main Results A Hankel matrix is a special case of the regular matrix; Let 𝑓 be 𝐿-integrable and periodic with period 2𝜋,andletthe that is, if 𝑎𝑛𝑘 =ℎ𝑛+𝑘 then the matrix is known as the Hankel Fourier series of 𝑓 be matrix. That is, a Hankel matrix is a square matrix (finite or infinite), constant on each diagonal orthogonal to the 1 ∞ main diagonal. Its (𝑛, 𝑘)th entry is a function of 𝑛+𝑘.The + ∑ (𝑎 𝑘𝑥+𝑏 𝑘𝑥) . 𝑎 𝑘 cos 𝑘 sin (1) Hankel transform of the sequence 𝑥=(𝑥𝑘) is defined as the 0 𝑘=1 2 Abstract and Applied Analysis

Then the series conjugated to it is 3. Proofs ∞ We will need the following lemma which is known as the ∑ (𝑏 𝑘𝑥 −𝑎 𝑘𝑥) , 𝑘 cos 𝑘 sin (2) Banach Weak Convergence Theorem5 [ ]. 𝑘=1 𝜋 Lemma 4. ∫ 𝑔 𝑑ℎ =0 ℎ ∈𝐵𝑉[0,𝜋] and the derived series is lim𝑛→∞ 0 𝑛 𝑥 for all 𝑥 if and only if ‖𝑔𝑛‖<∞for all 𝑛 and lim𝑛→∞𝑔𝑛 =0. ∞ ∑𝑘(𝑏 𝑘𝑥−𝑎 𝑘𝑥) . 𝑘 cos 𝑘 sin (3) Proof of Theorem 2. We have 𝑘=1 1 𝜋 𝑘 𝑆 (𝑥), 𝑆̃ (𝑥) 𝑆󸀠 (𝑥) 𝑆󸀠 (𝑥) = ∫ 𝜓 (𝑡) ( ∑ 𝑚 𝑚𝑡) 𝑑𝑡 Let 𝑛 𝑛 ,and 𝑛 denote the partial sums of 𝑘 𝜋 𝑥 sin series (1), (2), and (3)respectively.Wewrite 0 𝑚=1 1 𝜋 𝑑 (𝑘+1/2) 𝑡 sin (9) 𝑓 (𝑥+𝑡) −𝑓(𝑥−𝑡) , for 0<𝑡≤𝜋; =− ∫ 𝜓𝑥 (𝑡) ( )𝑑𝑡, 𝜓𝑥 (𝑡) =𝜓(𝑓,𝑡)={ 𝜋 0 𝑑𝑡 2 sin (𝑡/2) 𝑔 (𝑥) , for 𝑡=0, 2 𝜋 1 𝜓 (𝑡) =𝐼𝑘 + ∫ sin (𝑘 + )𝑡𝑑𝛽𝑥 (𝑡) , 𝑥 𝜋 0 2 𝛽𝑥 (𝑡) = , 4 sin (1/2) 𝑡 (4) where 1 𝜋 𝑡 sin (𝑘+1/2) 𝑡 where 𝑔(𝑥)=𝑓(𝑥+0)−𝑓(𝑥−0). 𝐼𝑘 = ∫ 𝛽𝑥 (𝑡) cos ( )𝑑𝑡. (10) 𝜋 0 2 sin (𝑡/2) Weproposetoprovethefollowingresults. Then Theorem 2. Let 𝑓(𝑥) be a function integrable in the sense of ∞ ∞ 2 𝜋 Lebesgue in [0, 2𝜋] and periodic with period 2𝜋.Let𝐻=(ℎ𝑛+𝑘) 󸀠 ∑ℎ𝑛+𝑘𝑆𝑘 (𝑥) = ∑ℎ𝑛+𝑘𝐼𝑘 + ∫ 𝐿𝑛 (𝑡) 𝑑𝛽𝑥 (𝑡) , (11) be a Hankel matrix. Then for each 𝛽𝑥(𝑡) ∈ 𝐵𝑉[0,,the 2𝜋] 𝜋 0 󸀠 𝑘=1 𝑘=1 Hankel matrix transform of the sequence (𝑆𝑘(𝑥)) is 𝛽𝑥(0+);that is, where ∞ ∞ 1 󸀠 ∑ℎ 𝑆 (𝑥) =𝛽 (0+) 𝐿𝑛 (𝑡) = ∑ℎ𝑛+𝑘 sin (𝑘 + )𝑡. (12) lim𝑛 𝑛+𝑘 𝑘 𝑥 (5) 2 𝑘=1 𝑘=1 𝛽 (𝑡) [0, 𝜋] 𝛽 (𝑡) → if and only if Since 𝑥 is of bounded variation on and 𝑥 𝛽𝑥(0+) as 𝑡→0,𝛽𝑥(𝑡) cos(𝑡/2) has also the same property. ∞ 1 Hence by Jordan’s convergence criterion for Fourier series ∑ℎ (𝑘 + )𝑡=0 𝐼 →𝛽(0+) 𝑘→∞ 𝑛→∞lim 𝑛+𝑘 sin 2 (6) 𝑘 𝑥 as . 𝑘=0 Since the Hankel matrix 𝐻=(ℎ𝑛+𝑘) is regular, we have for every 𝑡 ∈ (0, 𝜋],where𝐵𝑉[0, 2𝜋] denotes the set of all ∞ ∑ℎ 𝐼 =𝛽 (0+) . functions of bounded variations on [0, 2𝜋]. lim𝑛 𝑛+𝑘 𝑘 𝑥 (13) 𝑘=1

In the next result, we replace the Hankel matrix by an Now, it is enough to show that (6)holdsifandonlyif arbitrary nonnegative regular matrix in the result of Al- Homidan [2]. 𝜋 lim ∫ 𝐿𝑛 (𝑡) 𝑑𝛽𝑥 (𝑡) =0. (14) 𝑛 0 Theorem 3. Let 𝑓(𝑥) be a function integrable in the sense of Lebesgue in [0, 2𝜋] and periodic with period 2𝜋.Let𝐴= Hence, by Lemma 4,itfollowsthat(14)holdsifandonlyif (𝑎 ) 𝐴 𝑛𝑘 be a nonnegative regular matrix. Then -transform of the 󵄩 󵄩 ̃ 󵄩𝐿𝑛 (𝑡)󵄩 ≤𝑀 ∀𝑛,∀𝑡∈[0, 𝜋] , (15) sequence (𝑘𝑆𝑘(𝑥)) converges to 𝑔(𝑥)/𝜋;thatis and (6)holds.Since(15) is satisfied by Lemma 1(i), it follows ∞ 1 ∑𝑘𝑎 𝑆̃ (𝑥) = 𝑔 (𝑥) , that (14)holdsifandonlyif(6) holds. Hence the result follows lim𝑛 𝑛𝑘 𝑘 𝜋 (7) 𝑘=1 immediately. if and only if Proof of Theorem 3. We have 𝜋 ∞ ̃ 1 𝑆𝑛 (𝑥) = ∫ 𝜓𝑥 (𝑡) sin 𝑛𝑡 𝑑𝑡, ∑𝑎 𝑘𝑡 =0 𝜋 0 𝑛→∞lim 𝑛𝑘 cos (8) 𝑘=0 (16) 𝑔 (𝑥) 1 𝜋 = + ∫ cos 𝑛𝑡𝑥 𝑑𝜓 (𝑡) . for every 𝑡 ∈ (0, 𝜋],whereeach𝑎𝑘,𝑏𝑘 ∈ 𝐵𝑉[0, 2𝜋]. 𝑛𝜋 𝑛𝜋 0 Abstract and Applied Analysis 3

Therefore ∞ 𝑔 (𝑥) ∞ 1 𝜋 ∑𝑘𝑎 𝑆̃ (𝑥) = ∑𝑎 + ∫ 𝐾 (𝑡) 𝑑𝜓 (𝑡) , 𝑛𝑘 𝑘 𝜋 𝑛𝑘 𝜋 𝑛 𝑥 (17) 𝑘=1 𝑘=1 0 where ∞ 𝐾𝑛 (𝑡) = ∑𝑎𝑛𝑘 cos 𝑘𝑡. (18) 𝑘=1

Now, taking limit as 𝑛→∞on both sides of (17)andusing Lemmas 1 and 4 as in the proof of Theorem 2,wegetthe required result.

Remark 5. If we take 𝑎𝑛𝑘 =ℎ𝑛+𝑘,thenTheorem 2 is reduced to Theorem 4.1 of [2].

Remark 6. If we replace the matrix 𝐻 by an arbitrary nonnegative regular matrix 𝐴=(𝑎𝑛𝑘) in Theorem 2,weget Theorem 1 of Rao [6].

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 130-072-D1434. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References

[ 1 ] V. V. Pe l l e r, Hankel Operators and their Applications, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003. [2] S. Al-Homidan, “Hankel matrix transforms and operators,” Journal of Inequalities and Applications,vol.2012,article92, 2012. [3] M. A. Alghamdi and M. Mursaleen, “Hankel matrix transfor- mation of the Walsh—Fourier series,” Applied Mathematics and Computation,vol.224,pp.278–282,2013. [4] J. P. King, “Almost summable sequences,” Proceedings of the American Mathematical Society,vol.17,pp.1219–1225,1966. [5] S. Banach, Theorie´ des Operations Lineaires,Hafner,Warszawa, Poland, 1932. [6] A. S. Rao, “Matrix summability of a class of derived Fourier series,” Pacific Journal of Mathematics,vol.48,pp.481–484,1973. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 215612, 5 pages http://dx.doi.org/10.1155/2013/215612

Research Article Statistical Summability of Double Sequences through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

S. A. Mohiuddine and Abdullah Alotaibi

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to S. A. Mohiuddine; [email protected]

Received 21 September 2013; Accepted 26 October 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 S. A. Mohiuddine and A. Alotaibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to define some new types of summability methods for double sequences involving the ideas ofdela Vallee-Poussin´ mean in the framework of probabilistic normed spaces and establish some interesting results.

1. Introduction and Preliminaries forcardinalityoftheenclosedset.If𝛿2(𝐴) =2 𝛿 (𝐴),then N R |𝐴 (ℎ, 𝑙)| Throughout the paper, the symbols and will denote the 𝛿2 (𝐴) = (𝑃) lim (2) set of all natural and real numbers, respectively. The notion ℎ,𝑙 → ∞ ℎ𝑙 of convergence for double sequence was introduced by is called the double natural density of the set 𝐴.Inthesame Pringsheim [1]: we say that a double sequence 𝑥=(𝑥𝑗,𝑘)𝑗,𝑘∈N paper, using the notion of double natural density, they of reals is convergent to 𝐿 in Pringsheim’s sense (briefly,𝑃 ( ) extended the idea of statistical convergence from single to convergent) provided that given 𝜖>0there exists a positive double sequences (for recent work, see [18–23]). 𝑁 |𝑥 −𝐿|<𝜖 𝑗,𝑘≥𝑁 integer such that 𝑗,𝑘 whenever . The double sequence 𝑥=(𝑥𝑗,𝑘) is statistically convergent The idea of statistical convergence is a generalization of to the number 𝐿 if, for each 𝜖>0,theset{(𝑗,𝑘),𝑗≤ℎ,𝑘≤𝑙: convergence of real sequences which was first presented by |𝑥𝑗,𝑘 −𝐿|≥ 𝜖}has double natural density zero. We denote this Fast [2]andSteinhaus[3], independently. Some of its basic by 𝑆-lim 𝑥=𝐿(or 𝑥𝑗,𝑘 → 𝐿(𝑆)). properties and interesting concepts, especially, the notion of Mursaleen initiated the notion of 𝜆-statistical conver- statistically Cauchy sequence, were proved by Schoenberg gence (single sequences) with the help of de la Vallee-Poussin´ [4], Salˇ at´ [5], and Fridy [6]. See, for instance, [7–16]and mean, in [24]. For detail of 𝜆-statistical convergence, one can references therein. Mursaleen and Edely [17]introducedthe be referred to [25–31]andmanyothers.In[32], Mursaleen two-dimensional analogue of natural (or asymptotic) density et al. presented the notion of (𝜆, -statistical𝜇) convergence as follows: let 𝐴⊆N × N and 𝐴(ℎ,𝑙)={𝑗≤ℎ,𝑘≤𝑙:(𝑗,𝑘)∈ and (𝜆, -statistically𝜇) bounded for double sequences and 𝐴},whereℎ, 𝑙 ∈ N.Then showed that (𝜆, -statisticallyboundeddoublesequencesare𝜇) (𝜆, -statistical𝜇) convergence if and only if (𝜆, -statistical𝜇) |𝐴 (ℎ, 𝑙)| limit infimum of 𝑥=(𝑥𝑗,𝑘) is equal to (𝜆, -statistical𝜇) limit 𝛿 (𝐴) = (𝑃) , 2 lim sup ℎ𝑙 supremum of 𝑥 (also see [33]). ℎ,𝑙 → ∞ 𝜆=(𝜆 ) 𝜇=(𝜇) (1) Suppose that 𝑚 and 𝑛 are two nondecreas- |𝐴 (ℎ, 𝑙)| ing sequences of positive real numbers such that 𝛿2 (𝐴) = (𝑃) lim inf ℎ,𝑙 → ∞ ℎ𝑙 𝜆𝑚+1 ≤𝜆𝑚 +1, 𝜆1 =0, (3) 𝜇 ≤𝜇 +1, 𝜇 =0 are called the upper and lower asymptotic densities of a two- 𝑛+1 𝑛 1 dimensional set 𝐴, respectively, where the vertical bars stand and each tends to infinity. 2 Abstract and Applied Analysis

Recall that (𝜆, -𝜇) density of the set 𝐾⊆N × N is given by 2. Main Results 1 (𝜆, 𝜇) (𝜆, 𝜇) 𝛿 (𝐾) = (𝑃) We define the notions of -summable, statistically - 𝜆,𝜇 lim𝑚,𝑛 (𝜆, 𝜇) (𝜆, 𝜇) 𝜆𝑚𝜇𝑛 summable, statistically -Cauchy, and statistically - complete for double sequences with respect to PN-space and 󵄨 (4) × 󵄨{𝑚 − 𝜆𝑚 +1≤𝑗≤𝑚, establish some interesting results. 󵄨 𝑛− 𝜇 +1≤𝑘≤𝑛:(𝑗,𝑘)∈𝐾}󵄨 𝑛 󵄨 Definition 1. Adoublesequence𝑥=(𝑥𝑗,𝑘) is said to be (𝜆, -𝜇) summable in (𝑋, ],𝜏) (or, shortly, ](𝜆, -𝜇) 𝑠𝑢𝑚𝑚𝑎𝑏𝑙𝑒)to𝐿 if provided that the limit exists. for each 𝜖>0, 𝜃 ∈ (0, 1) there exists 𝑁∈N such that We remark, that, for 𝜆𝑚 =𝑚and 𝜇𝑛 =𝑛, the above den- ] (𝜖) > 1 − 𝜃 𝑚, 𝑛 ≥𝑁 𝑡𝑚,𝑛(𝑥)−𝐿 for all .Inthiscase,onewrites sity reduces to the double natural density. ](𝜆, -lim𝜇) 𝑥=𝐿. The generalized double de la Vallee-Poussin´ mean is de- fined as Definition 2. Adoublesequence𝑥=(𝑥𝑗,𝑘) is said to be 1 statistically (𝜆, -𝜇) summable in (𝑋, ],𝜏) (or, shortly, ](𝑆𝜆,𝜇)- 𝑡 (𝑥) = ∑ ∑ 𝑥 , 𝑚,𝑛 𝑗,𝑘 (5) summable) to 𝐿 if 𝛿2(𝐾𝜆,𝜇)=0,where𝐾𝜆,𝜇 ={(𝑚,𝑛)∈ 𝜆𝑚𝜇𝑛 𝑗∈𝐽𝑚 𝑘∈𝐼𝑛 N × N : ] (𝜖) ≤ 1 − 𝜃} 𝜖>0 𝑡𝑚,𝑛(𝑥)−𝐿 ;thatis,if,foreach , 𝜃 ∈ (0, 1), where 𝐽𝑚 =[𝑚−𝜆𝑚 +1,𝑚]and 𝐼𝑛 =[𝑛−𝜇𝑛 +1,𝑛]. 𝑥=(𝑥 ) (𝜆, 𝜇) 1 󵄨 󵄨 We say that 𝑗,𝑘 is -statistically convergent to 󵄨 󵄨 (𝑃) lim 󵄨{𝑚≤ℎ,𝑛≤𝑙:]𝑡 (𝑥)−𝐿 (𝜖) ≤1−𝜃}󵄨 =0 (7) the number 𝐿 if, for every 𝜖>0, ℎ,𝑙 ℎ𝑙 󵄨 𝑚,𝑛 󵄨 1 󵄨 󵄨 󵄨 󵄨 (𝑃) 󵄨{𝑗 ∈ 𝐽 ,𝑘∈𝐼 : 󵄨𝑥 −𝐿󵄨 ≥𝜖}󵄨 =0. or equivalently lim𝑚,𝑛 󵄨 𝑚 𝑛 󵄨 𝑗,𝑘 󵄨 󵄨 (6) 𝜆𝑚𝜇𝑛 1 󵄨 󵄨 󵄨 󵄨 (𝑃) lim 󵄨{𝑚≤ℎ,𝑛≤𝑙:]𝑡 (𝑥)−𝐿 (𝜖) >1−𝜃}󵄨 =1. (8) We denote this by 𝑆𝜆,𝜇-lim 𝑥=𝐿. ℎ,𝑙 ℎ𝑙 󵄨 𝑚,𝑛 󵄨 + The symbol Δ will denote the set of all distribution ](𝑆 ) 𝑥=𝐿 𝐿 functions (d.f.) 𝑓:R → [0, 1] which are nondecreasing, In this case, we write 𝜆,𝜇 -lim ,and is called the ](𝑆 ) 𝑥 left continuous on R,equaltozeroon[−∞, 0],andsuchthat 𝜆,𝜇 -limit of . + 𝑓(+∞) =.Thespace 1 Δ is partially ordered by the usual Definition 3. Adoublesequence𝑥=(𝑥𝑗,𝑘) is said to be pointwise ordering of functions. (𝜆, 𝜇) (𝑋, ],𝜏) ](𝑆 ) 𝑡 statistically -Cauchy in (or, shortly, 𝜆,𝜇 - A triangular norm (or a -norm) [34]isabinaryoperation 𝐶𝑎𝑢𝑐ℎ𝑦 𝜖>0 𝜃 ∈ (0, 1) 𝜏 : [0, 1] × [0, 1] → [0, 1] which satisfies the following )if,forevery and , there exist ℎ ,ℎ ,ℎ ∈ [0, 1] 𝑀, 𝑁 ∈ N such that, for all 𝑚, 𝑝, ≥𝑀 𝑛, 𝑞,theset ≥𝑀 conditions. For all 1 2 3 𝑆 (𝜆, 𝜇) = {(𝑚, 𝑛)∈ N × N : ] (𝜖) ≤ 1 − 𝜃} 𝜖 𝑡𝑚,𝑛(𝑥)−𝑡𝑝,𝑞(𝑥) has (i) 𝜏(𝜏(ℎ1,ℎ2), ℎ3)=𝜏(ℎ1,𝜏(ℎ2,ℎ3)), double natural density zero; that is,

(ii) 𝜏(ℎ1,ℎ2)=𝜏(ℎ2,ℎ1), 󵄨 󵄨 1 󵄨 󵄨 (𝑃) lim 󵄨{𝑚 ≤ ℎ, 𝑛 ≤ 𝑙: ]𝑡 (𝑥)−𝑡 (𝑥) (𝜖) ≤1−𝜃}󵄨 =0. (iii) 𝜏(ℎ1,ℎ3)≤𝜏(ℎ2,ℎ3) whenever ℎ1 ≤ℎ2, ℎ,𝑙 ℎ𝑙 󵄨 𝑚,𝑛 𝑝,𝑞 󵄨

(iv) 𝜏(ℎ1,1)=ℎ1. (9)

In the literature, we have two definitions of probabilistic Theorem 4. If a double sequence 𝑥=(𝑥𝑗,𝑘) is statistically normed space or, briefly, PN-space; the original one is given (𝜆, -summable𝜇) in (𝑋, ],𝜏),thatis,](𝑆𝜆,𝜇)-lim 𝑥=𝐿exists, by Serstnevˇ [35] in 1962 who used the concept of Menger [36] then ](𝑆𝜆,𝜇)-limit of (𝑥𝑗,𝑘) is unique. to define such space and the other one by Alsina et al. [37] ](𝑆 ) 𝑥=𝐿 ](𝑆 ) 𝑥= (for more details, see [38–40]). Proof. Assume that 𝜆,𝜇 -lim 1 and 𝜆,𝜇 -lim 𝐿 𝐿 =𝐿̸ 𝜖>0 According to Serstnevˇ [35], a probabilistic normed space 2.Wehavetoprovethat 1 2.Forgiven ,choose 𝑞>0 is a triple (𝑋, ],𝜏),where𝑋 is a real linear space, ] is the such that + probabilistic norm, that is, ] is a function from 𝑋 into Δ ,for 𝜏((1−𝑞),(1−𝑞))>1−𝜖. (10) 𝑥∈𝑋,thed.f.](𝑥) is denoted by ]𝑥, ]𝑥(𝑡) which is the value of ]𝑥 at 𝑡∈R,and𝜏 is a 𝑡-norm that satisfies the following Then, for any 𝑡>0, we define conditions: 𝑀󸀠 (𝜆, 𝜇) ={(𝑚, 𝑛) ∈ N × N : ] (𝑡) ≤1−𝑞}, (i) ]𝑥(0) = 0; 𝑞 𝑡𝑚,𝑛(𝑥)−𝐿1 (11) ] (𝑡) = 1 𝑡>0 𝑥=0 󸀠󸀠 (ii) 𝑥 for all if and only if ; 𝑀 (𝜆, 𝜇)( ={ 𝑚, 𝑛) ∈ N × N : ] (𝑡) ≤1−𝑞}. 𝑞 𝑡𝑚,𝑛(𝑥)−𝐿2 (iii) ]𝛼𝑥(𝑡) = ]𝑥(𝑡/|𝛼|) for all 𝑡>0, 𝛼∈R with 𝛼 =0̸ and 𝑥∈𝑋 󸀠 ; Since ](𝑆𝜆,𝜇)-lim 𝑥=𝐿1 implies 𝛿2(𝑀𝑞(𝜆, 𝜇)) =0 and sim- ] (𝑡 +𝑡)≥𝜏(] (𝑡 ), ] (𝑡 )) 𝑥, 𝑦 ∈𝑋 󸀠󸀠 󸀠 (iv) 𝑥+𝑦 1 2 𝑥 1 𝑦 2 for all and ilarly we have 𝛿2(𝑀𝑞 (𝜆, 𝜇)).Now,let =0 𝑀𝑞(𝜆, 𝜇)𝑞 =𝑀 𝑡 ,𝑡 ∈ R+ ={𝑥∈R :𝑥≥0} 󸀠󸀠 1 2 . (𝜆, 𝜇)𝑞 ∩𝑀 (𝜆, .Itfollowsthat𝜇) 𝛿2(𝑀𝑞(𝜆, 𝜇)) =0 and hence Abstract and Applied Analysis 3

𝑐 𝑐 the complement 𝑀𝑞(𝜆, 𝜇) is nonempty set and 𝛿2(𝑀𝑞(𝜆, 𝜇))= Proof. Assume that there exists a subset 𝐾 = {(𝑗𝑚,𝑘𝑛):𝑗1 < 𝑗 < ⋅⋅⋅ < 𝑗 <⋅⋅⋅ 𝑘 <𝑘 < ⋅⋅⋅ < 𝑘 < ⋅⋅⋅} ⊆ N × N 1.Now,if(𝑚, 𝑛) ∈ N × N \𝑀𝑞(𝜆, ,then𝜇) 2 𝑚 ; 1 2 𝑛 such 𝛿 (𝐾) = 1 ](𝜆, 𝜇) 𝑥 =𝐿 that 2 and -lim 𝑗𝑚,𝑘𝑛 . Then there exists 𝑡 𝑡 𝑁∈N such that ]𝐿 −𝐿 (𝑡) ≥𝜏(]𝑡 (𝑥)−𝐿 ( ),]𝑡 (𝑥)−𝐿 ( )) 1 2 𝑚,𝑛 1 2 𝑚,𝑛 2 2 (12) > 𝜏 ((1 − 𝑞) , (1 − 𝑞)) >1−𝜖. ] (𝑡) >1−𝜖 𝑡𝑚,𝑛(𝑥)−𝐿 (19) 𝜖>0 ] (𝑡) = 1 𝑡>0 Since was arbitrary, we obtain 𝐿1−𝐿2 for all . holds for all 𝑚, 𝑛 >𝑁.Put𝐾𝜖(𝜆, 𝜇) = {(𝑚, 𝑛)∈ N × N : 𝐿 =𝐿 ](𝑆 ) 󸀠 Hence 1 2.Thismeansthat 𝜆,𝜇 -limit is unique. ]𝑡 (𝑥)−𝜉(𝑡) ≤ 1−𝜖} and 𝐾 = {(𝑗𝑁+1,𝑘𝑁+1), (𝑗𝑁+2,𝑘𝑁+2),...}. 𝑗𝑚,𝑘𝑛 𝛿 (𝐾󸀠) 1 𝐾 (𝜆, 𝜇) ⊆ N −𝐾󸀠 Theorem 5. If a double sequence 𝑥=(𝑥𝑗,𝑘) is ](𝜆, -𝜇) Then 2 = and 𝜖 which implies that 𝛿 (𝐾 (𝜆, 𝜇)) =0 𝑥=(𝑥 ) (𝜆, 𝜇) summable to 𝐿,thenitis](𝑆𝜆,𝜇)-summable to the same limit. 2 𝜖 .Hence 𝑗,𝑘 is statistically - summable to 𝐿 in PN-space. Proof. Let us consider that ](𝜆, -lim𝜇) 𝑥=𝐿.Forevery𝜖>0 𝑡>0 𝑁 and , there exists a positive integer such that Conversely, suppose that 𝑥=(𝑥𝑗,𝑘) is ](𝑆𝜆,𝜇)-summable 𝐿 𝑞 = 1, 2, 3, . . 𝑡>0 ] (𝑡) >1−𝜖 to .For and ,write 𝑡𝑚,𝑛(𝑥)−𝐿 (13) 𝑚, 𝑛 ≥𝑁 1 holds for all .Since 𝐾𝑞 (𝜆, 𝜇) ={(𝑚, 𝑛) ∈ N × N : ]𝑡 (𝑥)−𝐿 (𝑡) ≤1− }, 𝑗𝑚,𝑘𝑛 𝑞 𝐾𝜖 (𝜆,) 𝜇 := {(𝑚, 𝑛) ∈ N × N : ]𝑡 (𝑥)−𝐿 (𝑡) ≤1−𝜖} (14) 𝑚,𝑛 1 𝑀 (𝜆, 𝜇)( ={ 𝑚, 𝑛) ∈ N × N : ] (𝑡) > }. 𝑞 𝑡𝑗 ,𝑘 (𝑥)−𝐿 is contained in N × N,hence𝛿2(𝐾𝜖(𝜆, 𝜇));thatis, =0 𝑥= 𝑚 𝑛 𝑞 (𝑥𝑗,𝑘) is ](𝑆𝜆,𝜇)-summable to 𝐿. (20)

Example 6. This example proves that the converse of Then 𝛿2(𝐾𝑞(𝜆, 𝜇)) =0 and Theorem 5 need not be true. We denote by (R,| ⋅ |) the set ofallrealnumberswiththeusualnormand𝜏(𝑎, 𝑏) =𝑎𝑏 for 𝑀1 (𝜆, 𝜇)2 ⊃𝑀 (𝜆,𝜇)⊃⋅⋅⋅𝑀𝑖 (𝜆, 𝜇)𝑖+1 ⊃𝑀 (𝜆, 𝜇) ⊃ ⋅ ⋅ ⋅, all 𝑎, 𝑏 ∈ [0, 1]. Assume that ]𝑥(𝑡) = 𝑡/(𝑡 + |𝑥|) for all 𝑥∈𝑋 (21) and all 𝑡>0.Here,weobservethat(R, ],𝜏)is a PN-space. The double sequence 𝑥=(𝑥𝑗,𝑘) is defined by 𝛿2 (𝑀𝑞 (𝜆,𝜇))=1, 𝑞=1,2,⋅⋅⋅. (22) 2 𝑚𝑛; if 𝑚, 𝑛 =𝑤 ,𝑤∈N 𝑡𝑚,𝑛 (𝑥) ={ (15) Now, we have to show that, for (𝑚, 𝑛)𝑞 ∈𝑀 (𝜆, ,𝜇) 𝑥= 0; otherwise. (𝑥 ) ](𝜆, 𝜇) 𝐿 𝑥=(𝑥 ) 𝑗𝑚,𝑘𝑛 is -summable to .Supposethat 𝑗𝑚,𝑘𝑛 is not ](𝜆, -summable𝜇) to 𝐿. Therefore, there is 𝜖>0such that For 𝜖>0and 𝑡>0,write ]𝑡 −𝐿(𝑡) ≤ 𝜖 for infinitely many terms. Let 𝑗𝑚,𝑘𝑛 𝐾 (𝜆,) 𝜇 = {(𝑚, 𝑛) ∈ N × N : ] (𝑡) ≤1−𝜖} . 𝜖 𝑡𝑚,𝑛(𝑥) (16) 𝑀𝜖 (𝜆, 𝜇)( ={ 𝑚, 𝑛) ∈ N × N : ]𝑡 −𝜉 (𝑡) >𝜖}, (23) It is easy to see that 𝑗𝑚,𝑘𝑛 𝑡 ] (𝑡) = and 𝜖>1/𝑞with 𝑞 = 1, 2, 3, ..Then . 𝑡𝑚,𝑛(𝑥) 󵄨 󵄨 𝑡+󵄨𝑡𝑚,𝑛 (𝑥)󵄨 𝛿 (𝑀 (𝜆,)) 𝜇 =0, 𝑡 (17) 𝜖 (24) { , 𝑚, 𝑛 =𝑤2,𝑤∈N; = 𝑡+𝑚𝑛 for { and by (21), 𝑀𝑞(𝜆, 𝜇)𝜖 ⊂𝑀 (𝜆, .Hence𝜇) 𝛿(𝑀𝑞(𝜆, 𝜇)), =0 1, otherwise; { which contradicts (22) and therefore 𝑥=(𝑥𝑗 ,𝑘 ) is ](𝜆, -𝜇) 𝐿 𝑚 𝑛 and hence summable to . 2 0, for if 𝑚, 𝑛 =𝑤 ,𝑤∈N; Theorem 8. If a double sequence 𝑥=(𝑥𝑗,𝑘) is statistically lim ]𝑡 (𝑥) (𝑡) ={ (18) (𝜆, 𝜇) (𝜆, 𝜇) 𝑚,𝑛 1, otherwise. -summable in PN-space, then it is statistically - Cauchy. We see that the sequence (𝑥𝑗,𝑘) is not (𝜆, -summable𝜇) in ](𝑆 ) 𝑥=𝐿 𝜖>0 (R, ],𝜏).Buttheset𝐾𝜖(𝜆, 𝜇) has double natural density zero Proof. Suppose that 𝜆,𝜇 -lim .Let be a given 𝑞>0 since 𝐾𝜖(𝜆, 𝜇) ⊂ {(1, 1), (4, 4), (9, 9), (16,.Fromhere, 16),..} number so that we choose such that we conclude that the converse of Theorem 5 need not be true. 𝜏((1−𝑞),(1−𝑞))>1−𝜖. (25) Theorem 7. A double sequence 𝑥=(𝑥𝑗,𝑘) is ](𝑆𝜆,𝜇)-summable to 𝐿 if and only if there exists a subset 𝐾={(𝑗𝑚,𝑘𝑛):𝑗1 <𝑗2 < Then, for 𝑡>0,wehave ⋅⋅⋅ < 𝑗𝑚 < ⋅⋅⋅ ;𝑘1 <𝑘2 <⋅⋅⋅<𝑘𝑛 < ⋅⋅⋅} ⊆ N × N such that 𝛿 (𝐾) = 1 ](𝜆, 𝜇) 𝑥 =𝐿 𝛿 (𝐴 (𝜆, 𝜇)) = 0, 2 and -lim 𝑗𝑚,𝑘𝑛 . 2 𝑞 (26) 4 Abstract and Applied Analysis

𝐴 (𝜆, 𝜇) = {(𝑚, 𝑛)∈ N × N : ] (𝑡/2) ≤ 1 − 𝑞} ] (𝑡/2) > (1 − 𝜖)/2 𝛿 (𝐸𝑐(𝜆, 𝜇)) =0 where 𝑞 𝑡𝑚,𝑛(𝑥)−𝐿 if 𝑡𝑚,𝑛(𝑥)−𝐿 . Therefore 2 𝜖 ;that which implies that is, 𝛿2(𝐸𝜖(𝜆, 𝜇)), =1 which leads to a contradiction, since 𝑥= (𝑥𝑗,𝑘) was ](𝑆𝜆,𝜇)-Cauchy. Hence 𝑥=(𝑥𝑗,𝑘) must be ](𝑆𝜆,𝜇)- 𝛿 (𝐴𝑐 (𝜆, 𝜇)) 2 𝑞 summable. 𝑡 To see that a probabilistic normed space is not complete =𝛿2 ({(𝑚, 𝑛) ∈ N × N : ]𝑡 (𝑥)−𝐿 ( )>1−𝑞}) 𝑚,𝑛 2 in general, we have the following example. =1. Example 11. Let 𝑋 = (0, 1] and ]𝑥(𝑡) = 𝑡/(𝑡 + |𝑥|) for (27) 𝑡>0.Then(𝑋, ],𝜏)is a probabilistic normed space but not 𝑐 complete, since the double sequence (1/𝑚𝑛) is Cauchy with Let (𝑓,𝑔) ∈𝐴 (𝜆, .Then𝜇) ]𝑡 (𝑥)−𝐿(𝑡/2) > 1 − 𝑞. 𝑞 𝑓,𝑔 respect to (𝑋, ],𝜏)but not 𝑃-convergent with respect to the Now, let present PN-space. 𝐵 (𝜆, 𝜇) ={(𝑚, 𝑛) ∈ N × N : ] (𝑡) ≤1−𝜖}. 𝜖 𝑡𝑚,𝑛(𝑥)−𝑡𝑓,𝑔 (𝑥) Conflict of Interests (28) The authors declare that there is no conflict of interests We need to show that 𝐵𝜖(𝜆, 𝜇) ⊂𝐴𝑞(𝜆, .Let𝜇) (𝑚, 𝑛)𝜖 ∈𝐵 regarding the publication of this paper. (𝜆, 𝜇)\𝐴 (𝜆, 𝜇) ] (𝑡) ≤ 1−𝜖 ] (𝑡/2) > 𝑞 .Then 𝑡𝑚,𝑛(𝑥)−𝑡𝑓,𝑔 (𝑥) , 𝑡𝑚,𝑛(𝑥)−𝐿 1−𝑞 ] (𝑡/2) > 1 − 𝑞 , and in particular 𝑡𝑓,𝑔 (𝑥)−𝐿 .Then Acknowledgments 1−𝜖≥] (𝑡) 𝑡𝑚,𝑛(𝑥)−𝑡𝑓,𝑔 (𝑥) This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under 𝑡 𝑡 Grant no. (303/130/1433). The authors, therefore, acknowl- ≥𝜏(]𝑡 (𝑥)−𝐿 ( ),]𝑡 (𝑥)−𝐿 ( )) (29) 𝑚,𝑛 2 𝑓,𝑔 2 edge with thanks DSR technical and financial support. > 𝜏 ((1 − 𝑞) , (1 − 𝑞)) >1−𝜖, References which is not possible. Hence 𝐵𝜖(𝜆, 𝜇) ⊂𝐴𝑞(𝜆, .𝜇) Therefore, by (26) 𝛿2(𝐵𝜖(𝜆, 𝜇)).Hence, =0 𝑥 is statistically (𝜆, -𝜇) [1] A. Pringsheim, “Zur theorie der zweifach unendlichen zahlen- Cauchy in PN-space. folgen,” Mathematische Annalen,vol.53,no.3,pp.289–321, 1900. Definition 9. Let (𝑋, ],𝜏)be a PN-space. Then, [2] H. Fast, “Sur la convergence statistique,” Colloquium Mathe- maticum,vol.2,pp.241–244,1951. (i) PN-space is said to be complete ifeveryCauchydouble 𝑃 (𝑋, ],𝜏) [3] H. Steinhaus, “Sur la convergence ordinaire et la convergence sequence is -convergent in ; asymptotique,” Colloquium Mathematicum,vol.2,pp.73–74, (ii) PN-space is said to be 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙𝑙𝑦- (𝜆,𝜇) 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 1951. (or, shortly, ](𝑆𝜆,𝜇)-𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒)ifeverystatistically [4] I. J. Schoenberg, “The integrability of certain function and (𝜆, -CauchysequenceinPN-spaceisstatistically𝜇) related summability methods,” The American Mathematical (𝜆, -summable.𝜇) Monthly,vol.66,pp.361–375,1959. [5] T. Salˇ at,´ “On statistically convergent sequences of real numbers,” Theorem 10. Every probabilistic normed space (𝑋, ],𝜏) is Mathematica Slovaca,vol.30,no.2,pp.139–150,1980. ](𝑆𝜆,𝜇)-complete but not complete in general. [6]J.A.Fridy,“Onstatisticalconvergence,”Analysis,vol.5,no.4, pp.301–313,1985. Proof. Suppose that 𝑥=(𝑥𝑗,𝑘) is ](𝑆𝜆,𝜇)-Cauchy but not [7] H. C¸ akalli and M. K. Khan, “Summability in topological spaces,” ](𝑆𝜆,𝜇)-summable. Then there exist 𝑀, 𝑁 ∈ N such that, for Applied Mathematics Letters,vol.24,no.3,pp.348–352,2011. all 𝑚, 𝑝, ≥𝑀 𝑛, 𝑞,theset ≥𝑀 𝐸𝜖(𝜆, 𝜇) = {(𝑚, 𝑛)∈ N × N : [8] B. Hazarika, “On generalized statistical convergence in random ] (𝑡) ≤ 1 − 𝜖} 2-normed spaces,” Scientia Magna,vol.8,no.1,pp.58–67,2012. 𝑡𝑚,𝑛(𝑥)−𝑡𝑝,𝑞(𝑥) has double natural density zero; that is, 𝛿2(𝐸𝜖(𝜆, 𝜇)) =0 and [9] S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,” 𝛿2 (𝐹𝜖 (𝜆, 𝜇)) Journal of Inequalities and Applications,vol.2012,article225, 2012. 𝑡 =𝛿2 ({(𝑚, 𝑛) ∈ N × N : ]𝑡 (𝑥)−𝐿 ( )>1−𝜖}) [10] S. A. Mohiuddine and M. Aiyub, “Lacunary statistical conver- 𝑚,𝑛 2 gence in random 2-normed spaces,” Applied Mathematics & =0. Information Sciences,vol.6,no.3,pp.581–585,2012. [11] S. A. Mohiuddine, H. S¸evli, and M. Cancan, “Statistical con- (30) vergence in fuzzy 2-normed space,” Journal of Computational 𝑐 Analysis and Applications,vol.12,no.4,pp.787–798,2010. This implies that 𝛿2(𝐹𝜖 (𝜆, 𝜇)),since =1 [12] M. Mursaleen, “On statistical convergence in random 2-normed 𝑡 spaces,” Acta Scientiarum Mathematicarum,vol.76,no.1-2,pp. ]𝑡 (𝑥)−𝑡 (𝑥) (𝑡) ≥2]𝑡 (𝑥)−𝐿 ( )>1−𝜖, (31) 𝑚,𝑛 𝑝,𝑞 𝑚,𝑛 2 101–109, 2010. Abstract and Applied Analysis 5

[13]M.MursaleenandO.H.H.Edely,“Generalizedstatistical [32] M. Mursaleen, C. C¸ akan, S. A. Mohiuddine, and E. Savas¸, convergence,” Information Sciences,vol.162,no.3-4,pp.287– “Generalized statistical convergence and statistical core of 294, 2004. double sequences,” Acta Mathematica Sinica,vol.26,no.11,pp. [14] M. Mursaleen, V. Karakaya, M. Erturk,¨ and F. Gursoy,¨ 2131–2144, 2010. “Weighted statistical convergence and its application to Ko- [33] E. Savas¸ and S. A. Mohiuddine, “𝜆-statistically convergent rovkin type approximation theorem,” Applied Mathematics and double sequences in probabilistic normed spaces,” Mathematica Computation,vol.218,no.18,pp.9132–9137,2012. Slovaca,vol.62,no.1,pp.99–108,2012. [15] A. S¸ahiner, M. Gurdal,¨ S. Saltan, and H. Gunawan, “Ideal [34] E. P.Klement, R. Mesiar, and E. Pap, Triangular Norms,vol.8of convergence in 2-normed spaces,” Taiwanese Journal of Math- Trends in Logic, Kluwer Academic Publishers, Dordrecht, The ematics, vol. 11, no. 5, pp. 1477–1484, 2007. Netherlands, 2000. ˇ [16] U. Yamancı and M. Gurdal,¨ “On lacunary ideal convergence [35] A. N. Serstnev, “Random normed spaces: problems of com- in random 2-normed space,” Journal of Mathematics,vol.2013, pleteness,” Kazanskij Gosudarstvennyj Universitet Imeni V.I. Article ID 868457, 8 pages, 2013. Ul’janova-Lenina. Uˇcenye Zapiski,vol.122,no.4,pp.3–20,1962. [17] M. Mursaleen and O. H. H. Edely, “Statistical convergence [36] K. Menger, “Statistical metrics,” Proceedings of the National of double sequences,” Journal of Mathematical Analysis and Academy of Sciences of the United States of America,vol.28,pp. Applications,vol.288,no.1,pp.223–231,2003. 535–537, 1942. [37] C. Alsina, B. Schweizer, and A. Sklar, “On the definition of a [18] M. Mursaleen and S. A. Mohiuddine, “Statistical convergence of probabilistic normed space,” Aequationes Mathematicae,vol.46, double sequences in intuitionistic fuzzy normed spaces,” Chaos, no. 1-2, pp. 91–98, 1993. Solitons & Fractals,vol.41,no.5,pp.2414–2421,2009. [38] B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific [19] S. A. Mohiuddine, H. S¸evli, and M. Cancan, “Statistical conver- Journal of Mathematics,vol.10,pp.313–334,1960. gence of double sequences in fuzzy normed spaces,” Filomat, vol.26,no.4,pp.673–681,2012. [39] B. Schweizer and A. Sklar, ProbabilisticMetric Spaces,Elsevier, New York, NY, USA, 1983. [20] R. C¸ olak and Y. Altin, “Statistical convergence of double [40] B. Schweizer and A. Sklar, Probabilistic Metric Spaces,Dover sequences of order 𝛼̃,” Journal of Function Spaces and Applica- Publication, Mineola, NY, USA, 2nd edition, 2005. tions, vol. 2013, Article ID 682823, 5 pages, 2013. [21] S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence of double sequences in locally solid Riesz spaces,” Abstract and Applied Analysis,vol.2012,ArticleID719729,9 pages, 2012. [22] S. A. Mohiuddine, B. Hazarika, and A. Alotaibi, “Double lacunary density and some inclusion results in locally solid Riesz spaces,” Abstract and Applied Analysis,vol.2013,Article ID 507962, 8 pages, 2013. [23] H. Dutta, “A characterization of the class of statistically pre- Cauchy double sequences of fuzzy numbers,” Applied Mathe- matics & Information Sciences,vol.7,no.4,pp.1437–1440,2013. [24] M. Mursaleen, “𝜆-statistical convergence,” Mathematica Slo- vaca,vol.50,no.1,pp.111–115,2000. [25] C. Belen and S. A. Mohiuddine, “Generalized weighted statis- tical convergence and application,” Applied Mathematics and Computation,vol.219,no.18,pp.9821–9826,2013. [26] R. C¸ olak and C. A. Bektas¸, “𝜆-statistical convergence of order 𝛼,” Acta Mathematica Scientia B, vol. 31, no. 3, pp. 953–959, 2011. 𝜆 𝑛 [27] H. Dutta and T. Bilgin, “Strongly (𝑉 ,𝐴,ΔV𝑚,𝑝)-summable sequence spaces defined by an Orlicz function,” Applied Math- ematics Letters,vol.24,no.7,pp.1057–1062,2011. [28] O. H. H. Edely, S. A. Mohiuddine, and A. K. Noman, “Korovkin type approximation theorems obtained through generalized statistical convergence,” Applied Mathematics Letters,vol.23,no. 11, pp. 1382–1387, 2010. [29] M. Et, M. C¸ ınar, and M. Karakas¸, “On 𝜆-statistical convergence of order 𝛼 of sequences of function,” Journal of Inequalities and Applications,vol.2013,article204,2013. [30] S. A. Mohiuddine and Q. M. D. Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space,” Chaos, Solitons & Fractals,vol.42,no.3,pp.1731–1737,2009. [31] S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “A new variant of statistical convergence,” Journal of Inequalities and Applications,vol.2013,article309,2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 968541, 5 pages http://dx.doi.org/10.1155/2013/968541

Research Article Existence of Periodic Solutions to Multidelay Functional Differential Equations of Second Order

Cemil Tunç and Ramazan Yazgan

Department of Mathematics, Faculty of Sciences, Yuz¨ unc¨ u¨ Yıl University, 65080 Van, Turkey

Correspondence should be addressed to Cemil Tunc¸; [email protected]

Received 23 August 2013; Accepted 3 October 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 C. Tunc¸ and R. Yazgan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature.

1. Introduction existence of an 𝜔-periodic solution. By this work, the authors proved that their theorems are better than Yoshizawa’s [2] It is well known that the problem of the existence of periodic periodic solutions theorem primarily by removing restric- solutions of retarded functional differential equations of tions of the size of the constant delay ℎ. An example of second order is not only very important in the background application was given at the end of the paper. Namely, in applications, but also of considerable significance in theory the same paper, the authors applied the following Theorem of differential equations. Besides, the scope of retarded A to discuss the existence of an 𝜔-periodic solution of the functional differential equations is very general. For exam- nonlinear delay differential equation of the second order: ple, it contains ordinary differential equations, differential- 󸀠󸀠 󸀠 difference equations, integrodifferential equations, and so on. 𝑥 (𝑡) +𝑎𝑥 (𝑡) +𝑔(𝑥 (𝑡−𝜏)) =𝑝(𝑡) , (1) The motivation of this paper is that in recent years the study of the existence of periodic solutions to various kinds of where 𝑝(𝑡) is an external force, 𝑔(𝑥(𝑡 − 𝜏)) is a delayed retarded functional differential equations of second order has restoring force, the delay 𝜏 is a positive constant and the become one of the most attractive topics in the literature. friction is proportional to the velocity, and 𝑎 is a positive Especially, by using the famous continuation theorem of constant. It should be noted that a feedback system with degreetheory(seeGainesandMawhin[1]), many authors friction proportional to velocity, an external force 𝑝(𝑡),and have made a lot of interesting contributions to the topic for a delayed restoring force 𝑔(𝑥(𝑡 ,− 𝜏)) (𝜏 > 0) may be written retarded functional differential equations of second order. as (1)(seeBurton[4]). Here,wewouldnotliketogivethedetailsoftheseworks. Consider the following general nonautonomous delay On the other hand, amongst the achieved excellent differential equation: results, one of them is the famous Yoshizawa’s theorem [2] 𝑥=𝐹(𝑡,𝑥̇ 𝑡), 𝑥𝑡 =𝑥(𝑡+𝜃) , for existence of periodic solutions of retarded functional (2) differential equations, which has vital influence and has been −ℎ≤𝜃≤0,𝑡≥0, widely used in the literature. This theorem has also been 𝑛 𝑛 generally one of the best results in the literature from the past where 𝐹:R ×𝐶 → R , 𝐶=𝐶([−ℎ,0],R ), ℎ is a positive till now. It should be noted that in 1994, Zhao et al. [3]proved constant, and we suppose that 𝐹 is continuous, 𝜔-periodic, 𝑛 four sufficiency theorems on the existence of periodic solu- and takes closed bounded sets into bounded sets of R ;and tions for a class of retarded functional differentials to have the such that solutions of initial value problems are unique, ℎ 2 Abstract and Applied Analysis can be either larger than 𝜔, or equal to or smaller than 𝜔. In this paper, we consider the following nonlinear differ- Here (𝐶, ‖ ⋅ ‖) is the Banach space of continuous function 𝜙: ential equation of second order with multiple constant delays, 𝑛 [−ℎ, 0] → R with supremum norm; ℎ>0, 𝐶𝐻 is the open 𝜏𝑖(> 0): 𝑛 𝐻-ball in 𝐶; 𝐶𝐻 :={𝜙∈𝐶([−ℎ,0],R ):‖𝜙‖<𝐻}.Standard 𝑥󸀠󸀠 (𝑡) +{𝑓(𝑥(𝑡) ,𝑥󸀠 (𝑡))+𝑔(𝑥(𝑡) ,𝑥󸀠 (𝑡))𝑥󸀠 (𝑡)}𝑥󸀠 (𝑡) existence theory, see Burton [4], shows that if 𝜙∈𝐶𝐻 and 𝑡≥0, then there is at least one continuous solution 𝑥(𝑡,0 𝑡 ,𝜙) [𝑡 ,𝑡 +𝛼) 𝑡>𝑡 𝑥 (𝑡, 𝜙) =𝜙 𝑛 (3) such that on 0 0 satisfying (2)for 0, 𝑡 +ℎ(𝑥 (𝑡)) + ∑ 𝑔 (𝑥 (𝑡 −𝜏 )) = 𝑝 (𝑡) , 𝛼 𝐵⊂𝐶 𝑖 𝑖 and is a positive constant. If there is a closed subset 𝐻 𝑖=1 such that the solution remains in 𝐵,then𝛼=∞.Further,the 𝑛 symbol |⋅|will denote a convenient norm in R with |𝑥|= where 𝜏𝑖 are fixed constants delay with 𝑡−𝜏𝑖 ≥0;theprimes 𝑛 max1≤𝑖≤𝑛|𝑥𝑖|. Let us assume that 𝐶(𝑡) = {𝜙 : [𝑡−𝛼, 𝑡]→ R | in (3) denote differentiation with respect to 𝑡∈R, 𝑓, 𝑔, ℎ, 𝑔𝑖, 𝜙 is continuous} and 𝜙𝑡 denotes the 𝜙 in the particular 𝐶(𝑡), and 𝑝 are continuous functions in their respective arguments 2 2 and that ‖𝜙𝑡‖=max𝑡−𝛼≤𝑠≤𝑡|𝜙(𝑡)|. Finally, by the periodicity, on R , R , R, R,andR, respectively, and also depend only we mean that that there is an 𝜔>0such that 𝐹(𝑡, 𝜙) is 𝜔- on the arguments displayed explicitly. The continuity of these periodic in the sense that if 𝑥(𝑡) is a solution of (2)sois functions is a sufficient condition for existence of the solution 𝑥(𝑡 + 𝜔). of (3). It is also assumed as basic that the functions 𝑓, 𝑔, ℎ, 󸀠 and 𝑔𝑖 satisfy a Lipschitz condition in 𝑥, 𝑥 ,𝑥(𝑡−1 𝜏 ), 𝑥(𝑡 − 𝑡=0 Definition 1. Solutions of (2)areuniformboundedat if 𝜏2),...,𝑥(𝑡𝑛 −𝜏 ). By this assumption, the uniqueness of 𝐵 𝐵 [𝜙∈𝐶,‖𝜙‖<𝐵,𝑡 ≥ 0] 󸀠 for each 1 there exists 2 such that 1 solutions of (3) is guaranteed. The derivatives 𝑑𝑔𝑖/𝑑𝑥≡𝑔𝑖 (𝑥) imply that |𝑥(𝑡, 0, 𝜙)|2 <𝐵 (see Burton [4]). exist and are continuous. It should be noted that throughout the paper, sometimes, 𝑥(𝑡) and 𝑦(𝑡) are abbreviated as 𝑥 and Definition 2. Solutions of (2) are uniform ultimate bounded 𝑦,respectively. 𝐵 𝑡=0 𝐵 >0 𝐾>0 for bound at if for each 3 there exists a We write (3) in system form as follows: such that [𝜙 ∈ 𝐶, ‖𝜙‖3 <𝐵 ,𝑡 ≥ 𝐾]imply that |𝑥(𝑡, 0, 𝜙)| <𝐵 (see Burton [4]). 𝑥󸀠 =𝑦,

𝑛 The first theorem given in Zhao et3 al.[ ]isthefollowing. 󸀠 𝑦 = − {𝑓 (𝑥, 𝑦) + 𝑔 (𝑥, 𝑦)𝑦}𝑦−ℎ (𝑥) − ∑ 𝑔𝑖 (𝑥) 𝑖=1 (4) Theorem A. If the solutions of (2) are ultimately bounded by the bound 𝐵,then 𝑛 0 󸀠 + ∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠+ 𝑝 (𝑡) , 𝑖=1 −𝜏𝑖 (i) equation (2) has an 𝜔-periodic solution and is bounded by 𝐵, 󸀠 where 𝑔𝑖 (𝑥(𝑡 + 𝑠))𝑖 =𝑑𝑔 /𝑑𝑥. It is clear that (3)isaparticularcaseof(2). It should (ii) if (2) is autonomous, then (2) has an equilibrium 𝐵 be noted that the reason or the motivation for taking into solution and is bounded by (see Zhao et al. [3]). consideration (3) comes from the following modified Lienard´ type equation of the form: Regarding (1)Zhaoetal.[3] proved the following theorem as example of application. 𝑥󸀠󸀠 (𝑡) +{𝑓(𝑥 (𝑡)) +𝑔(𝑥 (𝑡)) 𝑥󸀠 (𝑡)}𝑥󸀠 (𝑡) +ℎ(𝑥 (𝑡)) =𝑒(𝑡) . (5) Theorem B. Assume that the following conditions hold: These types of equations have great applications in theory and (1) 𝑝(𝑡) is an 𝜔-periodic continuous function, 𝑔 is a applications of the differential equations. Therefore, till now, continuous differentiable function, the qualitative behaviors, the stability, boundedness, global existence, existence of periodic solutions, and so forth, of 𝑔(𝑥) 𝑥=∞ (2) lim|𝑥| → ∞ sgn ,andthereisaboundedset thesetypedifferentialequationshavebeenstudiedbymany 󸀠 𝑐 𝑐 Ω containing the origin such that |𝑔 (𝑥)| ≤ 𝑐 on Ω ; Ω researchers, and the researches on these topics are still being is the complement of the set Ω, done in the literature. For example, we refer the readers to the 𝜏𝑐 <𝑎 books of Ahmad and Rao [5], Burton [4], Gaines and Mawhin (3) . [1], and the papers of Constantin [6], Graef [7], Huang and Yu [8], Jin [9], Liu and Huang [10], Napoles´ Valdes´ [11], Qian [12], Then, (1) has an 𝜔-periodic motion. Tunc¸[13–21], C. Tunc¸andE.Tunc¸[22], Zhao et al. [3], Zhou [23], and the references cited in these works. 𝑝(𝑡) =𝐾 𝐾 Moreover, when , is a constant, under the We here give certain sufficient conditions to guarantee 𝑥=𝑐 above conditions (1)hasaconstantmotion 0,andthe the existence of an 𝜔-periodic solution of (3). This paper is 𝑐 𝑔(𝑐 )=𝐾 constant 0 satisfies 0 . In fact, from the condition inspired by the mentioned papers and that in the literature. (2), there is a bounded set Ω1 containingtheoriginsuchthat 𝑥 Our aim is to generalize and improve the application given in |𝑔󸀠(𝑥)| ≤ 𝑐 ∫ 𝑔(𝑠)𝑑𝑠 >0 Ω𝑐 Ω𝑐 and 0 on 1; 1 is the complement [3]for(3). This paper has also a contribution to the inves- of the set Ω1. tigation of the qualitative behaviors of retarded functional Abstract and Applied Analysis 3

𝑛 differential equations of second order, and it may be useful for 2 +𝑦𝑝(𝑡) + ∑ (𝜆𝑖𝜏𝑖)𝑦 researchers who work on the above mentioned topics. Finally, 𝑖=1 without using the famous continuation theorem of degree theory, which belongs to Gaines and Mawhin [1], we prove 𝑛 0 − ∑ 𝜆 ∫ 𝑦2 (𝑡+𝑠) 𝑑𝑠. thefollowingmainresult.Thiscasemakesthetopicofthis 𝑖 𝑖=1 −𝜏𝑖 paper interesting. (7)

󸀠 2. Main Result By noting the assumption |𝑔𝑖 (𝑥)| ≤𝑖 𝑐 of the theorem and 2|𝛼𝛾|2 ≤𝛼 +𝛾2 Our main result is the following. the estimate ,onecanobtainthefollowing estimates: Theorem 3. We assume that there are positive constants 𝑎, 𝑎, 𝑛 0 𝑏 𝜏 𝑐 󸀠 , ,and 𝑖 such that the following conditions hold: 𝑦 ∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠 𝑖=1 −𝜏𝑖 (i) 𝑎 ≥ 𝑓(𝑥, 𝑦) + 𝑔(𝑥, 𝑦)𝑦≥𝑎, 𝑛 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ ∑ ∫ 󵄨𝑔󸀠 (𝑥 (𝑡+𝑠))󵄨 󵄨𝑦 (𝑡)󵄨 󵄨𝑦 (𝑡+𝑠)󵄨 𝑑𝑠 ℎ(𝑥) 𝑥=∞ 󵄨 𝑖 󵄨 󵄨 󵄨 󵄨 󵄨 (ii) lim|𝑥| → ∞ sgn ,andthereisaboundedset −𝜏 󸀠 𝑐 𝑐 𝑖=1 𝑖 Ω containing the origin such that |ℎ (𝑥)| ≤ 𝑏 on Ω ; Ω (8) Ω 𝑛 0 is the complement of the set , 1 2 2 ≤ ∑ ∫ 𝑐𝑖 (𝑦 (𝑡) +𝑦 (𝑡+𝑠))𝑑𝑠 2 −𝜏 (iii) lim|𝑥| → ∞ 𝑔𝑖(𝑥) sgn 𝑥=∞,andthereisaboundedset 𝑖=1 𝑖 Ω |𝑔󸀠(𝑥)| ≤ 𝑐 Ω𝑐 Ω𝑐 containing the origin such that 𝑖 𝑖 on ; 1 𝑛 1 𝑛 0 is the complement of the set Ω, ≤ ∑ (𝑐 𝜏 )𝑦2 + ∑ 𝑐 ∫ 𝑦2 (𝑡+𝑠) 𝑑𝑠. 2 𝑖 𝑖 2 𝑖 𝑖=1 𝑖=1 −𝜏𝑖 (iv) 𝑝(𝑡) is an 𝜔-periodic continuous function.

𝑛 Then, it follows that If 𝜏∑𝑖=1 𝑐𝑖 <𝑎,then(3) has an 𝜔-periodic solution. 𝑉̇ ≤ − {𝑓 (𝑥, 𝑦) + 𝑔 (𝑥, 𝑦)𝑦}𝑦2 +𝑦𝑝(𝑡) Moreover, 𝑝(𝑡) =𝐾, 𝐾-constant, under the above con- 𝑛 ditions (3)hasaconstantmotion𝑥=𝑐0,andtheconstant𝑐0 1 + ∑ (𝑐 +2𝜆)𝜏𝑦2 satisfies ℎ(𝑐0)+𝑔𝑖(𝑐0)=𝐾. 𝑖 𝑖 𝑖 2 𝑖=1 (9) 1 𝑛 0 Proof. Define the Lyapunov-Krasovskii functional 𝑉= + ∑ (𝑐 −2𝜆) ∫ 𝑦2 (𝑡+𝑠) 𝑑𝑠. 𝑉(𝑥 ,𝑦) 2 𝑖 𝑖 𝑡 𝑡 : 𝑖=1 −𝜏𝑖

𝑥 𝑛 𝑥 𝜆 =𝑐/2 𝜏= {𝜏 ,...,𝜏 } 1 2 Let 𝑖 𝑖 and max 1 𝑛 . In fact, these choices 𝑉= 𝑦 + ∫ ℎ (𝑠) 𝑑𝑠+ ∑ ∫ 𝑔𝑖 (𝑠) 𝑑𝑠 imply that 2 0 𝑖=1 0 (6) 𝑛 𝑛 0 𝑡 ̇ 󵄨 󵄨 󵄨 󵄨−1 2 2 𝑉≤−{𝑓(𝑥,𝑦)+𝑔(𝑥,𝑦)𝑦−𝜏∑ 𝑐𝑖 − 󵄨𝑝 (𝑡)󵄨 󵄨𝑦󵄨 }𝑦 + ∑ 𝜆𝑖 ∫ ∫ 𝑦 (𝜃) 𝑑𝜃 𝑑𝑠, 󵄨 󵄨 󵄨 󵄨 𝑖=1 𝑖=1 −𝜏𝑖 𝑡+𝑠 𝑛 𝜆 󵄨 󵄨 󵄨 󵄨−1 2 where 𝑖 are some positive constants to be determined later ≤−(𝑎−𝜏∑ 𝑐𝑖 − 󵄨𝑝 (𝑡)󵄨 󵄨𝑦󵄨 )𝑦 . in the proof. 𝑖=1 Evaluatingthetimederivativeof𝑉 along system (4), we (10) get In view of the continuity and periodicity of the function 𝑝 𝑎−𝜏∑𝑛 𝑐 >0 𝑉̇ = − {𝑓 (𝑥, 𝑦) + 𝑔 (𝑥,2 𝑦)𝑦}𝑦 and the assumption 𝑖=1 𝑖 , it follows that there is aboundedsetΩ2 ⊇Ω1 with Ω2 containing the origin and a 𝑛 0 positive constant 𝜇 such that +𝑦∑ ∫ 𝑔󸀠 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠 𝑖 󵄨 󵄨 𝑖=1 −𝜏𝑖 󵄨 󵄨 󵄨𝑝 (𝑡)󵄨 𝑐 𝜇≤𝑎−𝑐𝜏− 󵄨 󵄨 for R ×Ω2. (11) 𝑛 0 󵄨𝑦󵄨 2 2 󵄨 󵄨 +𝑦𝑝(𝑡) + ∑ 𝜆𝑖 ∫ (𝑦 (𝑡) −𝑦 (𝑡+𝑠))𝑑𝑠 −𝜏 𝑖=1 𝑖 Therefore, we can write = − {𝑓 (𝑥, 𝑦) + 𝑔 (𝑥,2 𝑦)𝑦}𝑦 ̇ 2 𝑐 𝑐 𝑉≤−𝜇𝑦 for (𝑡, 𝑥, 𝑦) ∈ R ×Ω2 ×Ω2. (12) 𝑛 0 󸀠 From the last estimate, we can arrive that the 𝑦-coor- +𝑦∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠 𝑖=1 −𝜏𝑖 dinate of the solutions of system (4)isultimatelybounded 4 Abstract and Applied Analysis

𝛽 𝑝 𝑛 𝑛 0 for a positive constant . On the other hand, since is a + ∑ (𝜆 𝜏 )𝑦2 − ∑ 𝜆 ∫ 𝑦2 (𝑡+𝑠) 𝑑𝑠 |𝑦| ≤ 𝛽 𝑉 =𝑉+𝑦 𝑖 𝑖 𝑖 continuous periodic function, if and 1 ,then, 𝑖=1 𝑖=1 −𝜏𝑖 subject to the assumptions of the theorem, it can be easily seen 𝑐 𝑛 that there is a constant 𝐾1 >0on R ×Ω2 such that +{𝑓(𝑥,𝑦)+𝑔(𝑥,𝑦)𝑦}𝑦+ℎ(𝑥) + ∑ 𝑔𝑖 (𝑥) 𝑖=1 ̇ ̇ 󸀠 𝑉1 = 𝑉+𝑦 𝑛 0 󸀠 2 − ∑ ∫ 𝑔 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠− 𝑝 (𝑡) = − {𝑓 (𝑥, 𝑦) + 𝑔 (𝑥, 𝑦)𝑦}𝑦 𝑖 𝑖=1 −𝜏𝑖 𝑛 0 𝑛 0 +𝑦∑ ∫ 𝑔󸀠 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠+ 𝑦𝑝 (𝑡) 2 󸀠 𝑖 ≤−𝑎𝑦+𝑦∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠 𝑖=1 −𝜏𝑖 𝑖=1 −𝜏𝑖 𝑛 𝑛 0 𝑛 𝑛 0 + ∑ (𝜆 𝜏 )𝑦2 − ∑ 𝜆 ∫ 𝑦2 (𝑡+𝑠) 𝑑𝑠 2 2 𝑖 𝑖 𝑖 +𝑦𝑝(𝑡) + ∑ (𝜆𝑖𝜏𝑖)𝑦 − ∑ 𝜆𝑖 ∫ 𝑦 (𝑡+𝑠) 𝑑𝑠 𝑖=1 𝑖=1 −𝜏𝑖 𝑖=1 𝑖=1 −𝜏𝑖

𝑛 𝑛 −{𝑓(𝑥,𝑦)+𝑔(𝑥,𝑦)𝑦}𝑦−ℎ(𝑥) − ∑ 𝑔 (𝑥) 󵄨 󵄨 𝑖 + 𝑎 󵄨𝑦󵄨 +ℎ(𝑥) + ∑ 𝑔𝑖 (𝑥) 𝑖=1 𝑖=1 𝑛 0 𝑛 0 + ∑ ∫ 𝑔󸀠 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠+ 𝑝 (𝑡) 󸀠 𝑖 − ∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠− 𝑝 (𝑡) 𝑖=1 −𝜏𝑖 𝑖=1 −𝜏𝑖 𝑛 0 2 󸀠 𝑛 ≤−𝑎𝑦 +𝑦∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠 ≤ℎ(𝑥) + ∑ 𝑔 (𝑥) +𝐾. −𝜏 𝑖 2 𝑖=1 𝑖 𝑖=1 𝑛 2 (15) +𝑦𝑝(𝑡) + ∑ (𝜆𝑖𝜏𝑖)𝑦 𝑖=1 Since ℎ(𝑥) → −∞ and 𝑔𝑖(𝑥) → −∞ as 𝑥→−∞,then itcanbechosenapositiveconstant𝐵2 such that 𝑛 0 󵄨 󵄨 𝑛 − ∑ 𝜆 ∫ 𝑦2 (𝑡+𝑠) 𝑑𝑠+ 𝑎 󵄨𝑦󵄨 −ℎ(𝑥) − ∑ 𝑔 (𝑥) ̇ 𝑖 󵄨 󵄨 𝑖 𝑉2 ≤ −0.5 for 𝑥≤−𝐵2. (16) 𝑖=1 −𝜏𝑖 𝑖=1

𝑛 0 + ∑ ∫ 𝑔󸀠 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠+ 𝑝 (𝑡) 𝑖 Then,wecanconcludethatthereisapositiveconstant 𝑖=1 −𝜏𝑖 𝛼2 such that the 𝑥-coordinateofthesolutionsofsystem(4) 𝑛 satisfies 𝑥≥−𝛼2 for |𝑦| ≤. 𝛽 On gathering the above whole ≤−ℎ(𝑥) − ∑ 𝑔𝑖 (𝑥) +𝐾1. discussion, one can see that the solutions of system (4)are 𝑖=1 ultimately bounded. Therefore, (3)hasan𝜔-periodic motion (13) (solution). When 𝑝(𝑡) =𝐾, 𝐾-constant, (3)hasaconstant motion 𝑥=𝑐0.From(3),itcanbeseenthattheconstant𝑥= Since ℎ(𝑥)→∞and 𝑔𝑖(𝑥)→∞as 𝑥→∞,thenitcan 𝑐0 is given by ℎ(𝑐0)+𝑔𝑖(𝑐0)=𝐾. be chosen a positive constant 𝐵1 such that

̇ Conflict of Interests 𝑉1 ≤ −0.5 for 𝑥≥𝐵1. (14) The authors declare that there is no conflict of interests Therefore, we can conclude that there is a positive constant regarding the publication of this paper. 𝛼1 such that the 𝑥-coordinateofthesolutionsofsystem(4) satisfies 𝑥≤𝛼1 for |𝑦| ≤. 𝛽 References In a similar manner, if |𝑦| ≤ 𝛽 and 𝑉2 =𝑉−𝑦,then, subject to the assumptions of the theorem, it can be easily [1] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,vol.568ofLecture Notes in followed by the time derivative of the functional 𝑉2 that there 𝑐 Mathematics, Springer, Berlin, Germany, 1977. is a constant 𝐾2 >0on R ×Ω2 such that [2] T. Yoshizawa, “Stability theory by Liapunov’s second method,” Publications of the Mathematical Society of Japan 9, The 𝑉̇ = 𝑉−𝑦̇ 󸀠 2 Mathematical Society of Japan, Tokyo, Japan, 1966. = − {𝑓 (𝑥, 𝑦) + 𝑔 (𝑥,2 𝑦)𝑦}𝑦 [3] J. M. Zhao, K. L. Huang, and Q. S. Lu, “The existence of periodic solutions for a class of functional-differential equations and 𝑛 0 their application,” Applied Mathematics and Mechanics. English 󸀠 +𝑦∑ ∫ 𝑔𝑖 (𝑥 (𝑡+𝑠)) 𝑦 (𝑡+𝑠) 𝑑𝑠+ 𝑦𝑝 (𝑡) Edition,vol.15,no.1,pp.49–59,1994,translatedfromApplied 𝑖=1 −𝜏𝑖 Mathematics and Mechanics,vol.15,no.1,pp.49–58,1994. Abstract and Applied Analysis 5

[4]T.A.Burton,Stability and Periodic Solutions of Ordinary and [22] C. Tunc¸ and E. Tunc¸, “On the asymptotic behavior of solutions Functional Differential Equations,Dover,Mineola,NY,USA, of certain second-order differential equations,” Journal of the 2005, Corrected Version of the 1985 Original. Franklin Institute,vol.344,no.5,pp.391–398,2007. [5]S.AhmadandM.R.Rao,TheoryofOrdinarydifferentialEqua- [23] J. Zhou, “Necessary and sufficient conditions for boundedness tions. With Applications in Biology and Engineering,Affiliated and convergence of a second-order nonlinear differential sys- East-West Press, New Delhi, India, 1999. tem,” Acta Mathematica Sinica. Chinese Series,vol.43,no.3,pp. [6] A. Constantin, “Anote on a second-order nonlinear differential 415–420, 2000. system,” Glasgow Mathematical Journal,vol.42,no.2,pp.195– 199, 2000. [7] J. R. Graef, “On the generalized Lienard´ equation with negative damping,” Journal of Differential Equations,vol.12,pp.34–62, 1972. [8] L. H. Huang and J. S. Yu, “On boundedness of solutions of generalized Lienard’s´ system and its application,” Annals of Differential Equations,vol.9,no.3,pp.311–318,1993. [9] Z. Jin, “Boundedness and convergence of solutions of a second- order nonlinear differential system,” Journal of Mathematical Analysis and Applications,vol.256,no.2,pp.360–374,2001. [10] B. Liu and L. Huang, “Boundedness of solutions for a class of retarded Lienard´ equation,” Journal of Mathematical Analysis and Applications,vol.286,no.2,pp.422–434,2003. [11] J. E. Napoles´ Valdes,´ “Boundedness and global asymptotic stability of the forced Lienard´ equation,” Revista de la Union´ Matematica´ Argentina,vol.41,no.4,pp.47–59,2000. [12] C. X. Qian, “Boundedness and asymptotic behaviour of solu- tions of a second-order nonlinear system,” The Bulletin of the London Mathematical Society,vol.24,no.3,pp.281–288,1992. [13] C. Tunc¸, “A note on the bounded solutions,” Applied Mathemat- ics and Information Sciences,vol.8,no.1,pp.393–399,2014. [14] C. Tunc¸, “A note on boundedness of solutions to a class of non- autonomous differential equations of second order,” Applicable Analysis and Discrete Mathematics,vol.4,no.2,pp.361–372, 2010. [15] C. Tunc¸, “Boundedness results for solutions of certain nonlinear differential equations of second order,” Journal of the Indonesian Mathematical Society,vol.16,no.2,pp.115–126,2010. [16] C. Tunc¸, “Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations,” Applied and Computational Mathematics,vol.10,no.3,pp.449–462, 2011. [17] C. Tunc¸, “On the boundedness of solutions of a non-auton- omous differential equation of second order,” Sarajevo Journal of Mathematics, vol. 7(19), no. 1, pp. 19–29, 2011. [18] C. Tunc¸, “Stability and uniform boundedness results for non- autonomous Lienard-type equations with a variable deviating argument,” Acta Mathematica Vietnamica,vol.37,no.3,pp.311– 325, 2012. [19] C. Tunc¸, “On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating arguments,” Afrika Matematika,vol.23, no. 2, pp. 249–259, 2012. [20] C. Tunc¸, “Stability to vector Lienard equation with constant deviating argument,” Nonlinear Dynamics,vol.73,no.3,pp. 1245–1251, 2013. [21] C. Tunc¸, “New results on the existence of periodic solutions for Rayleigh equation with state-dependent delay,” Journal of Mathematical and Fundamental Sciences,vol.45,no.2,pp.154– 162, 2013. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 364743, 10 pages http://dx.doi.org/10.1155/2013/364743

Research Article Sequence Spaces Defined by Musielak-Orlicz Function over 𝑛-Normed Spaces

M. Mursaleen,1 Sunil K. Sharma,2 and A. KJlJçman3

1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India 2 Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal 181122, Jammu and Kashmir, India 3 Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to A. Kılıc¸man; [email protected]

Received 21 July 2013; Accepted 16 September 2013

Academic Editor: Abdullah Alotaibi

Copyright © 2013 M. Mursaleen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the present paper we introduce some sequence spaces over n-normed spaces defined by a Musielak-Orlicz function M =(𝑀𝑘). We also study some topological properties and prove some inclusion relations between these spaces.

1. Introduction and Preliminaries is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M,the 𝑀 An Orlicz function is a function, which is continuous, Musielak-Orlicz sequence space 𝑡M and its subspace ℎM are 𝑀(0) = 0 𝑀(𝑥) >0 nondecreasing, and convex with , for defined as follows: 𝑥>0and 𝑀(𝑥) →∞ as 𝑥→∞. Lindenstrauss and Tzafriri [1]usedtheideaofOrlicz 𝑡M ={𝑥∈𝑤:𝐼M (𝑐𝑥) <∞for some 𝑐>0}, function to define the following sequence space. Let 𝑤 be the (4) ℎM ={𝑥∈𝑤:𝐼M (𝑐𝑥) <∞∀𝑐>0}, spaceofallrealorcomplexsequences𝑥=(𝑥𝑘);then

󵄨 󵄨 where 𝐼M is a convex modular defined by ∞ 󵄨𝑥 󵄨 ℓ ={𝑥∈𝑤:∑𝑀(󵄨 𝑘󵄨)<∞}, 𝑀 𝜌 (1) ∞ 𝑘=1 𝐼M (𝑥) = ∑ (𝑀𝑘)(𝑥𝑘), 𝑥=(𝑥𝑘)∈𝑡M. (5) 𝑘=1 whichiscalledasanOrliczsequencespace.Thespaceℓ𝑀 is a Banach space with the norm We consider 𝑡M equipped with the Luxemburg norm 𝑥 ∞ 󵄨 󵄨 󵄨𝑥 󵄨 ‖𝑥‖ = inf {𝑘 > 0 :M 𝐼 ( )≤1} (6) ‖𝑥‖ = {𝜌>0:∑𝑀 (󵄨 𝑘󵄨) ≤1} . 𝑘 inf 𝜌 (2) 𝑘=1 or equipped with the Orlicz norm Itisshownin[1] that every Orlicz sequence space ℓ𝑀 1 ℓ (𝑝 ≥ 1) Δ ‖𝑥‖0 = { (1+𝐼 (𝑘𝑥)):𝑘>0}. contains a subspace isomorphic to 𝑝 .The 2- inf 𝑘 M (7) condition is equivalent to 𝑀(𝐿𝑥) ≤ 𝑘𝐿𝑀(𝑥) for all values of 𝑥≥0and for 𝐿>1.AsequenceM =(𝑀𝑘) of Orlicz Let 𝑋 be a linear metric space. A function 𝑝 : 𝑋→R is functions is called a Musielak-Orlicz function (see [2, 3]). A called paranorm if sequence N =(𝑁𝑘) defined by (1) 𝑝(𝑥) ≥0 for all 𝑥∈𝑋,

𝑁𝑘 (V) = sup {|V| 𝑢−𝑀𝑘 (𝑢) :𝑢≥0} , 𝑘=1,2,..., (3) (2) 𝑝(−𝑥) = 𝑝(𝑥) for all 𝑥∈𝑋, 2 Abstract and Applied Analysis

(3) 𝑝(𝑥 + 𝑦) ≤ 𝑝(𝑥) +𝑝(𝑦) for all 𝑥, 𝑦 ∈𝑋, The concept of 2-normed spaces was initially developed by Gahler¨ [19]inthemid1960s,whileforthatof𝑛-normed (4) (𝜆𝑛) is a sequence of scalars with 𝜆𝑛 →𝜆 as 𝑛→ spaces one can see Misiak [20]. Since then, many others ∞ and (𝑥𝑛) is a sequence of vectors with 𝑝(𝑥𝑛 −𝑥) have studied this concept and obtained various results; see →0as 𝑛→∞;then𝑝(𝜆𝑛𝑥𝑛 −𝜆𝑥)→as 0 𝑛∈ 𝑛→∞. Gunawan ([21, 22]) and Gunawan and Mashadi [23]. Let N and let 𝑋 be a linear space over the field K,whereK is A paranorm 𝑝 for which 𝑝(𝑥) =0 implies 𝑥=0is thefieldofrealorcomplexnumbersofdimension𝑑,where 𝑛 called total paranorm and the pair (𝑋, 𝑝) is called a total 𝑑≥𝑛≥2.Arealvaluedfunction‖⋅,...,⋅‖on 𝑋 satisfying paranormed space. It is well known that the metric of any the following four conditions linear metric space is given by some total paranorm (see [4], 10.4.2 Theorem ,pp.183).Formoredetailsaboutsequence (1) ‖𝑥1,𝑥2,...,𝑥𝑛‖=0if and only if 𝑥1,𝑥2,...,𝑥𝑛 are spaces, see [5–12] and references therein. linearly dependent in 𝑋; A sequence of positive integers 𝜃=(𝑘𝑟) is called lacunary ‖𝑥 ,𝑥 ,...,𝑥 ‖ if 𝑘0 =0, 0<𝑘𝑟 <𝑘𝑟+1 and ℎ𝑟 =𝑘𝑟 −𝑘𝑟−1 →∞as 𝑟→ (2) 1 2 𝑛 is invariant under permutation; ∞. The intervals determined by 𝜃 will be denoted by 𝐼𝑟 = (3) ‖𝛼𝑥1,𝑥2,...,𝑥𝑛‖=|𝛼| ‖𝑥1,𝑥2,...,𝑥𝑛‖ for any 𝛼∈ (𝑘𝑟−1,𝑘𝑟) and 𝑞𝑟 =𝑘𝑟/𝑘𝑟−1.Thespaceoflacunarystrongly K; convergent sequences 𝑁𝜃 was defined by Freedman et al. [13] as 󸀠 󸀠 (4) ‖𝑥+𝑥 ,𝑥2,...,𝑥𝑛‖≤‖𝑥,𝑥2,...,𝑥𝑛‖+‖𝑥 ,𝑥2,...,𝑥𝑛‖ { 1 󵄨 󵄨 } 𝑁 = 𝑥∈𝑤: ∑ 󵄨𝑥 −𝑙󵄨 =0, 𝑙 . 𝑛 𝑋 (𝑋,‖⋅,...,⋅‖) 𝜃 { 𝑟→∞lim 󵄨 𝑘 󵄨 for some } (8) is called an -norm on ,andthepair is called ℎ𝑟 𝑘∈𝐼𝑟 an 𝑛-normed space over the field K. { } 𝑛 For example, if we may take 𝑋=R being Strongly almost convergent sequence was introduced and equipped with the 𝑛-norm ‖𝑥1,𝑥2,...,𝑥𝑛‖𝐸 =thevolume studied by Maddox [14] and Freedman et al. [13]. Parashar of the 𝑛-dimensional parallelepiped spanned by the vectors and Choudhary [15] have introduced and examined some 𝑥1,𝑥2,...,𝑥𝑛 whichmaybegivenexplicitlybytheformula properties of four sequence spaces defined by using an 𝑀 󵄨 󵄨 Orlicz function , which generalized the well-known Orlicz 󵄩 󵄩 󵄨 󵄨 󵄩𝑥1,𝑥2,...,𝑥𝑛󵄩𝐸 = 󵄨det (𝑥𝑖𝑗 )󵄨 , (13) sequence spaces [𝐶, 1, 𝑝], [𝐶, 1, 𝑝]0,and[𝐶, 1, 𝑝]∞.Itmaybe 󵄨 󵄨 notedherethatthespaceofstronglysummablesequenceswas 𝑛 discussed by Maddox [16]andrecentlyin[17]. where 𝑥𝑖 =(𝑥𝑖1,𝑥𝑖2,...,𝑥𝑖𝑛)∈R for each 𝑖 = 1,2,...,𝑛, Mursaleen and Noman [18] introduced the notion of 𝜆- leting (𝑋,‖⋅,...,⋅‖)be an 𝑛-normed space of dimension 𝑑≥ convergent and 𝜆-bounded sequences as follows. 𝑛≥2and {𝑎1,𝑎2,...,𝑎𝑛} be linearly independent set in 𝑋, 𝜆=(𝜆)∞ 𝑛−1 Let 𝑘 𝑘=1 be a strictly increasing sequence of then the following function ‖⋅,...,⋅‖∞ on 𝑋 defined by positive real numbers tending to infinity; that is, 󵄩 󵄩 󵄩𝑥1,𝑥2,...,𝑥𝑛−1󵄩 0<𝜆0 <𝜆1 <⋅⋅⋅ , 𝜆𝑘 󳨀→ ∞ as 𝑘󳨀→∞, (9) 󵄩 󵄩∞ (14) 󵄩 󵄩 = max {󵄩𝑥1,𝑥2,...,𝑥𝑛−1,𝑎𝑖󵄩 :𝑖=1,2,...,𝑛} and it is said that a sequence 𝑥=(𝑥𝑘)∈𝑤is 𝜆-convergent 󵄩 󵄩 to the number 𝐿,calledthe𝜆-limit of 𝑥 if Λ 𝑚(𝑥)→𝐿as 𝑚→∞,where defines an (𝑛 − 1)-norm on 𝑋 with respect to {𝑎1,𝑎2,...,𝑎𝑛}. Asequence(𝑥𝑘) in an 𝑛-normed space (𝑋,‖⋅,...,⋅‖) is 1 𝑚 𝜆 (𝑥) = ∑ (𝜆 −𝜆 ) 𝑥 . said to converge to some 𝐿∈𝑋if 𝑚 𝜆 𝑘 𝑘−1 𝑘 (10) 𝑚 𝑘=1 󵄩 󵄩 lim 󵄩𝑥𝑘 −𝐿,𝑧1,...,𝑧𝑛−1󵄩 =0 The sequence 𝑥=(𝑥𝑘)∈𝑤is 𝜆-bounded if 𝑘→∞ (15) sup𝑚|Λ 𝑚(𝑥)| <.Itiswellknown[ ∞ 18]thatiflim𝑚𝑥𝑚 =𝑎 for every 𝑧1,...,𝑧𝑛−1 ∈𝑋. in the ordinary sense of convergence, then

𝑚 1 󵄨 󵄨 Asequence(𝑥𝑘) in an 𝑛-normed space (𝑋,‖⋅,...,⋅‖) is ( (∑ (𝜆 −𝜆 ) 󵄨𝑥 −𝑎󵄨)) =0. lim𝑚 𝜆 𝑘 𝑘−1 󵄨 𝑘 󵄨 (11) said to be Cauchy if 𝑚 𝑘=1 󵄩 󵄩 This implies that lim 󵄩𝑥𝑘 −𝑥𝑝,𝑧1,...,𝑧𝑛−1󵄩 =0 𝑘→∞ 󵄩 󵄩 󵄨 󵄨 𝑝→∞ 󵄨 𝑚 󵄨 (16) 󵄨 󵄨 󵄨 1 󵄨 lim 󵄨Λ 𝑚 (𝑥) −𝑎󵄨 = lim 󵄨 ∑ (𝜆𝑘 −𝜆𝑘−1)(𝑥𝑘 −𝑎)󵄨 =0, 𝑚 󵄨 󵄨 𝑚 󵄨𝜆 󵄨 for every 𝑧1,...,𝑧𝑛−1 ∈𝑋. 󵄨 𝑚 𝑘=1 󵄨 (12) If every Cauchy sequence in 𝑋 converges to some 𝐿∈𝑋, which yields that lim𝑚Λ 𝑚 (𝑥) = 𝑎 and hence 𝑥=(𝑥𝑘)∈𝑤is then 𝑋 is said to be complete with respect to the 𝑛-norm. Any 𝜆-convergent to 𝑎. complete 𝑛-normed space is said to be 𝑛-Banach space. Abstract and Applied Analysis 3

𝑝 󵄩Λ (𝑥) −𝐿 󵄩 𝑘 Let M =(𝑀𝑘) be a Musielak-Orlicz function, and let −𝑠 󵄩 𝑘 󵄩 × ∑ 𝑘 [󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩] =0, 𝑝=(𝑝𝑘) be a bounded sequence of positive real numbers. We 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 define the following sequence spaces in the present paper: for some 𝐿, 𝜌 > 0, 𝑠 ≥0}, 𝜃 𝑤0 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝜃 𝑤∞ (Λ, 𝑝, 𝑠, ‖⋅,...,⋅‖) 1 ={𝑥=(𝑥)∈𝑤: 1 𝑘 𝑟→∞lim ℎ𝑟 ={𝑥=(𝑥𝑘)∈𝑤:sup 𝑟 ℎ𝑟 𝑝 󵄩 󵄩 𝑘 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 𝑝 × ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] =0, 󵄩Λ (𝑥) 󵄩 𝑘 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 −𝑠 󵄩 𝑘 󵄩 𝑘∈𝐼 󵄩 󵄩 × ∑ 𝑘 [󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩] <∞, 𝑟 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 𝜌 > 0, 𝑠 ≥ 0}, 𝜌 > 0, 𝑠 ≥ 0}.

𝜃 𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) (18) 1 𝑝=(𝑝)=1 𝑘∈N ={𝑥=(𝑥)∈𝑤: If we take 𝑘 for all ,wehave 𝑘 𝑟→∞lim ℎ𝑟 𝜃 𝑤0 (M,Λ,𝑠,‖⋅,...,⋅‖) 󵄩 󵄩 𝑝 󵄩Λ (𝑥) −𝐿 󵄩 𝑘 × ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 1 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 ={𝑥=(𝑥)∈𝑤: 𝑘∈𝐼 󵄩 󵄩 𝑘 𝑟→∞lim 𝑟 ℎ𝑟 󵄩 󵄩 =0, 𝐿, 𝜌 > 0, 𝑠 ≥0}, −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 for some × ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)]=0, 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 𝜃 𝑤∞ (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝜌 > 0, 𝑠 ≥ 0}, 1 ={𝑥=(𝑥)∈𝑤: 𝑘 supℎ 𝑟 𝑟 𝑤𝜃 (M,Λ,𝑠,‖⋅,...,⋅‖) 󵄩 󵄩 𝑝 󵄩Λ (𝑥) 󵄩 𝑘 × ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] <∞, 1 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 ={𝑥=(𝑥)∈𝑤: 𝑘∈𝐼 󵄩 󵄩 𝑘 𝑟→∞lim 𝑟 ℎ𝑟 󵄩Λ (𝑥) −𝐿 󵄩 𝜌 > 0, 𝑠 ≥ 0}. −𝑠 󵄩 𝑘 󵄩 × ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 (17) =0, for some 𝐿, 𝜌 > 0, 𝑠 ≥0}, M(𝑥) = 𝑥 If we take ,weget 𝜃 𝑤∞ (M,Λ,𝑠,‖⋅,...,⋅‖) 1 𝑤𝜃 (Λ, 𝑝, 𝑠, ‖⋅,...,⋅‖) ={𝑥=(𝑥𝑘)∈𝑤:sup 0 𝑟 ℎ𝑟 1 󵄩Λ 𝑥 󵄩 ={𝑥=(𝑥)∈𝑤: −𝑠 󵄩 𝑘 ( ) 󵄩 𝑘 𝑟→∞lim × ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)]<∞, ℎ𝑟 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 𝑝 󵄩 󵄩 𝑘 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 × ∑ 𝑘 [󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩] =0, 󵄩 𝜌 󵄩 𝜌 > 0, 𝑠 ≥ 0}. 𝑘∈𝐼𝑟 (19) 𝜌 > 0, 𝑠 ≥ 0}, The following inequality will be used throughout the 𝐻−1 paper. If 0≤𝑝𝑘 ≤ sup 𝑝𝑘 =𝐻, 𝐾=max(1, 2 ),then 𝑤𝜃 (Λ, 𝑝, 𝑠, ‖⋅,...,⋅‖) 󵄨 󵄨𝑝𝑘 󵄨 󵄨𝑝𝑘 󵄨 󵄨𝑝𝑘 󵄨𝑎𝑘 +𝑏𝑘󵄨 ≤𝐾{󵄨𝑎𝑘󵄨 + 󵄨𝑏𝑘󵄨 } (20) 1 ={𝑥=(𝑥)∈𝑤: 𝑝 𝐻 𝑘 𝑟→∞lim 𝑘 𝑎 ,𝑏 ∈ C |𝑎| 𝑘 ≤ (1, |𝑎| ) 𝑎∈C ℎ𝑟 for all and 𝑘 𝑘 .Also max for all . 4 Abstract and Applied Analysis

󵄩 󵄩 𝑝 1 󵄩Λ (𝑥) 󵄩 𝑘 In this paper, we introduce sequence spaces defined ≤𝐾 ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 𝑘 󵄩 1 2 𝑛−1󵄩 by a Musielak-Orlicz function over 𝑛-normed spaces. We ℎ𝑟 󵄩 𝜌1 󵄩 𝑘∈𝐼𝑟 study some topological properties and prove some inclusion 󵄩 󵄩 𝑝 relations between these spaces. 1 󵄩Λ (𝑦) 󵄩 𝑘 −𝑠 󵄩 𝑘 󵄩 +𝐾 ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] ℎ𝑟 󵄩 𝜌2 󵄩 𝑘∈𝐼𝑟 󵄩 󵄩 2. Main Results 󳨀→ 0 as 𝑟󳨀→∞. Theorem 1. M =(𝑀) Let 𝑘 be a Musielak-Orlicz (23) function, and let 𝑝=(𝑝𝑘) be a bounded sequence of 𝜃 𝛼𝑥+𝛽𝑦∈𝜃( 𝑤M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) positive real numbers, then the spaces 𝑤0(M,Λ,𝑝, Thus, we have 0 . 𝜃 𝜃 𝑤𝜃(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝑠, ‖⋅, . . . , ⋅‖), 𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖),and𝑤∞(M,Λ,𝑝, Hence, 0 is a linear space. 𝜃 𝑠,‖⋅,...,⋅‖)are linear spaces over the field of complex number Similarly, we can prove that 𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) and C 𝜃 . 𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)are linear spaces. 𝑥=(𝑥), 𝑦=(𝑦)∈𝑤𝜃(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) Proof. Let 𝑘 let 𝑘 0 , Theorem 2. Let M =(𝑀𝑘) be a Musielak-Orlicz function, 𝛼, 𝛽 ∈ C and let . In order to prove the result, we need to find and let 𝑝=(𝑝𝑘) be a bounded sequence of positive real 𝜌 𝜃 some 3 such that numbers. Then 𝑤0(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)is a topological linear space paranormed by 󵄩 󵄩 𝑝 1 󵄩Λ (𝛼𝑥+𝛽𝑦) 󵄩 𝑘 𝑔 (𝑥) ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 𝑟→∞lim 𝑘 󵄩 1 2 𝑛−1󵄩 ℎ𝑟 󵄩 𝜌3 󵄩 𝑘∈𝐼𝑟 󵄩 󵄩 { 𝑝 /𝐻 = 𝜌 𝑟 : =0. inf{ { (21) 󵄩 󵄩 1 −𝑠 󵄩Λ 𝑘 (𝑥) ( ∑𝑘 [𝑀 (󵄩 , 𝜃 ℎ 𝑘 󵄩 𝜌 𝑥=(𝑥), 𝑦 = (𝑦 )∈𝑤(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝑟𝑘∈𝐼 󵄩 Since 𝑘 𝑘 0 ,there 𝑟 [ 󵄩 exist positive numbers 𝜌1,𝜌2 >0such that 𝑝 1/𝐻 󵄩 𝑘 󵄩 } 𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ≤1 , 󵄩 󵄩 𝑝𝑘 1 2 𝑛−1󵄩 } 1 −𝑠 󵄩Λ (𝑥) 󵄩 󵄩 ∑ 𝑘 [𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] =0, 󵄩 ] } 𝑟→∞lim 𝑘 󵄩 1 2 𝑛−1󵄩 ℎ𝑟 󵄩 𝜌1 󵄩 𝑘∈𝐼𝑟 (24) 󵄩 󵄩 𝑝 1 󵄩Λ (𝑦) 󵄩 𝑘 where 𝐻=max(1, sup𝑘𝑝𝑘)<∞. ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] =0. 𝑟→∞lim 𝑘 󵄩 1 2 𝑛−1󵄩 ℎ𝑟 󵄩 𝜌2 󵄩 𝑘∈𝐼𝑟 Proof. Clearly 𝑔(𝑥) ≥0 for 𝑥=(𝑥𝑘)∈ 𝜃 (22) 𝑤0(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖).Since𝑀𝑘(0) = 0,weget𝑔(0) =. 0 Again if 𝑔(𝑥),then =0 𝜌 = (2|𝛼|𝜌 , 2|𝛽|𝜌 ) (𝑀 ) Define 3 max 1 2 .Since 𝑘 is nondecreas- { 𝑝 /𝐻 ing, convex function and by using inequality (20), we have inf{𝜌 𝑟 : { 󵄩 󵄩 𝑝 󵄩 1 󵄩Λ (𝛼𝑥+𝛽𝑦) 󵄩 𝑘 󵄩 ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 1 −𝑠[ 󵄩Λ 𝑘 (𝑥) 𝑘 󵄩 1 2 𝑛−1󵄩 ( ∑𝑘 𝑀𝑘(󵄩 , ℎ𝑟 󵄩 𝜌3 󵄩 ℎ 󵄩 𝜌 𝑘∈𝐼𝑟 𝑟𝑘∈𝐼 󵄩 𝑟 [ 󵄩 󵄩 󵄩 1 −𝑠 󵄩𝛼Λ 𝑘 (𝑥) 󵄩 𝑝 1/𝐻 ≤ ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩 󵄩 𝑘 ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 󵄩 } 𝑟 𝑘∈𝐼 󵄩 3 󵄩 󵄩 ] 𝑟 𝑧1,𝑧2,...,𝑧𝑛−1󵄩) ) ≤1}=0. 󵄩 󵄩 󵄩 𝑝 󵄩 ] } 󵄩𝛽Λ (𝑦) 󵄩 𝑘 󵄩 𝑘 󵄩 (25) + 󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] 󵄩 𝜌3 󵄩 This implies that for a given 𝜖>0, there exist some 𝜌𝜖(0 < 𝑝 󵄩 󵄩 𝑘 𝜌 <𝜖) 1 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 𝜖 such that ≤𝐾 ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] ℎ 2𝑝𝑘 󵄩 𝜌 󵄩 1/𝐻 𝑟 𝑘∈𝐼 󵄩 1 󵄩 󵄩 󵄩 𝑝 𝑟 1 󵄩Λ (𝑥) 󵄩 𝑘 ( ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ≤1. 𝑝 𝑘 󵄩 1 2 𝑛−1󵄩 󵄩 󵄩 𝑘 ℎ 󵄩 𝜌 󵄩 1 1 󵄩Λ (𝑦) 󵄩 𝑟 𝑘∈𝐼 󵄩 𝜖 󵄩 +𝐾 ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 𝑟 𝑝 𝑘 󵄩 1 2 𝑛−1󵄩 ℎ𝑟 2 𝑘 󵄩 𝜌2 󵄩 (26) 𝑘∈𝐼𝑟 󵄩 󵄩 Abstract and Applied Analysis 5

Thus, 1 𝜌 ≤( ∑ 𝑘−𝑠 [𝑀 ( 1 ) ℎ 𝑘 𝜌 +𝜌 𝑟 𝑘∈𝐼 1 2 𝑟 [ 1/𝐻 𝑝 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 󵄩 󵄩 ( ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) 󵄩Λ 𝑘 (𝑥) 󵄩 𝜌2 𝑘 󵄩 1 2 𝑛−1󵄩 ×[󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩]+( ) ℎ𝑟 󵄩 𝜖 󵄩 󵄩 1 2 𝑛−1󵄩 𝑘∈𝐼𝑟 󵄩 𝜌1 󵄩 𝜌1 +𝜌2 1/𝐻 𝑝 𝑝 1/𝐻 󵄩 󵄩 𝑘 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 󵄩 󵄩 ≤( ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) . 󵄩Λ 𝑘 (𝑦) 󵄩 ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 × [󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩]] ) 𝑟 𝑘∈𝐼 󵄩 𝜖 󵄩 󵄩 𝜌 1 2 𝑛−1󵄩 𝑟 [󵄩 2 󵄩]] (27) 𝜌 ≤( 1 ) 𝜌1 +𝜌2 Suppose that (𝑥𝑘) =0̸ for each 𝑘∈N. This implies that Λ (𝑥) =0̸ 𝑘∈N 𝜖→0 𝑘 for each .Let ,then 1/𝐻 󵄩 󵄩 𝑝 1 󵄩Λ (𝑥) 󵄩 𝑘 ×( ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 󵄩 󵄩 𝑟 𝑘∈𝐼 󵄩 1 󵄩 󵄩Λ 𝑘 (𝑥) 󵄩 𝑟 󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩 󳨀→ ∞. (28) 󵄩 𝜖 󵄩 𝜌 +( 2 ) 𝜌1 +𝜌2 It follows that 𝑝 1/𝐻 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑦) 󵄩 ×( ∑𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) 1/𝐻 ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 󵄩 󵄩 𝑝𝑘 𝑟𝑘∈𝐼 󵄩 2 󵄩 1 󵄩Λ (𝑥) 󵄩 𝑟 ( ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ℎ 𝑘 󵄩 𝜖 1 2 𝑛−1󵄩 𝑟 𝑘∈𝐼 󵄩 󵄩 (29) 𝑟 ≤1. (31) 󳨀→ ∞,

Since 𝜌,1 𝜌 ,and𝜌2 are nonnegative, we have which is a contradiction. Therefore, Λ 𝑘(𝑥) = 0 for each 𝑘,and thus (𝑥𝑘)=0for each 𝑘∈N.Let𝜌1 >0and 𝜌2 >0be the case such that 𝑔(𝑥+𝑦)

1/𝐻 𝑝 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 { ( ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ≤1, 𝑝 /𝐻 𝑘 󵄩 1 2 𝑛−1󵄩 = 𝜌 𝑟 : ℎ𝑟 󵄩 𝜌1 󵄩 inf{ 𝑘∈𝐼𝑟 { { 𝑝 1/𝐻 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑦) 󵄩 󵄩 ( ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ≤1. 1 󵄩Λ (𝑥+𝑦) ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 −𝑠 [ 󵄩 𝑘 𝑟 𝑘∈𝐼 󵄩 2 󵄩 ( ∑𝑘 𝑀𝑘(󵄩 , 𝑟 ℎ 󵄩 𝜌 𝑟 𝑘∈𝐼 󵄩 (30) 𝑟 [ 󵄩 𝑝 1/𝐻 󵄩 𝑘 󵄩 } 𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ≤1 Let 𝜌=𝜌1 +𝜌2; then, by using Minkowski’s inequality, we 1 2 𝑛−1󵄩 } 󵄩 } have 󵄩 ] }

{ 󵄩 󵄩 𝑝 1/𝐻 { 𝑝 /𝐻 1 󵄩Λ (𝑥+𝑦) 󵄩 𝑘 ≤ (𝜌 ) 𝑟 : −𝑠 󵄩 𝑘 󵄩 inf{ 1 ( ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] ) ℎ𝑟 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 󵄩 󵄩 { 󵄩 󵄩 󵄩 1 󵄩Λ (𝑥) +Λ (𝑦) 1 −𝑠 󵄩Λ 𝑘 (𝑥) −𝑠 [ 󵄩 𝑘 𝑘 ( ∑𝑘 [𝑀 (󵄩 , ≤( ∑ 𝑘 𝑀𝑘 ( 󵄩 , 𝑘 󵄩 ℎ 󵄩 𝜌 +𝜌 ℎ𝑟 󵄩 𝜌1 𝑟 𝑘∈𝐼 󵄩 1 2 𝑘∈𝐼𝑟 󵄩 𝑟 [ [ 1/𝐻 1/𝐻 𝑝𝑘 󵄩 𝑝𝑘 󵄩 󵄩 󵄩 } 󵄩 𝑧 ,𝑧 ,...,𝑧 󵄩)] ) ≤1 𝑧 ,𝑧 ,...,𝑧 󵄩)] ) 1 2 𝑛−1󵄩 } 1 2 𝑛−1󵄩 󵄩 } 󵄩 ] 󵄩 ] } 6 Abstract and Applied Analysis

𝑝 /𝐻 { 𝑟 { 𝑝 /𝐻 ×inf 𝑡 𝑟 : +inf{(𝜌2) : { { { { 󵄩 󵄩 󵄩 1 −𝑠 [ 󵄩Λ 𝑘 (𝑥) 󵄩 ( ∑𝑘 𝑀𝑘 (󵄩 , 1 −𝑠 󵄩Λ 𝑘 (𝑦) ℎ 󵄩 𝑡 ( ∑𝑘 [𝑀 (󵄩 , 𝑟 𝑘∈𝐼 󵄩 𝑘 󵄩 𝑟 [ 󵄩 ℎ𝑟 󵄩 𝜌2 𝑘∈𝐼𝑟 󵄩 𝑝 1/𝐻 [ 󵄩 𝑘 󵄩 𝑝 1/𝐻 𝑧 ,𝑧 ,...,𝑧 󵄩)] ) 󵄩 𝑘 1 2 𝑛−1󵄩 󵄩 } 󵄩 󵄩 ] 󵄩 ] 𝑧1,𝑧2,...,𝑧𝑛−1󵄩) ) ≤1} . 󵄩 } 󵄩 ] } } ≤1} . (35) (32) }

Therefore, 𝑔(𝑥+𝑦)≤ 𝑔(𝑥)+𝑔(𝑦).Finallyweprovethat So the fact that scalar multiplication is continuous follows the scalar multiplication is continuous. Let 𝜇 be any complex from the above inequality. This completes the proof of the number. By definition, theorem.

Theorem 3. Let M =(𝑀𝑘) be a Musielak-Orlicz func- 𝑔(𝜇𝑥) 𝑝𝑘 tion. If sup𝑘[𝑀𝑘(𝑥)] <∞for all fixed 𝑥>0,then 𝑤𝜃(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊆𝑤𝜃 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝑝 /𝐻 0 ∞ . =inf{𝜌 𝑟 : 𝑥=(𝑥)∈𝑤𝜃(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 󵄩 Proof. Let 𝑘 0 ; then there 1 󵄩Λ (𝜇𝑥) ( ∑𝑘−𝑠 [𝑀 (󵄩 𝑘 , exists positive number 𝜌1 such that 𝑘 󵄩 ℎ𝑟 𝑘∈𝐼 󵄩 𝜌 𝑝 𝑟 󵄩 󵄩 󵄩 𝑘 𝑝 1/𝐻 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 󵄩 𝑘 ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] =0. 󵄩 𝑟→∞lim 𝑘 󵄩 1 2 𝑛−1󵄩 󵄩 ℎ𝑟 󵄩 𝜌1 󵄩 𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] ) 𝑘∈𝐼𝑟 󵄩 󵄩 (36) ≤1}. Define 𝜌=2𝜌1.Since(𝑀𝑘) is nondecreasing and convex and by using inequality (20), we have (33) 󵄩 󵄩 𝑝 1 󵄩Λ (𝑥) 󵄩 𝑘 ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] sup 𝑘 󵄩 1 2 𝑛−1󵄩 𝑟 ℎ𝑟 󵄩 𝜌 󵄩 Thus, 𝑘∈𝐼𝑟 󵄩 1 󵄩Λ (𝑥) +𝐿−𝐿 𝑔(𝜇𝑥) = ∑ 𝑘−𝑠 [𝑀 (󵄩 𝑘 , sup 𝑘 󵄩 𝑟 ℎ𝑟 󵄩 𝜌 𝑘∈𝐼𝑟 𝑝 /𝐻 𝑝 { 𝑟 󵄩 𝑘 󵄨 󵄨 󵄩 =inf{ (󵄨𝜇󵄨 𝑡) : 𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] 󵄩 { 󵄩 1 −𝑠 1 󵄩Λ 𝑘 (𝑥) −𝐿 󵄩 ≤𝐾sup ∑ 𝑘 [𝑀𝑘 (󵄩 , 󵄩 𝑟 ℎ 2𝑝𝑘 󵄩 𝜌 1 −𝑠 󵄩Λ 𝑘 (𝑥) 𝑟 𝑘∈𝐼 󵄩 1 ( ∑ 𝑘 [𝑀 (󵄩 , 𝑟 ℎ 𝑘 󵄩 𝑡 󵄩 𝑝 𝑟 𝑘∈𝐼 󵄩 󵄩 𝑘 𝑟 [ 󵄩 󵄩 𝑧1,𝑧2,...,𝑧𝑛−1󵄩 )] 󵄩 𝑝 1/𝐻 󵄩 𝑘 󵄩 󵄩 󵄩 𝑝 󵄩 ] 1 1 󵄩 𝐿 󵄩 𝑘 𝑧1,𝑧2,...,𝑧𝑛−1󵄩) ) −𝑠 󵄩 󵄩 󵄩 +𝐾sup ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] 󵄩 𝑝𝑘 󵄩 󵄩 󵄩 𝑟 ℎ𝑟 2 󵄩𝜌1 󵄩 ] 𝑘∈𝐼𝑟 𝑝 󵄩 󵄩 𝑘 } 1 −𝑠 󵄩Λ 𝑘 (𝑥) −𝐿 󵄩 ≤1 , ≤𝐾sup ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] } 𝑟 ℎ 󵄩 𝜌 󵄩 𝑟 𝑘∈𝐼 󵄩 1 󵄩 } 𝑟 𝑝 (34) 󵄩 󵄩 𝑘 1 −𝑠 󵄩 𝐿 󵄩 +𝐾sup ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] 𝑟 ℎ 󵄩𝜌 󵄩 𝑝 sup 𝑝 𝑟 𝑘∈𝐼 󵄩 1 󵄩 where 1/𝑡 = 𝜌/|𝜇|.Since|𝜇| 𝑟 ≤ max(1, |𝜇| 𝑟 ),wehave 𝑟 <∞. 𝑔(𝜇𝑥) (37)

󵄨 󵄨sup 𝑝𝑟 𝜃 ≤ max (1, 󵄨𝜇󵄨 ) Hence, 𝑥=(𝑥𝑘)∈𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). Abstract and Applied Analysis 7

𝜃 Theorem 4. Let 0

󵄩 󵄩 𝑝𝑘 1 −𝑠 󸀠 󵄩Λ 𝑘 (𝑥) 󵄩 ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] =0. Theorem 5. Let 0<ℎ=inf 𝑝𝑘 =𝑝𝑘 < sup 𝑝𝑘 =𝐻<∞. 𝑟→∞lim 𝑘 󵄩 1 2 𝑛−1󵄩 ℎ𝑟 󵄩 𝜌 󵄩 M =(𝑀) 𝑘∈𝐼𝑟 Then for a Musielak-Orlicz function 𝑘 which satisfies Δ (38) 2-condition, one has 𝜃 𝜃 (i) 𝑤0(Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤0(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖); Let 𝜖>0and choose 𝛿 with 0<𝛿<1such that 𝑀𝑘(𝑡) < 𝜖 󸀠 𝜃 𝜃 for 0≤𝑡≤𝛿.Let(𝑦𝑘)=𝑀𝑘[‖Λ 𝑘(𝑥)/𝜌,1 𝑧 ,𝑧2,...,𝑧𝑛−1‖] for (ii) 𝑤 (Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖); all 𝑘∈N.Wecanwrite 𝜃 𝜃 (iii) 𝑤∞(Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). 1 𝑝 1 𝑝 ∑ 𝑘−𝑠(𝑀 [𝑦 ]) 𝑘 = ∑ 𝑘−𝑠(𝑀 [𝑦 ]) 𝑘 𝑘 𝑘 𝑘 𝑘 Proof . It is easy to prove, so we omit the details. ℎ𝑟 ℎ𝑟 𝑘∈𝐼𝑟 𝑘∈𝐼𝑟 𝑦 ≤𝛿 𝑘 Theorem 6. M =(𝑀) (39) Let 𝑘 be a Musielak-Orlicz function 1 𝑝 𝜃 −𝑠 𝑘 and let 0<ℎ=inf 𝑝𝑘.Then𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) ⊂ + ∑ 𝑘 (𝑀𝑘 [𝑦𝑘]) . ℎ 𝑤𝜃(Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝑟 𝑘∈𝐼𝑟 0 if and only if 𝑦𝑘≥𝛿 1 𝑝 ∑ 𝑘−𝑠(𝑀 (𝑡)) 𝑘 =∞ 𝑟→∞lim 𝑘 (45) So, we have ℎ𝑟 𝑘∈𝐼𝑟 1 𝑝 𝐻 1 𝑝 ∑ 𝑘−𝑠(𝑀 [𝑦 ]) 𝑘 ≤[𝑀 (1)] ∑ 𝑘−𝑠(𝑀 [𝑦 ]) 𝑘 𝑡>0 ℎ 𝑘 𝑘 𝑘 ℎ 𝑘 𝑘 for some . 𝑟 𝑘∈𝐼𝑟 𝑟 𝑘∈𝐼𝑟 𝑦𝑘≤𝛿 𝑦𝑘≤𝛿 𝜃 𝜃 Proof. Let 𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)0 ⊂𝑤 (Λ,𝑝,𝑠,‖⋅,...,⋅‖). 1 𝐻 −𝑠 𝑝𝑘 Suppose that (45) does not hold. Therefore, there are subin- ≤[𝑀𝑘 (2)] ∑ 𝑘 (𝑀𝑘 [𝑦𝑘]) . ℎ terval 𝐼𝑟(𝑗) of the set of interval 𝐼𝑟 and a number 𝑡0 >0,where 𝑟 𝑘∈𝐼𝑟 𝑦 ≤𝛿 𝑘 󵄩 󵄩 󵄩Λ 𝑘 (𝑥) 󵄩 (40) 𝑡0 = 󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩 ∀𝑘, (46) 󵄩 𝜌 󵄩 𝑦 >𝛿,𝑦 <𝑦/𝛿 < 1 + 𝑦 /𝛿 (𝑀 )󸀠𝑠 For 𝑘 𝑘 𝑘 𝑘 .Since 𝑘 are nonde- such that creasing and convex, it follows that 1 𝑝 = ∑ 𝑘−𝑠(𝑀 (𝑡 )) 𝑘 ≤𝐾<∞, 𝑚=1,2,3,.... 𝑦 1 1 2𝑦 ℎ 𝑘 0 𝑀 (𝑦 )<𝑀 (1 + 𝑘 )< 𝑀 (2) + 𝑀 ( 𝑘 ). 𝑟(𝑗) 𝑘∈𝐼 𝑘 𝑘 𝑘 𝛿 2 𝑘 2 𝑘 𝛿 (41) 𝑟(𝑗) (47) Since M =(𝑀𝑘) satisfies Δ 2-condition, we can write Let us define 𝑥=(𝑥𝑘) as follows: 1 𝑦𝑘 1 𝑦𝑘 𝑦𝑘 𝑀𝑘 (𝑦𝑘)< 𝑇 𝑀𝑘 (2) + 𝑇 𝑀𝑘 (2) =𝑇 𝑀𝑘 (2) . 𝜌𝑡0,𝑘∈𝐼𝑟(𝑗) 2 𝛿 2 𝛿 𝛿 Λ 𝑘 (𝑥) ={ (48) (42) 0, 𝑘 ∉𝑟(𝑗) 𝐼 . 𝜃 Hence, Thus, by (47), 𝑥∈𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖).But𝑥∉ 0 𝑤∞(Λ,𝑝,𝑠,‖⋅,...,⋅‖).Hence,(45)musthold. 1 −𝑠 𝑝 ∑ 𝑘 𝑀 [𝑦 ] 𝑘 Conversely, suppose that (45)holdsandlet𝑥∈ ℎ 𝑘 𝑘 𝑟 𝑘∈𝐼 𝜃 𝑟 𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). Then for each 𝑟, 𝑦𝑘≥𝛿

󵄩 󵄩 𝑝𝑘 𝐻 (43) 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 𝑀 (2) 1 𝑝 ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] ≤𝐾<∞. 𝑘 −𝑠 𝑘 𝑘 󵄩 1 2 𝑛−1󵄩 ≤ max (1, (𝑇 ) ) ∑ 𝑘 [𝑦𝑘] . ℎ𝑟 󵄩 𝜌 󵄩 𝛿 ℎ 𝑘∈𝐼𝑟 𝑟 𝑘∈𝐼𝑟 𝑦𝑘≤𝛿 (49) 8 Abstract and Applied Analysis

𝜃 𝜃 Suppose that 𝑥∉𝑤0(Λ,𝑝,𝑠,‖⋅,...,⋅‖).Thenforsomenumber Then 𝑥=(𝑥𝑘)∈𝑤0(Λ,𝑝,𝑠,‖⋅,...,⋅‖).Butby(57), 𝑥∉ 𝜖>0 𝑘 𝐼 𝜃 ,thereisanumber 0 such that for a subinterval 𝑟(𝑗),of 𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖), which contradicts (ii). Hence, (iii) the set of interval 𝐼𝑟, must holds. ⇒ 𝑥= 󵄩 󵄩 (iii) (i). Let (iii) hold and suppose that 󵄩Λ 𝑘 (𝑥) 󵄩 𝜃 󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩 >𝜖 for 𝑘≥𝑘0. (50) (𝑥𝑘)∈𝑤∞(Λ,𝑝,𝑠,‖⋅,...,⋅‖).Supposethat𝑥=(𝑥𝑘)∉ 󵄩 𝜌 󵄩 𝜃 𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖);then 𝑝 From properties of sequence of Orlicz functions, we obtain 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 sup ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] =∞. 󵄩 󵄩 𝑝𝑘 󵄩 󵄩 󵄩Λ (𝑥) 󵄩 𝑟 ℎ𝑟 󵄩 𝜌 󵄩 󵄩 𝑘 󵄩 𝑝𝑘 𝑘∈𝐼𝑟 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] ≥𝑀𝑘(𝜖) , (51) 󵄩 𝜌 󵄩 (59) which contradicts (45), by using (49). Hence, we get Let 𝑡=‖Λ𝑘 (𝑥)/𝜌,1 𝑧 ,𝑧2,...,𝑧𝑛−1‖ for each 𝑘;thenby(59),

1 𝑝 𝑤𝜃 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤𝜃 (Λ, 𝑝, 𝑠, ‖⋅,...,⋅‖). ∑ 𝑘−𝑠(𝑀 (𝑡)) 𝑘 =∞, ∞ 0 (52) sup 𝑘 (60) 𝑟 ℎ𝑟 𝑘∈𝐼𝑟 This completes the proof. which contradicts (iii). Hence, (i) must hold. Theorem 7. Let M =(𝑀𝑘) be a Musielak-Orlicz function. Then the following statements are equivalent: Theorem 8. Let M =(𝑀𝑘) be a Musielak-Orlicz function. Then the following statements are equivalent: 𝑤𝜃 (Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤𝜃 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) (i) ∞ ∞ ; 𝜃 𝜃 (i) 𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤 (Λ,𝑝,𝑠,‖⋅,...,⋅‖); 𝑤𝜃(Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤𝜃 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖) 0 0 (ii) 0 ∞ ; 𝑤𝜃(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖)⊂𝑤𝜃 (Λ,𝑝,𝑠,‖⋅,...,⋅‖) −𝑠 𝑝𝑘 (ii) 0 ∞ ; (iii) sup𝑟1/ℎ𝑟 ∑𝑘∈𝐼 𝑘 (𝑀𝑘(𝑡)) <∞for all 𝑡>0. 𝑝 𝑟 1/ℎ ∑ 𝑘−𝑠(𝑀 (𝑡)) 𝑘 >0 𝑡>0 (iii) inf𝑟 𝑟 𝑘∈𝐼𝑟 𝑘 for all . ⇒ Proof . (i) (ii). Let (i) hold. To verify (ii), it is enough to ⇒ prove Proof. (i) (ii). It is obvious. (ii) ⇒ (iii). Let (ii) hold. Suppose that (iii) does not hold. 𝜃 𝜃 Then 𝑤0 (Λ, 𝑝, 𝑠, ‖⋅,...,⋅‖)⊂𝑤∞ (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). (53) 1 −𝑠 𝑝𝑘 𝜃 ∑ 𝑘 (𝑀 (𝑡)) =0 𝑡>0, 𝑥=(𝑥)∈𝑤(Λ,𝑝,𝑠,‖⋅,...,⋅‖) 𝜖>0 inf𝑟 ℎ 𝑘 for some (61) Let 𝑘 0 .Thenfor ,there 𝑟 𝑘∈𝐼 exists 𝑟≥0,suchthat 𝑟 󵄩 󵄩 𝑝 and we can find a subinterval 𝐼𝑟(𝑗),ofthesetofinterval𝐼𝑟, 1 󵄩Λ (𝑥) 󵄩 𝑘 ∑ 𝑘−𝑠[󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩] <𝜖. such that 󵄩 1 2 𝑛−1󵄩 (54) ℎ𝑟 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 1 𝑝 1 ∑ 𝑘−𝑠(𝑀 (𝑗)) 𝑘 < , 𝑗=1,2,3,... ℎ 𝑘 𝑗 (62) 𝑟(𝑗) 𝑘∈𝐼 Hence, there exists 𝐾>0such that 𝑟(𝑗) 𝑝 󵄩 󵄩 𝑘 𝑥=(𝑥) 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 Let us define 𝑘 as follows: sup ∑ 𝑘 [󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩] <𝐾. 𝑟 ℎ 󵄩 𝜌 󵄩 (55) 𝑟 𝑘∈𝐼 󵄩 󵄩 𝑟 𝜌𝑗, 𝑘 ∈𝐼𝑟(𝑗) Λ 𝑘 (𝑥) ={ (63) 𝜃 0, 𝑘 ∉𝑟(𝑗) 𝐼 . So, we get 𝑥=(𝑥𝑘)∈𝑤∞(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). ⇒ 𝜃 (ii) (iii). Let (ii) hold. Suppose (iii) does not hold. Then Thus, by (62), 𝑥=(𝑥𝑘)∈𝑤0(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖),but𝑥= for some 𝑡>0 𝜃 (𝑥𝑘)∉𝑤∞(Λ,𝑝,𝑠,‖⋅,...,⋅‖), which contradicts (ii). Hence, 1 −𝑠 𝑝 (iii) must hold. ∑ 𝑘 (𝑀 (𝑡)) 𝑘 =∞, sup 𝑘 (56) ⇒ 𝑥=(𝑥)∈ 𝑟 ℎ𝑟 (iii) (i). Let (iii) hold. Suppose that 𝑘 𝑘∈𝐼𝑟 𝜃 𝑤0(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖).Then 𝐼 𝑝 and therefore we can find a subinterval 𝑟(𝑗),ofthesetof 󵄩 󵄩 𝑘 1 −𝑠 󵄩Λ 𝑘 (𝑥) 󵄩 interval 𝐼𝑟,suchthat ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 󳨀→ 0 ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 𝑟 𝑘∈𝐼 󵄩 󵄩 𝑝 𝑟 (64) 1 1 𝑘 ∑ 𝑘−𝑠(𝑀 ( )) >𝑗, 𝑗=1,2,3,... 𝑘 (57) 𝑟󳨀→∞. ℎ𝑟(𝑗) 𝑗 as 𝑘∈𝐼𝑟(𝑗) 𝜃 Again suppose that 𝑥=(𝑥𝑘)∉𝑤0(Λ,𝑝,𝑠,‖⋅,...,⋅‖);forsome Let us define 𝑥=(𝑥𝑘) as follows: number 𝜖>0and a subinterval 𝐼𝑟(𝑗),ofthesetofinterval𝐼𝑟, 𝜌 we have { ,𝑘∈𝐼 𝑗 𝑟(𝑗) 󵄩Λ (𝑥) 󵄩 Λ 𝑘 (𝑥) = { (58) 󵄩 𝑘 󵄩 󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩 ≥𝜖 ∀𝑘. (65) {0, 𝑘 ∉𝑟(𝑗) 𝐼 . 󵄩 𝜌 󵄩 Abstract and Applied Analysis 9

Then from properties of the Orlicz function, we can write Theorem 10. (i) If 0

Consequently, by (64), we have 𝜃 𝜃 𝑤 (M,Λ,𝑠,‖⋅,...,⋅‖) ⊆𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). (75) 1 𝑝 ∑ 𝑘−𝑠(𝑀 (𝜖)) 𝑘 =0, 𝑟→∞lim 𝑘 (67) 𝜃 ℎ𝑟 Proof. (i) Let 𝑥=(𝑥𝑘)∈𝑤(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖);then 𝑘∈𝐼𝑟 󵄩 󵄩 𝑝 1 󵄩Λ (𝑥) −𝐿 󵄩 𝑘 which contradicts (iii). Hence, (i) must hold. ∑ 𝑘−𝑠[𝑀 (󵄩 𝑘 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] =0. 𝑟→∞lim 𝑘 󵄩 1 2 𝑛−1󵄩 ℎ𝑟 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 Theorem 9. Let 0≤𝑝𝑘 ≤𝑞𝑘 for all 𝑘 and let (𝑞𝑘/𝑝𝑘) be (76) bounded. Then 𝜃 𝜃 Since 00,wehave ℎ 𝑘 ℎ 𝑘 𝑟 𝑘∈𝐼 𝑟 𝑘∈𝐼 󵄩 𝑟 𝑟 1 󵄩Λ (𝑥) −𝐿 ∑ 𝑘−𝑠 [𝑀 (󵄩 𝑘 , 𝑟→∞lim 𝑘 󵄩 𝑘 ℎ𝑟 󵄩 𝜌 Now for each , 𝑘∈𝐼𝑟 (80) 𝜇 1−𝜇 𝑝 1 𝜇 1 1 󵄩 𝑘 ∑ V = ∑ ( V ) ( ) 󵄩 𝑘 𝑘 𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] =0<1. ℎ𝑟 ℎ𝑟 ℎ𝑟 󵄩 𝑘∈𝐼𝑟 𝑘∈𝐼𝑟 󵄩 𝜇 1≤𝑝 ≤ 𝑝 <∞ 1 𝜇 1/𝜇 Since 𝑘 sup 𝑘 ,wehave ≤(∑ [( V𝑘) ] ) 𝑝 ℎ 󵄩 󵄩 𝑘 𝑘∈𝐼 𝑟 1 −𝑠 󵄩Λ 𝑘 (𝑥) −𝐿 󵄩 𝑟 ∑ 𝑘 [𝑀 (󵄩 ,𝑧 ,𝑧 ,...,𝑧 󵄩)] 𝑟→∞lim ℎ 𝑘 󵄩 𝜌 1 2 𝑛−1󵄩 (72) 𝑟 𝑘∈𝐼 󵄩 󵄩 1/(1−𝜇) 1−𝜇 𝑟 1 1−𝜇 ×(∑ [( ) ] ) 󵄩 󵄩 1 −𝑠 󵄩Λ 𝑘 (𝑥) −𝐿 󵄩 ℎ𝑟 ≤ lim ∑ 𝑘 [𝑀𝑘 (󵄩 ,𝑧1,𝑧2,...,𝑧𝑛−1󵄩)] 𝑘∈𝐼𝑟 𝑟→∞ 󵄩 󵄩 ℎ𝑟 󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 𝜇 1 =( ∑ V ) , =0 ℎ 𝑘 𝑟 𝑘∈𝐼 𝑟 <1. and so (81) 𝜇 𝜃 1 𝜇 1 1 Therefore, 𝑥=(𝑥𝑘)∈𝑤(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖),foreach𝜌>0. ∑ V𝑘 ≤ ∑ 𝑡𝑘 +( ∑ V𝑘) . (73) ℎ𝑟 ℎ𝑟 ℎ𝑟 Hence, 𝑘∈𝐼𝑟 𝑘∈𝐼𝑟 𝑘∈𝐼𝑟 𝜃 𝜃 𝜃 𝑤 (M,Λ,𝑠,‖⋅,...,⋅‖) ⊆𝑤 (M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). (82) Hence, 𝑥=(𝑥𝑘)∈𝑤(M,Λ,𝑝,𝑠,‖⋅,...,⋅‖). This completes the proof of the theorem. This completes the proof of the theorem. 10 Abstract and Applied Analysis

Theorem 11. If 0

Research Article Strongly Almost Lacunary 𝐼-Convergent Sequences

Adem KJlJçman1 and Stuti Borgohain2

1 Department of Mathematics and Institute for Mathematical Research, Faculty of Science, Universiti Putra Malaysia, 43400Serdang,Selangor,Malaysia 2 Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India

Correspondence should be addressed to Adem Kılıc¸man; [email protected]

Received 16 July 2013; Revised 12 September 2013; Accepted 2 October 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 A. Kılıc¸man and S. Borgohain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study some new strongly almost lacunary 𝐼-convergent generalized difference sequence spaces defined by an Orlicz function. We give also some inclusion relations related to these sequence spaces.

1. Introduction The following space of strongly almost convergent sequence was introduced by Maddox [3]: The notion of ideal convergence was first introduced by [𝑐̂] ={𝑥∈ℓ : 𝑡 (|𝑥−𝐿𝑒|) Kostyrko et al. [1] as a generalization of statistical convergence ∞ lim𝑚 𝑚,𝑛 which was later studied by many other authors. (2) By a lacunary sequence, we mean an increasing integer exists uniformly in 𝑛 for some 𝐿} , sequence 𝜃=(𝑘𝑟) such that 𝑘0 =0and ℎ𝑟 = 𝑘𝑟 −𝑘𝑟−1 →∞ 𝑒=(1,1,...) as 𝑟→∞. where . ℓ (Δ) Throughout this paper, the intervals determined by 𝜃 will Kızmaz [4] studied the difference sequence spaces ∞ , 𝑐(Δ),and𝑐0(Δ) of crisp sets. The notion is defined as follows: be denoted by 𝐽𝑟 =(𝑘𝑟−1,𝑘𝑟],andtheratio𝑘𝑟/𝑘𝑟−1 will be defined by 𝜙𝑟. 𝑍 (Δ) ={𝑥=(𝑥𝑘):(Δ𝑥𝑘) ∈ 𝑍} , (3) 𝑀:[0,∞)→[0,∞) An Orlicz function is a function , for 𝑍=ℓ∞,𝑐,and𝑐0,whereΔ𝑥 = (Δ𝑥𝑘) = (𝑥𝑘 −𝑥𝑘+1),forall 𝑀(0) = which is continuous, nondecreasing, and convex with 𝑘∈𝑁. 0 𝑀(𝑥) >0 𝑥>0 𝑀(𝑥) →∞ 𝑥→∞ , ,for and ,as . The above spaces are Banach spaces, normed by ℓ 𝑐 𝑐 󵄨 󵄨 󵄨 󵄨 Let ∞, ,and 0 be the Banach space of bounded, 𝑥 = 󵄨𝑥 󵄨 + 󵄨Δ𝑥 󵄨 . 𝑥=(𝑥) ‖ ‖Δ 󵄨 1󵄨 sup 󵄨 𝑘󵄨 convergent, and null sequences 𝑘 ,respectively,with 𝑘 (4) the usual norm ‖𝑥‖ = sup𝑛|𝑥𝑛|. 𝑥∈ℓ Tripathy et al. [5] introduced the generalized difference Asequence ∞ is said to be almost convergent if all of sequence spaces which are defined as, for 𝑚≥1and 𝑛≥1, its Banach limits coincide. Let 𝑐̂denote the space of all almost 𝑍 (Δ𝑛 ) = {𝑥=(𝑥 ) : (Δ𝑛 𝑥 ) ∈𝑍} , 𝑍=ℓ ,𝑐,𝑐 . convergent sequences. 𝑚 𝑘 𝑚 𝑘 for ∞ 0 Lorentz [2] introduced the following sequence space (5) This generalized difference has the following binomial 𝑐=̂ {𝑥∈ℓ : 𝑡 (𝑥) 𝑛} , representation: ∞ lim𝑚 𝑚,𝑛 exists uniformly in (1) 𝑛 𝑛 Δ𝑛 𝑥 = ∑(−1)𝑟 ( ) 𝑥 . 𝑚 𝑘 𝑟 𝑘+𝑟𝑚 (6) where 𝑡𝑚,𝑛(𝑥) = (𝑥𝑛 +𝑥𝑛+1 +⋅⋅⋅+𝑥𝑚+𝑛)/(𝑚 + 1). 𝑟=0 2 Abstract and Applied Analysis

2. Definitions and Preliminaries 3. Main Results Kostyrko et al. [1] introduced the following three definitions. Esi [12] introduced the strongly almost ideal convergent 𝑋 Let 𝑋 be a nonempty set. Then a family of sets 𝐼⊆2 sequence spaces in 2-normed spaces. In this paper we (power sets of 𝑋)issaidtobe𝑖𝑑𝑒𝑎𝑙 if 𝐼 is additive, that is, introduced the strongly almost lacunary ideal convergent 𝐴,𝐵∈𝐼⇒𝐴∪𝐵∈𝐼,andhereditary,thatis,𝐴∈𝐼,𝐵⊆𝐴⇒ sequence spaces using generalized difference operator and 𝐵∈𝐼. Orlicz function. 𝐼 𝑁 𝑀 Asequence(𝑥𝑘) in a normed space (𝑋, ‖ ⋅ ‖) is said to be Let be an admissible ideal of , an Orlicz function, 𝑥 ∈𝑋 𝜀>0 and 𝜃=(𝑘𝑟) a lacunary sequence. Further, let 𝑠=(𝑠𝑘) I-convergent to 0 if for each ,theset 𝑞 be a bounded sequence of positive real numbers and Δ 𝑝 a 󵄩 󵄩 𝐸 (𝜀) ={𝑘∈𝑁:󵄩𝑥𝑘 −𝑥0󵄩 ≥𝜀} belongs to 𝐼. (7) generalized difference operator. For every 𝜀>0and for some 𝜌>0,wehaveintroduced Asequence(𝑥𝑘) in a normed space (𝑋, ‖ ⋅ ‖) is said to be the following sequence spaces: I-bounded if there exists 𝑀>0such that the set {𝑘 ∈ 𝑁: 𝐼 𝑞 𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃) ‖𝑥𝑘‖>𝑀}belongs to 𝐼. 𝑁 Freedman et al. [6] defined the space 𝜃. For any lacunary { 𝜃=(𝑘) sequence 𝑟 , = { (𝑥𝑘): { { −1 󵄨 󵄨 } 󵄨 󵄨 󵄨 𝑞 󵄨 𝑠𝑘 𝑁𝜃 = (𝑥𝑘): lim ℎ ∑ 󵄨𝑥𝑘 −𝐿󵄨 =0, for some 𝐿 . (8) 󵄨 󵄨 { 𝑟→∞ 𝑟 󵄨 󵄨 } { 1 󵄨Δ 𝑝𝑡𝑘𝑚 (𝑥) −𝑙󵄨 } 𝑘∈𝐽 𝑟∈𝑁: ∑ {𝑀 ( )} ≥𝜀 ∈𝐼, { 𝑟 } { ℎ 𝜌 } 𝑟 𝑘∈𝐽 { 𝑟 } The space 𝑁𝜃 is a 𝐵𝐾 space with the norm } 󵄩 󵄩 −1 󵄨 󵄨 𝑚∈𝑁,for some 𝑙∈𝑅 , 󵄩(𝑥𝑘)󵄩𝜃 = sup ℎ𝑟 ∑ 󵄨𝑥𝑘󵄨 . } 𝑟 (9) 𝑘∈𝐽𝑟 } 𝑤̂𝐼 (𝑀, Δ𝑞 ,𝑠,𝜃) The notion of lacunary ideal convergence of real 0 𝑝 sequences introduced by Tripathy et al. in [7, 8]andHazarika { [9, 10] introduced the lacunary ideal convergent sequences of = (𝑥 ): fuzzy real numbers and studied some properties. In [5, 7], the { 𝑘 lacunary ideal convergence is defined as follows. { 𝜃=(𝑘) (𝑥 ) Let 𝑟 be a lacunary sequence. Then a sequence 𝑘 󵄨 𝑞 󵄨 𝑠𝑘 { 1 󵄨Δ 𝑝𝑡 (𝑥)󵄨 } is said to be lacunary 𝐼-convergent if for every 𝜀>0,suchthat 𝑟∈𝑁: ∑ {𝑀 (󵄨 𝑘𝑚 󵄨)} ≥𝜀 ∈𝐼, { ℎ 𝜌 } 𝑟 𝑘∈𝐽 { 𝑟 } { 󵄨 󵄨 } 𝑟∈𝑁:ℎ−1 ∑ 󵄨𝑥 −𝑥󵄨 ≥𝜀 ∈𝐼, { 𝑟 󵄨 𝑘 󵄨 } (10) } 𝑘∈𝐽 { 𝑟 } 𝑚∈𝑁} , } we write 𝐼𝜃 − lim 𝑥𝑘 =𝑥. 𝐼 𝑞 Lindenstrauss and Tzafriri [11]usedtheideaofOrlicz 𝑤̂∞ (𝑀, Δ 𝑝,𝑠,𝜃) function to construct the sequence space: { ∞ |𝑥| = { (𝑥𝑘): ℓ ={(𝑥)∈𝑤:∑𝑀( )<∞, 𝜌>0}. 𝑀 𝑘 𝜌 for some { 𝑘=1 󵄨 󵄨 𝑠 (11) 󵄨 𝑞 󵄨 𝑘 { 1 󵄨Δ 𝑝𝑡 (𝑥)󵄨 } 𝑟∈𝑁:∃𝐾>0 . . ∑{𝑀 (󵄨 𝑘𝑚 󵄨)} ≥𝐾 { s t ℎ 𝜌 } ℓ 𝑟 𝑘∈𝐽 The space 𝑀 with the norm { 𝑟 } 󵄨 󵄨 ∞ 󵄨𝑥 󵄨 } ‖𝑥‖ = {𝜌>0:∑𝑀 (󵄨 𝑘󵄨) ≤1} inf 𝜌 (12) ∈𝐼,𝑚∈𝑁} . 𝑘=1 } (13) becomes a Banach space which is called an Orlicz sequence space. Particular Cases. Consider the following. In this paper, we defined some new generalized difference 𝜃=(2𝑟) 𝑤̂𝐼(𝑀, Δ𝑞 ,𝑠,𝜃) = 𝑤̂𝐼(𝑀, Δ𝑞 lacunary 𝐼-convergent sequence spaces defined by Orlicz (1) If ,wehave 𝑝 𝑝, 𝑠) 𝑤̂𝐼(𝑀, Δ𝑞 ,𝑠,𝜃)=𝑤̂𝐼(𝑀, Δ𝑞 ,𝑠) 𝑤̂𝐼 (𝑀, Δ𝑞 ,𝑠 function. We also introduce and examine some new sequence , 0 𝑝 0 𝑝 ,and ∞ 𝑝 , 𝐼 𝑞 spaces and study their different properties. 𝜃) = 𝑤̂∞(𝑀, Δ 𝑝,𝑠). Abstract and Applied Analysis 3

𝐼 𝑞 𝐼 𝑞 𝐼 𝑞 (2) If 𝑀(𝑥) =𝑥,then𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃) = 𝑤̂ (Δ 𝑝,𝑠,𝜃), Then, for 𝑥∈𝑤̂ (𝑀, Δ 𝑝,𝑠),wehave 𝐼 𝑞 𝐼 𝑞 𝐼 𝑞 𝑤̂0(𝑀, Δ 𝑝,𝑠,𝜃) = 𝑤̂0(Δ 𝑝,𝑠,𝜃),and𝑤̂∞(𝑀, Δ 𝑝,𝑠, 𝑞 𝐼 𝑠 𝜃) = 𝑤̂∞(Δ 𝑝,𝑠,𝜃). 𝑞 𝑘 { |Δ 𝑝𝑡𝑘𝑚(𝑥) − 𝑙| } 𝑟 𝑚∈𝑁: ∑ (𝑀 ( )) ≥𝜀 ∈𝐼. (15) (3) If 𝑠𝑘 =1for all 𝑘∈𝑁, 𝑀(𝑥) =𝑥,and𝜃=(2),then { 𝜌 } 𝑞 𝑞 𝑞 𝑞 𝑘∈𝐽 𝐼 𝐼 𝐼 𝐼 { 𝑟 } 𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃)=𝑤̂ (Δ 𝑝), 𝑤̂0(𝑀, Δ 𝑝,𝑠,𝜃)=𝑤̂ (Δ 𝑝), 𝐼 𝑞 𝐼 𝑞 and 𝑤̂∞(𝑀, Δ 𝑝,𝑠,𝜃)=𝑤̂ (Δ 𝑝). Let Theorem 1. Let the sequence (𝑠𝑘) be bounded; then 𝐼 𝑞 𝐼 𝑞 𝐼 𝑞 󵄨 󵄨 𝑠 𝑤̂ (𝑀, Δ ,𝑠,𝜃)⊂𝑤̂ (𝑀, Δ ,𝑠,𝜃)⊂𝑤̂ (𝑀, Δ ,𝑠,𝜃) 󵄨 𝑞 󵄨 𝑘 0 𝑝 𝑝 ∞ 𝑝 . 1 󵄨Δ 𝑝𝑡 (𝑥) −𝑙󵄨 𝐴 = ∑ (𝑀 (󵄨 𝑘𝑚 󵄨)) 𝑟 ℎ 𝜌 𝐼 𝑞 𝑟 𝑘∈𝐽 Proof. Let 𝑥∈𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃).Then,forsome𝜌>0,wehave 𝑟

𝑘 𝑞 𝑠𝑘 󵄨 𝑞 󵄨 𝑠𝑘 𝑟 󵄨 󵄨 1 |Δ 𝑝𝑡𝑘𝑚(𝑥) − 𝑙| 1 󵄨Δ 𝑝𝑡 (𝑥)󵄨 ∑ (𝑀 (󵄨 𝑘𝑚 󵄨)) = ∑ (𝑀 ( )) ℎ𝑟 𝜌 ℎ𝑟 2𝜌 𝑘=1 𝑘∈𝐽𝑟 𝑠 𝑘 󵄨 𝑞 󵄨 𝑘 󵄨 󵄨 𝑠 𝑟−1 󵄨Δ 𝑡 (𝑥) −𝑙󵄨 󵄨 𝑞 󵄨 𝑘 1 󵄨 𝑝 𝑘𝑚 󵄨 𝐷 1 󵄨Δ 𝑝𝑡𝑘𝑚 (𝑥) −𝑙󵄨 − ∑ (𝑀 ( )) ≤ ∑ (𝑀 ( )) ℎ 𝜌 (16) 𝑝 𝑟 𝑘=1 ℎ𝑟 2 𝑘 𝜌 𝑘∈𝐽𝑟 𝑠 𝑘 󵄨 𝑞 󵄨 𝑘 𝑠 𝑟 󵄨Δ 𝑡 (𝑥) −𝑙󵄨 𝐷 1 |𝑙| 𝑘 𝑘𝑟 1 󵄨 𝑝 𝑘𝑚 󵄨 + ∑ (𝑀 ( )) = ( ∑ (𝑀 ( )) ) 𝑝 (14) ℎ 𝑘 𝜌 ℎ𝑟 2 𝑘 𝜌 𝑟 𝑟 𝑘=1 𝑘∈𝐽𝑟

󵄨 𝑞 󵄨 𝑠𝑘 󵄨 𝑞 󵄨 𝑠𝑘 𝑘𝑟−1 󵄨 󵄨 󵄨 󵄨 𝑘 1 󵄨Δ 𝑝𝑡𝑘𝑚 (𝑥) −𝑙󵄨 𝐷 󵄨Δ 𝑝𝑡 (𝑥) −𝑙󵄨 𝑟−1 󵄨 󵄨 ≤ ∑ (𝑀 (󵄨 𝑘𝑚 󵄨)) − ( ∑ (𝑀 ( )) ). ℎ𝑟 𝑘𝑟−1 𝜌 ℎ𝑟 𝜌 𝑘=1 𝑘∈𝐽𝑟 |𝑙| 𝐻 Since ℎ𝑟 =𝑘𝑟 −𝑘𝑟−1,wehave𝑘𝑟/ℎ𝑟 ≤(1+𝛿)/𝛿and +𝐷max {1, sup (𝑀 ( )) }, 𝜌 𝑘𝑟−1/ℎ𝑟 ≤1/𝛿. So, for 𝜀>0and for some 𝜌>0, 𝐻−1 where sup𝑘𝑠𝑘 =𝐻and 𝐷=max(1, 2 ). 𝐼 𝑞 Hence, 𝑥∈𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃). 𝑘 𝑞 𝑠𝑘 ∞ 𝑘 1 𝑟 |Δ 𝑡 (𝑥) − 𝑙| 𝑤̂𝐼(𝑀, Δ𝑞 ,𝑠,𝜃) ⊂ 𝑤̂𝐼(𝑀, Δ𝑞 ,𝑠,𝜃) {𝑟 ∈ 𝑁: 𝑟 ( ∑(𝑀 ( 𝑝 𝑘𝑚 )) )≥𝜀}∈𝐼, The inclusion 0 𝑝 𝑝 is obvi- ℎ 𝑘 𝜌 ous. 𝑟 𝑟 𝑘=1 𝑚∈𝑁, 𝑙∈𝑅, Theorem 2. Let the sequence (𝑠𝑘) be bounded; then for some 𝐼 𝑞 𝐼 𝑞 𝐼 𝑞 𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃), 𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃),and𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃) 0 ∞ { are closed under the operations of addition and scalar {𝑟∈𝑁: multiplication. { 𝑠 Theorem 3. Let 𝑀1,𝑀2 be Orlicz functions; then we have 𝑘 󵄨 𝑞 󵄨 𝑘 𝑟−1 󵄨Δ 𝑡 (𝑥) −𝑙󵄨 } 𝑘𝑟−1 1 󵄨 𝑝 𝑘𝑚 󵄨 𝐼 𝑞 𝐼 𝑞 𝐼 𝑞 ( ∑(𝑀 ( )) )≥𝜀} ∈𝐼, (1) 𝑤̂0(𝑀1,Δ𝑝,𝑠,𝜃)∩𝑤̂0(𝑀2,Δ𝑝,𝑠,𝜃)⊂𝑤̂0(𝑀1+𝑀2,Δ𝑝, ℎ 𝑘 𝜌 𝑟 𝑟−1 𝑘=1 𝑠, 𝜃), } 𝐼 𝑞 𝐼 𝑞 𝐼 𝑞 (2) 𝑤̂ (𝑀1,Δ𝑝,𝑠,𝜃)∩𝑤̂ (𝑀2,Δ𝑝,𝑠,𝜃)⊂𝑤̂ (𝑀1+𝑀2,Δ𝑝, 𝑚∈𝑁,for some 𝑙∈𝑅. 𝑠, 𝜃), (17) 𝐼 𝑞 𝐼 𝑞 𝐼 (3) 𝑤̂∞(𝑀1,Δ𝑝,𝑠,𝜃)∩𝑤̂∞(𝑀2,Δ𝑝,𝑠,𝜃)⊂𝑤̂∞(𝑀1 +𝑀2, 𝑞 Δ ,𝑠,𝜃) 𝐼 𝑞 𝐼 𝑞 𝑝 . Hence, 𝑤̂ (𝑀, Δ 𝑝,𝑠)⊂𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃). Next, suppose that lim sup𝑟𝑞𝑟 <∞. Then, there exists 𝛽> Theorem 4. Let 0<𝑠𝑘 ≤𝑢𝑘 for all 𝑘∈𝑁,andlet(𝑢𝑘/𝑠𝑘) be 0 𝑞 <𝛽 𝑟≥1 𝐼 𝑞 𝐼 𝑞 ,suchthat, 𝑟 for all . bounded; then we have 𝑤̂ (𝑀1,Δ𝑝,𝑢,𝜃)⊆𝑤̂ (𝑀1,Δ𝑝,𝑠,𝜃). 𝐼 𝑞 Let 𝑥∈𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃)and 𝜀>0. There exists 𝑅>0such that for every 𝑗≥𝑅, Theorem 5. Let 𝜃=(𝑘𝑟) be a lacunary sequence with 1< lim inf𝑟𝑢𝑟 ≤ sup𝑟𝑢𝑟 <∞. Then, for any Orlicz function 𝑀, 𝐼 𝑞 𝐼 𝑞 𝑞 𝑝𝑘 𝑤̂ (𝑀, Δ 𝑝,𝑠)=𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃). { 1 |Δ 𝑡 (𝑥) − 𝑙| } 𝐴 = 𝑟∈𝑁: ∑ (𝑀 ( 𝑝 𝑘𝑚 )) ≥𝜀 ∈𝐼. 𝑗 { ℎ 𝜌 } 𝑟 𝑘∈𝐽 Proof. Suppose lim inf𝑟𝑢𝑟 >1then there exists 𝛿>0such { 𝑗 } that 𝑢𝑟 =𝑘𝑟/𝑘𝑟−1 ≥1+𝛿for all 𝑟≥1. (18) 4 Abstract and Applied Analysis

Let 𝐾>0such that 𝐴𝑗 ≤𝐾for all 𝑗 = 1, 2, ..Nowlet . 𝑛 Let 𝜌=max(2𝜌1,2𝜌2).Thenwehave be any integer with 𝑘𝑟−1 <𝑛≤𝑘𝑟,where𝑟>𝑅.Then, 󵄨 󵄨 𝑠𝑘 󵄨 󵄨 𝑠 1 󵄨𝑥0 −𝑥1󵄨 𝑛 󵄨Δ𝑞 𝑡 (𝑥) −𝑙󵄨 𝑘 ∑ (𝑀 ( )) 1 󵄨 𝑝 𝑘𝑚 󵄨 ℎ 𝜌 ∑ (𝑀 ( )) 𝑟 𝑘∈𝐽 𝑛 𝜌 𝑟 𝑘=1 󵄨 𝑞 󵄨 𝑠𝑘 󵄨 𝑞 󵄨 𝑝𝑘 󵄨Δ 𝑡 (𝑥) −𝑥 󵄨 𝑘𝑟 󵄨 󵄨 𝐷 󵄨 𝑝 𝑘𝑚 0󵄨 1 󵄨Δ 𝑝𝑡𝑘𝑚 (𝑥) −𝑙󵄨 ≤ ∑ (𝑀 ( )) ≤ ∑ (𝑀 (󵄨 󵄨)) (22) ℎ𝑟 𝜌1 𝑘 𝜌 𝑘∈𝐽𝑟 𝑟−1 𝑘=1

𝑝 󵄨 𝑞 󵄨 𝑠𝑘 󵄨 𝑞 󵄨 𝑘 󵄨 󵄨 { 󵄨Δ 𝑡 (𝑥) −𝑙󵄨 𝐷 󵄨Δ 𝑝𝑡𝑘𝑚 (𝑥) −𝑥1󵄨 1 󵄨 𝑝 𝑘𝑚 󵄨 + ∑ (𝑀 (󵄨 󵄨)) , = { ∑ (𝑀 ( )) 𝑘 𝜌 ℎ𝑟 𝜌2 𝑟−1 𝑘∈𝐽 𝑘∈𝐽𝑟 { 1 󵄨 󵄨 𝑝 𝐻−1 󵄨 𝑞 󵄨 𝑘 𝑠 =𝐻 𝐷= (1, 2 ) 󵄨Δ 𝑝𝑡 (𝑥) −𝑙󵄨 where sup𝑘 𝑘 and max . + ∑ (𝑀 (󵄨 𝑘𝑚 󵄨)) 𝜌 Thus, from (21), we get 𝑘∈𝐽2 { 𝑠𝑘 } 󵄨 𝑞 󵄨 𝑝𝑘 1 |𝑥0 −𝑥1| 󵄨Δ 𝑡 (𝑥) −𝑙󵄨 } (19) 𝑟∈𝑁: ∑ (𝑀 ( )) ≥𝜀 ∈𝐼. 󵄨 𝑝 𝑘𝑚 󵄨 { ℎ 𝜌 } (23) +⋅⋅⋅+ ∑ (𝑀 ( )) 𝑟 𝑘∈𝐽 1 𝜌 } { 𝑟 } 𝑘∈𝐽 𝑟 } 𝑠𝑘 𝑠 Further, 𝑀(|𝑥0 −𝑥1|/𝜌) → 𝑀(|𝑥0 −𝑥1|/𝜌) as 𝑘→ 𝑘1 𝑘2 −𝑘1 𝑘𝑅 −𝑘𝑅−1 = 𝐴1 + 𝐴2 +⋅⋅⋅+ 𝐴𝑅 ∞,and,therefore, 𝑘𝑟−1 𝑘𝑟−1 𝑘𝑟−1 𝑠𝑘 𝑠 |𝑥0 −𝑥1| |𝑥0 −𝑥1| 𝑘𝑟 −𝑘𝑟−1 𝑀( ) 󳨀→ 𝑀 ( ) =0. (24) +⋅⋅⋅+ 𝐴𝑟 𝜌 𝜌 𝑘𝑟−1

Hence, 𝑥0 =𝑥1. 𝑘𝑅 𝑘𝑟 −𝑘𝑅 =(sup𝐴𝑗) + sup (𝐴𝑗) 𝑗≥1 𝑘𝑟−1 𝑗≥𝑅 𝑘𝑟−1 4. Conclusion 𝑘 <𝐾 𝑅 +𝜀𝛽. 𝐼 𝑘 The concept of lacunary -convergence has been studied by 𝑟−1 various mathematicians. In this paper, we have introduced Since 𝑘𝑟−1 →∞as 𝑟→∞,itfollowsthat some fairly wide classes of strongly almost lacunary 𝐼- 󵄨 󵄨 𝑠 convergent sequences of real numbers using Orlicz function 𝑛 󵄨 𝑞 󵄨 𝑘 { 1 󵄨Δ 𝑝𝑡𝑘𝑚 (𝑥) −𝑙󵄨 } with the generalized difference operator. Giving particular {𝑚∈𝑁: ∑(𝑀 ( )) ≥𝜀} ∈𝐼. 𝑛 𝜌 values to the sequence 𝜃=(𝑘𝑟) and 𝑀,weobtainsomenew 𝑘=1 { } sequence spaces which are the special cases of the sequence (20) spaces we have defined. There are lots more to be investigated 𝐼 𝑞 𝐼 𝑞 in the future. Hence, 𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃)⊂𝑤̂ (𝑀, Δ 𝑝,𝑠). Theorem 6. 𝑠 >0 𝑥 If lim 𝑘 and is strongly almost lacunary Acknowledgments convergent to 𝑥0,withrespecttotheOrliczfunction𝑀,thatis, 𝐼 𝑞 (𝑥𝑘)→𝑙(𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃)),then𝑥0 is unique. First of all, the authors sincerely thank the referees for the valuable comments. The first author gratefully acknowledges Proof. Let lim 𝑠𝑘 =𝑠>0and suppose that 𝑥𝑘 → that part of this research was partially supported by the 𝐼 𝑞 𝐼 𝑞 𝑥1(𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃)), 𝑥𝑘 →𝑥0(𝑤̂ (𝑀, Δ 𝑝,𝑠,𝜃)). University Putra Malaysia under the ERGS Grant Scheme Then there exist 𝜌1 and 𝜌2 such that having Project no. 5527068. The work of the second author was carried under the Postdoctoral Fellow under National 𝑞 𝑠𝑘 { 1 |Δ 𝑝𝑡𝑘𝑚(𝑥) −0 𝑥 | } Board of Higher Mathematics, DAE (Government of India), 𝑟∈𝑁: ∑ (𝑀 ( )) ≥𝜀 ∈𝐼, { ℎ 𝜌 } Project no. NBHM/PDF.50/2011/64. 𝑟 𝑘∈𝐽 1 { 𝑟 } 𝑚∈𝑁, References

𝑞 𝑠𝑘 [1] P. Kostyrko, T. Salˇ at,´ and W. Wilczynski,´ “𝐼-convergence,” Real { 1 |Δ 𝑝𝑡 (𝑥) − 𝑥 | } 𝑟∈𝑁: ∑ (𝑀 ( 𝑘𝑚 1 )) ≥𝜀 ∈𝐼, Analysis Exchange,vol.26,no.2,pp.669–685,2001. { ℎ 𝜌 } 𝑟 𝑘∈𝐽 1 { 𝑟 } [2] G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica,vol.80,pp.167–190,1948. 𝑚∈𝑁. [3] I. J. Maddox, “Spaces of strongly summable sequences,” The (21) Quarterly Journal of Mathematics,vol.18,pp.345–355,1967. Abstract and Applied Analysis 5

[4] H. Kızmaz, “On certain sequence spaces,” Canadian Mathemat- ical Bulletin, vol. 24, no. 2, pp. 169–176, 1981. [5]B.C.Tripathy,A.Esi,andB.Tripathy,“Onanewtype of generalized difference Cesaro` sequence spaces,” Soochow Journal of Mathematics,vol.31,no.3,pp.333–340,2005. [6] A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesaro-` type summability spaces,” Proceedings of the London Mathemat- ical Society,vol.37,no.3,pp.508–520,1978. [7] B. C. Tripathy, B. Hazarika, and B. Choudhary, “Lacunary 𝐼-convergent sequences,” in Proceedings of the Real Analysis Exchange Summer Symposium,pp.56–57,2009. [8] B. C. Tripathy, B. Hazarika, and B. Choudhary, “Lacunary 𝐼- convergent sequences,” Kyungpook Mathematical Journal,vol. 52,no.4,pp.473–482,2012. [9] B. Hazarika, “Lacunary 𝐼-convergent sequence of fuzzy real numbers,” Pacific Journal of Science and Technology,vol.10,no. 2, pp. 203–206, 2009. [10] B. Hazarika, “Fuzzy real valued lacunary 𝐼-convergent sequences,” Applied Mathematics Letters,vol.25,no.3,pp. 466–470, 2012. [11] J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,” Israel Journal of Mathematics,vol.10,pp.379–390,1971. [12] A. Esi, “Strongly almost summable sequence spaces in 2- normed spaces defined by ideal convergence and an Orlicz function,” Studia. Universitatis Babes¸-Bolyai Mathematica,vol. 57,no.1,pp.75–82,2012. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 147409, 9 pages http://dx.doi.org/10.1155/2013/147409

Research Article The Existence and Attractivity of Solutions of an Urysohn Integral Equation on an Unbounded Interval

Mohamed Abdalla Darwish,1 Józef BanaV,2 and Ebraheem O. Alzahrani3

1 Mathematics Department, Science Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Rzeszow´ University of Technology, al. Powstanc´ ow´ Warszawy 8, 35-959 Rzeszow,´ Poland 3 Mathematics Depertment, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Correspondence should be addressed to Jozef´ Bana´s; [email protected]

Received 21 August 2013; Accepted 4 September 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 Mohamed Abdalla Darwish et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove a result on the existence and uniform attractivity of solutions of an Urysohn integral equation. Our considerations are conducted in the Banach space consisting of real functions which are bounded and continuous on the nonnegative real half axis. The main tool used in investigations is the technique associated with the measures of noncompactness and a fixed point theorem of Darbo type. An example showing the utility of the obtained results is also included.

1. Introduction existence results, but also allows us to look for solutions of mentioned equations having some desired properties such as The theory of nonlinear functional integral equations creates monotonicity, attractivity, and asymptotic stability (cf. [16– an important branch of the modern nonlinear analysis. The 19], for instance). largepartofthattheorydescribesalotofclassicalnonlinear In the paper, we will use the above described approach integral equations such as nonlinear Volterra integral equa- associated with the technique of measures of noncompact- tions, Hammerstein integral equations, and Urysohn integral ness in order to obtain a result on the existence of solu- equations with solutions defined on a bounded interval (cf. tions of a quadratic Urysohn integral equation. Applying [1–4]). the mentioned technique in conjunction with a fixed point Nevertheless,moreimportantandsimultaneously,amore theoremofDarbotype,weshowthattheequationinquestion difficult part of that theory is connected with the study of has solutions defined, continuous, and bounded on the solutions of the mentioned integral equations defined on nonnegative real half axis R+ which are uniformly attractive an unbounded domain. Obviously, there are some known (asymptotic stable) on R+. results concerning the existence of solutions of those integral The results obtained in this paper generalize several equations in such a setting but, in general, they are mostly results obtained earlier in numerous papers treating non- obtained under rather restrictive assumptions [2, 4–8]. linear functional integral equations, which were quoted On the other hand, the use of some tools of nonlinear above. Particularly, we generalize the results concerning the analysis enables us to obtain several valuable results under Urysohn or Hammerstein integral equations obtained in the less restrictive assumptions (cf. [1, 9–15]). It turns out that papers [9, 10, 20]. the technique of measures of noncompactness creates a very convenienttoolforthestudyofthesolvabilityofnonlinear functional integral equations of various types. It is caused 2. Notation, Definitions, and Auxiliary Facts bythefactthattheapproachtothestudyofsolutionsof those equations with the use of the technique of measures In this section, we establish some notations, and we collect of noncompactness gives not only the possibility to obtain auxiliary facts which will be used in the sequel. 2 Abstract and Applied Analysis

𝑇 By the symbol R wedenotethesetofrealnumbers,whileR+ by 𝜔 (𝑥, 𝜀) the modulus of continuity of the function 𝑥 on the stands for the half axis [0, ∞). Further, assume that (𝐸, ‖⋅‖) is interval [0, 𝑇];thatis, 𝜃 𝑥∈𝐸 a given real Banach space with the zero element .For 𝑇 and for a fixed 𝑟>0, denote by 𝐵(𝑥, 𝑟) theclosedballcentered 𝜔 (𝑥,) 𝜀 = sup {|𝑥 (𝑡) −𝑥(𝑠)| :𝑡,𝑠∈[0, 𝑇] , |𝑡−𝑠| ≤𝜀} . at 𝑥 and with radius 𝑟.Wewrite𝐵𝑟 in order to denote the ball (2) 𝐵(𝜃,. 𝑟) Moreover, if 𝑋 and 𝑌 are nonempty subsets of 𝐸 and 𝜆∈ Next, we put R, then we denote by 𝑋+𝑌, 𝜆𝑋, the usual algebraic operations 𝑇 𝑇 𝜔 (𝑋, 𝜀) = sup {𝜔 (𝑥,) 𝜀 :𝑥∈𝑋}, on sets. If 𝑋 is a subset of 𝐸 then the symbols 𝑋 and Conv 𝑋 denote the closure and closed convex hull of 𝑋,respectively. 𝜔𝑇 (𝑋) = 𝜔𝑇 (𝑋, 𝜀) , 0 lim (3) Apart from this, we denote by M𝐸 the family of all nonempty 𝜀→0 and bounded subsets of 𝐸 and by N𝐸 its subfamily consisting 𝑇 𝜔0 (𝑋) = lim 𝜔 (𝑋) . of all relatively compact sets. 𝑇→∞ 0 In what follows, we will accept the following definition of 𝑡∈R the concept of a measure of noncompactness [21]. Further, for a fixed number + let us put 𝑋 (𝑡) = {𝑥 (𝑡) :𝑥∈𝑋} , Definition 1. Amapping𝜇:M𝐸 → R+ is said to be a (4) measure of noncompactness in 𝐸 if it satisfies the following 󵄨 󵄨 diam 𝑋 (𝑡) = sup {󵄨𝑥 (𝑡) −𝑦(𝑡)󵄨 :𝑥,𝑦∈𝑋}. conditions. ∘ Finally, let us consider the function 𝜇 defined on the family (1 )Thefamilyker𝜇={𝑋∈M𝐸 :𝜇(𝑋)=0}is nonempty M𝐵𝐶(R ) by the formula and ker 𝜇⊂N𝐸. + ∘ 𝑋⊂𝑌⇒𝜇(𝑋)≤𝜇(𝑌) 𝜇 (𝑋) =𝜔 (𝑋) + 𝑋 (𝑡) . (2 ) . 0 lim sup diam (5) ∘ 𝑡→∞ (3 ) 𝜇(𝑋) = 𝜇(Conv 𝑋) = 𝜇(𝑋). ∘ 𝜇 (4 ) 𝜇(𝜆𝑋 + (1 − 𝜆)𝑌) ≤ 𝜆𝜇(𝑋) + (1−𝜆)𝜇(𝑌) for 𝜆∈[0,1]. It can be shown [21]thatthefunction is a measure of noncompactness in the space 𝐵𝐶(R+).Moreover,the ∘ (𝑋 ) M (5 )If 𝑛 is a sequence of closed sets from 𝐸 such that kernel ker 𝜇 of this measure consists of all nonempty and 𝑋𝑛+1 ⊂𝑋𝑛 for 𝑛 = 1, 2, . . and if lim𝑛→∞𝜇(𝑋𝑛)=0, ∞ bounded subsets 𝑋 of 𝐵𝐶(R+) such that functions from 𝑋 𝑋∞ = ⋂ 𝑋𝑛 then the intersection set 𝑛=1 is nonempty. are locally equicontinuous on R+, and the thickness of the ∘ 𝑋 The family ker 𝜇 appearing in 1 is called the kernel of the bundle formed by functions from tendstozeroatinfinity. measure of noncompactness 𝜇. This property in combination with Remark 3 permits us to ∘ characterize solutions of the integral equation considered in Observe that the set 𝑋∞ from the axiom 5 is a member of the sequel. the family ker 𝜇. Indeed, since 𝜇(𝑋∞)≤𝜇(𝑋𝑛) for any natural For further purposes, we introduce now the concept of number 𝑛,weinferthat𝜇(𝑋∞)=0.Consequently,𝑋∞ ∈ ker 𝜇. This simple observation will be essential in our further attractivity (stability) of solutions of operator equations in 𝐵𝐶(R ) Ω investigations. the space + . To this end, assume that is a nonempty 𝐵𝐶(R ) 𝐹 Now, we formulate a fixed point theorem of Darbo type subset of the space + .Moreover,let be an operator Ω 𝐵𝐶(R ) which will be used further on [21]. defined on with values in + . Let us consider the operator equation of the form Theorem 2. Let Ω be a nonempty, bounded, closed, and convex 𝑥 (𝑡) = (𝐹𝑥)(𝑡) ,𝑡∈R+. (6) subset of the Banach space 𝐸,andlet𝐹:Ω →be Ω a 𝑘∈ continuous mapping. Assume that there exists a constant Definition 4. We say that solutions of (6)areattractive (or [0, 1) 𝜇(𝐹𝑋) ≤ 𝑘𝜇(𝑋) 𝑋 such that for any nonempty subset of locally attractive) if there exists a ball 𝐵(𝑥0,𝑟) in the space Ω 𝐹 Ω .Then, has a fixed point in the set . 𝐵𝐶(R+) such that 𝐵(𝑥0,𝑟)∩Ω=0̸ , and for arbitrary solutions 𝑥=𝑥(𝑡) 𝑦=𝑦(𝑡) 𝐵(𝑥 ,𝑟)∩Ω 𝐹 , of (6)belongingtotheset 0 we Remark 3. Denote by Fix the set of all fixed points of the have that operator 𝐹 belonging to Ω.Itcanbeeasilyseen[21]thatthe 𝐹 𝜇 (𝑥 (𝑡) −𝑦(𝑡))=0. set Fix belongs to the family ker . 𝑡→∞lim (7) In what follows, we will work in the Banach space 𝐵𝐶(R+) consisting of all real functions 𝑥=𝑥(𝑡)defined, continuous, Inthecasewhenlimit(7)isuniformwithrespecttotheset and bounded on R+.Thisspacewillbeendowedwiththe 𝐵(𝑥0,𝑟)∩Ω,thatis,whenforeach𝜀>0there exists 𝑇>0 standard supremum norm such that 󵄨 󵄨 ‖𝑥‖ = sup {|𝑥 (𝑡)| :𝑡∈R+}. (1) 󵄨𝑥 (𝑡) −𝑦(𝑡)󵄨 ≤𝜀 (8)

Now,werecalltheconstructionofameasureofnoncompact- for all 𝑥, 𝑦 ∈0 𝐵(𝑥 ,𝑟)∩Ωbeing solutions of (6)andforany ness in the space 𝐵𝐶(R+) which was introduced in [21]. To 𝑡≥𝑇,wewillsaythatsolutionsof(6)areuniformly attractive this end, fix a nonempty and bounded subset 𝑋 of the space (or asymptotically stable). 𝐵𝐶(R+) andpositivenumbers𝜀>0, 𝑇>0.For𝑥∈𝑋, denote Notice that the previous definition comes from16 [ , 19]. Abstract and Applied Analysis 3

3. Main Result Remark 5. It is worthwhile mentioning that in the theory of improper Riemann integral with a parameter there has We will consider the existence and asymptotic behaviour of been considered the concept of the uniform convergence of solutions of the quadratic Urysohn integral equation having the improper integral with respect to that parameter (cf. [22]). the form In order to recall this concept, suppose that there is a given ∞ function 𝑧(𝑡, 𝑠) =𝑧: R+ × R+ → Rsuch that the improper 𝑥 (𝑡) =𝑎(𝑡) +𝑓(𝑡, 𝑥 (𝑡)) ∫ 𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) 𝑑𝑠 (9) integral 0 ∞ for 𝑡∈R+. ∫ 𝑧 (𝑡, 𝑠) 𝑑𝑠 (15) In our study, we will impose the following assumptions. 0

does exist for every fixed 𝑡∈R+. (i) 𝑎∈𝐵𝐶(R+). We say that the integral (15)isuniformly convergent with (ii) 𝑓:R+ × R → R is a continuous function, and respect to 𝑡∈R+ if the function 𝑡→𝑓(𝑡,0)isamemberofthespace 𝑇 ∞ 𝐵𝐶(R+). lim ∫ 𝑧 (𝑡, 𝑠) 𝑑𝑠 = ∫ 𝑧 (𝑡, 𝑠) 𝑑𝑠 (16) (iii) The function 𝑓=𝑓(𝑡,𝑥)satisfies the Lipschitz 𝑇→∞ 0 0 condition with respect to the second variable; that is, uniformly with respect to 𝑡∈R+. there exists a constant 𝑘>0such that Equivalently (cf. [10]), the integral (15)isuniformly 󵄨 󵄨 󵄨 󵄨 convergent with respect to 𝑡∈R+ if 󵄨𝑓 (𝑡, 𝑥) −𝑓(𝑡, 𝑦)󵄨 ≤𝑘󵄨𝑥−𝑦󵄨 (10) ∞ for 𝑥, 𝑦 ∈ R+ and 𝑡∈R+. lim {sup ∫ 𝑧 (𝑡, 𝑠) 𝑑𝑠} = 0. 𝑇→∞ (17) 𝑡∈R+ 𝑇 (iv) 𝑢:R+ × R+ × R → R is a continuous function, and 𝑔:R × R → there exists a continuous function + + Let us observe that if integrals appearing in assumption R ℎ: + and a continuous, nondecreasing function (v), that is, the integrals R+ → R+ with lim𝜀→0ℎ(𝜀) =,suchthat 0 ∞ ∞ 󵄨 󵄨 󵄨 󵄨 󵄨𝑢 (𝑡, 𝑠, 𝑥) −𝑢(𝑡, 𝑠, 𝑦)󵄨 ≤𝑔(𝑡, 𝑠) ℎ (󵄨𝑥−𝑦󵄨) ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠, ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 (18) 󵄨 󵄨 󵄨 󵄨 (11) 0 0 𝑡, 𝑠 ∈ R 𝑥, 𝑦 ∈ R for + and . are uniformly convergent with respect to 𝑡∈R+,then theinequalitiesfromassumption(vi)aresatisfied(cf.[10]). (v) For each 𝑡∈R+,thefunctions𝑠→𝑔(𝑡,𝑠)and 𝑠→ Indeed, this conclusion follows easily from the inequalities |𝑢(𝑡, 𝑠, 0)| are integrable R+ and ∞ ∞ ∞ ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 = 0. sup ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 ≤ sup ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠, lim (12) 𝑡∈[0,𝑇] 𝑇 𝑡∈R 𝑇 𝑡→∞ 0 + ∞ ∞ (19) ∞ 𝑡→∫|𝑢(𝑡, 𝑠, 0)|𝑑𝑠 ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 ≤ ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠, Moreover, the function 0 is sup sup 𝑡∈[0,𝑇] 𝑇 𝑡∈R+ 𝑇 bounded on R+. (vi) The following equalities hold: which are valid for any 𝑇>0. It may be also shown that the converse implications are, ∞ in general, not valid [10]. lim {sup {∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 : 𝑡∈ [0, 𝑇]}} = 0, 𝑇→∞ 𝑇 (13) Remark 6. It can be also shown [10] that the requirements ∞ 𝑔(𝑡, 𝑠) lim {sup {∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 : 𝑡∈ [0, 𝑇]}} = 0. concerning the function imposedinassumption(v) 𝑇→∞ 𝑇 are independent, that is, there exist functions 𝑔𝑖 : R+ × R+ → R+ (𝑖 = 1, 2) such that for each 𝑡∈R+ ∞ ∫ 𝑔 (𝑡, 𝑠)𝑑𝑠 (𝑖 =1,2) Let us observe that in view of assumptions (ii) and (v) we there exist the integrals 0 𝑖 and such ∞ ∫ 𝑔 (𝑡, 𝑠)𝑑𝑠 can define the following finite constants: that the integral 0 1 is uniformly convergent but ∞ ∞ ∫ 𝑔 (𝑡, 𝑠)𝑑𝑠 =0̸ ∫ 𝑔 (𝑡, 𝑠)𝑑𝑠 =0 󵄨 󵄨 lim𝑡→∞ 0 1 while lim𝑡→∞ 0 2 , 𝑓= {󵄨𝑓 (𝑡,) 0 󵄨 :𝑡∈R }, ∞ sup 󵄨 󵄨 + ∫ 𝑔 (𝑡, 𝑠)𝑑𝑠 but the integral 0 2 is not uniformly convergent. ∞ Now, we formulate our last assumption. 𝑔=sup {∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 : 𝑡∈ R+}, 0 (14) (vii) The inequality ∞ 𝑢=sup {∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 : 𝑡∈ R+}. 0 ‖𝑎‖ +𝑘𝑔𝑟ℎ (𝑟) +𝑘𝑢𝑟+𝑓𝑔ℎ (𝑟) + 𝑓𝑢 ≤𝑟 (20) 4 Abstract and Applied Analysis

𝑇 󵄨 󵄨 has a positive solution 𝑟0 such that ≤𝜔 (𝑎, 𝜀) +[󵄨𝑓 (𝑡, 𝑥 (𝑡)) −𝑓(𝑡, 𝑥 (𝑠))󵄨 󵄨 󵄨 + 󵄨𝑓 (𝑡, 𝑥 (𝑠)) −𝑓(𝑠, 𝑥 (𝑠))󵄨] 𝑘(𝑔ℎ (𝑟 )+𝑢) < 1. 0 (21) ∞ × ∫ |𝑢 (𝑡, 𝜏, 𝑥 (𝜏))| 𝑑𝜏 0 𝑟 >0 󵄨 󵄨 󵄨 󵄨 Remark 7. Assume that 0 satisfies the first inequality +[󵄨𝑓 (𝑠, 𝑥 (𝑠)) −𝑓(𝑠,) 0 󵄨 + 󵄨𝑓 (𝑠,) 0 󵄨] from assumption (vii); that is, 󵄨 󵄨 󵄨 󵄨 ∞ × ∫ |𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) −𝑢(𝑠, 𝜏, 𝑥 (𝜏))| 𝑑𝜏 0 ‖𝑎‖ +𝑘𝑔𝑟0ℎ(𝑟0)+𝑘𝑢𝑟0 + 𝑓𝑔ℎ 0(𝑟 )+𝑓𝑢≤𝑟0. (22) 𝑇 𝑇 ≤𝜔 (𝑎, 𝜀) +[𝑘|𝑥 (𝑡) −𝑥(𝑠)| +𝜔‖𝑥‖ (𝑓,𝜀)] Then, we obtain ∞ × ∫ [|𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) −𝑢(𝑡, 𝜏,) 0 | 0 ‖𝑎‖ 𝑓𝑔ℎ (𝑟 ) 𝑓𝑢 + |𝑢 (𝑡, 𝜏,) 0 |] 𝑑𝜏 𝑘(𝑔ℎ (𝑟 )+𝑢) ≤ 1 − − 0 − . (23) 0 𝑟 𝑟 𝑟 0 0 0 󵄨 󵄨 +[𝑘|𝑥 (𝑠)| + 󵄨𝑓 (𝑠,) 0 󵄨] 𝑇 Thus, the second inequality from assumption (vii) is satisfied ×{∫ |𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) −𝑢(𝑠, 𝜏, 𝑥 (𝜏))| 𝑑𝜏 provided at least one of the quantities ‖𝑎‖, 𝑓𝑔ℎ(𝑟0),and𝑓𝑢 0 does not vanish. ∞ Now,wearepreparedtoformulateourmainresult. + ∫ |𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) −𝑢(𝑠, 𝜏, 𝑥 (𝜏))| 𝑑𝜏} 𝑇 Theorem 8. Under assumptions (i)-(vii), (9) has at least one 𝑇 𝑇 𝑇 ≤𝜔 (𝑎, 𝜀) +[𝑘𝜔 (𝑥,) 𝜀 +𝜔‖𝑥‖ (𝑓,𝜀)] solution 𝑥=𝑥(𝑡)in the space 𝐵𝐶(R+). Moreover, all solutions of (9) are uniformly attractive. ∞ × ∫ [𝑔 (𝑡, 𝜏) ℎ (|𝑥 (𝜏)|) + |𝑢 (𝑡, 𝜏,) 0 |]𝑑𝜏 0 Proof. Consider the operator 𝑈 defined on the space 𝐵𝐶(R+) by the formula 𝑇 { 𝑇 +(𝑘‖𝑥‖ + 𝑓) { ∫ 𝜔‖𝑥‖ (𝑢,) 𝜀 𝑑𝜏 0 ∞ { (𝑈𝑥)(𝑡) =𝑎(𝑡) +𝑓(𝑡, 𝑥 (𝑡)) ∫ 𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) 𝑑𝑠, 𝑡∈ R . + ∞ 0 + ∫ |𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) −𝑢(𝑡, 𝜏,) 0 (24) 𝑇 +𝑢(𝑡, 𝜏,) 0 −𝑢(𝑠, 𝜏,) 0 Notice that in view of assumptions (i), (ii), (iv), and (v), the function 𝑡 → (𝑈𝑥)(𝑡) is well defined on the interval R+.We } show that this function is continuous on R+.Tothisend,fix +𝑢(𝑠, 𝜏,) 0 −𝑢(𝑠, 𝜏, 𝑥 (𝜏))| 𝑑𝜏} arbitrarily 𝑇>0and 𝜀>0. Next, take arbitrary numbers 𝑡, 𝑠 ∈ } [0, 𝑇] with |𝑡 − 𝑠| ≤𝜀. Then, in view of imposed assumptions ≤𝜔𝑇 (𝑎, 𝜀) +[𝑘𝜔𝑇 (𝑥,) 𝜀 +𝜔𝑇 (𝑓,𝜀)] we obtain ‖𝑥‖ ∞ × ∫ [𝑔 (𝑡, 𝜏) ℎ (‖𝑥‖) + |𝑢 (𝑡, 𝜏,) 0 |]𝑑𝜏 |(𝑈𝑥)(𝑡) − (𝑈𝑥)(𝑠)| 0 𝑇 𝑇 ≤ |𝑎 (𝑡) −𝑎(𝑠)| +(𝑘‖𝑥‖ + 𝑓) {∫ 𝜔‖𝑥‖ (𝑢,) 𝜀 𝑑𝜏 0 󵄨 󵄨 ∞ ∞ + 󵄨𝑓 (𝑡, 𝑥 (𝑡)) ∫ 𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) 𝑑𝜏 󵄨 + ∫ |𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) −𝑢(𝑡, 𝜏,) 0 | 𝑑𝜏 󵄨 0 𝑇 ∞ 󵄨 ∞ 󵄨 −𝑓 (𝑠, 𝑥 (𝑠)) ∫ 𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) 𝑑𝜏󵄨 + ∫ |𝑢 (𝑠, 𝜏, 𝑥 (𝜏)) −𝑢(𝑠, 𝜏,) 0 | 𝑑𝜏 0 󵄨 𝑇 󵄨 ∞ ∞ 󵄨 + 󵄨𝑓 (𝑠, 𝑥 (𝑠)) ∫ 𝑢 (𝑡, 𝜏, 𝑥 (𝜏)) 𝑑𝜏 + ∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏 󵄨 0 𝑇 ∞ 󵄨 ∞ 󵄨 −𝑓 (𝑠, 𝑥 (𝑠)) ∫ 𝑢 (𝑠, 𝜏, 𝑥 (𝜏)) 𝑑𝜏󵄨 + ∫ |𝑢 (𝑠, 𝜏,) 0 | 𝑑𝜏} 0 󵄨 𝑇 Abstract and Applied Analysis 5

𝑇 𝑇 𝑇 ≤𝜔 (𝑎, 𝜀) +[𝑘𝜔 (𝑥,) 𝜀 +𝜔‖𝑥‖ (𝑓,𝜀)] Hence, we obtain the following evaluation

×(𝑔ℎ (‖𝑥‖) + 𝑢) + (𝑘 ‖𝑥‖ + 𝑓) |(𝑈𝑥)(𝑡)| ≤ ‖𝑥‖ +(𝑘‖𝑥‖ + 𝑓)𝑔ℎ ( (‖𝑥‖) + 𝑢) , ∞ (30) 𝑇 ×{𝑇𝜔‖𝑥‖ (𝑢,) 𝜀 +2∫ 𝑔 (𝑡, 𝑠) ℎ (‖𝑥‖) 𝑑𝑠 𝑇 𝑈𝑥 R ∞ ∞ which implies that the function is bounded on +. + ∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏 + ∫ |𝑢 (𝑠, 𝜏,) 0 | 𝑑𝜏} , Combining this fact with the continuity of the function 𝑈𝑥 𝑇 𝑇 on R+,weconcludethattheoperator𝑈 transforms the space (25) 𝐵𝐶(R+) into itself. Further, observe that from (30)weget where we denoted 󵄨 󵄨 𝜔𝑇 (𝑓,𝜀) = {󵄨𝑓 (𝑡, 𝑥) −𝑓(𝑠, 𝑥)󵄨 : 𝑑 sup 󵄨 󵄨 ‖𝑈𝑥‖ ≤ ‖𝑎‖ +𝑘𝑔 ‖𝑥‖ ℎ (‖𝑥‖) 𝑡, 𝑠 ∈ [0, 𝑇] , |𝑡−𝑠| ≤𝜀,|𝑥| ≤𝑑} , (31) +𝑘𝑢 ‖𝑥‖ + 𝑓𝑔ℎ (‖𝑥‖) + 𝑓𝑢. 𝑇 𝜔𝑑 (𝑢,) 𝜀 = sup {|𝑢 (𝑡, 𝜏, 𝑥) −𝑢(𝑠, 𝜏, 𝑥)| : 𝑡, 𝑠, 𝜏∈ [0, 𝑇] , |𝑡−𝑠| ≤𝜀,|𝑥| ≤𝑑} . Linking the previous inequality with assumption (vii), we 𝑈 𝐵 (26) deduce that the operator maps the ball 𝑟0 into itself, where 𝑟0 >0is a number indicated in assumption (vii). Obviously, in the previous performed calculations we should Now, let us take a nonempty subset 𝑋 of the ball 𝐵𝑟 . ‖𝑥‖ 𝑑 0 put in place of . Fix numbers 𝜀>0and 𝑇>0andchooseanarbitrary Further, let us notice that in view of assumptions (ii) and function 𝑥∈𝑋. Then, in virtue of estimate25 ( ), for an 𝑓 [0, 𝑇]× (iv), the function is uniformly continuous on the set arbitrary 𝑡∈[0,𝑇]we obtain [−‖𝑥‖, ‖𝑥‖], while the function 𝑢 is uniformly continuous on the set [0, 𝑇] × [0, 𝑇] × [−‖𝑥‖, ‖𝑥‖]. Hence, we derive that 𝜔𝑇 (𝑓,𝜀)→0 𝜔𝑇 (𝑢, 𝜀) →0 𝜀→0 𝑇 𝑇 𝑇 ‖𝑥‖ and ‖𝑥‖ as . Next, let 𝜔 (𝑈𝑥, 𝜀) ≤𝜔 (𝑎, 𝜀) +(𝑔ℎ 0(𝑟 )+𝑢) 𝑘𝜔 (𝑥,) 𝜀 us observe that based on assumption (vi) we can choose a number 𝑇 so large that the last terms in estimate (25), that +(𝑔ℎ (𝑟 )+𝑢) 𝜔𝑇 (𝑓,𝜀) + (𝑘𝑟 + 𝑓) 0 𝑟0 0 is, the integrals ∞ ∞ 𝑇 ×{𝑇𝜔𝑟 (𝑢,) 𝜀 +2ℎ(𝑟0) ∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏, ∫ |𝑢 (𝑠, 𝜏,) 0 | 𝑑𝜏, (27) 0 𝑇 𝑇 (32) ∞ aresufficientlysmall.Thesameisalsotruewithregardtothe × sup [∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 : 𝑡∈ [0, 𝑇]] integral 𝑇 ∞ ∞ +2 [∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏 : 𝑡∈ [0, 𝑇]]} . ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠. (28) sup 𝑇 𝑇 Thus, taking into account the all facts established above and estimate (25), we infer that the function 𝑈𝑥 is continuous on Hence, we derive the following estimate the whole interval [0, 𝑇] for each 𝑇>0being big enough. This implies that 𝑈𝑥 is continuous on the whole interval R+. 𝑇 𝑇 𝑇 In what follows, we show that the function 𝑈𝑥 is bounded 𝜔 (𝑈𝑋, 𝜀) ≤𝜔 (𝑎, 𝜀) +(𝑔ℎ 0(𝑟 )+𝑢) 𝑘𝜔 (𝑋, 𝜀) on R+. Indeed, keeping in mind our assumptions, for an 𝑡∈R +(𝑔ℎ (𝑟 )+𝑢)𝑇 𝜔 (𝑓,𝜀) + (𝑘𝑟 + 𝑓) arbitrary fixed +, we get the following estimates: 0 𝑟0 0 󵄨 󵄨 ∞ |(𝑈𝑥)(𝑡)| ≤ |𝑎 (𝑡)| + 󵄨𝑓 (𝑡, 𝑥 (𝑡))󵄨 ∫ |𝑢 (𝑡, 𝑠, 𝑥 (𝑠))| 𝑑𝑠 𝑇 󵄨 󵄨 ×{𝑇𝜔𝑟 (𝑢,) 𝜀 +2ℎ(𝑟0) 0 0

󵄨 󵄨 󵄨 󵄨 ∞ ≤ |𝑎 (𝑡)| +[󵄨𝑓 (𝑡, 𝑥 (𝑡)) −𝑓(𝑡,) 0 󵄨 + 󵄨𝑓 (𝑡,) 0 󵄨] × sup [∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 : 𝑡∈ [0, 𝑇]] ∞ 𝑇 × ∫ [|𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) −𝑢(𝑡, 𝑠, 0)| + |𝑢 (𝑡, 𝑠, 0)|] 𝑑𝑠 ∞ 0 +2 sup [∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏 : 𝑡∈ [0, 𝑇]]} . ≤ |𝑎 (𝑡)| +(𝑘‖𝑥‖ + 𝑓) 𝑇 (33) ∞ ∞ ×{∫ 𝑔 (𝑡, 𝑠) ℎ (|𝑥 (𝑠)|) 𝑑𝑠 + ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠} . 0 0 Further, keeping in mind the properties of the terms of the (29) right hand side of the perviously obtained inequality, which 6 Abstract and Applied Analysis

∞ 󵄨 󵄨 were mentioned earlier (cf. assumptions (ii) and (iv)), we ≤𝑘󵄨𝑥 (𝑡) −𝑦(𝑡)󵄨 {∫ 𝑔 (𝑡, 𝑠) ℎ (|𝑥 (𝑠)|) 𝑑𝑠 deduce that the following estimate holds 0 ∞ + ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠} 𝜔𝑇 (𝑈𝑋) 0 0 󵄨 󵄨 󵄨 󵄨 +(𝑘󵄨𝑦 (𝑡)󵄨 + 󵄨𝑓 (𝑡,) 0 󵄨) 𝑇 󵄨 󵄨 󵄨 󵄨 ≤𝑘(𝑔ℎ 0(𝑟 )+𝑢)0 𝜔 (𝑋) ∞ 󵄨 󵄨 ∞ × ∫ 𝑔 (𝑡, 𝑠) ℎ(󵄨𝑥 (𝑠) −𝑦(𝑠)󵄨)𝑑𝑠 0 +(𝑘𝑟0 + 𝑓) {[∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏 : 𝑡∈ [0, 𝑇]]2ℎ(𝑟0) 𝑇 ∞ 󵄩 󵄩 󵄨 󵄨 ≤𝑘󵄩𝑥−𝑦󵄩 ℎ (‖𝑥‖) ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 +𝑘𝑢 󵄨𝑥 (𝑡) −𝑦(𝑡)󵄨 ∞ 0 × [∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 : 𝑡∈ [0, 𝑇]] sup ∞ 𝑇 󵄩 󵄩 󵄩 󵄩 +(𝑘󵄩𝑦󵄩 + 𝑓) ℎ (󵄩𝑥−𝑦󵄩) ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠. ∞ 0 +2sup [∫ |𝑢 (𝑡, 𝜏,) 0 | 𝑑𝜏 : 𝑡∈ [0, 𝑇]]} . (36) 𝑇 (34) Hence, we get

diam (𝑈𝑋)(𝑡) ≤𝑘𝑢 diam 𝑋 (𝑡) Consequently, in view of assumption (vi) we have +{2𝑘𝑟ℎ(𝑟 )+(𝑘𝑟 + 𝑓) ℎ (2𝑟 )} 0 0 0 0 (37) 𝜔 (𝑈𝑋) ≤𝑘(𝑔ℎ (𝑟 )+𝑢) 𝜔 (𝑋) . ∞ 0 0 0 (35) × ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠. 0

In what follows assume, as previously, that 𝑋 is a fixed Taking into account assumption (v), from the previous 𝐵 inequality we derive the following one: nonempty subset of the ball 𝑟0 . Next, take arbitrary elements 𝑥, 𝑦 ∈𝑋. Then, for an arbitrarily fixed 𝑡∈R+,invirtueof lim sup diam (𝑈𝑋)(𝑡) ≤𝑘𝑢 lim sup diam 𝑋 (𝑡) . imposed assumptions, we obtain: 𝑡→∞ 𝑡→∞ (38)

Obviously, the previous inequality implies the following 󵄨 󵄨 󵄨(𝑈𝑥)(𝑡) −(𝑈𝑦)(𝑡)󵄨 estimate: 󵄨 ∞ (𝑈𝑋)(𝑡) 󵄨 lim sup diam ≤ 󵄨𝑓 (𝑡, 𝑥 (𝑡)) ∫ 𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) 𝑑𝑠 𝑡→∞ 󵄨 0 (39) ≤𝑘(𝑔ℎ 0(𝑟 )+𝑢) lim sup diam 𝑋 (𝑡) . ∞ 󵄨 󵄨 𝑡→∞ −𝑓 (𝑡, 𝑦 (𝑡)) ∫ 𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) 𝑑𝑠󵄨 0 󵄨 Finally, let us observe that by combining (35)and(39)wehave 󵄨 ∞ 󵄨 + 󵄨𝑓(𝑡,𝑦(𝑡)) ∫ 𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) 𝑑𝑠 𝜇 (𝑈𝑋) ≤𝑘(𝑔ℎ 0(𝑟 )+𝑢) 𝜇 (𝑋) , (40) 󵄨 0 ∞ 󵄨 where 𝜇 is the measure of noncompactness defined by 󵄨 −𝑓 (𝑡, 𝑦 (𝑡)) ∫ 𝑢(𝑡,𝑠,𝑦(𝑠))𝑑𝑠󵄨 󵄨 formula (5). 0 󵄨 In the last step of our proof, we show that the operator ∞ 𝑈 is continuous on the ball 𝐵𝑟 .Tothisend,fixanarbitrary 󵄨 󵄨 0 ≤ 󵄨𝑓 (𝑡, 𝑥 (𝑡)) −𝑓(𝑡,𝑦(𝑡))󵄨 ∫ |𝑢 (𝑡, 𝑠, 𝑥 (𝑠))| 𝑑𝑠 𝜀>0 𝑥, 𝑦 ∈𝐵 ‖𝑥 − 𝑦‖ ≤𝜀 󵄨 󵄨 number and take 𝑟0 such that .Then, 0 from estimate (36)weobtain 󵄨 󵄨 ∞ 󵄨 󵄨 + 󵄨𝑓(𝑡,𝑦(𝑡))󵄨 ∫ 󵄨𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) −𝑢(𝑡,𝑠,𝑦(𝑠))󵄨 𝑑𝑠 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩𝑈𝑥−𝑈𝑦󵄩 ≤𝑘𝑔ℎ 0(𝑟 )𝜀+𝑘𝑢𝜀 + (𝑘𝑟0 + 𝑓) 𝑔ℎ (𝜀) . (41) 0

∞ 󵄨 󵄨 The previous inequality in conjunction with assumption (iv) ≤𝑘󵄨𝑥 (𝑡) −𝑦(𝑡)󵄨 ∫ [|𝑢 (𝑡, 𝑠, 𝑥 (𝑠)) −𝑢(𝑡, 𝑠, 0)| implies that the operator 𝑈 is a continuous self-mapping of 0 𝐵 the ball 𝑟0 . + |𝑢 (𝑡, 𝑠, 0)|] 𝑑𝑠 Finally, using the above established facts and (40)and taking into account assumption (vii) and Theorem 2,weinfer 󵄨 󵄨 󵄨 󵄨 +[󵄨𝑓(𝑡,𝑦 𝑡 )−𝑓 𝑡, 0 󵄨 + 󵄨𝑓 𝑡, 0 󵄨] 𝑈 𝑥 𝐵 󵄨 ( ) ( )󵄨 󵄨 ( )󵄨 that the operator hasatleastonefixedpoint in the ball 𝑟0 . Obviously, every function 𝑥=𝑥(𝑡)being a fixed point of the ∞ 󵄨 󵄨 operator 𝑈, is a solution of (9). Moreover, keeping in mind × ∫ 𝑔 (𝑡, 𝑠) ℎ(󵄨𝑥 (𝑠) −𝑦(𝑠)󵄨)𝑑𝑠 𝑈 0 Remark 3,weconcludethatthesetFix of all fixed points Abstract and Applied Analysis 7

of the operator 𝑈 belonging to the ball 𝐵𝑟 (equivalently: the which is satisfied for all 𝑥, 𝑦 ∈ R and for any fixed 𝑎, 𝑎≥0, 𝑈 0 𝐵 set Fix of all solutions of (9)belongingtotheball 𝑟0 ) we obtain isamemberofker𝜇. Hence, in view of the description of 2/3 󵄨 󵄨 (𝑥−𝑦) the kernel ker 𝜇 given in Section 2 we infer that all solutions 󵄨𝑢 (𝑡, 𝑠, 𝑥) −𝑢(𝑡,𝑠,𝑦)󵄨 ≤ . (47) 𝐵 󵄨 󵄨 1+𝑡+𝑠2 of (9)belongingtotheball 𝑟0 are uniformly attractive (asymptotically stable). This implies that the function 𝑢(𝑡, 𝑠, 𝑥) satisfies assumption The proof is complete. 2/3 2 (iv) with ℎ(𝑟) = 𝑟 and 𝑔(𝑡, 𝑠) = 1/(1 +𝑡𝑠 ).Obviously, ℎ:R → R It is worthwhile mentioning that the above result gener- the function + + and is continuous and increasing on R+, while the function 𝑔(𝑡, 𝑠) transforms R+ × R+ into R+ alizes those obtained in [3, 5, 8, 10, 17, 20], among others. R × R Now, we are going to illustrate the result contained in and is continuous on + +. Theorem 8 by an example. Moreover, we get ∞ 𝐴 Example 9. Let us consider the following quadratic Urysohn ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 = lim ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 𝐴→∞ integral equation: 0 0 1 𝐴 𝑑𝑠 3 3/2 2 = lim ∫ 𝛼𝑒𝑡 𝑡2 + 𝑥 (𝑡) ∞ √𝑡 +𝑥 (𝑠) 𝐴→∞𝑡+1 2 𝑥 (𝑡) = +𝛽 sin ∫ 𝑑𝑠, (42) 0 1+(𝑠/√1+𝑡) 𝑡 2 2 1+𝑒 1+𝑡 0 1+𝑡+𝑠 (48) √1+𝑡 𝐴 𝑡∈R 𝛼, 𝛽 = lim arctan where + and are positive constants. 𝐴→∞ 1+𝑡 √1+𝑡 Observe that this equation is a special case of (9)ifweput √1+𝑡 𝜋 𝛼𝑒𝑡 = ⋅ . 𝑎 (𝑡) = , 1+𝑡 2 1+𝑒𝑡 Thus, the function 𝑔(𝑡, 𝑠) is integrable over R+. Next, we 𝑡2 + 𝑥 𝑓 (𝑡, 𝑥) =𝛽 sin , obtain 1+𝑡2 (43) ∞ √1+𝑡𝜋 3 lim ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 = lim =0. (49) √𝑡3/2 +𝑥2 𝑡→∞ 𝑡→∞ 1+𝑡 2 𝑢 (𝑡, 𝑠, 𝑥) = . 0 1+𝑡+𝑠2 Apart from this, in view of (48) we can easily to obtain that It is easily to check that for the previous functions there are ∞ 𝜋 𝑎= 𝑔=sup {∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 : 𝑡∈ R+}= . (50) satisfied assumptions of Theorem 8. Indeed, the function 0 2 𝑎(𝑡) is an element of the space 𝐵𝐶(R+) and ||𝑎|| = 𝛼.This means that there is verified assumption (i). Next, notice that Further, let us fix arbitrarily a number 𝑇>0.Then,similarly 2 2 𝑓 is continuous on R+ × R and 𝑓(𝑡, 0) = 𝛽𝑡 /(1 + 𝑡 ).Thus, as above, we get the function 𝑡→𝑓(𝑡,0)belongs to 𝐵𝐶(R+),andthereis ∞ satisfied assumption (ii). Moreover, we have that 𝑓=𝛽.Apart sup ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠 𝑡∈[0,𝑇] t from this, for 𝑥, 𝑦 ∈ R and for 𝑡∈R+ we obtain 󵄨 󵄨 √1+𝑡 𝜋 𝑇 󵄨 󵄨 󵄨sin 𝑥−sin 𝑦󵄨 = { ( − )} (51) 󵄨𝑓 (𝑡, 𝑥) −𝑓(𝑡,𝑦)󵄨 ≤𝛽 sup 1+𝑡 2 arctan √ 󵄨 󵄨 1+𝑡2 𝑡∈[0,𝑇] 𝑡+1 󵄨 󵄨 (44) 𝜋 𝑇 󵄨𝑥−𝑦󵄨 󵄨 󵄨 ≤𝛽󵄨 󵄨 ≤𝛽󵄨𝑥−𝑦󵄨 . ≤ − arctan . 1+𝑡2 󵄨 󵄨 2 √𝑇+1

This implies that the function 𝑓(𝑡, 𝑥) satisfies assumption (iii) Thisallowsustodeducethefollowingequality: with 𝑘=𝛽. ∞ Further, let us note that the function 𝑢(𝑡, 𝑠, 𝑥) is continu- lim { sup ∫ 𝑔 (𝑡, 𝑠) 𝑑𝑠} = 0. (52) 𝑇→∞ 𝑡∈[0,𝑇] 𝑇 ous on the set R+ ×R+ ×R. For arbitrarily fixed 𝑡, 𝑠 ∈ R+,and 𝑥, 𝑦 ∈ R we obtain Finally,letustakeintoaccountthefunction 󵄨 󵄨 󵄨√3 𝑡3/2 +𝑥2 − √3 𝑡3/2 +𝑦2󵄨 √𝑡 󵄨 󵄨 󵄨 󵄨 𝑢 (𝑡, 𝑠, 0) = |𝑢 (𝑡, 𝑠, 0)| = . (53) 󵄨𝑢 (𝑡, 𝑠, 𝑥) −𝑢(𝑡,𝑠,𝑦)󵄨 ≤ 󵄨 󵄨. (45) 1+𝑡+𝑠2 󵄨 󵄨 1+𝑡+𝑠2 Calculating the indefinite integral of the function 𝑢(𝑡, 𝑠, 0),we Hence, using the inequality (cf. [9]) obtain 󵄨 󵄨 √𝑡2 +𝑡 𝑠 󵄨 3 3 󵄨 3 2 󵄨√𝑎+𝑥2 − √𝑎+𝑦2󵄨 ≤ √(𝑥−𝑦) , (46) ∫ 𝑢 (𝑡, 𝑠, 0) 𝑑𝑠 = arctan . (54) 󵄨 󵄨 𝑡+1 √𝑡+1 8 Abstract and Applied Analysis

Hence, we get Acknowledgment ∞ ∞ ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 = ∫ 𝑢 (𝑡, 𝑠, 0) 𝑑𝑠 This project was funded by the Deanship of Scientific 0 0 Research (DSR), King Abdulaziz University, Jeddah, under 𝐴 Grant no. (470/363/1433). The authors, therefore, acknowl- = lim ∫ 𝑢 (𝑡, 𝑠, 0) 𝑑𝑠 edge with thanks DSR technical and financial support. 𝐴→∞ 0 (55) √𝑡2 +𝑡 𝐴 Conflict of Interests = lim arctan 𝐴→∞ 𝑡+1 √𝑡+1 The authors declare that there is no conflict of interests in the √𝑡2 +𝑡𝜋 submitted paper. = . 𝑡+1 2 Hence, taking into account that References √𝑡2 +𝑡 [1] J. Bana´s and K. Sadarangani, “Compactness conditions in sup =1, (56) the study of functional, differential, and integral equations,” 𝑡∈R 𝑡+1 + Abstract and Applied Analysis, vol. 2013, Article ID 819315, 14 we derive the following equality: pages, 2013. [2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, ∞ 𝜋 𝑢= ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 = . Germany, 1985. sup 2 (57) 𝑡∈R+ 0 [3]M.A.Krasnosel’skii,P.P.Zabrejko,J.I.Pustylnik,andP.I. 𝑇>0 Sobolevskii, IntegralOperatorsinSpacesofSummableFunctions, Now, fix arbitrarily a number .Then,from(54)we Nordhoff, Leyden, Mass, USA, 1976. get [4] P.P.Zabrejko,A.I.Koshelev,M.A.Krasnosel’skii,S.G.Mikhlin, ∞ √𝑡2 +𝑡 𝜋 𝑇 L. S. Rakovschik, and V. J. Stetsenko, Integral Equations,Nord- ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 = ( − arctan ) . (58) hoff, Leyden, Mass, USA, 1975. 𝑇 𝑡+1 2 √𝑡+1 [5] I. J. Cabrera and K. B. Sadarangani, “Existence of solutions This allows us to derive the following estimate: of a nonlinear integral equation on an unbounded interval,” ∞ 𝜋 𝑇 Dynamic Systems and Applications,vol.18,no.3-4,pp.551–570, ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠 ≤ − . 2009. sup 2 arctan √ (59) 𝑡∈[0,𝑇] 𝑇 𝑇+1 [6] C. Corduneanu, Intergral Equations and Applications,Cam- bridge University Press, Cambridge, UK, 1991. Consequently, this yields that the following equality holds: [7] N. Dunford and J. T. Schwartz, Linear Operators I,International ∞ Publishing, Leyden, The Netherlands, 1963. lim { sup ∫ |𝑢 (𝑡, 𝑠, 0)| 𝑑𝑠} = 0. (60) 𝑇→∞ 𝑡∈[0,𝑇] 𝑇 [8] D. O’Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Next, observe that taking into account (48), (52), (55), Dordrecht, The Netherlands, 1998. and (60), we conclude that the functions 𝑔(𝑡, 𝑠) and 𝑢(𝑡, 𝑠, 0) [9]R.P.Agarwal,J.Bana´s, K. Bana´s, and D. O’Regan, “Solvability satisfy assumptions (v) and (vi). of a quadratic Hammerstein integral equation in the class of Now, we are coming to the last assumption of Theorem 8, functions having limits at infinity,” Journal of Integral Equations that is, assumption (vii). Notice, that in the case of our and Applications, vol. 23, no. 2, pp. 157–181, 2011. Equation (42), in view of estimates (44), (50), and (55), the [10] J. Bana´s and L. Olszowy, “On solutions of a quadratic Urysohn first inequality from assumption (vii) has the form integral equation on an unbounded interval,” Dynamic Systems 𝜋 𝜋 𝜋 𝜋 and Applications,vol.17,no.2,pp.255–270,2008. 𝛼+𝛽⋅ 𝑟⋅𝑟2/3 +𝛽 𝑟+𝛽⋅ 𝑟2/3 +𝛽 ≤𝑟. 2 2 2 2 (61) [11] M. A. Darwish and J. Henderson, “Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type,” The Hence, after some simplification, we obtain Rocky Mountain Journal of Mathematics,vol.42,no.2,pp.545– 𝜋 566, 2012. 𝛼+ 𝛽(𝑟5/3 +𝑟+𝑟2/3 +1)≤𝑟. 2 (62) [12] M. A. Darwish and K. Sadarangani, “Nondecreasing solutions of a quadratic Abel equation with supremum in the kernel,” Thus, if for fixed positive 𝛼 and 𝛽 there exists a number 𝑟 >0 Applied Mathematics and Computation,vol.219,no.14,pp. 0 satisfying inequality (62), then the second inequality 7830–7836, 2013. from assumption (vii) is automatically satisfied. Indeed, it is [13] M. Gil and S. Wedrychowicz, “Schauder-Tychonoff fixed-point an immediate consequence of Remark 7.Insuchacase(42) theorem in the theory of superconductivity,” Journal of Function 𝐵 has solutions belonging to the ball 𝑟0 which are uniformly Spaces and Applications,vol.2013,ArticleID692879,12pages, attractive. 2013. 𝛼=𝛽=1/8 For example, it is easy to check that if we take [14] L. Olszowy, “Fixed point theorems in the Frechet´ space C(R+) then the number 𝑟0 =4/5satisfies inequality (62). This means and functional integral equations on an unbounded interval,” that (42)with𝛼=𝛽=1/8has solutions belonging to the ball Applied Mathematics and Computation, vol. 218, no. 18, pp. 𝐵4/5 being uniformly attractive. 9066–9074, 2012. Abstract and Applied Analysis 9

[15] L. Olszowy, “Nondecreasing solutions of a quadratic integral equation of Urysohn type on unbounded interval,” Journal of Convex Analysis,vol.18,no.2,pp.455–464,2011. [16] B. C. Dhage and V. Lakshmikantham, “On global existence and attractivity results for nonlinear functional integral equations,” Nonlinear Analysis: Theory, Methods & Applications,vol.72, A no. 5, pp. 2219–2227, 2010. [17] A. Aghajani and N. Sabzali, “Existence and local attractivity of solutions of a nonlinear quadratic functional integral equation,” Iranian Journal of Science and Technology, Transaction A,vol.36, no. 4, pp. 453–460, 2012. [18] M. A. Darwish, “Monotonic solutions of a convolution functional-integral equation,” Applied Mathematics and Com- putation,vol.219,no.22,pp.10777–10782,2013. [19] X. Hu and J. Yan, “The global attractivity and asymptotic stability of solution of a nonlinear integral equation,” Journal of Mathematical Analysis and Applications,vol.321,no.1,pp.147– 156, 2006. [20] R. Stanczy,´ “Hammerstein equations with an integral over a noncompact domain,” Annales Polonici Mathematici,vol.69,no. 1, pp. 49–60, 1998. [21] J. Bana´sandK.Goebel,Measures of Noncompactness in Banach Spaces,vol.60ofLecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. [22] G. M. Fichtenholz, Differential and Integral Calculus,vol.2, WydawnictwoNaukowePWN,Warsaw,Poland,2007,(Polish). Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 890657, 14 pages http://dx.doi.org/10.1155/2013/890657

Research Article On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

M. De la Sen

Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus de Leioa, P.O. Box 644, 48080 Bilbao, Spain

Correspondence should be addressed to M. De la Sen; [email protected]

Received 24 July 2013; Accepted 6 September 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite- dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

1. Introduction forth.Also,ithasbeenstudiedtherelevanceofthetheory in properties in both general theory and applications such Compact operators in infinite-dimensional separable Hilbert as the existence and uniqueness of solutions in differential, spaces are of relevance in the study of certain relevant applied difference, and hybrid equations as well as in continuous- problems in control theory and signal theory, [1]. A natural time, discrete-time, and hybrid dynamic systems, stability property of such operators is that they can be represented theory in the above problems [3–7], the existence/uniqueness with expansions using two orthogonal or orthonormal recip- of fixed points and best proximity points, and the bound- rocal bases of the separable Hilbert space. If the bases are edness of iterated sequences being constructed through the orthonormal then both of them coincide so that this basis is mapsandtheconvergenceofsuchiteratedcalculationsto autoreciprocal and then the formal study is facilitated [1, 2]. limit points. See, for instance, [3–6, 8–15] and the references Many of the involved operators in mapping map an input therein. The investigation of existence and uniqueness of spaceintoanoutputspaceintheaboveproblemsarein common fixed points and best proximity points for several addition self-adjoint. Another property of such operators is mappings and related properties is also important [10– that they admit truncations using a finite number of the 12]. The study of fixed and best proximity points has also members of the orthonormal basis so that the truncated inherent study of convergence of sequences to such points. operators are also compact in a natural way, [1, 2]. The Other studies of properties of convergence of sequences and truncated operator describes a natural orthogonal projection operator sequences have been described in different problems of the involved vectors of the Hilbert space into a finite- as, for instance, the research on approximating operators dimensional space whose dimension is deceased as the and approximation theorems that of sigma convergence of number of members of the basis used for its representation double sequences or that of lamda-statistical convergence decreases. On the other hand, important attention is being and summability. See, for instance, [13–17] and the references devoted to many aspects of fixed point theory in metric, therein. Banach, and more general spaces including the study of map- This paper is devoted to the investigation of self-adjoint pings being contractive, nonexpansive, asymptotically con- compact operators in separable Hilbert spaces, their finite- tractive, asymptotically nonexpansive, quasi-nonexpansive, dimensional truncated counterparts, and the relations in- Kannan and Meir-Keeler and cyclic-type contractions, and so between the corresponding properties for the norms of the 2 Abstract and Applied Analysis mutual errors end the errors in-between the corresponding (vii) If 𝑉=𝐻is a separable infinite-dimensional Hilbert fixed points and their respective convergence properties space and 𝑎𝑛 =⟨𝑥,𝑒𝑛⟩,for𝑛∈N,then when iterated calculations through the operators are per- formed. Some examples of interest in signal theory and 󵄩 𝑀 󵄩2 ∞ 󵄩 󵄩 󵄨 󵄨2 controltheoryarealsogiven.Theoperatorsandtheiterated 󵄩𝑥−∑ 𝑎 𝑒 󵄩 ≤(∑ 󵄨𝑎 󵄨 )≤‖𝑥‖2. 󵄩 𝑛 𝑛󵄩 󵄨 𝑛󵄨 (3) sequences constructed through them are studied by using 󵄩 𝑛=1 󵄩 𝑛=1 the expansions of the operators and their finite dimensional truncated versions by using a numerable orthonormal basis of the involved Hilbert space. 𝑀 2 If, in addition, ‖𝑥−∑𝑛=1 𝑎𝑛𝑒𝑛‖ <+∞,then𝑎𝑛 →0as 𝑛(∈ 2. Preliminaries and Main Results N)→∞. If, furthermore, there is some integer 𝛼≥𝑀such that the real sequence {|𝑎𝑛|}𝑛≥𝛼 converges to zero exponentially 𝑛 The following result includes some properties related to the according to |𝑎𝑛|≤𝜌 ≤𝜌<1,for𝑛∈N,then approximations of 𝑥∈𝑉and 𝑥∈𝐻through orthonor- 𝑀 2 ‖𝑥 − ∑ 𝑎𝑛𝑒𝑛‖ ≤ 𝐶(𝛼)+ 𝜌/(1−𝜌) for any given 𝑥∈𝐻with mal systems of different dimensions, complete orthonormal 𝑛=1 𝜌 ∈ (0, 1) being some real constant and 𝐶(𝛼) being a bounded systems in 𝐻, and orthonormal basis, that is, a maximal constant dependent on 𝛼 satisfying 𝐶(𝑀). =0 orthonormal system; that is, it is not a proper subset of any 𝐻 𝑉 𝐻 orthonormal system of ,where and are an inner Proof. Properties (i)-(ii) follow from the best approximation product space and a Hilbert space, respectively. Note that lemma since in the case where 𝐻 is separable, a complete orthonormal system is always an orthonormal basis and vice versa. 󵄩 󵄩2 󵄩 󵄩2 󵄩 𝑁 󵄩 󵄩 𝑀 𝑀 󵄩 󵄩 󵄩 󵄩 󵄩 Lemma 1. 𝑉 󵄩𝑥−∑ 𝑎𝑛𝑒𝑛󵄩 = 󵄩(𝑥 − ∑ 𝑎𝑛𝑒𝑛)−∑ 𝑎𝑛𝑒𝑛󵄩 Let be an inner product space of inner product 󵄩 󵄩 󵄩 󵄩 ⟨⋅, ⋅⟩ : 𝐻 × 𝐻 → C (or R)endowedwithanorm‖⋅‖:𝑉 → 󵄩 𝑛=1 󵄩 󵄩 𝑛=𝑀+1 𝑛=1 󵄩 R ‖𝑥‖ = ⟨𝑥,1/2 𝑥⟩ 𝑥∈𝐻 R = 0+ defined by for any ,where 0+ 󵄩 𝑁 󵄩2 𝑀 𝑁 𝑁 󵄩 󵄩 󵄨 󵄨2 {𝑧 ∈ R :𝑧≥0},let{𝑒𝑛}𝑛=1 and {𝑎𝑛}𝑛=1 be a finite orthonormal = 󵄩𝑥− ∑ 𝑎 𝑒 󵄩 − ∑ 󵄨⟨𝑥, 𝑒 ⟩󵄨 󵄩 𝑛 𝑛󵄩 󵄨 𝑛 󵄨 system in 𝑉 and a given finite or numerable sequence of scalars, 󵄩 𝑛=𝑀+1 󵄩 𝑛=1 respectively, and let 𝑀 and 𝑁 be given integers fulfilling 1≤ 𝑁 𝑀 2 𝑀≤𝑁≤∞.If𝑁=∞then {𝑒𝑛}𝑛=1 is, in addition, assumed 󵄨 󵄨 + ∑ 󵄨𝑎𝑛 −⟨𝑥,𝑒𝑛⟩󵄨 , to be numerable. Then, the following properties hold for any 𝑛=1 𝑥∈𝐻. (4) 󵄩 󵄩2 󵄩 󵄩2 2 2 󵄩 𝑀 󵄩 󵄩 𝑀 𝑁 󵄩 ‖𝑥 − ∑𝑁 𝑎 𝑒 ‖ =‖𝑥−∑𝑁 𝑎 𝑒 ‖ − ∑𝑀 |⟨𝑥, 󵄩 󵄩 󵄩 󵄩 (i) 𝑛=1 𝑛 𝑛 𝑛=𝑀+1 𝑛 𝑛 𝑛=1 󵄩𝑥−∑ 𝑎𝑛𝑒𝑛󵄩 = 󵄩(𝑥 + ∑ 𝑎𝑛𝑒𝑛)−∑ 𝑎𝑛𝑒𝑛󵄩 𝑀 󵄩 󵄩 󵄩 󵄩 2 2 󵄩 𝑛=1 󵄩 󵄩 𝑛=𝑀+1 𝑛=1 󵄩 𝑒𝑛⟩| + ∑𝑛=1 |𝑎𝑛 −⟨𝑥,𝑒𝑛⟩| . 󵄩 󵄩 󵄩 󵄩 𝑀 2 𝑁 2 𝑁 󵄩 󵄩2 ‖𝑥 − ∑ 𝑎 𝑒 ‖ =‖𝑥+∑ 𝑎 𝑒 ‖ − ∑ |⟨𝑥, 󵄩 𝑁 󵄩 𝑁 (ii) 𝑛=1 𝑛 𝑛 𝑛=𝑀+1 𝑛 𝑛 𝑛=1 󵄩 󵄩 󵄨 󵄨2 2 𝑁 2 = 󵄩𝑥− ∑ 𝑎𝑛𝑒𝑛󵄩 − ∑ 󵄨⟨𝑥,𝑛 𝑒 ⟩󵄨 𝑒𝑛⟩| +∑ |𝑎𝑛 −⟨𝑥,𝑒𝑛⟩| . 󵄩 󵄩 𝑛=1 󵄩 𝑛=𝑀+1 󵄩 𝑛=1 𝑁 2 𝑀 2 𝑁 (iii) ‖𝑥−∑𝑛=1 𝑎𝑛𝑒𝑛‖ −‖𝑥−∑𝑛=1 𝑎𝑛𝑒𝑛‖ =∑𝑛=𝑀+1(|⟨𝑥, 𝑁 𝑒 ⟩|2 +|⟨𝑥,𝑒 ⟩−𝑎 |2) 󵄨 󵄨2 𝑛 𝑛 𝑛 . + ∑ 󵄨𝑎𝑛 −⟨𝑥,𝑒𝑛⟩󵄨 . 𝑗 2 𝑗 2 𝑛=1 (iv) ‖ ∑𝑛=𝑖 𝑎𝑛𝑒𝑛‖ = ∑𝑛=𝑖 |𝑎𝑛| any integers 𝑖, 𝑗(≥ 𝑖)∈ 𝑁= {1,2,...,𝑁}. 𝑉=𝐻 Property (iii) is a direct consequence of subtracting both (v) If is a finite-dimensional Hilbert space of sides of the relations in Properties (i)-(ii). Property (iv) is 𝑁 𝑎 =⟨𝑥,𝑒⟩ 𝑛∈𝑁 dimension and 𝑛 𝑛 ,forall and Pythagoras theorem in inner product spaces. Property (v) 𝑁=𝑀 ,then (Bessel’s inequality) follows directly from Property (i) with 󵄩 󵄩2 {𝑒 }𝑁 𝐻 󵄩 𝑁 󵄩 𝑁 the orthonormal system 𝑛 𝑛=1 in the Hilbert space being 󵄩 󵄩 󵄨 󵄨2 2 𝐻 󵄩𝑥−∑⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛󵄩 + ∑ 󵄨⟨𝑥,𝑛 𝑒 ⟩󵄨 = ‖𝑥‖ abasisof . Property (vi) follows from Properties (ii)–(iii) 󵄩 󵄩 󵄨 󵄨 𝑁 󵄩 𝑛=1 󵄩 𝑛=1 󵄩 󵄩 with 𝑎𝑛 =⟨𝑥,𝑒𝑛⟩; 𝑛∈𝑁 and the orthonormal system {𝑒𝑛}𝑛=1 (1) 𝐻 𝐻 𝑁 𝑀 in being an orthonormal basis of since one gets from 󵄨 󵄨2 󵄨 󵄨2 Property (i) ≥ ∑ 󵄨⟨𝑥,𝑛 𝑒 ⟩󵄨 ≥ ∑ 󵄨⟨𝑥,𝑛 𝑒 ⟩󵄨 . 𝑛=1 𝑛=1

󵄩 𝑁 󵄩2 󵄩 𝑁 󵄩2 𝑀 𝑉=𝐻 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨2 (vi) If is a finite-dimensional Hilbert space of 󵄩𝑥−∑ 𝑎 𝑒 󵄩 = 󵄩𝑥− ∑ 𝑎 𝑒 󵄩 − ∑ 󵄨⟨𝑥, 𝑒 ⟩󵄨 𝑁 𝑎 =⟨𝑥,𝑒⟩ 𝑛∈𝑁 󵄩 𝑛 𝑛󵄩 󵄩 𝑛 𝑛󵄩 󵄨 𝑛 󵄨 dimension and 𝑛 𝑛 ,for ,then 󵄩 𝑛=1 󵄩 󵄩 𝑛=𝑀+1 󵄩 𝑛=1

󵄩 𝑀 󵄩2 𝑁 𝑀 𝑀 󵄩 󵄩 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄩𝑥−∑ 𝑎 𝑒 󵄩 ≤2∑ 󵄨𝑎 󵄨 −3∑ 󵄨𝑎 󵄨 . + ∑ 󵄨𝑎 −⟨𝑥,𝑒 ⟩󵄨 󵄩 𝑛 𝑛󵄩 󵄨 𝑛󵄨 󵄨 𝑛󵄨 (2) 󵄨 𝑛 𝑛 󵄨 󵄩 𝑛=1 󵄩 𝑛=1 𝑛=1 𝑛=1 Abstract and Applied Analysis 3

󵄩 𝑁 󵄩2 𝑀 𝐻 󵄩 󵄩 󵄨 󵄨2 via an orthonormal system in of smaller dimension than = 󵄩𝑥− ∑ 𝑎 𝑒 󵄩 − ∑ 󵄨𝑎 󵄨 󵄩 𝑛 𝑛󵄩 󵄨 𝑛󵄨 that of such a space. Property (vii) relies on the approximation 󵄩 𝑛=𝑀+1 󵄩 𝑛=1 of an element in an infinite-dimensional separable Hilbert 󵄩 𝑁 󵄩2 𝑀 space by using a numerable orthonormal basis of 𝐻. 󵄩 󵄩 󵄨 󵄨2 =0󳨐⇒󵄩𝑥− ∑ 𝑎 𝑒 󵄩 = ∑ 󵄨𝑎 󵄨 󵄩 𝑛 𝑛󵄩 󵄨 𝑛󵄨 Lemma 2. 𝑇:𝐻 →𝐻 󵄩 𝑛=𝑀+1 󵄩 𝑛=1 Let be a linear, closed, and compact self-adjoint operator in an infinite-dimensional sep- (5) arable Hilbert space 𝐻 with a numerable orthonormal basis ∞ of generalized eigenvectors {𝑒𝑛}𝑛=1 𝑇:𝐻.Then,the →𝐻 and from (5), Property (ii), and 𝑎𝑛 =⟨𝑥,𝑒𝑛⟩, 𝑛∈𝑁 following properties hold: 󵄩 𝑀 󵄩2 󵄩 𝑁 𝑁 󵄩2 𝑀 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨2 𝑁 ∞ 𝑁 󵄩𝑥−∑ 𝑎 𝑒 󵄩 = 󵄩𝑥+ ∑ 𝑎 𝑒 − ∑ 𝑎 𝑒 󵄩 − ∑ 󵄨𝑎 󵄨 (i) 𝑇 𝑥=∑𝑛=1 𝜆𝑛 (𝑇)⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛, 󵄩 𝑛 𝑛󵄩 󵄩 𝑛 𝑛 𝑛 𝑛󵄩 󵄨 𝑛󵄨 󵄩 𝑛=1 󵄩 󵄩 𝑛=𝑀+1 𝑛=1 󵄩 𝑛=1 for all 𝑥∈𝐻for any 𝑁∈N,where𝜆𝑛(𝑇) ∈ 𝜎(𝑇);thespectrum 󵄩 󵄩2 󵄩 󵄩2 𝑇 𝜆 (𝑇) = ⟨𝑇𝑒 ,𝑒 ⟩ 𝑛∈N 󵄩 𝑁 󵄩 󵄩 𝑁 󵄩 of the operator is defined by 𝑛 𝑛 𝑛 ,forall 󵄩 󵄩 󵄩 󵄩 𝑁 𝑁 𝑁 𝑁 ≤2󵄩𝑥−∑ 𝑎𝑛𝑒𝑛󵄩 +2󵄩 ∑ 𝑎𝑛𝑒𝑛󵄩 and 𝜆𝑛 (𝑇) = ⟨𝑇𝑒𝑛,𝑒𝑛⟩ ∈𝜎(𝑇 ) with |⟨𝑇𝑒𝑛,𝑒𝑛⟩| →0as 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛=1 󵄩 󵄩 𝑛=𝑀+1 󵄩 𝑛→∞,forall𝑁∈N. P 𝐻 𝑀 If 𝑛 is the orthogonal projection operator of on the one- 󵄨 󵄨2 󵄨 󵄨 dimensional subspace 𝐷𝑛 generated by the eigenvector 𝑒𝑛 then − ∑ 󵄨𝑎𝑛󵄨 𝑛=1 𝑁 lim (P𝑛 (𝑇 𝑥)) = {0} (∈ 𝐷𝑛); ∀𝑛,𝑁∈N,∀𝑥∈𝐻. 󵄩 󵄩2 𝑛→∞ 󵄩 𝑁 󵄩 𝑀 󵄩 󵄩 󵄨 󵄨2 (9) =2󵄩 ∑ 𝑎𝑛𝑒𝑛󵄩 − ∑ 󵄨𝑎𝑛󵄨 󵄩 󵄩 󵄩 𝑛=𝑀+1 󵄩 𝑛=1 P 𝐻 𝑛 If Ω 𝑖 is the orthogonal projection operator of on the Ω𝑖 - 𝑁 𝑀 dimensional eigensubspace Ω𝑖,then 󵄨 󵄨2 󵄨 󵄨2 =2 ∑ 󵄨𝑎𝑛󵄨 − ∑ 󵄨𝑎𝑛󵄨 𝑁 𝑛=𝑀+1 𝑛=1 lim (PΩ (𝑇 𝑥)) = {0} (∈ Ω𝑖); ∀𝑁∈N,∀𝑥∈𝐻 𝑖→∞ 𝑖 (10) 𝑁 𝑀 󵄨 󵄨2 󵄨 󵄨2 =2∑ 󵄨𝑎 󵄨 −3∑ 󵄨𝑎 󵄨 . 𝑇𝑁𝑥=𝑃 (𝑇𝑁)(𝑥) ⊕I ( −𝑃 (𝑇𝑁))(𝑥) 𝑃 (𝑇𝑁) 󵄨 𝑛󵄨 󵄨 𝑛󵄨 with Ω𝑖 Ω𝑖 where Ω𝑖 𝑛=1 𝑛=1 𝑁 (𝑥) ≡Ω 𝑃 (𝑇 𝑥),forall𝑛, 𝑁 ∈ N,forall𝑥∈𝐻. (6) 𝑖 (ii) If, in addition, ‖𝑇‖ ≤ 𝛼,then <1 Hence, Property (vi). Property (vii) follows from the assump- tion that the infinite-dimensional Hilbert space is separable 󵄨 󵄨𝑁 lim 󵄨𝜆𝑛 (𝑇)󵄨 =0, and Property (vi) leads to 𝑁→∞ 2 ∞ 𝑁 󵄩 𝑀 󵄩 ∞ ∞ 󵄨 󵄨𝑁 𝛼 󵄩 󵄩 󵄨 󵄨2 󵄨 󵄨2 2 ∑ 󵄨𝜆 (𝑇)󵄨 ≤ <∞; ∀𝑛∈N, 󵄩𝑥−∑ 𝑎 𝑥 󵄩 = ∑ 󵄨𝑎 󵄨 ≤ ∑ 󵄨𝑎 󵄨 ≤ ‖𝑥‖ 󵄨 𝑛 󵄨 1−𝛼𝑁 󵄩 𝑛 𝑛󵄩 󵄨 𝑛󵄨 󵄨 𝑛󵄨 𝑛=1 󵄩 𝑛=1 󵄩 𝑛=𝑀+1 𝑛=1 (11) 𝑁 lim (P𝑖 (𝑇 𝑥)) = {0} (∈ 𝐷𝑖), <+∞󳨐⇒(𝑎𝑛 󳨀→ 0 as 𝑛 (∈ N) 󳨀→ ∞ ) 𝑁→∞ (7) (P (𝑇𝑁𝑥)) = {0} (∈ Ω ); ∀𝑖∈N. lim Ω𝑖 𝑖 which holds under, perhaps, eventual reordering of the 𝑁→∞ elements of the orthonormal basis of 𝐻 which is a complete orthonormal system for the separable Hilbert space 𝐻.If Proof. Note that there is a numerable orthonormal basis for 𝐻 since 𝐻 is separable and infinite dimensional. Such a basis there is some integer 𝛼≥𝑀such that the real sequence ∞ {𝑒𝑛}𝑛=1 can be chosen as the set of generalized eigenvectors {|𝑎𝑛|}𝑛≥𝛼 converges to zero exponentially, then of the linear self-adjoint 𝑇:𝐻since →𝐻 it is closed and 󵄩 𝑀 󵄩2 ∞ 𝛼 𝛼 compact and then bounded 󵄩 󵄩 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄩𝑥−∑ 𝑎 𝑒 󵄩 = ∑ 󵄨𝑎 󵄨 = ∑ 󵄨𝑎 󵄨 + ∑ 󵄨𝑎 󵄨 󵄩 𝑛 𝑛󵄩 󵄨 𝑛󵄨 󵄨 𝑛󵄨 󵄨 𝑛󵄨 ∞ 󵄩 𝑛=1 󵄩 𝑛=𝑀+1 𝑛=𝑀+1 𝑛=𝛼+1 (8) 𝑥=∑⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛;∀𝑥∈𝐻. (12) 𝜌 ≤𝐶(𝛼) + , 𝑛=1 1−𝜌 Also, since the linear operator 𝑇:𝐻is →𝐻 closed and 𝑛 𝜎(𝑇) 𝑇:𝐻 →𝐻 where |𝑎𝑛|≤𝜌 ≤𝜌<1,forall𝑛(∈ N)≥𝛼with 𝐶(𝛼) = compact, the spectrum of is a proper 𝛼 2 nonempty (since 𝑇:𝐻is →𝐻 infinite dimensional and ∑ |𝑎𝑛| <+∞being dependent on 𝛼 such that 𝐶(𝛼). =0 𝑛=𝑀+1 C Hence, Property (vii). bounded since it is compact) subset of and numerable and it satisfies 𝜎(𝑇)𝑝 =𝜎 (𝑇) ∪ {0},with𝜎𝑐(𝑇) ∪𝑟 𝜎 (𝑇) = {0}, Note that Property (vi) of Lemma 1 quantifies an approxi- where 𝜎𝑝(𝑇), 𝜎𝑐(𝑇),and𝜎𝑟(𝑇) are the punctual, continuous, mation of an element of a finite-dimensional Hilbert space 𝐻 and residual spectra of 𝑇:𝐻,respectively.Notethat →𝐻 4 Abstract and Applied Analysis

{0} ∈ 𝜎(𝑇) 𝜎(𝑇) ∞ isalsoanaccumulationpointofthespectrum = ∑ 𝜆𝑁+1 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 𝐻 𝑇:𝐻 →𝐻 𝑛 𝑛 𝑛 since is infinite dimensional and is compact. 𝑛=1 Also, since 𝐻 is separable, the spectrum of 𝑇:𝐻is →𝐻 ∞ numerable, and ⟨𝑒𝑗,𝑒𝑛⟩=𝛿𝑗𝑛;forall𝑗, 𝑛 ∈ N,onegets 𝑁+1 = ∑ ⟨𝑇𝑒𝑛,𝑒𝑛⟩ ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛, ∞ ∞ 𝑛=1 𝑇𝑒𝑛 = ∑⟨𝑇𝑒𝑛,𝑒𝑗⟩𝑒𝑛 = ∑⟨𝑇𝑒𝑛,𝑒𝑗⟩𝑒𝑛𝛿𝑗𝑛 (17) 𝑗=0 𝑗=0 (13) 𝑁 𝑁 where 𝛿𝑗𝑛 is the Kronecker delta. Then, 𝜆𝑛 (𝑇) = ⟨𝑇𝑒𝑛,𝑒𝑛⟩ ∈ =⟨𝑇𝑒𝑛,𝑒𝑛⟩𝑒𝑛 =𝜆𝑛 (𝑇) 𝑒𝑛;∀𝑛∈N, 𝑁 𝑁 𝜎(𝑇 ).Furthermore,𝑇 :𝐻 →is 𝐻 compact as it follows 𝑁 𝜆 (𝑇) = ⟨𝑇𝑒 ,𝑒 ⟩ 𝑇:𝐻 →𝐻 by complete induction as follows. Assume that 𝑇 :𝐻 → 𝐻 where 𝑛 𝑛 𝑛 is an eigenvalue of ;that 𝑁 𝜆 (𝑇) ∈ 𝜎(𝑇) 𝑒 is compact, then it is bounded. Note also that 𝑇 :𝐻 →is 𝐻 is, 𝑛 , associated with the eigenvector 𝑛 since 𝑁+1 self-adjoint by construction and then normal. Thus, 𝑇 = 𝑁 𝜆𝑛 (𝑇) 𝑒𝑛 =𝜆𝑛 (𝑇) ⟨𝑒𝑛,𝑒𝑛⟩𝑒𝑛 =⟨𝜆𝑛 (𝑇) 𝑒𝑛,𝑒𝑛⟩𝑒𝑛 𝑇(𝑇 ):𝐻 →is 𝐻 compact since it is a composite operator 𝑁 (14) of a bounded operator 𝑇 :𝐻 →with 𝐻 a compact =⟨𝑇𝑒,𝑒 ⟩=⟨𝑇𝑒,𝑒 ⟩𝑒 𝑁 𝑛 𝑛 𝑛 𝑛 𝑛 operator 𝑇:𝐻. →𝐻 Then, by complete induction, 𝜆𝑛 (𝑇) = 𝑁 𝑁 ⟨𝑇𝑒𝑛,𝑒𝑛⟩ → 0 (∈ 𝜎(𝑇 )) as 𝑛→∞,forany𝑁∈N since so that 𝑁 𝑇 :𝐻 →is 𝐻 compact and 𝐻 is infinite dimensional. Also, ∞ ∞ ∞ 𝑁 𝑁−1 𝑁 𝑇𝑥 = ∑⟨𝑇𝑥,𝑛 𝑒 ⟩𝑒𝑛 = ∑ ⟨𝑇 ( ∑ ⟨𝑥,𝑗 𝑒 ⟩𝑒𝑗),𝑒𝑛⟩𝑒𝑛𝛿𝑗𝑛 P𝑛 (𝑇 𝑥) = ⟨𝑇𝑒𝑛,𝑒𝑛⟩ P𝑛 (𝑥) =⟨𝑇𝑒𝑛,𝑒𝑛⟩ 𝑒𝑛 𝑛=1 𝑛=1 𝑗=1 =𝜆𝑁 𝑇 𝑒 󳨀→ 0 𝑛󳨀→∞; ∞ 𝑛 ( ) 𝑛 as (18) = ∑ ⟨𝑇 (⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛),𝑒𝑛⟩𝑒𝑛 ∀𝑁 ∈ N;∀𝑥∈𝐻, 𝑛=1

∞ where P𝑛 is the projection operator of 𝐻 on the one- = ∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 ;∀𝑥∈𝐻, 𝑛 𝑛 𝑛 dimensional subspace 𝐷𝑛 generated by the eigenvector 𝑒𝑛 so 𝑛=1 that P𝑛𝑥=⟨𝑥,𝑥𝑛⟩𝑥𝑛 →0as 𝑛→∞,forall𝑥∈𝐻. (15) Thus, Property (i) has been proved. To prove Property (ii), take an orthonormal basis associated with the set of finite- so that, except perhaps for reordering, |𝜆𝑛(𝑇)| ≥ |𝜆𝑛+1(𝑇)|, 𝑛∈N {𝜆 (𝑇)} → 0 𝐻 dimensional eigenspaces of the respective eigenvalues. Note for all with 𝑛 since is separable and from Cauchy-Schwarz inequality that 𝜎(𝑇) is numerable. Assume that for any positive integer 𝑁 󵄨 󵄨𝑁 󵄨 󵄨𝑁 󵄩 󵄩𝑁 the following identity is true: 󵄨𝜆 (𝑇)󵄨 = 󵄨⟨𝑇𝑒 ,𝑒 ⟩󵄨 ≤ ‖𝑇‖𝑁󵄩𝑒 󵄩 󵄨 𝑛 󵄨 󵄨 𝑛+𝑞𝑛 𝑛+𝑞𝑛 󵄨 󵄩 𝑛+𝑞𝑛 󵄩 ∞ (19) 𝑁 𝑁 𝑁 𝑁 𝑇 𝑥=∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛. (16) ≤ ‖𝑇‖ ≤𝛼 <1; ∀𝑛,𝑁∈N 𝑛=1 for some real constant 𝛼 ∈ (0, 1),where{𝑞𝑛}𝑛∈N is {𝑒 }∞ Then, since 𝑛 𝑛=1 is an orthonormal basis of generalized a nondecreasing sequence of finite nonnegative integers eigenvectors, 𝑖−1 defined by 𝑞𝑖 = ∑𝑗=1 𝑝𝑗 being built such that each 𝑞𝑛 ∞ for 𝑛∈N accountsforthetotalofthedimensions𝑝𝑗 𝑁+1 𝑁 𝑁 Ω 𝑇 𝑥=𝑇(𝑇 𝑥) = ∑ 𝜆𝑛 (𝑇) ⟨𝑇 𝑥,𝑛 𝑒 ⟩𝑒𝑛 of the eigenspaces 𝑗 associated with the set of eigenval- 𝑛=1 ues {𝜆1(𝑇), 𝜆2(𝑇),...,𝜆𝑛−1(𝑇)} previous to 𝜆𝑛(𝑇) for 𝑛∈ N after eventual reordering by decreasing moduli. Then, ∞ ∞ 𝑁 𝑁 lim𝑁→∞|𝜆𝑛(𝑇)| =0,forall𝑛∈N,and = ∑ 𝜆𝑛 (𝑇) ⟨∑ 𝜆𝑗 (𝑇) 𝑛=1 𝑗=1 󵄨 󵄨 󵄩 𝑁 󵄩 󵄨 𝑁󵄨 󵄩 󵄩 󵄩P𝑖 (𝑇 𝑥)󵄩 = 󵄨⟨𝑇𝑒𝑖+𝑞 ,𝑒𝑖+𝑞 ⟩ 󵄨 󵄩P𝑖 (𝑥)󵄩 󵄩 󵄩 󵄨 𝑖 𝑖 󵄨 󵄩 󵄩 󵄩 󵄩 ×⟨𝑥,𝑒𝑗⟩𝑒𝑗,𝑒𝑛⟩𝑒𝑛 󵄩 𝑝𝑖−1 󵄩 󵄩 𝑁 (𝑗) (𝑗) 󵄩 = 󵄩⟨𝑇𝑒 ,𝑒 ⟩ ( ∑ 𝛾 𝑒 )󵄩 (20) 󵄩 𝑖+𝑞𝑖 𝑖+𝑞𝑖 𝑖+𝑞𝑖 𝑖+𝑞𝑖 󵄩 󵄩 𝑗=0 󵄩 ∞ ∞ 󵄩 󵄩 𝑁 = ∑ ∑ 𝜆 (𝑇) 𝜆 (𝑇) ⟨⟨𝑥, 𝑒 ⟩𝑒 ,𝑒 ⟩𝑒 󵄩 𝑁 󵄩 𝑁 𝑗 𝑛 𝑗 𝑗 𝑛 𝑛 = 󵄩𝜆 (𝑇) 𝑒 󵄩 ≤𝑃𝑝 𝛼 , 𝑗=1 𝑛=1 󵄩 𝑖 𝑖+𝑞𝑖 󵄩 𝑖 𝑖

∞ ∞ (𝑗) 𝑁 {𝑒 : 𝑗 = 0,1,...,𝑝 } 𝑝 where 𝑖+𝑞𝑖 𝑖−1 is now a set of 𝑖 linearly = ∑ ∑ 𝜆𝑗 (𝑇) 𝜆𝑛 (𝑇) 𝑗=1 𝑛=1 independent elements belonging to the orthonormal basis of 𝐻 that generate the eigenspace Ω𝑖 associated with 𝜆𝑖(𝑇) with × ⟨⟨𝑥, 𝑒 ⟩𝑒 ,𝑒 ⟩𝛿 𝑒 𝑒(0) =𝑒 {𝛾(𝑗) :𝑗=0,1,...,𝑝 } 𝑗 𝑗 𝑛 𝑗𝑛 𝑛 𝑖+𝑞𝑖 𝑖+𝑞𝑖 being an eigenvector and 𝑖+𝑞𝑖 𝑖−1 Abstract and Applied Analysis 5

𝑁 is a set of complex coefficients. Then, 𝛼 𝑒𝑖 →0as →∞, pointofsuchaspectrum𝜎(𝑇) since the Hilbert space is finite- 𝑁 ⟨𝑇𝑒 ,𝑒 ⟩→0 𝑛→∞ for all 𝑖∈N from (20), so that lim𝑁→∞(P𝑖(𝑇 𝑥)) = {0}(∈ dimensional. Therefore, the result 𝑛 𝑛 as 𝐷𝑖). If there are some multiple eigenvalues, with all being of Lemma 1 does not hold. Then, Property (i) follows directly of finite multiplicity since the operator 𝑇:𝐻is →𝐻 from the above considerations. Property (ii) follows from the compact, the above expression may be reformulated with relations projections on the finite-dimensional eigenspaces associated 𝑝 𝑝 󵄨 󵄨𝑁 󵄨 󵄨𝑁 to each of the eventually repeated eigenvalues leading to ∑ 󵄨𝜆𝑛 (𝑇)󵄨 = ∑ 󵄨⟨𝑇𝑒𝑛+𝑞 ,𝑒𝑛+𝑞 ⟩󵄨 𝑁 󵄨 󵄨 󵄨 𝑛 𝑛 󵄨 (P (𝑇 𝑥)) = {0}(∈ Ω ) 𝑖∈N 𝑛=1 𝑛=1 lim𝑁→∞ Ω𝑖 𝑖 ,forall .Notethat 𝑝𝑖 (23) Ω𝑖 ≡𝐷𝑞 ×𝐷𝑞 ... × 𝐷𝑞 𝑝𝑖 ≥ 𝑝 𝑁 𝑁(𝑝+1) 𝑖 𝑖 𝑖 is the finite ( 1)-dimension of 𝜂(𝛼 −𝛼 ) 𝑛𝑁 the eigenspace Ω𝑖 associated with 𝜆𝑖(𝑇),where𝑝𝑖 is one- ≤ ∑ 𝜂𝛼 = <∞. 1−𝛼𝑁 dimensional if 𝜆𝑖 ∈ 𝜎(𝑇) is single. Finally, it follows from (19) 𝑛=1 that Remark 4. It turns out that Lemma 2 (ii) and Lemma 3 (ii) ∞ ∞ 𝑇:𝐻 →𝐻 󵄨 󵄨𝑁 󵄨 󵄨𝑁 also hold if is not self-adjoint since the ∑ 󵄨𝜆𝑛 (𝑇)󵄨 = ∑ 󵄨⟨𝑇𝑒𝑛+𝑞 ,𝑒𝑛+𝑞 ⟩󵄨 󵄨 󵄨 󵄨 𝑛 𝑛 󵄨 corresponding mathematical proofs are obtained by using an 𝑛=1 𝑛=1 (21) orthonormal basis formed by all linearly independent vectors ∞ 𝛼𝑁 generating each of the subspaces. However, if the operator is ≤ ∑ 𝛼𝑛𝑁 = <∞ 1−𝛼𝑁 not self-adjoint or if it is infinite dimensional while being self- 𝑛=1 adjoint, the set of (nongeneralized) eigenvectors is not always an orthogonal basis of the Hilbert space. and Property (ii) has been proved. In the following, we relate the properties of operators on Lemma 2 becomes modified for compact operators ona 𝐻 with their degenerate versions obtained via truncations of finite-dimensional Hilbert space as follows. their expanded expansions. Lemma 3. Let 𝑇:𝐻be →𝐻 a linear closed and com- Theorem 5. Let 𝐻 be a separable Hilbert space and let pact self-adjoint operator in a finite-dimensional Hilbert space 𝑇(𝑝) : 𝐻 →𝐻 be a linear degenerated 𝑝-finite-dimensional 𝐻 of finite dimension 𝑝 with a finite orthonormal basis of {𝑒 }𝑝 𝑇:𝐻 →𝐻 approximating operator of the linear closed and compact self- eigenvectors 𝑛 𝑛=1 of .Then,thefollowing adjoint operator 𝑇:𝐻.Then,thefollowingproperties →𝐻 properties hold. hold. 𝑁 𝑝 𝑁 𝑁 𝑁 (i) 𝑇 𝑥=∑𝑛=1 𝜆𝑛 (𝑇)⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 (i) Assume that ‖𝑇‖ ≤𝜂𝛼,forall𝑁∈N for some real constants 𝛼 ∈ (0, 1) and 𝜂≥1,where for any 𝑁∈N,where𝜆𝑛(𝑇) ∈ 𝜎(𝑇); the spectrum of the ∞ {𝑒𝑛}𝑛=1 is a numerable orthonormal basis of generalized operator 𝑇 is defined by 𝜆𝑛(𝑇) = ⟨𝑇𝑒𝑛,𝑒𝑛⟩,forall𝑛∈𝑝 and 𝑇:𝐻 →𝐻 𝑁 𝑁 𝑁 eigenvectors of .Then, 𝜆𝑛 (𝑇) = ⟨𝑇𝑒𝑛,𝑒𝑛⟩ ∈𝜎(𝑇 ),forall𝑁∈N. 𝑁(𝑝+1) 󵄩 𝑁 𝑁 󵄩 𝜂𝛼 𝑁 𝑁 (󵄩𝑇 𝑥−𝑇 (𝑝) 𝑥󵄩 : ‖𝑥‖ ≤1)≤ , (ii) If, in addition, ‖𝑇‖ ≤𝜂𝛼 for some real constants sup 󵄩 󵄩 1−𝛼𝑁 𝛼 ∈ (0, 1) and 𝜂≥1,then (24) 󵄩 𝑁 𝑁 󵄩 󵄩𝑇 𝑥−𝑇 (𝑝) 𝑥󵄩 󳨀→ 0 𝑎 𝑠 𝑁 󳨀→ ∞ ;∀𝑥∈𝐻. 𝑁 󵄩 󵄩 󵄨 󵄨𝑁 𝜂𝛼 󵄨𝜆 (𝑇)󵄨 ≤ <∞, ∀𝑁∈N, 󵄨 𝑛 󵄨 𝑁 1−𝛼 (ii) Assume that there is a finite 𝑛0 ∈ N such that ∑∞ |𝜆 (𝑇)| ≤ 𝑀 <+∞ 󵄨 󵄨𝑁 𝑛=𝑛 𝑛 0 for some positive real 󵄨𝜆 (𝑇)󵄨 󳨀→ 0 𝑎 𝑠 𝑁 󳨀→ ∞;∀𝑛∈ 𝑝 0 󵄨 𝑛 󵄨 constant 𝑀0 =𝑀0(𝑛0).Thus,foranygivenpositive 𝜀≤1 𝑝 𝜂(𝛼𝑁 −𝛼𝑁(𝑝+1)) real constant ,therearenonnegativefiniteintegers 󵄨 󵄨𝑁 (22) 𝑝 =𝑝(𝜀, 𝑛 )>𝑛 𝑁 =𝑁(𝑝 ,𝜀) ∑ 󵄨𝜆 (𝑇)󵄨 ≤ <∞; ∀𝑁∈N, 0 0 0 0 and 0 0 0 such 󵄨 𝑛 󵄨 𝑁 𝑛=1 1−𝛼 that for any finite 𝑝(≥𝑝0)-dimensional degenerated approximating operator 𝑇(𝑝) : 𝐻 →𝐻 of 𝑇:𝐻 → 𝑝 󵄨 󵄨𝑁 𝐻,thefollowinginequalityholds ∑ 󵄨𝜆 (𝑇)󵄨 󳨀→ 0 𝑎 𝑠 𝑁 󳨀→ ∞,∀𝑝∈ N. 󵄨 𝑛 󵄨 󵄩 󵄩 󵄩 󵄩 𝑛=1 󵄩 𝑁 󵄩 󵄩 𝑁 󵄩 󵄩𝑇 (𝑝) 𝑥󵄩 ≤ 󵄩𝑇 𝑥󵄩 +𝜀‖𝑥‖ 󵄩 󵄩 (25) 𝑇:𝐻 →𝐻 󵄩 𝑁󵄩 Outline of Proof. First note that the spectrum of ≤(󵄩𝑇 󵄩 +𝜀)‖𝑥‖ ;∀𝑁>𝑁0 ∀𝑥 ∈ 𝐻. is nonempty since the operator is self-adjoint. Note also that, since the Hilbert space is finite-dimensional Hilbert space, Furthermore, any set of normalized linearly independent eigenvectors of a 󵄩 𝑁 𝑁 󵄩 lim (󵄩𝑇 𝑥−𝑇 (𝑝) 𝑥󵄩) = 0; ∀𝑥 ∈ 𝐻 self-adjoint operator is an orthonormal basis of such a Hilbert 𝑁→∞ 󵄩 󵄩 (26) space [1]. Property (i) is a direct counterpart of Property (i) of Lemma 2 except that {0} can be a value of the punctual for any 𝑇(𝑝) : 𝐻 →𝐻 linear degenerated 𝑝(≥𝑝0)-finite- spectrum of 𝑇:𝐻but →𝐻 it is not an accumulation dimensional approximating operator of the linear closed and 6 Abstract and Applied Analysis

𝑁 𝑝 𝑁 compact self-adjoint operator 𝑇:𝐻and →𝐻 some finite so that the assumption 𝑇 (𝑝)𝑥 = ∑𝑛=1 𝜆𝑛 (𝑇)⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 is true 𝑝0 ∈ N. as it has been proved from (30) by complete induction. The 𝑁 {𝑇𝑁𝑥}→𝑧 𝑁→∞ 𝑥, 𝑧 ∈𝐻 following properties are also direct for any 𝑥∈𝐻if ‖𝑇‖ ≤ (iii) If as for some such that 𝑁 (‖𝑇𝑁𝑥−𝑇𝑁(𝑝)𝑥‖) =0 {𝑇𝑁(𝑝)𝑥} →𝑧 𝜂𝛼 <1for some real constants 𝛼 ∈ (0, 1) and 𝜂≥1;forall lim𝑁→∞ ,then 𝑁≥𝑁 𝑁 ∈ N as 𝑁→∞.Furthermore,sucha𝑧 isafixedpointof 0 and some finite 0 ,wehave both 𝑇:𝐻and →𝐻 𝑇(𝑝) : 𝐻 →𝐻. 󵄩 𝑝 󵄩 󵄩 󵄩 󵄩 ∞ 󵄩 Proof. The operator 𝑇:𝐻is →𝐻 represented as follows: 󵄩 𝑁 󵄩 󵄩 𝑁 𝑁 󵄩 󵄩𝑇 𝑥󵄩 = 󵄩∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 + ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛󵄩 ∞ ∞ ∞ 󵄩 󵄩 󵄩 󵄩 󵄩𝑛=1 𝑛=𝑝+1 󵄩 𝑇𝑥 = ∑ ⟨𝑇𝑥, 𝑒 ⟩𝑒 = ∑ ∑ ⟨⟨𝑇𝑥, 𝑒 ⟩𝑒 ,𝑒 ⟩𝑒 𝑛 𝑛 𝑗 𝑗 𝑛 𝑛 󵄩 󵄩 𝑛=1 𝑛=1 𝑗=1 󵄩 𝑝 󵄩 𝑁(𝑝+1) 󵄩 𝑁 󵄩 𝜂𝛼 (27) ≤ 󵄩∑ 𝜆 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛󵄩 + ‖𝑥‖ ∞ 󵄩 𝑛 󵄩 1−𝛼𝑁 󵄩𝑛=1 󵄩 = ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛. 󵄩 𝑝 󵄩 𝑁(𝑝+1) 𝑛=1 󵄩 󵄩 𝜂𝛼 = 󵄩∑ 𝜆𝑁 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 + ‖𝑥‖ The associated degenerated 𝑝-finite-dimensional operator is 󵄩 𝑛 𝑛 𝑛󵄩 1−𝛼𝑁 󵄩𝑛=1 󵄩 𝑝 𝑝 𝑁(𝑝+1) 𝑇(𝑝)𝑥= ∑ ⟨𝑇𝑥,𝑛 𝑒 ⟩𝑒𝑛 = ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 (28) 󵄩 𝑁 󵄩 𝜂𝛼 = 󵄩𝑇 (𝑝) 𝑥󵄩 + ‖𝑥‖ 𝑛=1 𝑛=1 󵄩 󵄩 1−𝛼𝑁 󵄩 󵄩 so that 󵄩 ∞ 󵄩 𝑝 ∞ 󵄩 𝑁 𝑁 󵄩 󵄩 𝑁 󵄩 󵄩𝑇 𝑥−𝑇 (𝑝)󵄩 𝑥 = 󵄩 ∑ ⟨𝑇 𝑥, 𝑒 ⟩𝑒 󵄩 𝑇𝑁 (𝑝) 𝑥 = ∑ ⟨𝑇𝑁𝑥, 𝑒 ⟩𝑒 =𝑇𝑁𝑥− ∑ ⟨𝑇𝑁𝑥, 𝑒 ⟩𝑒 󵄩 󵄩 󵄩 𝑛 𝑛 󵄩 𝑛 𝑛 𝑛 𝑛 󵄩𝑛=𝑝+1 󵄩 𝑛=1 𝑛=𝑝+1 𝜂𝛼𝑁(𝑝+1) ∞ ∞ ≤ ,∀𝑥∈𝐻 ‖𝑥‖ ≤1 𝑁 𝑁 𝑁 with = ∑ ⟨𝑇 𝑥,𝑛 𝑒 ⟩𝑒𝑛 − ∑ ⟨𝑇 𝑥,𝑛 𝑒 ⟩𝑒𝑛. 1−𝛼 𝑛=1 𝑛=𝑝+1 󵄩 󵄩 󵄩𝑇𝑁𝑥−𝑇𝑁 (𝑝) 𝑥󵄩 󳨀→ 0 𝑁󳨀→∞;∀𝑥∈𝐻. (29) 󵄩 󵄩 as 𝑁 𝑝 𝑁 (31) Thus, assume that 𝑇 (𝑝)𝑥 =∑𝑛=1 𝜆𝑛 (𝑇)⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛.Then, 𝑝 𝑇𝑁+1 (𝑝) 𝑥 = ∑ ⟨𝑇𝑁+1𝑥, 𝑒 ⟩𝑒 Property (i) has been proved. On the other hand, if 𝑛 𝑛 ∞ 𝑛=1 ∑ |𝜆 (𝑇)| ≤ 𝑀 <+∞ 𝑛 ∈ N 𝑛=𝑛0 𝑛 0 for some finite 0 and 𝑝 ∞ some 𝑀0 ∈ R+,thenforanygivenreal𝜀(≤1) ∈ R+,there 𝑁+1 𝑝 =𝑝(𝜀) > 𝑛 = ∑ ∑ ⟨⟨𝑇 𝑥,𝑗 𝑒 ⟩𝑒𝑗,𝑒𝑛⟩𝑒𝑛 is a positive finite integer 0 0 0 such that for 𝑛=1 𝑗=1 any ](∈ R+)≤𝜀/𝑀0 and any 𝑝(≥𝑝0) ∈ N,thefollowing inequalities hold: 𝑝 𝑁+1 = ∑ ⟨⟨𝑇 𝑥,𝑛 𝑒 ⟩𝑒𝑛,𝑒𝑛⟩𝑒𝑛 𝑛=1 ∞ 󵄨 󵄨 ∞ 󵄨 󵄨 ∑ 󵄨𝜆 (𝑇)󵄨 ≤ ∑ 󵄨𝜆 (𝑇)󵄨 ≤ ]𝑀 ≤𝜀 𝑝 󵄨 𝑛 󵄨 󵄨 𝑛 󵄨 0 𝑛=𝑝+1 𝑛=𝑝 +1 = ∑ ⟨⟨𝜆 (𝑇) 𝑇𝑁𝑥, 𝑒 ⟩𝑒 ,𝑒 ⟩𝑒 0 𝑛 𝑛 𝑛 𝑛 𝑛 (32) 𝑛=1 ∞ 󵄨 󵄨 ∞ 󵄨 󵄨 ≤ ∑ 󵄨𝜆 (𝑇)󵄨 < ∑ 󵄨𝜆 (𝑇)󵄨 ≤𝑀 𝑝 󵄨 𝑛 󵄨 󵄨 𝑛 󵄨 0 𝑛=𝑛 𝑁 𝑛=𝑛0+1 0 = ∑ ⟨⟨𝑇 𝑥,𝑛 𝑒 ⟩𝑒𝑛,𝑒𝑛⟩𝜆𝑛 (𝑇) 𝑒𝑛 (30) 𝑛=1

𝑝 since |𝜆𝑛(𝑇)| ≥ |𝜆𝑛+1(𝑇)|,forall𝑛∈N, 𝜆𝑛(𝑇)→0as 𝑛→ 𝑁 = ∑ 𝜆𝑛 (𝑇) ⟨𝑇 𝑥,𝑛 𝑒 ⟩𝑒𝑛 ∞, 0∈𝜎(𝑇),and|𝜆𝑛(𝑇)| ≤,forall 𝜀 𝑛(≥𝑛0) ∈ N.Notethat ∞ 𝑛=1 𝑀 ∈ R ∑ |𝜆 (𝑇)| ≤ 𝑀 <+∞ since 0 + exists such that 𝑛=𝑛0 𝑛 0 for 𝑝 some finite 𝑛0 ∈ N,then,foranygiven𝜀(≤1) ∈ R+,(32)holds 𝑁 ∗ 𝑝 ≥𝑝 ∈ N 𝑝 =𝑝(𝜀) > 𝑛 = ∑ 𝜆𝑛 (𝑇) ⟨𝑥, (𝑇 ) 𝑒𝑛⟩𝑒𝑛 for any ( 0) and some 0 0 0.Then,onegets 𝑛=1 via complete induction for any 𝑁(>0 𝑁 )∈N 𝑝 = ∑ 𝜆 (𝑇) ⟨𝑥, (𝑇∗)𝑁𝑒 ⟩𝑒 𝑛 𝑛 𝑛 ∞ 󵄨 󵄨 󵄨 󵄨 ∞ 󵄨 󵄨 𝑛=1 ∑ 󵄨𝜆𝑁 (𝑇)󵄨 ≤ 󵄨𝜆 (𝑇) 󵄨 ( ∑ 󵄨𝜆𝑁−1 (𝑇)󵄨) 󵄨 𝑛 󵄨 󵄨 𝑝+1 󵄨 󵄨 𝑛 󵄨 𝑝 𝑛=𝑝+1 𝑛=𝑝+1 (33) 𝑁+1 = ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 𝑁 𝑛=1 ≤𝜀 <1 Abstract and Applied Analysis 7 󵄨 󵄨 ∑∞ |𝜆𝑁(𝑇)| → 0 𝑁→∞ 𝜀<1 𝑝 ≥𝑝 󵄨 󵄩 󵄩 󵄩 󵄩󵄨 and 𝑛=𝑝+1 𝑛 as if ,forall ( 0) 󵄨 󵄩 𝑁 󵄩 󵄩̃ 󵄩󵄨 ≥ 󵄨lim sup (󵄩𝑧−𝑇 (𝑝) 𝑥󵄩)− lim 󵄩𝑧𝑁󵄩󵄨 ∈ N.Thus,onegetsfromLemma 1 (iv) 󵄨 𝑁→∞ 𝑁→∞ 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 𝑁 󵄩 󵄩 𝑝 ∞ 󵄩 = lim sup (󵄩𝑧−𝑇 (𝑝) 𝑥󵄩) 󵄩 𝑁 󵄩 󵄩 𝑁 𝑁 󵄩 𝑁→∞ 󵄩𝑇 𝑥󵄩 = 󵄩∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 + ∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 󵄩 󵄩 󵄩 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛󵄩 󵄩𝑛=1 𝑛=𝑝+1 󵄩 (37) 󵄩 󵄩 󵄩 𝑝 󵄩 󵄩 ∞ 󵄩 ∃ (‖𝑧 −𝑁 𝑇 (𝑝)𝑥‖) =0 𝑇:𝐻 → 󵄩 󵄩 󵄩 󵄩 and then lim𝑁→∞ .Also, ≤ 󵄩∑ 𝜆𝑁 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 + 󵄩 ∑ 𝜆𝑁 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 󵄩 𝑛 𝑛 𝑛󵄩 󵄩 𝑛 𝑛 𝑛󵄩 𝐻 is bounded, since it is compact, and it is then continuous 󵄩 󵄩 󵄩 󵄩 󵄩𝑛=1 󵄩 󵄩𝑛=𝑝+1 󵄩 since it is linear and bounded. Also, 𝑇(𝑝) : 𝐻 →𝐻 is of 󵄩 𝑝 󵄩 ∞ finite-dimensional and closed image, then compact, and then 󵄩 󵄩 󵄨 󵄨 𝑁 ≤ 󵄩∑ 𝜆𝑁 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 +( ∑ 󵄨𝜆𝑁 (𝑇)󵄨) bounded and continuous since it is linear. Thus, ‖𝑧−𝑇 𝑥‖ → 󵄩 𝑛 𝑛 𝑛󵄩 󵄨 𝑛 󵄨 𝑁 󵄩𝑛=1 󵄩 𝑛=𝑝+1 0, ‖𝑧 − 𝑇 (𝑝)𝑥‖ →0 as 𝑁→∞leads to 󵄩 󵄩 󵄩 󵄩 󵄩 𝑁+1 󵄩 󵄩 𝑁 󵄩 󵄩 ∞ 󵄩 󵄩 𝑝 󵄩 0←󳨀󵄩𝑇 𝑥−𝑧󵄩 = 󵄩𝑇(𝑇 𝑥) −󵄩 𝑧 󵄩 󵄩 󵄩 𝑁 󵄩 󵄩 󵄩 󵄩 󵄩 × 󵄩 ∑ ⟨𝑥, 𝑒 ⟩𝑒 󵄩 ≤ 󵄩∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 󵄩 𝑛 𝑛󵄩 󵄩 𝑛 𝑛 𝑛󵄩 󵄩𝑛=𝑝+1 󵄩 󵄩𝑛=1 󵄩 󳨀→ ‖𝑇𝑧‖ −𝑧 as 𝑁󳨀→∞ ∞ 󵄩 ∞ 󵄩 󵄩 𝑁+1 󵄩 󵄨 󵄨 󵄩 󵄩 𝑧=𝑇𝑧0←󳨀󵄩𝑇 (𝑝) 𝑥󵄩 −𝑧 +( ∑ 󵄨𝜆𝑁 (𝑇)󵄨) 󵄩∑ ⟨𝑥, 𝑒 ⟩𝑒 󵄩 implying 󵄩 󵄩 󵄨 𝑛 󵄨 󵄩 𝑛 𝑛󵄩 󵄩 󵄩 󵄩 󵄩 (38) 𝑛=𝑝+1 󵄩𝑛=1 󵄩 󵄩 𝑁 󵄩 = 󵄩𝑇(𝑝)(𝑇 (𝑝) 𝑥) −𝑧󵄩 󵄩 𝑝 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩∑ 𝜆𝑁 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 +𝜀‖𝑥‖ ,∀𝑥∈𝐻 󳨀→ 󵄩𝑇(𝑝)𝑧−𝑧󵄩 𝑁󳨀→∞ 󵄩 𝑛 𝑛 𝑛󵄩 󵄩 󵄩 as 󵄩𝑛=1 󵄩 (34) implying 𝑧=𝑇(𝑝)𝑧, and Property (iii) has been proved. for any 𝑝(≥𝑝0) ∈𝑁and for all 𝑁(>𝑁0) ∈ N.Furthermore, note from (32)that] →0and 𝑝0 →∞as 𝜀→0and the Note that Theorem 5 (ii) cannot be generalized, in the function ] = ](𝜀) is nonincreasing. Also, a strictly monotone general case, for the case of a finite dimensional approximat- decreasing positive real sequence ]𝑛 = ](𝜀𝑛) canbebuilt ing linear operator 𝑇(𝑝) : 𝐻 →𝐻 of smaller dimension with {𝜀𝑛}→0since there are infinite many values of the 𝑝<𝑞to any linear degenerated operator 𝑇:𝐻 →𝐻 spectrum 𝜎(𝑇) such that the inequality |𝜆𝑛(𝑇)| ≥ |𝜆𝑛+1(𝑇)| of (finite) dimension 𝑞.Thereasonisthatthepropertythat is strict since, otherwise, the convergence of the sequence 0∈𝜎(𝑇)does not any longer hold, in general if 𝑇:𝐻 →𝐻 {|𝜆𝑛(𝑇)|} to zero would be impossible. Then, from (34)and is finite dimensional. On the other hand, a way of describing ∞ 𝑁 ∑𝑛=𝑝+1 |𝜆𝑛 (𝑇)| → 0 as 𝑁→∞if 𝜀<1,forall𝑝(≥𝑝0) ∈ N, the operator 𝑇:𝐻and →𝐻 its approximating finite- there are subsequences of positive real and positive integers dimensional counterpart 𝑇(𝑝) : 𝐻 →𝐻 is through the {𝜀 }→ 0 {𝑝 (𝜀 )} → +∞ 𝑁→∞ ̃ 𝑝0 and 𝑜 𝑝0 ,respectively,as absolute error operator 𝑇𝑝(≡ 𝑇 − :𝑇(𝑝)) 𝐻→𝐻.This such that the following subsequent relation holds: is useful if either 𝑇:𝐻is →𝐻 finite dimensional of dimension 𝑞>𝑝where 𝑝 is the dimension of 𝑇(𝑝) : 𝐻→ 󵄩 𝑝≥𝑝 󵄩 󵄩 󵄩 󵄩 0 󵄩 󵄩 ∞ 󵄩 󵄩 𝑁 𝑁 󵄩 󵄩 𝑁 󵄩 𝐻 or if 𝑇:𝐻is →𝐻 nondegenerated. Another useful 󵄩𝑇 𝑥− ∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 = 󵄩 ∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 󵄩 󵄩 𝑛 𝑛 𝑛󵄩 󵄩 𝑛 𝑛 𝑛󵄩 𝑇(𝑝)̃ : 󵄩 𝑛=1 󵄩 󵄩 𝑛=𝑝+1 󵄩 characterization is the use of the relative error operator 𝐻→𝐻satisfying the operator identity 𝑇(𝑝) = 𝑇(I + ̃ ≤𝜀𝑝 ‖𝑥‖ ,∀𝑥∈𝐻 𝑇(𝑝)). Another alternative operator identity 𝑇=𝑇(𝑝)(I + 0 ̃ (35) 𝑇1(𝑝)) cannot be used properly if 𝑇:𝐻is →𝐻 infinite dimensional since 𝑇(𝑝) : 𝐻 →𝐻 is degenerated of finite for all 𝑁(>0 𝑁 )∈N.Then, dimension 𝑝. We discuss some properties of the operator ̃ identity 𝑇(𝑝) = 𝑇(I + 𝑇(𝑝)) through the subsequent result. 󵄩 𝑝≥𝑝 󵄩 󵄩 0 󵄩 󵄩 𝑁 𝑁 󵄩 lim sup (󵄩𝑇 𝑥− ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛󵄩)≤0 (36) Lemma 6. Let 𝐻 be a separable Hilbert space and let 𝑇: 𝑁→∞ 󵄩 󵄩 󵄩 𝑛=1 󵄩 𝐻→𝐻be a nonnull and nondegenerated (i.e., of infinite- dimensional image) linear closed and compact operator and let and Property (ii) follows directly. 𝑇(𝑝) : 𝐻 →𝐻 be the linear degenerated 𝑝-finite-dimensional {𝑇𝑁𝑥} → 𝑧 (‖𝑇𝑁𝑥−𝑇𝑁(𝑝)𝑥‖) =0 𝑁→ If and lim𝑁→∞ as approximating operator of 𝑇:𝐻.Then,thereisan →𝐻 ∞ 𝑥, 𝑧 ∈𝐻 ∃{𝑧̃ } ̃ for some ,then 𝑁 which converges to zero operator 𝑇(𝑝) : 𝐻 →𝐻 such that 𝑇(𝑝) can be represented by such that ̃ ̃ ̃ 𝑇(𝑝) = 𝑇(I + 𝑇(𝑝)), Dom(𝑇(𝑝)) ⊆ Dom(𝑇),andIm(𝑇(𝑝)) ⊆ 󵄩 𝑁 𝑁 󵄩 󵄩 𝑁 󵄩 Dom(𝑇) with the following properties. 0= lim 󵄩𝑇 𝑥−𝑇 (𝑝) 𝑥󵄩 = 󵄩𝑧+𝑧̃𝑁 −𝑇 (𝑝) 𝑥 󵄩 𝑁→∞ 󵄩 󵄩 󵄩 󵄩 𝑇(𝑝)̃ : 󵄨󵄩 󵄩 󵄨 (i) There exists an (in general, nonunique) operator 󵄨󵄩 𝑁 󵄩 󵄩̃ 󵄩󵄨 ̃ ̃ ≥ 󵄨󵄩𝑧−𝑇 (𝑝) 𝑥󵄩 − 󵄩𝑧𝑁󵄩󵄨 𝐻→𝐻,restrictedto𝑇(𝑝) : Dom 𝑇(𝑝)| Dom 𝑇→ 8 Abstract and Applied Analysis

∞ ∞ ∞ 𝑇(𝑝)̃ ⊆ 𝑇 𝑇(𝑝) : 󵄨 󵄨2 Im Dom for each approximating = ∑ ∑ ∑ 𝜆 (𝑇) 𝜆 (𝑇(𝑝))̃ 󵄨𝛾 󵄨 ⟨𝑥, 𝑒 ⟩𝑒 𝛿 𝐻→𝐻 𝑝 𝑛 𝑘 󵄨 𝑘𝑗 󵄨 𝑗 𝑗 𝑗𝑛 of given dimension . 𝑛=1 𝑘=1 𝑗=1 𝑇𝑇(𝑝)̃ : 𝐻 →𝐻 (ii) The operator is nondegenerated, ∞ ∞ unique, and compact. ̃ 󵄨 󵄨2 = ∑ 𝜆𝑛 (𝑇) (∑ 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑛󵄨 )⟨𝑥,𝑒𝑛⟩𝑒𝑛. ̃ 𝑛=1 𝑘=1 (iii) The minimum modulus of 𝑇:𝐻is →𝐻 𝜇(𝑇(𝑝)) =0 so that if it is invertible, its inverse is not bounded. (40) 𝑇:𝐻 →𝐻 If is degenerated, that is, finite ̃ dimensional of dimension 𝑞>𝑝, injective with closed Then, 𝑇𝑇(𝑝) : 𝐻 →𝐻 is a unique nondegenerated image then its minimum modulus is positive and finite. compactoperatorfromitsrepresentation(40). It follows that ̃ If, furthermore, 𝑇:𝐻is →𝐻 invertible then the operator identity 𝑇(𝑝) = 𝑇(I + 𝑇(𝑝)) holds on 𝐻 if and ̃ ̃ 𝑇(𝑝) : 𝐻 →𝐻 is a compact operator with bounded only if 𝑇(𝑝)𝑥=𝑇(Ι + 𝑇(𝑝))𝑥;forall𝑥∈𝐻and, equivalently, ̃ ̃ minimum modulus 𝜇(𝑇(𝑝)). since 𝑇 and 𝑇(𝑝) are linear, ̃ ∞ ∞ Proof. The existence of such an operator 𝑇(𝑝) : 𝐻 →𝐻 is ̃ 󵄨 󵄨2 ∑ 𝜆𝑛 (𝑇) (1 + ∑ 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑛󵄨 )⟨𝑥,𝑒𝑛⟩𝑒𝑛 proved by construction. Let {𝑒𝑛}𝑛∈N be an orthonormal basis 𝑇:𝐻 →𝐻 {V } 𝑛=1 𝑘=1 of generalized eigenvectors of and 𝑛 𝑛∈N an (41) ̃ 𝑝 orthonormal basis of 𝑇(𝑝) : 𝐻 →𝐻,respectively.Then,one = ∑ 𝜆 (𝑇) ⟨𝑥, 𝑒 ⟩𝑒 . gets for some sequences of complex coefficients {𝛾𝑛𝑗 } ,for 𝑛 𝑛 𝑛 𝑗∈N 𝑛=1 all 𝑛∈N, Since the vectors in {𝑒𝑛}𝑛∈N form an orthonormal basis, ∞ ̃ V = ∑ 𝛾 𝑒 (41), if the following constraints defining the operator 𝑇(𝑝) : 𝑛 𝑛𝑗 𝑗 ̃ ̃ 𝑗=1 𝐻→𝐻,restrictedas𝑇(𝑝) : Dom 𝑇(𝑝) | Dom 𝑇→ 𝑇(𝑝)̃ ⊆ 𝑇 𝑇:𝐻 →𝐻 ∞ ∞ Im Dom , hold for a nonnull operator

𝑇𝑥 = ∑ ⟨𝑇𝑥,𝑛 𝑒 ⟩𝑒𝑛 = ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 𝑝 ∞ 𝑛=1 𝑛=1 ̃ 󵄨 󵄨2 ∑ ∑ 𝜆𝑛 (𝑇) 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑛󵄨 ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 𝑝 𝑝 𝑛=1 𝑘=1

𝑇(𝑝)𝑥= ∑ ⟨𝑇𝑥,𝑛 𝑒 ⟩𝑒𝑛 = ∑ 𝜆𝑛 (𝑇) ⟨𝑥,𝑛 𝑒 ⟩𝑒𝑛 ∞ ∞ 𝑛=1 𝑛=1 ̃ 󵄨 󵄨2 + ∑ 𝜆𝑛 (𝑇) (1 + ∑ 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑛󵄨 )⟨𝑥,𝑒𝑛⟩𝑒𝑛 =0 ∞ ∞ 𝑛=𝑝+1 𝑘=1 ̃ ̃ ̃ 𝑇(𝑝)𝑥= ∑ ⟨𝑇(𝑝)𝑥,V𝑛⟩ V𝑛 = ∑ 𝜆𝑛 (𝑇(𝑝))⟨𝑥,V𝑛⟩ V𝑛 (42) 𝑛=1 𝑛=1

∞ ∞ so that (42)holdsifandonlyif ̃ = ∑ ∑ 𝜆𝑛 (𝑇(𝑝))⟨𝑥,𝛾𝑛𝑗 𝑒𝑗⟩ V𝑛 ∞ 𝑛=1 𝑗=1 ̃ 󵄨 󵄨2 ∑ 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑛󵄨 =0 for 𝑛∈𝑝; ∞ ∞ ∞ 𝑘=1 ̃ (43) = ∑ ∑ ∑ 𝜆𝑛 (𝑇(𝑝))⟨𝑥,𝛾𝑛𝑗 𝑒𝑗⟩𝛾𝑛𝑘𝑒𝑘𝛿𝑗𝑘 ∞ 𝑛=1 𝑗=1 𝑘=1 ̃ 󵄨 󵄨2 1+∑ 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑛󵄨 =0 for 𝑛>𝑝 ∞ ∞ 𝑘=1 = ∑ ∑ 𝜆 (𝑇(𝑝))⟨𝑥,𝑒̃ ⟩ 𝛾 𝛾 𝑒 𝑛 𝑗 𝑛𝑗 𝑛𝑗 𝑗 {𝑒 } 𝑛=1 𝑗=1 since the elements of 𝑛 are linearly independent. Then (43) holds under infinitely many combinations of constraints on ∞ ∞ ̃ 󵄨 󵄨2 𝑇(𝑝) : 𝐻 →𝐻 ̃ 󵄨 󵄨 the spectrum of .Inparticular,(43)holdsif = ∑ ∑ 𝜆𝑛 (𝑇(𝑝))󵄨𝛾𝑛𝑗 󵄨 ⟨𝑥,𝑗 𝑒 ⟩𝑒𝑗 𝑛=1 𝑗=1 ̃ 𝜆𝑛 (𝑇 (𝑝)) = 0, ∀𝑛 (∈ N) =𝑝+1;̸ (39) 1 ∞ ∞ 𝜆 (𝑇(𝑝))=−̃ ; 󵄨 󵄨2 𝑝+1 󵄨 󵄨2 ̃ ̃ 󵄨 󵄨 󵄨𝛾 󵄨 𝑇𝑇(𝑝)𝑥=𝑇(∑ ∑ 𝜆𝑘 (𝑇(𝑝))󵄨𝛾𝑘𝑗 󵄨 ⟨𝑥,𝑗 𝑒 ⟩𝑒𝑗) 󵄨 𝑝+1,𝑛󵄨 (44) 𝑘=1 𝑗=1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝛾𝑝+1,𝑛󵄨 =0 for 𝑛∈𝑝 󵄨𝛾𝑝+1,𝑛󵄨 =𝛾=0̸ ∞ ∞ ∞ 󵄨 󵄨2 ̃ 󵄨 󵄨 = ∑ 𝜆𝑛 (𝑇) ⟨∑ ∑ 𝜆𝑘 (𝑇(𝑝))󵄨 𝛾𝑘𝑗 󵄨 for 𝑛 (∈ N) >𝑝+1. 𝑛=1 𝑘=1 𝑗=1

Equations (43) are also satisfied with 𝛾𝑘𝑛 =0,forall𝑛∈𝑝, ∞ ̃ 2 ×⟨𝑥,𝑒⟩𝑒 ,𝑒 ⟩𝑒 𝛿 for all 𝑘∈N,and1+∑𝑘=1 𝜆𝑘(𝑇(𝑝))|𝛾𝑘𝑛| =0for 𝑛>𝑝which 𝑗 𝑗 𝑛 𝑛 𝑗𝑛 2 2 ∞ ̃ holds, for instance, if |𝛾𝑘𝑛| =|𝛾𝑛| =−1/(∑𝑘=1 𝜆𝑘(𝑇(𝑝))) for Abstract and Applied Analysis 9

̃ all 𝑛>𝑝.Thus,𝑇:𝐻is →𝐻 then non-unique, in general. instance, [1, 2, 7, 9, 16, 17, 19, 20] and the references therein. Properties (i)-(ii) have been proved. Two examples with the use of the above formalism to dynamic Now, let 𝜇(Γ) = {inf ‖Γ𝑥‖:𝑥∈𝐻,‖𝑥‖=1}be the systems and control issues are now discussed in detail. minimum modulus of the linear operator Γ:𝐻 →.If 𝐻 ‖𝑥‖ =,thenif 1 𝑇:𝐻is →𝐻 injective with closed image Example 1. Consider the forced linear time-invariant differ- (this implies that such an image is finite dimensional), then ential system of real coefficients and 𝑛th as 𝜇(𝑇) >0 and since 𝑇, 𝑇:𝐻are →𝐻 both bounded since 𝑠 𝑑𝑖𝑦 (𝑡) they are compact, one gets ∑ 𝛼 =𝛽𝑢(𝑡) 𝑛 𝑖 (47) 𝑖=0 𝑑𝑡 𝜇(𝑇(𝑝))≤𝜇(𝑇̃ 𝑇(𝑝))𝜇̃ −1 ( 𝑇) under a piecewise continuous square-integrable forcing func- 󵄩 󵄩 −1 2 ≤ max 󵄩𝑇𝑥−𝑇(𝑝)𝑥󵄩 𝜇 (𝑇) tion 𝑢:R0+ → R;thatis,𝑢∈𝐿(0, ∞),with𝛼𝑛 =0̸.The ‖𝑥(∈𝐻)‖=1 𝑖 𝑖 (45) unique solution for any given initial conditions (𝑑𝑦 (0))/𝑑𝑡 󵄩 󵄩 −1 𝑖=0,1,...,𝑠−1 ≤ 󵄩𝑇−𝑇(𝑝)󵄩 𝜇 (𝑇) for is 𝑡 󵄩 󵄩 −1 𝑇 𝐴𝑡 𝛽 𝐴(𝑡−𝜏) ≤(‖𝑇‖ + 󵄩𝑇(𝑝)󵄩)𝜇 (𝑇) <∞. 𝑦 (𝑡) =𝑐 (𝑒 𝑥 (0) + ∫ 𝑒 𝑏𝑢 (𝜏) 𝑑𝜏) , (48) 𝛼𝑛 0 𝑇:𝐻 →𝐻 𝜇(𝑇) =0 If is infinite dimensional, then 𝑇 𝑐, 𝑏 ∈ R𝑠 and it cannot then have bounded inverse. If 𝑇:𝐻 →𝐻 where the superscript stands for transposition, are 𝑞=𝑝 𝑇(𝑝)̃ Euclidean vectors of, respectively, first and last components is degenerated of dimension ,then is the null 𝑥(𝑡) = 𝜇(𝑇)̃ = 0 𝑇:𝐻 →𝐻 being unity and the remaining ones being zero operator with .If is degenerated of 𝑠−1 𝑠−1 𝑇 −1 −1 ∗ (𝑦(𝑡),(𝑑𝑦(𝑡))/𝑑𝑡,...,𝑑 𝑦(𝑡)/𝑑𝑡 ) dimension 𝑞>𝑝and invertible, then 𝜇 (𝑇) = 𝜇 (𝑇 )= ,and −1 ̃ −1 ‖𝑇 ‖<∞and ‖𝑇(𝑝)‖ ≤ ‖𝑇 ‖‖𝑇 − 𝑇‖ < ∞ so that 010⋅⋅⋅ 00 𝑇(𝑝)̃ : 𝐻 →𝐻 [ ] is bounded and compact since it is a [ 0010 ⋅⋅⋅ 0 ] (𝑇 − 𝑇(𝑝)) 𝐻 [ ] composite operator of a compact operator on 𝐴=[ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 01 ] . (49) 𝑇−1 𝐻 [ 𝛼 𝛼 𝛼 ] andaboundedoperator on . Property (iii) has been − 𝑛−1 − 𝑛−2 ⋅⋅⋅ − 0 𝛼 proved. [ 𝛼𝑛 𝛼𝑛 𝑛 ]

𝑇, 𝑇(𝑝) : 𝐻→ 𝐴𝑡 Example 7. Assume that are two degen- The matrix function 𝑒 is a 𝐶0-semigroup generated by erated finite-dimensional operators on a separable Hilbert the infinitesimal generators 𝐴,respectively[17, 19]. Using a 𝐻 space of, respectively, dimensions two and one defined by sampling period of length 𝜃,wecanwritefrom(48)fortime 𝑇𝑥=𝜆(𝑇)⟨𝑥, 𝑒 ⟩𝑒 +𝜆(𝑇)⟨𝑥, 𝑒 ⟩𝑒 𝑥∈𝐻 1 1 1 2 2 2;forall and instants being integer multiples of the sampling period 𝑇(𝑝)𝑥=𝜆1(𝑇)⟨𝑥,1 𝑒 ⟩𝑒1;forall𝑥∈𝐻.Thus,theconstraints ̃ (42) hold for an incremental operator 𝑇(𝑝) : 𝐻 →𝐻 of 𝑥𝑛+1 := 𝑥 ((𝑛+1) 𝜃) =𝑇(𝑢, 𝜃, 𝑛) 𝑥𝑛 ̃ ̃ 2 spectrum defined by 𝜆1(𝑇(𝑝)), =0 𝜆2(𝑇(𝑝)) = −1/|𝛾22| with =𝑇ℎ (𝜃) 𝑥𝑛 +𝑇𝑓 (𝜃) 𝑢 (𝑛,) 𝜃 𝛾22 =0̸ and 𝛾21 =0.Then, 𝜃 −1 𝐴𝜃 𝛽 −𝐴𝜏 𝑇(𝑝)𝑥=̃ ⟨𝑥, 𝑒 ⟩𝑒 ; =𝑒 (𝑥𝑛 + ∫ 𝑒 𝑏𝑢 󵄨 󵄨2 2 2 𝑔 0 󵄨𝛾22󵄨 ̃ ̃ 󵄨 󵄨2 (46) 𝑇𝑇(𝑝)𝑥=−𝜆2 (𝑇) 𝜆2 (𝑇(𝑝))󵄨𝛾22󵄨 ⟨𝑥,2 𝑒 ⟩𝑒2 × (𝑛𝜃) +𝜏 𝑑𝜏 ) , ∀𝑛∈ N0 = N ∪ {0} , =−𝜆 𝑇 ⟨𝑥, 𝑒 ⟩𝑒 . 2 ( ) 2 2 𝑇 𝑇 𝐴𝜃 𝑦𝑛+1 := 𝑐 𝑥 ((𝑛+1) 𝜃) =𝑐 𝑒 Remark 8. If 𝑇:𝐻is →𝐻 infinite dimensional and ̃ 𝛽 𝜃 invertible, then 𝑇(𝑝) : 𝐻 →𝐻 is not compact, since ×(𝑥 + ∫ 𝑒−𝐴𝜏𝑏𝑢 (𝑛𝜃) +𝜏 𝑑𝜏) , ∀𝑛∈ N , −1 −1 𝑛 0 𝑇 :𝐻 →is 𝐻 unbounded, since 𝜇(𝑇) = 0 ⇔𝜇 (𝑇) = 𝑔 0 −1 ‖𝑇 ‖=∞. (50)

where 𝑥𝑛 =𝑥(𝑛𝜃)and 𝑔=𝛼𝑛 provided that the input is 𝑢𝑛 = 3. Examples 𝑢𝑛(𝜃) =𝑛 𝑢 (𝜏),forall𝜏 ∈ [𝑛𝜃, (𝑛.Thematrixfunction +1)𝜃) 𝑒𝐴𝜃 Hilbert spaces for the formulation of equilibrium points, can be expanded as follows: stability, controllability [16, 18, 19], boundedness, and square 𝜇−1 𝜇−1 ]𝑘−1 𝜗 𝐴𝜃 𝑘 𝑗 𝜆ℓ𝜃 𝑘 integrability (or summability in the discrete formalism) of the 𝑒 = ∑ 𝛼𝑘 (𝜃) 𝐴 = ∑ ( ∑ ∑ 𝛾𝑗𝑘𝜃 𝑒 ) 𝐴 , (51) solution in the framework of square-integrable (or square- 𝑘=0 𝑘=0 𝑗=0 ℓ=0 summable) control and output functions are of relevant importance in signal processing and control theory and in where 𝜎(𝐴) = {𝜆𝑘 : 𝑘 = 0,1,...,𝜗−1}is the spectrum of general formulations of dynamic systems, in general. See, for 𝐴,thatis,setof𝜗 distinct eigenvalues of 𝐴 with respective 10 Abstract and Applied Analysis

multiplicities ]𝑘 for 𝑘 = 0,1,...,𝜇 −1 in the minimal Proposition 2 (constant piecewise constant open-loop con- 𝜗 (𝜆) < 0 𝜆∈𝜎(𝐴) polynomial of 𝐴 where 𝜇=∑ ]𝑘 is the degree of the trol). Assume that Re ,forall , and consider a 𝑘=1 𝑢 =𝑢 𝑛∈N minimal polynomial of 𝐴,andthen1≤𝜇≤𝑠and 𝛾𝑗𝑘 constant open-loop control 𝑛 0,forall .Thefollowing are complex constants. The above 𝛼𝑘(𝑡); 𝑘 = 0,1,...,𝜇− 1 properties hold. are everywhere continuous and linearly independent time- (i) The sequence {𝑦𝑛}𝑛∈N satisfies 𝑦𝑛+1 =𝑇𝑦𝑛 =𝑇𝑛+1𝑦0, differentiable functions on R. Then, the unique solution (or 0+ subject to 𝑦0 =𝑢0ℎ0,forall𝑛∈N0, where the operator output) of (47) for zero initial conditions is 𝑇:N0 × R → R is defined as the sequence of scalar 𝑛+1 𝑛 ∞ gains {∑𝑖=0 ℎ𝑛+1−𝑖/∑𝑖=0 ℎ𝑛−𝑖},forall𝑛∈N in the 𝑦 (𝑡) =(Λ𝑐𝑢) (𝑡) = ∫ ℎ (𝑡, 𝜏) 𝑢 (𝜏) 𝑑𝜏 Banach space (R,| ⋅ |) which is the Euclidean Hilbert 0 space for the product of real numbers being an inner 𝑡 (52) ∗ product. Furthermore, {𝑦𝑛}𝑛∈N →𝑦. = ∫ ℎ (𝑡−𝜏) 𝑢 (𝜏) 𝑑𝜏 0 0 (ii) Assume that 𝑝(∈ N)≥𝑝0 for some given 𝑝0 ∈ N,and 𝑝0−1 𝑁 2 2 |𝑢0| 𝑡) ∈ R0+. and 0 such that 𝑛 𝑛 ,forall N )≥𝑝≥𝑝 𝑢 ∈ R Thus, such an operator is normal, since it is time invariant 0 0.Also,foreachgiven 0 satisfying 𝑁 [1], and then self-adjoint. Now, define the sequence of samples ∃ lim𝑁→∞(|𝑢0|max0≤𝑖≤𝑝−1(|ℎ𝑛−𝑖|)) =0,forall𝑛(∈ N)≥𝑝−1 {𝑦𝑛 := 𝑦(𝑛𝜃)}𝑛∈N for a sampling period 𝜃 as ,itfollowsthat 󵄨 󵄨𝑁 𝑁 𝑁 lim 󵄨𝑦𝑛 − 𝑦 (𝑝)󵄨 =0, lim (𝑦 − 𝑦 (𝑝)) = 0, 𝑦𝑛 := Λ𝑢̂𝑛 =(Λ𝑐𝑢) (𝑛𝜃) 𝑁→∞󵄨 𝑛 󵄨 𝑁→∞ 𝑛 𝑛 𝑛𝜃 (53) ∀𝑛 (∈ N) ≥𝑝−1. = ∫ ℎ (𝑛𝜃, 𝜏) 𝑢 (𝜏) 𝑑𝜏; ∀𝑛 ∈ N 0 (56) 𝑛 with the operator Λ being defined from Λ 𝑐 on the space Proof. Property (i) follows from 𝑦𝑛 =𝑢0(∑𝑖=0 ℎ𝑛−𝑖),orequiv- ℓ2(0, ∞) 𝑢̂ := 𝑛+1 𝑛 of square-summable sequences ,where 𝑛 alently, 𝑦𝑛+1 =(∑𝑖=0 ℎ𝑛+1−𝑖)/(∑𝑖=0 ℎ𝑛−𝑖)𝑦𝑛,forall𝑛∈N0 (𝑢0,𝑢1,...,𝑢𝑛−1),forall𝑛∈N. Assume that the forcing input subject to an initial condition 𝑦0 =𝑢0ℎ0.Since{ℎ𝑛} is 𝑢(𝑡) =𝑢 =𝑢(𝑛𝜃) 𝑛∈N 𝑛 𝑛 is piecewise constant, for all ,forall bounded, {ℎ𝑛}→0as 𝑛→∞,and∑𝑖=0 ℎ𝑖 =𝐻<+∞, 𝑡 ∈ [𝑛𝜃, (𝑛+1)𝜃) ℎ =0 ℎ (𝑠) 𝑛 ∗ ∞ .Notethatif 0 ,then 𝐿 ,theunilateral then 𝑦𝑛 =𝑢0( ∑𝑖=0 ℎ𝑛−𝑖)→𝑦 =𝑢0𝐻=𝑢0(∑𝑖=0 ℎ𝑛−𝑖)< Laplace transform of ℎ(𝑡), is strictly proper; that is, it has +∞ {𝑦 } 𝑦 = .Thus,thesequence 𝑛 𝑛∈N0+ is generated as 𝑛+1 more poles than zeros. In the case that ℎ0 = ℎ(0) =0̸, ℎ𝐿(𝑠) is 𝑛+1 𝑇𝑦𝑛 =𝑇 𝑦0,subjectto𝑦0 =𝑢0ℎ0,forall𝑛∈N0, proper by not strictly proper; that is, it has the same number where the operator 𝑇:N0 × R → R is defined in of poles and zeros. It turns out that we can define an operator (R,| ⋅ |) ̂ 2 2 the Banach space asthesequenceofscalargains sequence 𝑇𝑛 :ℓ[0, ∞) → ℓ [0, 𝑛 + 1]:forall𝑛∈N,withthe {(∑𝑛+1 ℎ )/(∑𝑛 ℎ )} 𝑛∈N 𝑃 ℓ2[0, 𝑛 + 1] 𝑖=0 𝑛+1−𝑖 𝑖=0 𝑛−𝑖 ,forall which is the second one being a natural projection 𝑛+1 on of EuclideanHilbertspacefortheproductofrealnumbers 𝑇̂ ℓ2[0, ∞] 𝑇̂ :ℓ2[0, ∞) → ∗ an operator on so that, by using 𝑛 being an inner product. Furthermore, {𝑦𝑛}𝑛∈N →𝑦. ℓ2 [0, 𝑛 + 1] 𝑛∈N 0 ;forall ,onegets: Property (i) has been proved. On the other hand, since |𝑢0|< 𝑝 −1 (1/𝐻, 1/(∑ 0 |ℎ |)) |𝑦∗|≤|𝑢𝐻| < 1 𝑦̂ = 𝑇̂ 𝑦̂ = 𝑇̂𝑦;̂ ∀𝑛 ∈ N min 𝑖=0 𝑛−𝑖 ,itfollowsthat 0 𝑛+1 𝑛 𝑛 (54) and then 󵄨 󵄨 󵄨 𝑛 󵄨 𝑇 󵄨 𝑛 󵄨 󵄨 󵄨 𝑦̂ =(𝑦,𝑦 ,...,𝑦 ,0,0...) 𝑛∈N 𝑦=(𝑦̂ ,𝑦 , 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨∑𝑖=0 ℎ𝑛−𝑖󵄨 with 𝑛 0 1 𝑛 ;forall , 0 1 󵄨𝑦 󵄨 = 󵄨𝑢 󵄨 (󵄨∑ ℎ 󵄨)< 󵄨 󵄨 ≤1 ∀𝑛∈N, 𝑇 󵄨 𝑛󵄨 󵄨 0󵄨 󵄨 𝑛−𝑖󵄨 ∑∞ 󵄨ℎ 󵄨 ...,𝑦𝑛,𝑦𝑛+1,𝑦𝑛+2,...) , 𝑦̂0 =𝑦0 =𝑇0𝑦0,with𝑇0 being the 󵄨𝑖=0 󵄨 𝑖=0 󵄨 𝑖󵄨 identity operator. One has from (51)that 󵄨 󵄨 󵄨 ∗󵄨 (57) 󵄨𝑦𝑛󵄨 󳨀→ 󵄨𝑦 󵄨 <1 as 𝑛󳨀→∞, ℎ =ℎ(𝑛𝜃) 𝑛 󵄨 󵄨𝑁 󵄨𝑦𝑛󵄨 󳨀→ 0 as 𝑁󳨀→∞,∀𝑛∈N, 𝜇−1 ] −1 𝑇 𝑘 𝜗 (55) 𝛽𝑐 𝑗 𝜆 𝑛𝜃 𝑘 󵄨 󵄨 𝑝−1 󵄨 󵄨 = (∑ ∑ ∑ 𝛾 (𝑛𝜃) 𝑒 ℓ 𝐴 )𝑏; ∀𝑛∈N 󵄨𝑝−1 󵄨 ∑ 󵄨ℎ 󵄨 𝑔 𝑗𝑘 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖=0 󵄨 𝑛−𝑖󵄨 𝑘=0 𝑗=0 ℓ=0 󵄨𝑦𝑛 (𝑝)󵄨 = 󵄨𝑢0󵄨 󵄨∑ ℎ𝑛−𝑖󵄨 < 󵄨 󵄨 𝑝0−1 󵄨 󵄨 󵄨 𝑖=0 󵄨 ∑𝑖=0 󵄨ℎ𝑛−𝑖󵄨 and ℎ𝑛 →0as 𝑛→∞if Re(𝜆𝑖)<0; ℓ=0,1,...,𝜗−1.Some 𝑝−1 󵄨 󵄨 ∑ 󵄨ℎ 󵄨 (58) particular cases are discussed below under the assumption ≤ 𝑖=0 󵄨 𝑛−𝑖󵄨 =1, ∀𝑛∈N, {𝑢 }⊂ℓ2[0, ∞) (𝜆 )<0 ℓ = 0,1,...,𝜗−1 𝑝−1 󵄨 󵄨 𝑛 and Re 𝑖 ; implying ∑𝑖=0 󵄨ℎ𝑛−𝑖󵄨 2 𝑛 {ℎ𝑛}⊂ℓ[0, ∞), {|ℎ𝑛|} ⊂ ℓ[0, ∞),sothat∑𝑖=0 |ℎ𝑖|=𝐻<+∞ 𝑛 󵄨 󵄨𝑁 and ∑𝑖=0 ℎ𝑖 =𝐻<+∞,since{ℎ𝑛} is bounded. 󵄨𝑦𝑛 (𝑝) 󵄨 󳨀→ 0 as 𝑁󳨀→∞,∀𝑛∈N. Abstract and Applied Analysis 11

𝑛 Property (ii) has been proved. Now, note that for any vector in R with its 𝑖th component being one, such that 𝜀∈R 𝑢 ∈ R 𝑝 =𝑝(𝜀, 𝑢 )∈N 1/2 given + and 0 ,thereis 0 0 0 such the set {|𝑢0| √|ℎ𝑖|V𝑖}𝑖∈N is an orthonormal basis so that that for any 𝑝(≥𝑝0) ∈ N 1/2 1/2 ⟨|𝑢0| √|ℎ𝑖|V𝑖,|𝑢0| √|ℎ𝑗|V𝑗⟩=𝛿𝑖𝑗 as 󵄨 󵄨 󵄨 𝑛 󵄨 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑦 − 𝑦 (𝑝)󵄨 = 󵄨𝑢 󵄨 󵄨 ∑ ℎ 󵄨 ≤ 󵄨𝑢 󵄨 ( ∑ 󵄨ℎ 󵄨) 󵄨 𝑛 𝑛 󵄨 󵄨 0󵄨 󵄨 𝑛−𝑖󵄨 󵄨 0󵄨 󵄨 𝑛−𝑖󵄨 𝑛 󵄨𝑖=𝑝+1 󵄨 𝑖=𝑝0+1 𝑦𝑛 (𝑝) = ∑ ℎ𝑛−𝑖𝑢𝑖 (59) 𝑖=𝑛−𝑝+1 ∞ 󵄨 󵄨 󵄨 󵄨 (63) ≤ 󵄨𝑢0󵄨 ( ∑ 󵄨ℎ𝑛−𝑖󵄨)≤𝜀 𝑛−1 𝑖 𝑖=𝑝 +1 1 0 = ( ∑ ∑ ℎ𝑛−𝑖𝑠𝑖𝑗 𝑦𝑗), ∀𝑛∈N. 1−ℎ0𝑠𝑛𝑛 𝑖=𝑛−𝑝+1 𝑗=0 𝑝 =𝑝(𝜀, 𝑢 )∈N (∑∞ |ℎ |) ≤ 𝜀|𝑢 |−1 for 0 0 0 satisfying 𝑖=𝑝0+1 𝑛−𝑖 0 if 𝑢0 =0̸ and such a 𝑝0 exists since {|ℎ𝑛|} ⊂ ℓ[0, ∞).Notethat if 𝑢0 =0,then|𝑦𝑛 − 𝑦𝑛(𝑝)| = 0 so that |𝑦𝑛 − 𝑦𝑛(𝑝)| ≤ 𝜀 𝑝 =𝑝(𝜀) ∈ N for any 0 0 . The first part of Property (iii) has Example 2. Consider again (47)withRe(𝜆) <,forall 0 𝜆∈ ∃ |𝑦𝑁 − 𝑦𝑁(𝑝)| = 0 been proved. Note that lim𝑁→∞ 𝑛 𝑛 .Then, 𝜎(𝐴). If one measures some more state variables than just the the second part of Property (iii) follows since solution, then an extended solution (48)oftheform 󵄨 󵄨 󵄨𝑦𝑁 − 𝑦𝑁 (𝑝)󵄨 lim sup 󵄨 𝑛 𝑛 󵄨 𝑇 𝑁→∞ 𝑧 (𝑡) =(𝑦(𝑡) ,𝑥0𝑇 (𝑡)) (60) 󵄨 󵄨 󵄨 󵄨𝑁−1 󵄨 󵄨𝑁−1 𝑡 (64) ≤𝐻󵄨𝑢0󵄨 lim sup (󵄨𝑢0󵄨 max 󵄨ℎ𝑛−𝑖 󵄨 )=0. 𝐴𝑡 𝐴(𝑡−𝜏) 𝑁→∞ 0≤𝑖≤ 𝑝 =𝐶(𝑒 𝑥 (0) + ∫ 𝑒 𝐵𝑢 (𝜏) 𝑑𝜏) 0 ̂ Property (iii) follows from Theorem 5 with the operator 𝑇𝑛 : 2 2 𝑠0 ℓ [0, 𝑛] → ℓ [0,𝑛+1] 𝑛∈N is built with 𝑧:R0+ → R which is the output; 1≤𝑠0 ≤ ,forall of (58)andits 0 ̂ 2 𝑠, 𝑧(𝑡) = 𝐶𝑥(𝑡) and 𝑥 (𝑡) isformedbyallorsomeofthe degenerated finite truncation 𝑇𝑛(𝑝) : ℓ [0, 𝑝],forall𝑛∈N in 𝑠 ×𝑠 𝑠×𝑠 𝑥(𝑡) 𝑦(𝑡) 𝐶∈R 0 𝐵∈R 𝑚 the subsequent way components of except , ,and where 𝑠𝑚 ≥1is the dimension of the piecewise-continuous 󵄨 󵄨 󵄩 󵄩 𝑠𝑚 2 0 󵄨 󵄨 󵄩 󵄩 input 𝑢:R0+ → R which is in 𝐿𝑠 [0, ∞).If𝑥 (𝑡) is not 󵄨𝑦𝑛 − 𝑦𝑛 (𝑝)󵄨 = 󵄩𝑦̂𝑛 − 𝑦̂𝑛 (𝑝)󵄩 𝑚 𝑧(𝑡) = 𝑦(𝑡) 𝑠 =1 𝑧(𝑡) = 𝑥(𝑡) 󵄩 󵄩 used to (64), then and 0 .If ,then 󵄩̂ ̂ ̂ ̂ 󵄩 𝑠 =𝑠 = 󵄩𝑇𝑛−1𝑦𝑛−1 − 𝑇𝑛−1 (𝑝) 𝑦𝑛−1󵄩 0 .Equation(64) can be expressed as ∞ 󵄨 𝑛 󵄨 󵄨 𝑛 󵄨 󵄨 󵄨1/2 𝑧 (𝑡) =(Λ 𝑥 ) (𝑡) +(Λ 𝑢) (𝑡) = ∑ ⟨󵄨𝑇 𝑦0 − 𝑇 (𝑝)0 𝑦 󵄨 , 󵄨𝑢0󵄨 𝑔𝑖V𝑖⟩ ℎ 0 𝑓 𝑖=1 ∞ 󵄨 󵄨1/2 × 󵄨𝑢 󵄨 𝑔 V , =𝑇ℎ (𝑡) 𝑥0 + ∫ 𝑇𝑓 (𝑡, 𝜏) 𝑢 (𝜏) 𝑑𝜏 (65) 󵄨 0󵄨 𝑖 𝑖 0 󵄨 󵄨 󵄩 󵄩𝑁 (61) 󵄨𝑦𝑁 − 𝑦𝑁 (𝑝)󵄨 = 󵄩𝑇̂ 𝑦̂ − 𝑇̂ (𝑝) 𝑦̂ 󵄩 =𝑇 (𝑡) 𝑥 +(𝑇̂ 𝑢 ) (𝑡) ,∀𝑡∈R 󵄨 𝑛 𝑛 󵄨 󵄩 𝑛−1 𝑛−1 𝑛−1 𝑛−1󵄩 ℎ 0 𝑓 𝑒 0+ ∞ 󵄨 𝑛 󵄨 = ∑ ⟨ 󵄨𝑇𝑛𝑦 − 𝑇 (𝑝) 𝑦 󵄨 , 𝐴𝑡 2 󵄨 0 0󵄨 with 𝑥0 = 𝑥(0), 𝑇ℎ(𝑡) = 𝐶𝑒 and the operators 𝑇𝑓 : R0+ × 𝑖=1 𝑠 𝑠 2 2 R 𝑚 → R 0 𝑇̂ :𝐿 (−∞, ∞) → 𝐿 (−∞, ∞) and 𝑓 𝑠𝑚 𝑠0 are 󵄨 𝑛 󵄨𝑁−1 󵄨 𝑛 󵄨 defined as 󵄨𝑇 𝑦0 − 𝑇 (𝑝)0 𝑦 󵄨 󵄨 󵄨1/2 󵄨 󵄨1/2 𝐴(𝑡−𝜏) × 󵄨𝑢0󵄨 𝑔𝑖V𝑖⟩ 󵄨𝑢0󵄨 𝑔𝑖V𝑖, 𝑇𝑓 (𝑡, 𝜏) =𝑇𝑓 (𝑡−𝜏) =𝐶𝑒 𝐵; ∀𝑡 ∈ R0+ (66) ∞ 𝑇:R → R ̂ 𝐴(𝑡−𝜏) where maps each element of the sequence (𝑇𝑓𝑢𝑒) (𝑡) = ∫ 𝐶𝑒 𝐵𝑢𝑒 (𝜏) 𝑤𝑡 (𝜏) 𝑑𝜏 {𝑦 } −∞ 𝑛 𝑛∈N0 , which is strictly ordered according to the time occurrence, to its next consecutive one, ∞ = ∫ 𝑇𝑓 (𝑡−𝜏) 𝑢 (𝜏) 1 (𝑡−𝜏) 𝑑𝜏 0 √ ℎ𝑖 if ℎ𝑖 ≥0 𝑔𝑖 ={ 󵄨 󵄨 (62) ∞ 𝑖√󵄨ℎ 󵄨 ℎ <0 󵄨 𝑖󵄨 if 𝑖 = ∫ 𝑇𝑓 (𝑡−𝜏) 𝑢𝑒 (𝜏) 1 (𝑡−𝜏) 𝑑𝜏 −∞ 2 ∞ (then 𝑔𝑖 =−ℎ̸ 𝑖 inthesecondpartof(62)), for all 𝑖∈ N {V } V = = ∫ 𝑇𝑓 (𝑡−𝜏) 𝑤𝑡 (𝜏) 𝑢𝑒 (𝜏) 𝑑𝜏; ∀𝑡 ∈ R0+ 0,and 𝑖 𝑖∈N is a basis of orthogonal vectors 𝑖 −∞ −1/2 |𝑢0ℎ𝑖| 𝑒𝑖 if 𝑢0ℎ𝑖 =0̸ and V𝑖 =0,where𝑒𝑖 is the 𝑖th unit (67) 12 Abstract and Applied Analysis

𝑇 (𝑡, 𝜏) =0 𝜏>𝑡 𝐿2 (−∞, ∞) so that 𝑓 for is a convolution operator, where 𝑠0 and one has from (67)to(69)that 2 𝑠 𝑢 : R →𝐿 (−∞, ∞)∩𝑃𝐶(R, R 𝑚 ) 𝑒 𝑠𝑚 is piecewise continuous on R and square integrable defined as 𝑢𝑒(𝑡) = 𝑢(𝑡) for 𝑡∈R0+ 𝑢(𝑡) =0 𝑤 ∞ 󵄩 󵄩2 and ;otherwise, 𝑡 is a truncated multiplicative (or ∬ 󵄩(𝑇 (𝑡−𝜏) −𝑇 (𝑝, 𝑡 − 𝜏)) 𝑤 (𝜏)󵄩 𝑑𝜏𝑑𝑡 (−∞, 𝑡] ∩ R (0, 1) 󵄩 𝑓 𝑓 𝑡 󵄩 truncated gate) from to for each defined as −∞ (71) 𝑤𝑡(𝜏) = 1 for 0≤𝜏≤𝑡and 𝑤𝑡(𝜏) =,otherwise,forall 0 2 󳨀→ 0 𝑝󳨀→∞ 𝑡∈R.Notethat𝑤𝑡(𝜏) = 1(𝑡 − 𝜏)1(𝜏),forall(𝑡, 𝜏) ∈ R .The as multiplicative (or gate) operator 𝑤 from R to (0, 1) is defined ∞ 𝑤(𝑡) =1 𝑡≥0 𝑤(𝑡) =1 𝑡∈R as for and ,otherwise;forall . 𝑇𝑓 (𝑡−𝜏) = ∑ 𝜓𝑖 (𝑡) 𝜃𝑖 (𝜏) , Now, 𝑖=1 𝑝

𝑝 𝑇𝑓 (𝑝, 𝑡 − 𝜏) = ∑ 𝜓𝑖 (𝑡) 𝜃𝑖 (𝜏) , ̂ ̂ 𝑖=1 (𝑇𝑓 (𝑝)𝑒 𝑢 ) (𝑡) = ∑ ⟨(𝑇𝑓𝑢) (𝑡) ,𝜃𝑖 (𝑡)⟩𝜑𝑖 (𝑡) 𝑖=1 ∞ (72) 𝜓𝑖 (𝑡) = ∫ 𝑇𝑓 (𝑡−𝜏) 𝜑𝑖 (𝜏) 𝑑𝜏, ∀𝑡 ∈ R0+ ∞ −∞ = ∫ 𝑇𝑓 (𝑝, 𝑡 − 𝜏)𝑒 𝑢 (𝜏) 1 (𝑡−𝜏) 𝑑𝜏 󵄩 󵄩 −∞ 󵄩 ̂ ̂ 󵄩 󵄩((𝑇𝑓 − 𝑇𝑓 (𝑝))𝑒 𝑢 ) (𝑡)󵄩 ∞ 󵄩 󵄩 󵄩 ∞ 󵄩 = ∫ 𝑇𝑓 (𝑝, 𝑡 − 𝜏)𝑡 𝑤 (𝜏) 𝑢𝑒 (𝜏) 𝑑𝜏, 󵄩 󵄩 −∞ = 󵄩 ∑ ⟨(𝑇̂ 𝑢 ) (𝑡) ,𝜃 (𝑡)⟩𝜑 (𝑡)󵄩 ,∀𝑡∈R 󵄩 𝑓 𝑒 𝑖 𝑖 󵄩 0+ (68) 󵄩𝑖=𝑝+1 󵄩 󵄩 󵄩 󵄩 ∞ 󵄩 󵄩 ̂ ̂ 󵄩 󵄩 ̂ 󵄩 2 sup 󵄩(𝑇𝑓 − 𝑇𝑓 (𝑝))𝑒 𝑢 󵄩 = sup 󵄩 ∑ ⟨(𝑇𝑓𝑢𝑒),𝜃𝑖⟩𝜑𝑖󵄩 where ⟨⋅, ⋅⟩ is the inner product on 𝐿𝑠 (−∞, ∞), 𝑇𝑓(𝑝, 𝑡 − 𝜏) 󵄩 󵄩 󵄩 󵄩 0 ‖𝑢‖=1 ‖𝑢‖=1 󵄩𝑖=𝑝+1 󵄩 (𝑇̂ (𝑝))(𝑡) {𝜃 } {𝜑 } 𝑖∈𝑝 is the kernel of 𝑓 ; 𝑖 and 𝑖 ; are two 󵄨 󵄨 𝑝 󵄨 ̂ 󵄨 reciprocal orthogonal bases of the th dimensional subspace = 󵄨𝜆𝑝+1 (𝑇𝑓)󵄨 𝑀 𝐿2 (−∞, ∞) 𝑇̂ (𝑝) 𝑢 ∈𝐿2 (−∞, ∞) 𝑝 of 𝑠0 and 𝑓 maps 𝑒 𝑠𝑚 (73) ̂ in the orthogonal projection of (𝑇𝑓𝑢𝑒)(𝑡) on 𝑀𝑝,forall ̂ 𝑡∈R0+.Notethat𝑇𝑓(𝑝) is a self-adjoint, since it is 𝑢 ∈𝐿2 (−∞, ∞) 𝜆 (𝑇̂ )∈𝜎(𝑇̂ ) time invariant (and convolution), compact operator since with 𝑒 𝑠𝑛 , 𝑝+1 𝑓 𝑓 , being nonzero for its image is finite dimensional. On the other hand, note 𝑝 𝜓 : R →𝐿2 (−∞, ∞) 𝑖∈N any finite , 𝑖 𝑠0 ;forall is a linearly that ̂ independent set, since the kernel 𝑇𝑓(𝑡 − 𝜏) of 𝑇𝑓 is bounded 2 and 𝜓𝑖 : R →𝐿𝑠 (−∞, ∞),forall𝑖∈N,andwherethe ∞ 0 󵄩 󵄩2 2 ∬ 󵄩𝑇 (𝑡−𝜏) 𝑤 (𝜏)󵄩 𝑑𝜏𝑑𝑡 norm is associated with the inner product on 𝐿𝑠 (−∞, ∞). 󵄩 𝑓 𝑡 󵄩 0 −∞ Equation (70) describes the truncated error norm on (−∞, 𝑡] ∞ ̂ ̂ 󵄩 󵄩2 of (𝑇𝑓 − 𝑇𝑓(𝑝))𝑢𝑒 in (71), for all 𝑡∈R0+ via the formula (69) 󵄩 󵄩 󵄨 󵄨2 (69) = ∬ 󵄩𝑇𝑓 (𝑡−𝜏)󵄩 󵄨𝑤𝑡 (𝜏)󵄨 𝑑𝜏𝑑𝑡 (−∞, ∞) −∞ while (71) refers to the whole real interval .From 󵄩 󵄩 𝑝 =𝑝(𝜀) ‖𝑇̂ −𝑇̂ (𝑝)‖ ≤ 𝜀 𝑝≥ 󵄩 󵄩 (73), there is 0 0 such that 𝑓 𝑓 for any = ‖w‖ 󵄩t𝑓󵄩 <+∞ ̂ 𝑝0 and any prefixed 𝜀∈R+.Since(𝑇𝑓𝑢𝑒)(𝑡) = 𝑧(𝑡)ℎ −𝑇 (𝑡)𝑥0, ̂ ̂ (𝑇𝑓(𝑝)𝑢𝑒)(𝑡) =𝑝 𝑧 (𝑡)−𝑇ℎ(𝑡)𝑥0,forall𝑡∈R0+,and|𝜆𝑛(𝑇𝑓)| → ̂ 2 2 ̂ 2 2 𝑇 :𝐿 (−∞, ∞) → 𝐿 (−∞, ∞) 0 as 𝑛→∞since 𝑇𝑓 :𝐿𝑠 (−∞, ∞) → 𝐿𝑠 (−∞, ∞) is so that 𝑓 𝑠𝑚 𝑠0 has a square- 𝑚 0 integrable kernel so that it is a Hilbert-Schmidt operator, then compact and then compact, and also self-adjoint since it is time invariant. Note that ‖𝑢𝑒‖ = ‖𝑢‖.Thus, 󵄩 󵄩 󵄩 󵄩 󵄩 ∞ 󵄩 󵄩 ̂ ̂ 󵄩 󵄩 ̂ 󵄩 󵄩(𝑇𝑓 − 𝑇𝑓 (𝑝))𝑒 𝑢 󵄩 = 󵄩 ∑ ⟨(𝑇𝑓𝑢𝑒),𝜃𝑖⟩𝜑𝑖󵄩 ∞ 󵄩 󵄩 ̂ 𝐴(𝑡− 𝜏) 󵄩𝑖=𝑝+1 󵄩 (𝑇𝑓𝑢𝑒) (𝑡) = ∫ 𝐶𝑒 𝐵𝑢𝑒 (𝜏) 𝑤𝑡 (𝜏) 𝑑𝜏 −∞ 󵄨 ̂ 󵄨 = 󵄨𝜆𝑝+1 (𝑇𝑓)󵄨 ‖𝑢‖ ∞ (70) 󵄨 󵄨 = ∑ ⟨(𝑇̂ 𝑢 ) (𝑡) ,𝜃 (𝑡)⟩𝜑 (𝑡) , 𝑓 𝑒 𝑖 𝑖 ≤𝜀‖𝑢‖ ,∀𝑝≥𝑝0,∀𝑡∈R0+, 𝑖=1 󵄩 󵄩 󵄩 󵄩 󵄩 ̂ ̂ 󵄩 󵄩 󵄩 lim sup 󵄩(𝑇𝑓 − 𝑇𝑓 (𝑝))𝑒 𝑢 (𝑡)󵄩 = lim sup 󵄩𝑧 (𝑡) −𝑧𝑝 (𝑡)󵄩 𝑡→∞ 𝑡→∞ where {𝜃𝑖} and {𝜑𝑖}; 𝑖∈N are two orthogonal complete 󵄨 󵄨 󵄨 ̂ 󵄨 systems of the infinite-dimensional separable Hilbert space ≤ 󵄨𝜆𝑝+1 (𝑇𝑓)󵄨 ‖𝑢‖ , Abstract and Applied Analysis 13

󵄩 󵄩 ( 󵄩(𝑇̂ − 𝑇̂ (𝑝)) 𝑢 𝑡 󵄩) it follows that 𝑥∈𝐿∞ with sup𝑡∈R ‖𝑥(𝑡)‖2 ≤𝑀<+∞,and lim lim sup 󵄩 𝑓 𝑓 𝑒 ( )󵄩 0+ 𝑝→∞ 𝑡→∞ then 󵄩 󵄩 󵄩 󵄩 = lim (lim sup 󵄩𝑧 (𝑡) −𝑧𝑝 (𝑡)󵄩) 𝑝→∞ 𝑡→∞ 󵄩 ∗󵄩 𝐾𝐴 lim sup‖𝑥 (𝑡)‖2 ≤ 󵄩𝑧 󵄩2 + (‖𝑢‖) ,∀𝑡∈R0+, (77a) 󵄨 󵄨 𝑡→∞ 𝜌 󵄨 ̂ 󵄨 𝐴 ≤ lim 󵄨𝜆𝑝+1 (𝑇𝑓)󵄨 =0, 𝑝→∞ 󵄩 ∗󵄩 lim sup‖𝑧 (𝑡)‖2 ≤ 󵄩𝑧 󵄩2 (74) 𝑡→∞ 𝛿 ‖𝐶‖‖𝐴‖ (77b) + 2 ‖𝑥 (𝑡)‖ ,∀𝑡∈R , concludingthefollowing:(a)thetrueandapproximateforced sup 2 0+ 𝐾𝐴𝜌𝐴 𝑡∈R andcompletesolutionsmightbemadeascloseassuited, 0+ in terms of difference of norms, by using a finite-range 󵄩 󵄩 󵄩 󵄩 𝛿 ‖𝐶‖‖𝐴‖ 𝑀 󵄩𝑧 (𝑡)󵄩 ≤ 󵄩𝑧∗󵄩 +𝜀(𝑝)‖𝑢‖ + 2 , operator approximant of sufficiently large range dimension; lim sup󵄩 𝑝 󵄩2 󵄩 󵄩2 ∗ 𝑡→∞ 𝐾𝐴𝜌𝐴 (b) if the true asymptotic solution is a fixed point 𝑧 = ∞ (77c) ∫ 𝐶𝑒𝐴(𝑡−𝜏)𝐵𝑢(𝜏)𝑑𝜏 0 ,then 󵄩 󵄩 󵄩 ∗󵄩 𝛿 ‖𝐶‖‖𝐴‖2𝑀 lim (lim sup 󵄩𝑧𝑝 (𝑡)󵄩 )≤󵄩𝑧 󵄩 + 󵄩 󵄩 󵄩 󵄩 𝑝→∞ 󵄩 󵄩2 󵄩 󵄩2 𝐾 𝜌 (77d) 󵄩𝑧 (𝑡)󵄩 ≤ 󵄩𝑧∗󵄩 +𝜀(𝑝)‖𝑢‖ , 𝑡→∞ 𝐴 𝐴 lim sup 󵄩 𝑝 󵄩 󵄩 󵄩2 𝑡→∞ 2 𝛿∈R ‖𝐴‖̃ ≤𝛿<1/‖𝐴−1‖ =𝜆 (𝐴𝑇𝐴) 󵄩 󵄩 󵄨󵄩 ∗󵄩 󵄨 for any + satisfying 2 2 min , 󵄩𝑧 (𝑡)󵄩 ≥ 󵄨󵄩𝑧 󵄩 −𝜀(𝑝)‖𝑢‖󵄨 , 𝐴𝑡 lim𝑡→∞ inf 󵄩 𝑝 󵄩2 󵄨󵄩 󵄩2 󵄨 (75) and 𝐾𝐴 ≥1and 𝜌𝐴 >0are real constants such that ‖𝑒 ‖2 ≤ −𝜌𝐴𝑡 𝐾𝐴𝑒 ,forall𝑡∈R0+.Inparticular,(−𝜌𝐴) is the stability 󵄩 ∗󵄩 (󵄩𝑧 (𝑡) −𝑧 󵄩 )=0, 𝐴 𝑝→∞lim 𝑡→∞lim 󵄩 𝑝 󵄩2 abscissa of the dominant eigenvalue of if it is either simple or it has an associate diagonal Jordan block, or a number ∗ arbitrarily close to it but larger. so that 𝑧𝑝(𝑡) → 𝑧 as 𝑝(∈ N), 𝑡(∈ R)→∞,where ‖⋅‖2 denotes the spectral norm for vector and matrices. Now, assume that the dynamics is perturbed with a parametrical Acknowledgments 𝐴̃ 𝐴 disturbance in the matrix , which is nonsingular since The author is very grateful to the Spanish Government for its 𝜆<0 𝜆∈𝜎(𝐴) 𝐴󸀠 =𝐴+𝐴=𝐴(𝐼+𝐴̃ −1𝐴)̃ Re ,forall to give , support of this research through Grant DPI2012-30651 and 𝐼 𝑛 𝐴󸀠 with being the th identity matrix. Thus, is also a stability to the Basque Government for its support of this research ̃ −1 matrix if 1>‖𝐴‖‖𝐴 ‖ for any matrix norm since from through Grants IT378-10 and SAIOTEK S-PE12UN015. He 󸀠−1 −1 −1 ̃ Banach perturbation lemma ‖𝐴 ‖≤‖𝐴 ‖/(1 − ‖𝐴 ‖‖𝐴‖) is also grateful to the University of Basque Country for its 󸀠 −1 −1 −1 −1 [7, 19, 21, 22], since 𝐴 = (𝐼+𝐴 𝐴)̃ 𝐴 , exists and its financial support through Grant UFI 2011/07. maximum modulus eigenvalues do not cross the imaginary complex axis from the continuity of the eigenvalues with References respect to the matrix entries. Thus, the perturbed dynamic system has the following solution: [1] L. E. Franks, Signal Theory, Prentice Hall, Englewood Cliffs, NJ, USA, 1969. [2] J. M. Berezanskii, “Expansion of eigenvectors of self-adjoint 𝑡 𝐴󸀠𝑡 𝐴(𝑡−𝜏) ̃ operators,” Translation of Mathematical Monographs,vol.17, 𝑥 (𝑡) =𝑒 𝑥 (0) + ∫ 𝑒 (𝐵𝑢 (𝜏) + 𝐴𝑥 (𝜏))𝑑𝜏 1968. 0 [3] M. De la Sen, “Some fixed point properties of self-maps 𝑡 (76a) 𝐴󸀠𝑡 𝐴󸀠(𝑡−𝜏) constructed by switched sets of primary self-maps on normed =𝑒 𝑥 (0) + ∫ 𝑒 𝐵𝑢 (𝜏) 𝑑𝜏; ∀𝑡 ∈ R0+, 0 linear spaces,” Fixed Point Theory and Applications,vol.2010, Article ID 438614, 25 pages, 2010. 𝑇 𝑧 (𝑡) =(𝑦(𝑡) ,𝑥0𝑇 (𝑡)) [4] M. De la Sen, “Fixed and best proximity points of cyclic jointly accretive and contractive self-mappings,” Journal of Applied 𝑡 󸀠 Mathematics, vol. 2012, Article ID 817193, 29 pages, 2012. =𝐶(𝑒𝐴 𝑡𝑥 (0) + ∫ 𝑒𝐴(𝑡−𝜏) (𝐵𝑢 (𝜏) + 𝐴𝑥̃ (𝜏))𝑑𝜏) 0 [5]M.DelaSen,“AboutrobuststabilityofCaputolinearfractional dynamic systems with time delays through fixed point theory,” 𝑡 𝐴󸀠𝑡 𝐴󸀠(𝑡−𝜏) Fixed Point Theory and Applications,vol.2011,ArticleID =𝐶(𝑒 𝑥 (0) + ∫ 𝑒 𝐵𝑢 (𝜏) 𝑑𝜏) , ∀𝑡∈ R0+. 0 867932, 19 pages, 2011. (76b) [6] M. De la Sen, “Total stability properties based on fixed point theoryforaclassofhybriddynamicsystems,”Fixed Point Theory and Applications,vol.2009,ArticleID826438,19pages, 𝑧∗ If the nominal (i.e., unperturbed) solution is a fixed point 2009. 𝐴󸀠𝑡 󸀠 and, since ‖𝑒 ‖2 →0as 𝑡→∞since 𝐴 is a stability [7] P. N. Anh, “A hybrid extragradient method for pseudomono- matrix, then applying Holder’s inequality to (76a)and(76b), tone equilibrium problems and fixed point problems,” Bulletin 14 Abstract and Applied Analysis

of the Malaysian Mathematical Sciences Society,vol.36,no.1,pp. 107–116, 2013. [8] H.-P. Kunzi¨ and O. Olela Otafudu, “q-hyperconvexity in quasipseudometric spaces and fixed point theorems,” Journal of Function Spaces and Applications,vol.2012,ArticleID765903, 18 pages, 2012. [9] M. G. Cojocaru and S. Pia, “Nonpivot and implicit projected dynamical systems on Hilbert spaces,” Journal of Function Spaces and Applications, vol. 2012, Article ID 508570, 23 pages, 2012. [10] C.M.Chen,T.H.Chang,andK.S.Juang,“Commonfixedpoint theorems for the stronger Meir-Keeler cone-type function in cone ball-metric spaces,” Applied Mathematics Letters,vol.25, no. 4, pp. 692–697, 2012. [11] C.-M. Chen, “Common fixed-point theorems in complete generalized metric spaces,” Journal of Applied Mathematics,vol. 2012, Article ID 945915, 14 pages, 2012. [12] P. Kumam and P. Katchang, “Viscosity approximations with weak contraction for finding a common solution of fixed points and a general system of variational inequalities for two accretive operators,” Journal of Computational Analysis and Applications, vol.14,no.7,pp.1269–1287,2012. [13] M. Mursaleen and S. A. Mohiuddine, “Some new double se- quence spaces of invariant means,” Glasnik Matematiˇcki,vol.45, no.65,pp.139–153,2010. [14] O. Duman, M. K. Khan, and C. Orhan, “A-statistical conver- gence of approximating operators,” Mathematical Inequalities & Applications,vol.6,no.4,pp.689–699,2003. [15] A. D. Gadjiev and C. Orhan, “Some approximation theorems via statistical convergence,” The Rocky Mountain Journal of Mathematics,vol.32,no.1,pp.129–138,2002. [16] S. A. Mohiuddine and A. Alotaibi, “Some spaces of double sequences obtained through invariant mean and related con- cepts,” Abstract and Applied Analysis,vol.2013,ArticleID 507950, 11 pages, 2013. [17] C. Belen and S. A. Mohiuddine, “Generalized weighted statis- tical convergence and application,” Applied Mathematics and Computation,vol.219,no.18,pp.9821–9826,2013. [18] A. Ashyralyev and M. E. Koksal, “Stability of a second order of accuracy difference scheme for hyperbolic equation in a Hilbert space,” Discrete Dynamics in Nature and Society,vol. 2007, Article ID 57491, 25 pages, 2007. [19] M. de la Sen, “The reachability and observability of hybrid multirate sampling linear systems,” Computers & Mathematics with Applications,vol.31,no.1,pp.109–122,1996. [20] A. Ashyralyev and Y. A. Sharifov, “Optimal control problem for impulsive systems with integral boundary conditions,” in Proceedings of the 1st International Conference on Analysis and Applied Mathematics (ICAAM ’12),vol.1470ofBook Series AIP Conference Proceedings, pp. 8–11, 2012. [21] M. de la Sen, “On some structures of stabilizing control laws for linear and time-invariant systems with bounded point delays and unmeasurable states,” International Journal of Control,vol. 59,no.2,pp.529–541,1994. [22] J. M. Ortega, Numerical Analysis, Academic Press, New York, NY, USA, 1972.