Proceeding Book 2020

7th International Conference on Recent Advances in Pure and Applied Mathematics Goddess of Bodrum Isis Hotel, Bodrum/Muğla, TURKEY September 25-28, 2020

Proceeding Book of ICRAPAM (2020)

Editör Ekrem SAVAS

Associate Editors Mahpeyker OZTURK, Emel Aşıcı Veli CAPALI

Date of Publication: 20.12.2020

ISBN Number:978-625-409-857-4

1 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Goddess of Bodrum Isis Hotel Bodrum/Muğla, TURKEY ICRAPAM CONFERENCE PROCEEDING

PREFACE

International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2020) was held in Bodrum Muğla, Turkey, at the Goddess of Bodrum Isis Hotel Bodrum/Muğla Turkey, from September 25-28, 2020. It was the 7th edition of such conferences. The chairman of the Organizing Committee of ICRAPAM 2020 was Associate Prof. Mahpeyker Ozturk, and the Scientic Committee consisted of mathematicians from 20 countries. 200 participants from 130 countries attended the conference and 120 papers have been presented, including 6 plenary lectures. The conference was devoted to almost all fields of mathematics and variety of its applications. This issue of the proceeding contains 12 papers presented at the conference and selected by the usual editorial procedure of scientific committee. We would like to express our gratitude to the authors of articles published in this issue and to the referees for their kind assistance and help in evaluation of contributions. I would like to thank to the following my colleagues and students who helped us at every stage of International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2020).

Editor: Ekrem SAVAS

Usak University, Usak – Turkey

AIM OF THE CONFERENCE

International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2020) is aimed to bring researchers and professionals to discuss recent developments in both theoretical and applied mathematics and to create a professional knowledge exchange platform between mathematicians.

SCOPUS

Prospective authors are invited to submit their one-page abstracts on the related, but not limited, following topics of interest:

Numerical Analysis, Ordinary and Partial Differential Equations, Scientific computing, Boundary Value Problems, Approximation Theory, Sequence Spaces and Summability, Real Analysis, Functional Analysis, Fixed Point Theory, Optimization, Geometry, Computational Geometry, Differential Geometry, Applied Algebra, Combinatorics, Complex Analysis, Flow Dynamics, Control, Mathematical modelling in scientific disciplines, Computing Theory, Numerical and Semi-Numerical Algorithms, Game Theory, Operations Research, Optimization Techniques, Fuzzy sequence spaces, Symbolic Computation, Fractals and Bifurcations, Analysis and design tools, Cryptography, Number Theory and Mathematics Education, Finance Mathematics, Fractional Dynamics, Fuzzy systems and fuzzy control, Dynamical systems and chaos, Biomathematics & modeling. Soft Computing, Cryptology & Security Analysis, Image Processing, etc.

PROCEEDING BOOK

The full texts contained in this proceeding book contain all oral presentations in ICRAPAM 2020 Conference.

2 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Goddess of Bodrum Isis Hotel Bodrum/Muğla, TURKEY ICRAPAM CONFERENCE PROCEEDING

COMMITTEE

Honorary Chair

Prof. Dr. Ekrem SAVAŞ Rector of Uşak University, Turkey

Prof. Dr. Vatan KARAKAYA Rector of Ahi Evran University, Turkey

Prof. Dr. Hamdullah ŞEVLİ Rector of Yüzüncü Yıl University, Turkey

Prof. Dr. Billy E. RHOADES Indiana University Bloomington, USA

Prof. Dr. Gradimir V. MILOVANOVİĆ Serbian Academt OF Science and Arts,SERBIA

Scientific Committee

Prof. Dr. Abdelmejid Bayad Prof. Dr. Pratulananda Das (France) (India) Prof. Dr. Abedallah M. Rababah Prof. Reza Saadati (Jordan) (Iran)

Prof. Dr. Alberto Manuel Tavares Prof. Roman Dwilewicz (Portugal) (Usa)

Prof. Dr. Amiran Gogatishvili Prof. Dr. S.sadiq Basha (Czech Republic) (India)

Prof. Dr. Ants Aasma Tallinn Prof. Dr. Taras Banakh (Estonia) (Ukraınıan)

Prof. Dr. Hüseyin Çakallı Prof.dr. Wasfi Shatanawi (Turkey) (Jordan)

Prof. Dr. Hemen Dutta Prof. Dr. Wutiphol Sintunavarat (India) (Thailand) Prof. Dr. Emine Misirli Prof. Dr. Varga Kalantarov (Turkey) (Turkey) Prof. Dr. Erdal Karapinar Prof. Dr. Vasil Angelov (Taiwan) (Bulgaria) Prof. Dr. Fahreddin Abdullayev Prof. Dr. Vasile Berinde (Turkey) (Romania)

Prof. Dr. Fairouz Tchier Prof. Dr. Yilmaz Şimşek (Saudi Arabia) (turkey)

3 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Goddess of Bodrum Isis Hotel Bodrum/Muğla, TURKEY ICRAPAM CONFERENCE PROCEEDING

Prof. Dr. Ibrahim Çanak Prof. Dr. Zoran D. Mitrovic (Turkey) (Vietnam) Prof. Dr. Oktay Duman Prof. Dr. Eman Samir Bhaya (Turkey) (Iraq) Prof. Dr. Ishak Altun Prof. Dr. Ayhan Şerbetçi (Turkey) (Turkey) Prof. Dr. Jeff Connor Prof. Dr. Ismail Ekincioğlu (Usa) (Turkey) Prof. Dr. Lubomira Softova Prof. Dr. Erhan Deniz (Italy) (Turkey) Prof. Dr. Manuel De La Sen Prof. Dr. Rifat Çolak (Spain) (Turkey) Prof. Dr. Martin Bohner Prof. Dr. Ammar Isam Edress (Usa) (Iraq) Prof. Dr. Marc Lassonde Prof. Dr. Sadek Bouroubi (Guadeloupe) (Algeria) Prof.dr. Mehmet Dik Prof. Dr. Hamlet Guliyev (Usa) (Azerbaijan) Prof.dr. Metin Başarir Prof. Dr. Akbar Aliyev (Turkey) (Azerbaijan) Prof.dr. Mikail Et Assoc.prof. Yusuf Zeren (Turkey) (Turkey) Prof. Dr. M. Mursaleen Assist. Prof.dr. Arzu Akgül (India) (Turkey) Prof.dr. Naseer Shahzad Assoc. Prof. Dr. Şükran Konca (Saudi Arabia) (Turkey) Prof.dr. Necip Şimşek Assoc.prof. Dr. Ulaş Yamanci (Turkey) (Turkey) Prof. Dr. Poom Kumam Assist. Prof. Dr. Vuqar Mehrabov (Thailand) (Azerbaijan)

4 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Goddess of Bodrum Isis Hotel Bodrum/Muğla, TURKEY ICRAPAM CONFERENCE PROCEEDING

Organizing Committee

Prof. Dr. Debasis Giri Assoc. Prof. Dr. Emel Aşici (India) (Turkey) Prof. Dr. Maria Zelster Assoc. Prof. Dr. Mahpeyker Öztürk (Estonia) (Turkey) Prof. Dr. Mehmet Gurdal Assoc. Prof. Dr. Moosa Gabeleh (Turkey) (Iran) Prof. Dr. Ram Mohapatra Assoc. Prof. Dr. Murat Kirişçi (Usa) (Turkey) Prof. Dr. Richard F. Patterson Assoc.prof. Dr. Narin Petrot (Usa) (Thailand) Prof. Dr. Werner Varnhorn Assoc prof. Dr. Olivier Olela Otafudu (Germany) (Witwatersrand South Africa) Prof. Dr. Hari Mohan Srivastava Assoc. Prof. Dr. Rahmet Savaş (Canada) (Turkey) Prof. Dr. Naim Braha Assist. Prof. Dr. Veli Çapali (Chuang) (Turkey) Prof. Dr. Mujahid Abbas Assist. Prof. Dr. Banu Güntürk (Pakistan) (Turkey) Prof. Dr. Ali Akhmedov Assist. Prof. Dr. Serkan Araci (Azerbaijan) (Turkey) Prof. Dr. Ziyatkhan Aliyev Dr. Abdurrahman Büyükkaya (Azerbaijan) (Turkey) Assoc. Prof. Dr. Azhar Hussain Dr. Melek Erişçi Büyükkaya (Pakistan) (Turkey) Assoc. Prof. Bancha Panyanak Dr. Rabia Savaş (Thailand) (Turkey) Assoc. Prof. Dr. Edixon M. Rojas Dr. Sefa Anil Sezer (Colombia) (Turkey)

5 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Goddess of Bodrum Isis Hotel Bodrum/Muğla, TURKEY ICRAPAM CONFERENCE PROCEEDING

INVITED SPEAKERS

6 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Goddess of Bodrum Isis Hotel Bodrum/Muğla, TURKEY ICRAPAM CONFERENCE PROCEEDING

7 /

CONTENT

Preface 2

Committee 3

Invited Speakers 6

Content 8

Some investigations on nullnorms on bounded lattices Emel Aşıcı 9

On solving the symmetric regularized long-wave equation by (1 / Gꞌ)- expansion method 12 Gizem Aydin, Esin Ilhan, Haci Mehmet Baskonus, Hasan Bulut The Bounds for the Length Between Dirichlet and the Periodic Eigenvalues of Hill’s Equation with Symmetric Single Well Potential 19 Elif Başkaya On Solving the (2+1)-Dimensional Zoomeron Equation by (1/Gꞌ) -Expansion Method 24 Hasan Bulut,Gizem Aydin

Some Comparisons of BLUPs under General Linear Random Effects Models Melek Eriş Büyükkaya, Nesrin Güler 29

On Generalized Statistical Convergence via Ideal in Cone Metric Spaces 36 Isıl Açık Demirci, Mehmet Gürdal

I-Statistical Limit Superior and I-Statistical Limit Inferior of Triple Sequences Mualla Birgül Huban, Mehmet Gürdal and Ekrem Savaş 42

I-Statistically Localized Sequences of Weighted g via Modulus Functions in 2-Normed Space Mualla Birgül Huban, Mehmet Gürdal and Ekrem Savaş 50

Green’s Function for Hill’s Equation with Symmetric Single Well Potential 57 Ayşe Kabataş N-dimensional bound state solutions of the hyperbolic potential function in approximate analytic form 63 Aysel Özfidan Almost Lacunary P- Bounded Variation and Matrix Transformation 69 Rabia Savaş

A new sequence spaces defined by bounded variation in n- normed spaces 73 Ekrem Savaş

Hankel matrices for blind image deconvolution 78 Belhaj Skander, Alsulami Abdulrahman

Asymptotically Lacunary Statistical Equivalent Functions on Time Scales 84 Bayram Sözbir, Selma Altundağ

8 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

Some investigations on nullnorms on bounded lattices

Emel A¸sıcı1 1Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, 61830 Trabzon, Turkey

Keywords Abstract: In this paper, we study an order induced by nullnorms on bounded lattices. Bounded lattice, We study on the set of incomparable elements with respect to the F-partial order which Nullnorm, L is denoted by KF . Also, we give some examples for clarity. Finally, we obtain some conclusions for t-norms and t-conorms on bounded lattices.

1. Introduction and motivation

Triangular norms (briefly t-norms) and triangular conorms (briefly t-conorms) were introduced by Schweizer and Sklar [19] in the study of probabilistic metric spaces as a special kind of associative functions defined on the unit interval [0,1]. Although the t-norms and t-conorms were strictly defined on the unit interval [0,1], they were mostly studied on bounded lattices. Nullnorms and t-operators were introduced in [15] and [8], respectively, which are also generalizations of t-norms and t-conorms. It has been pointed out that nullnorms and t-operators are equivalent [16]. Nullnorms are generalization of triangular norms and triangular conorms with a zero element in the interior of the unit interval and have to satisfy an additional condition. Such nullnorms are interesting not only from a theoretical point of view, but also for their applications, since they have been proved to be useful in several fields like expert systems, neural networks, fuzzy quantifiers. Also, nullnorms that has to be used as aggregators or in fuzzy logic maintain as many logical properties as possible. A natural order for semigroups was defined in [17]. Similarly, a partial order defined by means of t-norms on a bounded lattice was introduced in [13]: x T y :⇔ T(`,y) = x for some ` ∈ L, where L is a bounded lattice, x,y of a bounded lattice L and T is a t-norm on L. This partial order T is called a T-partial order on L. Then, an order F induced by a nullnorm on a bounded lattice L was defined in [6]. Also the set of incomparable elements with respect to the F-partial order for any nullnorm on [0,1] was defined and properties of the introduced set was investigated. In this paper, we investigate some properties of an order induced by nullnorms on bounded lattices. The paper is organized as follows. We shortly recall some basic notions in Section 2. In Section 3, we study on the set of incomparable elements L with respect to the F-partial order which is denoted by KF . Then, we give example and counter example. Finally, by using Proposition 3.2, we obtain some conclusions for t-norms and t-conorms on bounded lattices.

2. Preliminaries

A lattice [2, 3, 7, 9–11] is a partially ordered set (L,≤) in which each two element subset {x,y} has an infimum, denoted as x ∧ y and a supremum, denoted as x ∨ y. A bounded lattice (L,≤,0,1) is a lattice that has the bottom and top elements written as 0 and 1, respectively. Given a bounded lattice (L,≤,0,1) and a,b ∈ L, if a and b are incomparable, in this case, we use the notation a k b. We denote the set of elements which are incomparable with a by Ia. So Ia = {x ∈ L | x k a}. Given a bounded lattice (L,≤,0,1) and a,b ∈ L, a ≤ b, a subinterval [a,b] of L is defined as [a,b] = {x ∈ L | a ≤ x ≤ b} Similarly, [a,b) = {x ∈ L | a ≤ x < b}, (a,b] = {x ∈ L | a < x ≤ b} and (a,b) = {x ∈ L | a < x < b}. Definition 2.1. [4, 18] Let (L,≤,0,1) be a bounded lattice. A t-norm T is a binary operation on L which is commutative, associative, increasing with respect to both variables and it satisfies T(x,1) = x for all x ∈ L.

Example 2.2. [14] The following are the four basic t-norms TM, TP, TL and TD on the real unit interval [0,1] given by, respectively,

∗ Corresponding author: [email protected] 9 TM(x,y) = min(x,y), TP(x,y) = xy, TL(x,y) = max(x + y − 1,0), ( 0 if (x,y) ∈ [0,1)2 , TD(x,y) = min(x,y) otherwise. Extremal t-norms T∧ and TW on L are deﬁned as follows, respectively:  x y = 1 ,  T∧(x,y) = x ∧ y TW (x,y) = y x = 1, 0 otherwise.

Deﬁnition 2.3. [18] Let (L,≤,0,1) be a bounded lattice. A triangular conorm S is a binary operation on L which is commutative, associative, increasing with respect to both variables and it satisﬁes S(x,0) = x for all x ∈ L.

Example 2.4. [3, 14] The following are the four basic t-conorms SM, SP, SL and SD on the real unit interval [0,1] given by, respectively, SM(x,y) = max(x,y), SP(x,y) = x + y − xy, SL(x,y) = min(x + y,1), ( 1 if (x,y) ∈ (0,1]2 , SD(x,y) = max(x,y) otherwise. Similarly, extremal t-conorms S∨ and TW on L are defined on bounded lattice. Example 2.5. [14] The t-norm T nM on [0,1] is defined as follows: ( 0 x + y ≤ 1 , T nM(x,y) = min(x,y) otherwise. is called nilpotent minimum t-norm. The t-norm T ∗ on [0,1] is defined as follows: ( 0 (x,y) ∈ (0,k)2 , T ∗(x,y) = 0 < k < 1 min(x,y) otherwise. Definition 2.6. [5, 8, 12] A nullnorm is a binary operator F : L2 → L which is commutative, associative, non-decreasing in each variable and there exists some element a ∈ L such that F(x,0) = x for all x ≤ a, F(x,1) = x for all x ≥ a. Clearly, F is a t-norm if a = 0 and a t-conorm a = 1. It is easy to show that F(x,a) = a for all x ∈ L. Therefore, a ∈ L is the zero (absorbing) element for F. It is clear that F(1,0) = a. In this study, for the sake of brevity, the set [0,a) × (a,1] ∪ (a,1] × [0,a) for a ∈ L\{0,1} is denoted by Da, i.e., Da = [0,a) × (a,1] ∪ (a,1] × [0,a) for a ∈ L\{0,1}. Proposition 2.7. [12] Let (L,≤,0,1) be a bounded lattice, a ∈ L \{0,1} and F be a nullnorm with zero element a on L. Then ∗ 2 (i) S = F |[0,a]2 : [0,a] → [0,a] is a t-conorm on [0,a]. ∗ 2 (ii) T = F |[a,1]2 : [a,1] → [a,1] is a t-norm on [a,1]. Definition 2.8. [13] Let L be a bounded lattice and T be a t-norm on L. The order defined as follows is called a T− partial order (triangular order) for t-norm T:

x T y :⇔ T(`,y) = x for some ` ∈ L. Deﬁnition 2.9. [6] Let (L,≤,0,1) be a bounded lattice and F be a nullnorm with zero element a on L. Deﬁne the following relation, for x,y ∈ L, as  if x,y ∈ [0,a] and there exist k ∈ [0,a] such that F(x,k) = y or,  x F y :⇔ if x,y ∈ [a,1] and there exist ` ∈ [a,1] such that F(y,`) = x or, (1) if (x,y) ∈ L∗ and x ≤ y, ∗ where Ia = {x ∈ L | x k a} and L = [0,a] × [a,1] ∪ [0,a] × Ia ∪ [a,1] × Ia ∪ [a,1] × [0,a] ∪ Ia × [0,a] ∪ Ia × [a,1] ∪ Ia × Ia.

Note: The partial order F in (1) is called F-partial order on L.

Lemma 2.10. [6] Let (L,≤,0,1) be a bounded lattice. For all nullnorms F and all x ∈ L it holds that 0 F x, x F x and x F 1.

Proposition 2.11. [6] Let (L,≤,0,1) be a bounded lattice and F be a nullnorm on L. If x F y for any x,y ∈ L, then x ≤ y. 10 L 3. About the set KF on any bounded lattice

L In this section we study on the set of incomparable elements with respect to the F-partial order which is denoted by KF . We give Proposition 3.2 and by using Proposition 3.2, we obtain some conclusions for t-norms and t-conorms on bounded lattices.

L Deﬁnition 3.1. [1] Let F be a nullnorm on (L,≤,0,1) with zero element a and let KF be deﬁned by L KF = {x ∈ L\{0,1} | for some y ∈ L\{0,1}, [x < y implies x F y]or

[y < x implies y F x]orx k y}. Proposition 3.2. Let (L,≤,0,1) be a bounded lattice and F be a nullnorm on L with zero element a. If there exist two L elements in L which are incomparable with each other, then KF 6= /0. Example 3.3. Consider the lattice (L = {0, p,m,a,n,1},≤,0,1) whose lattice diagram is displayed in Figure 1. Consider

Figure 1. The order ≤ on L the nullnorm on L with zero element a deﬁned as follows:

x ∨ y ,(x,y) ∈ [0,a]2  y ,x = 1 and y ≥ a F(x,y) = x ,y = 1 and x ≥ a  a ,otherwise

It is clear that KF = {m,a,n}. L Remark 3.4. The converse of Proposition 3.2 is not correct. That is, although the set KF 6= /0, it need not be the case that all elements in L are incomparable. To illustrate this claim, we shall give the following example.

2 1 Example 3.5. Consider the nullnorm F1 : [0,1] → [0,1] with zero element deﬁned by 2 2

 2 max(x,y) ,(x,y) ∈ [ , 1 ]  0 2  1 1 2 F1 (x,y) = 2 ,(x,y) ∈ [ 2 ,1) ∪ D 1 2  2 min(x,y) ,otherwise.

1 Then, KF1 = ( 2 ,1), but since L = [0,1] is a chain all elements are comparable with each other in L. Let us show that 2 1 1 K 1 . Let x ∈ KF1 . We will show that x ∈ ( ,1). Suppose that x ∈/ ( ,1). Since x ∈ KF1 , there exists an element F1 =( 2 ,1) 2 2 2 2 2 y ∈ (0,1) such that x < y and x F1 y or y < x and y F1 x. 2 2 1 1 Let x < y and x F1 y. Since x ∈/ ( 2 ,1), we have x ≤ 2 . Since x < y, we have that max(x,y) = y. By the deﬁnition of F1 , 2 2 we obtain that y = max(x,y) = F1 (x,y) 2

Then, it is obtained that x F1 y, a contradiction. 2 If y < x, then similarly it can be obtained y x, a contradiction. So, we have x ∈ ( ,1 ). Thus, it is obtained that F1 21 2 11 1 KF1 ⊆ ( 2 ,1). 2 1 1 Let x ∈ ( 2 ,1) and x < y < 1. Then x F1 y. Suppose that x F1 y. Then, there exists an element k ∈ [ 2 ,1] such that 2 2 1 2 x = F1 (y,k). Since x 6= y, it is not possible k = 1. Since (y,k) ∈ [ ,1) , we have that 2 2 1 x = F1 (y,k) = , 2 2

1 a contradiction. So, we have that x F1 y, x ∈ KF1 . So it is obtained that ( 2 ,1) ⊆ KF1 . Therefore, we have that 2 2 2 1 KF1 = ( 2 ,1). 2 Corollary 3.6. Let (L,≤,0,1) be a bounded lattice and T be a t-norm on L. If there exist two elements in L which are L incomparable with each other, then KT 6= /0. Corollary 3.7. Let (L,≤,0,1) be a bounded lattice and S be a t-conorm on L. If there exist two elements in L which are L incomparable with each other, then KS 6= /0. References

[1] A¸sıcı,E. 2019. An extension of the ordering based on nullnorms. Kybernetika, 55(2) (2019), 217-232. [2] A¸sıcı,E. 2019. The equivalence of uninorms induced by the U-partial order. Hacettepe Journal of Mathematics and Statistics. 48 (2019), 439-450. [3] A¸sıcı,E., Mesiar, R. 2020. New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice. International Journal of General Systems. 49(2) (2020), 143-160. [4] A¸sıcı,E., Mesiar, R. 2021. On the construction of uninorms on bounded lattices. Fuzzy Sets and Systems. doi: 10.1016/j.fss.2020.02.007, in press. [5] A¸sıcı,E., Mesiar, R. 2019. Direct product of nullnorms on bounded lattices. Journal of Intelligent and Fuzzy Systems. 36(6) (2019), 5745-5756. [6] A¸sıcı,E. 2017. An order induced by nullnorms and its properties. Fuzzy Sets and Systems. 325 (2017), 35-46. [7] Birkhoff, G. 1967. Lattice Theory. 3 rd edition, Providence, 1967. [8] Calvo, T. De Baets, B., Fodor, J. 2001. The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets and Systems. 120 (2001) 385-394. [9] Çaylı, G.D. 2020. Nullnorms on bounded lattices derived from t-norms and t-conorms. Information Sciences. 512 (2020), 1134-1154. [10] Çaylı, G.D. 2019. Alternative approaches for generating uninorms on bounded lattices. Information Sciences. 488 (2019), 111-139. [11] Çaylı, G.D. 2019. New methods to construct uninorms on bounded lattices. International Journal of Approximate Reasoning. 115 (2019), 254-264. [12] Karaçal, F., Ince, M.A., Mesiar, R. 2015. Nullnorms on bounded lattices. Information Sciences. 325(2015), 227-236. [13] Karaçal, F., Kesicioglu,˘ M.N. 2011. A T-partial order obtained from t-norms. Kybernetika. 47(2011) 300-314. [14] Klement E.P, Mesiar, R., Pap, E. 2000. Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. [15] Mas, M., Mayor, G., Torrens, J. 1999. t-operators. International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems. 7 (1999) 31-50. [16] Mas, M., Mayor, G., Torrens, J. 2002. The distributivity condition for uninorms and t-operators. Fuzzy Sets and Systems. 128 (2002) 209-225. [17] Mitsch, H. 1986. A natural partial order for semigroups, Proceedings of the American Mathematical Society, 97 (1986) 384-388. [18] Saminger, S. 2006. On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems. 157 (2006), 1403-1416. [19] Schweizer, B., Sklar, A. Statistical metric spaces. Paciﬁc Journal of Mathematics. 10 (1960), 313-334.

12 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERENCE PROCEEDING

1 On solving the symmetric regularized long-wave equation by (G 0 ) expansion method

Gizem Aydin1, Esin Ilhan2, Haci Mehmet Baskonus∗3, Hasan Bulut1 1Department of Mathematics, Firat University, Elazig, Turkey 2Kirsehir Ahi Evran University, Kirsehir, Turkey 3Faculty of Education, Harran University, Sanliurfa, Turkey

1 Keywords Abstract: In this article, the ( G0 ) expansion method is considered to ﬁnd new wave The symmetric regularized solutions for nonlinear partial differential equations named symmetric regularized long- long-wave equation, wave equation. Many new traveling wave solutions to the given equation are found. 1 The ( G0 ) expansion method, In addition, 2D and 3D graphs are plotted by selecting the values of the appropriate Analytical solutions, parameters. Traveling wave solutions.

1. Introduction

It is well known that nonlinear partial differential equations (NLPDEs) define a wide variety of phenomena not only in physics but also in biology, applied science, chemistry, engineering, and many other fields. The investigation of moving wave solutions for NLPDEs plays an important role in the investigation of nonlinear physical phenomena. In recent years, many scientists have applied various methods to obtain analytical and numerical solutions for the nonlinear partial G 1 differential equations in mathematical physics, such as the ( G0 ) expansion method [1-2], the ( G0 ) expansion method [3-4], the tanh-coth method [5], the sine-Gordon expansion method [6-8], the Laplace perturbation method [9], the extended simple equation method [10], the integral approximation [11], the hyperbolic tangent function expansion method [12], the generalized expansion method [13], the Bernoulli sub-equation method [14-17], the Hirota’s bilinear transformation method [18], the extended sinh-Gordon method[19-20], the modified simple equation method[21], a direct algebraic method[22], an inverse spectral transform technique[23], the multiplier method [24], the generalized Kudryashov method [25], the sechp −tanhp functions method [26], the homotopy method [27-28], and many other numerical and analytical methods 1 [29-40]. The main purpose of this work is to apply the ( G0 ) expansion method to the symmetric regularized long-wave (SRLW) equation, u2 u − u + ( ) − u = 0, (1) tt xx 2 xx xxtt which was first introduced by Seyler and Fenstermacher in 1983 [41] from a weakly nonlinear analysis of the cold-electron plasma equations suitable for a strongly magnetized non-relativistic electron beam such that the fluid motion is restricted to one direction [41,42]. The SRLW equation represents a weekly nonlinear ion-acoustic and space-charge waves, and the real-valued u(x,t) refers to the dimensionless fluid velocity with a decay condition limx→∞ u = 0 . The SRLW equation is explicitly symmetry in the and derivatives and is quite similar to the regularized-long-wave equation that represents shallow-water waves and plasma drift waves [41, 42]. The radial basis functions collocation method has been used to study the single solitary wave solution, the interaction of two positive solitary waves, and the clash of two solitary waves [43, 44].The existence and uniqueness of solutions for the SRLW have been established, e.g., in [45,46]. On the other hand, many authors tried to find the approximation and exact solution to the SRLW equation, such as the radial basis functions collocation method[47]. The modified cubic B-Splines has been implemented, then very accurate solutions in different settings of parameters have been provided [48]. The four-level linear implicit finite difference has been used G to obtain some solitary wave solutions [49]. The ( G0 ) -expansion method has been utilized to find exact solutions to the SRLW equation [50]. The Lie symmetry approach along with the simplest equation and exp-function methods are used to obtain solutions of the SRLW equation [51]. The existence and nonlinear stability of periodic traveling wave solutions of the cnoidal type of the SRLW equation have been investigated in [52].We organized the paper as follows: In Section 2, 1 the basic concepts of the ( G0 ) -expansion method are addressed. In Section 3, the application of the used method to the equation under consideration is presented. The Conclusion is given in Section 4.

∗ Corresponding author: [email protected] 12 /

1 2. The( G0 ) expansion method

1 In this portion, the general details of the ( G0 ) - expansion method are provided. Let us consider a three-variable general form of the nonlinear partial differential equation (PDE) as

P(u,ut , uux, uuy, uuxx,···) = 0, (2) transform equation (2) with the aid of u(x,y,t) = U(ξ),ξ = κ1x+κ2y−κ3t in which κ1, κ2 and κ3 are non-zero constants. After transformation, we have a nonlinear ordinary differential equation (ODE) as follows

N(U, U0, U00, U000,....) = 0. (3)

The solution of (3) shall be taken in the following form

m 1 i U(ξ) = a0 + ∑ ai( 0 ) . (4) i=1 G

Here G = G(ξ) is a function that veriﬁes the following second-order ordinary differential equation

G00 + λG0 + µ = 0, (5) here a0,a1,a2,··· ,λ, µ are constants and m is a term of balance. The balance term is a fixed number obtained between the linear term and the non-linear term in any nonlinear ODE that has the highest order. This number is written in place of (4) and then the derivatives required for the solution are obtained. Here we gain a system of algebraic equations by equalizing 1 i the coefficients of ( G0 ) ,(i = 1,2,3,.....m) terms to zero. This algebraic equation scheme is solved automatically or with the aid of a computer program depending on the degree of difficulty. These solutions (4) given by the solution function and moving wave transformation is written in place (2) given by the solution of the PDE is found [41].

The general solution of the ODE (5) could be reported as below

µξ G(ξ) = − −C eλξ +C , (6) λ 1 2 here C1 and C2 are ﬁxed. Depending on the variable ξ the derivative of the (6) solution function is taken and necessary arrangements are made, we have 1 1 = (7) G0 µ λξ − λ +C1e When we transform the algebraic expression provided in (7) to a trigonometric function may write as

1 λ 0 = . (8) G −µλC1(Cosh(ξλ) − Sinh(ξλ))

1 3. Implementation of ( G0 ) expansion method

1 In this sub-section, we apply the ( G0 ) expansion method to the SRLW equation. We start by inserting u(x,t) = U(ξ),ξ = x − ct, (9) into Eq. (9), we get

U2 (c2 − 1)U00 − c( )00 − c2U0000 = 0. (10) 2 Twice integrate the Eq. (10) and setting to zero of constants of integrations, we have c (c2 − 1)U − U2 − c2U00 = 0. (11) 2 Balancing among U2 and U00 in Equation (11), we get .In this case, the Eq. (4) takes the form 1 1 U(ξ) = a + a ( ) + a ( )2, (12) 0 1 G0 2 G0

13 /

1 i using Eq. (12) together with its derivative of Eq. (11). Then, summing the ( G0 ) coefﬁcients with the likely powers and setting every summation to zero, we get a system of algebraic equations as

1 ca2 ( )0 : −a + c2a − 0 = 0, G0 0 0 2 1 ( )1 : −a + c2a − c2λ 2a − ca a = 0, G0 1 1 1 0 1 1 ca2 ( )2 : −3c2λ µa − 1 − a + c2a − 4c2λ 2a − ca a = 0, G0 1 2 2 2 2 0 2 1 ( )3 : −2c2µ2a − 10c2λ µa − ca a = 0, G0 1 2 1 2 1 ca2 ( )4 : −6c2µ2a − 2 = 0, (13) G0 2 2 we solve the above system of equations by the computer package program, we achieve the following families of solution. 12λ µ 12µ2 1 Case-1 When a0 = 0, a1 = −√ , a2 = −√ , c = √ , λ 2−1 λ 2−1 λ 2−1 we can gain the following soliton solution

12µ2 u(x,t) = √ 2 2 µ 1 1 λ − 1 −C1cosh(λ(x − √ )) +C1sinh(λ(x − √ )) λ λ 2−1t λ 2−1t (14) 12λ µ + . √ 2 2 µ 1 1 λ − 1 −C1cosh(λ(x − √ )) +C1sinh(λ(x − √ )) λ λ 2−1t λ 2−1t

2λ 2 12λ µ 12µ2 1 Case-2 When a0 = √ , a1 = √ , a2 = √ , c = −√ , 1+λ 2 1+λ 2 1+λ 2 1+λ 2 we obtain the following soliton solution

2λ 2 12µ2 u(x,t) =√ + 1 + 2 √ 2 λ 2 1 1 µ 1 + λ C1cosh(λ(x + √ )) −C1sinh(λ(x + √ )) − 1+λ 2t 1+λ 2t λ (15) 12λ µ + . √ 2 2 1 1 µ 1 + λ C1cosh(λ(x + √ )) −C1sinh(λ(x + √ )) − 1+λ 2t 1+λ 2t λ √ √ 2 2 2 −1+c Case-3 When a0 = 0, a1 = −12 −1 + c , a2 = 12cµ , λ = c , we have 12cµ2 u(x,t) = √ √ 2 cµ c2−1(x−ct) c2−1(x−ct) √ +C1sinh( ) −C1cosh( ) c2−1 c c √ (16) 12 c2 − 1µ + . √ √ 2 cµ c2−1(x−ct) c2−1(x−ct) √ −C1sinh( ) +C1cosh( ) c2−1 c c √ √ 2 2 2 2 c −1 Case-4 When a0 = −( c ), a1 = −12 +c − 1µ, a2 = −12cµ , λ = c , we get 2 12cµ2 u(x,t) = − + 2c − √ √ 2 c c2−1(x−ct) c2−1(x−ct) cµ C1cosh( ) −C1cosh( ) − √ c c c2−1 √ (17) 12 c2 − 1µ − . √ √ 2 c2−1(x−ct) c2−1(x−ct) cµ C1cosh( ) −C1cosh( ) − √ c c c2−1

14 /

, Figure 1. The topological type of solution to the SRLW equation plotted when λ = −0.5, µ = 0.3,C1 = 0.4 and t = 1 for 2D ﬁgure.

, Figure 2. The non-topological type of solution to the SRLW equation plotted when c = 3, µ = 4,C1 = 0.2 and t = 1 for 2D ﬁgure.

, Figure 3. The singular type of solution to the SRLW equation plotted when c = 3, µ = 4,C1 = 0.2 and t = 1 for 2D ﬁgure.

15 /

, Figure 4. The non-topological type of solution to the SRLW equation plotted when c = −0.9, µ = 0.3,C1 = 0.2and t = 1 for 2D ﬁgure.

4. Conclusion

1 In this study, we have successfully obtained some new exact traveling wave solutions by using the ( G0 ) -expansion method to the SRLW equation, which has a very important place in applied sciences. All solutions obtained in this article are 1 veriﬁed by the PDE. Chand and Malik [50] have used the ( G0 ) -expansion method and kink-antikink shaped, bell shaped soliton and periodic wave solutions to the SRLW equation have been constructed. In Ref. [51], the Lie symmetry approach, the exp-function method and the simplest equation method have been utilized to construct some solutions to the studied equation. Comparing our results with the results reported in Refs. [51,52], it is concluded that our results are new and different. In the solutions presented, constants are given speciﬁc values to obtain graphics that represent the wave. Figures (1-4) symbolize the traveling wave behaviors of the governing model. As these constants change, the position of speed, amplitude, and wavelength of the wave may change.

References

G [1] A. Yokus, M. Tuz, An application of a new version of ( G0 )-expansion method,In AIP Conference Proceedings 2017 Jan 27 (Vol. 1798, No. 1, p. 020165). AIP Publishing LLC. [2] A.Yokus, D. Kaya, Traveling wave solutions of some nonlinear partial differential equations by using extended (G’/G)-expansion method, Istanbul Ticaret Universitesi Fen Bilimleri Dergisi 14, no. 28 (2015): 85. [3] A. Yokus, Comparison of Caputo and conformable derivatives for time-fractional Korteweg-de Vries equation via the finite difference method, Int. J. Mod. Phys. B, 2018. 1 [4] A. Yokus and H. Durur, Complex hyperbolic traveling wave solutions of KuramotoSivashinsky equation using ( G0 ) expansion method for nonlinear dynamic theory, Balikesir Universitesi Fen Bilim. Enstitusu Derg., vol. 21, no. 2, pp. 590-599,2019. [5] A.Irshad,andS.T.Mohyud-Din,Solitary wave solutions for Zoomeron equation, Walailak J. Sci. Technol., vol. 10, no. 2, pp. 201-208, 2012. [6] K. K. Ali, R. Yilmazer, and H. Bulut, Analytical Solutions to the Coupled Boussinesq-Burgers Equations via Sine-Gordon Expansion Method, in 4th International Conference on Computational Mathematics and Engineering Sciences(CMES-2019), p. 233. [7] H. F. Ismael, H. Bulut, and H. M. Baskonus, Optical soliton solutions to the Fokas-Lenells equation via sine-Gordon G0 expansion method and (m+ ( G )))-expansion method, Pramana - J. Phys., vol. 94, no. 1, Dec.2020. [8] K. K. Ali, R. Yilmazer, A. Yokus, and H. Bulut,Analytical solutions for the (3+ 1)-dimensional nonlinear extended quantum Zakharov-Kuznetsov equation in plasma physics, Phys. A Stat. Mech. its Appl., p. 124327,2020. [9] M. Yavuz, N. Ozdemir, and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, Eur. Phys. J.Plus, 2018. [10] E. M. E. Zayed, R. M. A. Shohib, and A.G. Al-Nowehy, On solving the (3+ 1)-dimensional NLEQZK equation and the (3+ 1)-dimensional NLmZK equation using the extended simplest equation method, Comput. Math. with Appl.,2019. [11] H. F. Ismael, Linear Stability of Thin Liquid Films flows down on an Inclined Plane using Integral Approximation, Raf. J. Comp. Math’s, vol. 11, no. 2, pp. 13-24,2014.

16 /

[12] S. Liu, Z. Fu, S. Liu, and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. Sect. A Gen. At. Solid State Phys.,2001. [13] R. Sabry, W. M. Moslem, F. Haas, S. Ali, and P. K. Shukla, Nonlinear structures:Explosive, soliton, and shock in a quantum electron-positron-ion magnetoplasma,Phys. Plasmas,2008. [14] T. A. Sulaiman, A. Yokus, N. Gulluoglu, and H. M. Baskonus, Regarding the numerical solutions of the Sharma- Tasso-Olver equation, in ITM Web of Conferences,2018, vol. 22, p. 1036. [15] H. H. Abdulkareem, H. F. Ismael, E. S. Panakhov, and H. Bulut, Some Novel Solutions of the Coupled Whitham- Broer-Kaup Equations, in International Conference on Computational Mathematics and Engineering Sciences, 2019, pp. 200-208. [16] K. K. Ali, R. Yilmazer, H. M. Baskonus, and H. Bulut,Modulation instability analysis andanalytical solutions to the system of equations for the ionsoundand Langmuir waves, Phys. Scr., 2020. [17] H. F. Ismael and H. Bulut, On the Solitary Wave Solutions to the (2+ 1)-Dimensional Davey-Stewartson Equations, in International Conference on Computational Mathematics and Engineering Sciences, 2019, pp.156-165. [18] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vriesequation,Phys. Lett. A, vol. 85, no. 8-9, pp. 407-408, 1981. [19] W. Gao, H. F. Ismael, S. A. Mohammed, H. M. Baskonus, and H. Bulut, Complex and Real Optical Soliton Properties of the Paraxial Non-linear Schrodinger Equation in Kerr Media With M-Fractional, Front. Phys., vol. 7, Nov.2019. [20] H. Dutta, H. Gunerhan, K. K. Ali, and R. Yilmazer, Exact soliton solutions to the cubic-quartic nonlinear Schrodinger equation with conformable derivative," Front. Phys., vol. 8, p. 62,2020. [21] K. Khan and M. Ali Akbar, "Traveling wave solutions of the (2 + 1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp- function method, Ain Shams Eng. J.,2014. [22] A. R. Seadawy, Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma, Comput. Math. with Appl., vol. 67, no. 1, pp. 172-180, 2014. [23] F. Calogero and A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform.-I, Nuovo Cim. B Ser. 11,1976. [24] T. Motsepa, C. Khalique, and M. Gandarias, Symmetry Analysis and Conservation Laws of the Zoomeron Equation, Symmetry (Basel).,2017. [25] E. Aksoy, A. C. Cevikel, and A. Bekir, Soliton solutions of (2+1)-dimensional time- fractional Zoomeron equation, Optik (Stuttg).,2016. [26] M. Alquran and K. Al-Khaled, Mathematical methods for a reliable treatment of the (2+1)-dimensional Zoomeron equation, Math. Sci.,2013. [27] M. Yavuz, Dynamical Behaviors of Separated Homotopy Method Defined by Conformable Operator, Konuralp J. Math., vol. 7, no. 1, pp.1-6. [28] M. A. Yousif, B. A. Mahmood, K. K. Ali, and H. F. Ismael, Numerical simulation using the homotopy perturbation method for a thin liquid film over an unsteady stretching sheet, Int. J. Pure Appl. Math., vol. 107, no. 2, 2016. [29] H.F.Ismael and K.K.Ali, MHD cass on flow over a nunsteady stretching sheet, Adv. Appl. Fluid Mech., vol. 20, no. 4, pp. 533-541, Oct. 2017. [30] H. F. Ismael, Carreau-Casson fluids flow and heat transfer over stretching plate with internal heat source/sink and radiation, Int. J. Adv. Appl. Sci. J., vol. 6, no. 2, pp. 81- 86, 2017. [31] A. Yokus, "Numerical Solutions of Time Fractional Korteweg-de Vries Equation and Its Stability Analysis," Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., vol. 68, no. 1, pp. 353-361, 2019. [32] K. K. Ali and A. Varol, "Weissenberg and Williamson MHD Flow over a stretching surface with thermal radiation and chemical reaction,"2019. [33] A. Zeeshan, H. F. Ismael, M. A. Yousif, T. Mahmood, and S. U. Rahman, "Simultaneous effects of slip and wall stretching/shrinking on radiative flow of magneto nanofluid through porous medium," J. Magn., vol. 23, no. 4, pp. 491-498, 2018. [34] A. Yokus, B. Kuzu, and U. Demiroglu, "Investigation of solitary wave solutions for the (3+ 1)-dimensional Zakharov- Kuznetsov equation," Int. J. Mod. Phys. B, vol. 33, no. 29, p. 1950350, 2019. [35] K. K. Ali, H. F. Ismael, B. A. Mahmood, and M. A. Yousif, "MHD Casson fluid with heat transfer in a liquid film over unsteady stretching plate," Int. J. Adv. Appl. Sci., vol. 4, no. 1, pp. 55-58, 2017.

17 /

[36] W. Gao, H. F. Ismael, A. M. Husien, H. Bulut, and H. M. Baskonus, "Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrodinger and Resonant Nonlinear Schrodinger Equation with the Parabolic Law," Appl. Sci., vol. 10, no. 1, p. 219, Dec. 2019. [37] M. A. Yousif, M. Hatami, and H. F. Ismael, "Heat Transfer Analysis of MHD Three Dimensional Casson Fluid Flow Over a Porous Stretching Sheet by DTM-Pade," Int. J. Appl. Comput. Math., vol. 3, no. S1, pp. 813-828, 2017. [38] H. F. Ismael and N. M. Arifin, "Flow and heat transfer in a maxwell liquid sheet overa stretching surface with thermal radiation and viscous dissipation," JP J. Heat Mass Transf., vol. 15, no. 4, 2018. [39] A. Yokus and S. Gulbahar, "Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation," Appl. Math. Nonlinear Sci.,2019. [40] W. Gao and H. F. I. and H. B. and H. M. Baskonus, "Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media," Phys. Scr.,2019. [41] Seyler, C. E., and D. L. Fenstermacher. "A symmetric regularized?long?wave equation." The Physics of fluids 27, no. 1 (1984): 4-7. [42] Dong, Xuanchun. "Numerical solutions of the symmetric regularized-long-wave equation by trigonometric integrator pseudospectral discretization." arXiv preprint arXiv:1109.0764 (2011). [43] Albert, John. "On the decay of solutions of the generalized Benjamin-Bona-Mahony equation." Journal of Mathemati- cal Analysis and Applications 141, no. 2 (1989): 527-537. [44] Amick, Charles J., Jerry L. Bona, and Maria E. Schonbek. "Decay of solutions of some nonlinear wave equations." Journal of Differential Equations 81, no. 1 (1989): 1-49. [45] Bo-Ling, Guo. "The spectral method for symmetric regularized wave equations." Journal of Computational Mathe- matics 5, no. 4 (1987): 297-306. [46] Lin, Chen. "Stability and instability of solitary waves for generalized symmetric regularized-long-wave equations." Physica D: Nonlinear Phenomena 118, no. 1-2 (1998): 53-68. [47] Kaplan, Ayse Gull, and Yilmaz Dereli. "Numerical solutions of the symmetric regularized long wave equation using radial basis functions." Comput. Model. Eng. Sci 84 (2012): 423-438. [48] Mittal, R. C., and A. Tripathi. "Numerical solutions of symmetric regularized long wave equations using collocation of cubic B-splines finite element." Int. J.Computational Methods in Engineering Science and Mechanics 16, no. 2 (2015): 142-150. [49] Yimnet, S., B. Wongsaijai, Thaned Rojsiraphisal, and K. Poochinapan. "Numerical implementation for solving the symmetric regularized long wave equation." Applied Mathematics and Computation 273 (2016): 809-825. [50] F. Chand and A. K. Malik, "Exact Traveling Wave Solutions of Some Nonlinear Equations Using G’/G -expansion Method methods The SRLW equation," International Journal of Nonlinear Science, vol. 14, no. 4, pp. 416-424,2012. [51] Mhlanga, Isaiah Elvis, and Chaudry Masood Khalique. "Exact solutions of the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations." In Abstract and Applied Analysis, vol. 2014. Hindawi, 2014. [52] Brango, Carlos Banquet. "The symmetric regularized-long-wave equation: Well-posedness and nonlinear stability." Physica D: Nonlinear Phenomena 241, no. 2 (2012): 125-133.

18 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

The Bounds for the Length Between Dirichlet and the Periodic Eigenvalues of Hill’ s Equation with Symmetric Single Well Potential

Elif Ba¸skaya∗1 1Karadeniz Technical University, Science Faculty, Department of Mathematics, Turkey

Keywords Abstract: A symmetric single well potential on [0,a] is deﬁned as symmetric with Hill’ s equation, respect to the midpoint a/2 and nonincreasing on [0,a/2]. In a symmetric hydrogen bond Symmetric single well that occurs in many biological structures such as DNA and water, the proton free energy potential, landscape is a symmetric single well potential. Besides, especially in recent years, since Dirichlet eigenvalues, quantum mechanic has gained importance, there are a lot of studies on eigenvalues of The periodic eigenvalues Hill’ s equation and Schrödinger’ s operator with symmetric single well potential such as anharmonic oscillator. The eigenvalues of these equations represent excitation energy and eigenfunctions are named as wavefunction in physics. In this study, the upper and lower bounds for the length between Dirichlet and the periodic eigenvalues are determined for Hill’ s equation when the potential is symmetric single well.

1. Introduction

In this study, we deal with

y00 (x) + [λ − q(x)]y(x) = 0, x ∈ [0,a] (1) where λ is a real parameter; q(x) is a real- valued, continuous and periodic function of periodic a. This equation is known as the standart form of Hill’ s equation. Let y1 and y2 be the linearly independent solutions of Hill’ s equation and initial conditions be as following for these solutions:

0 0 y1 (0,λ) = y2 (0,λ) = 1, y1 (0,λ) = y2 (0,λ) = 0.

D(λ) that is an analytic function of λ , is deﬁned by

0 D(λ) = y1 (a,λ) + y2 (a,λ) is called the discriminant of Hill’ s equation. The periodic problem of (1) is deﬁned with boundary conditions y(0) = y(a) 0 0 and y (0) = y (a). The eigenvalues of this problem are zeros of D(λ) − 2 and shown by {λn}. The semi-periodic problem of (1) is described with boundary conditions y(0) = −y(a) and y0 (0) = −y0 (a). The eigenvalues of this problem are the zeros of D(λ) + 2 and shown by {µn}. The periodic and semi-periodic eigenvalues {λn} and {µn} occur in the order [1]:

−∞ < λ0 < µ0 ≤ µ1 < λ1 ≤ λ2 < µ2 ≤ µ3 < ...

Besides, the eigenvalues of Hill’ s equation with y(0) = y(a) = 0 are denoted by Λn and the eigenvalues of Hill’ s equations 0 0 with y (0) = y (a) = 0 are denoted by νn. These conditions are named as Dirichlet and Neumann boundary conditions, respectively. It is also proven in [1] that for n = 0,1,2,...

µ2n ≤ Λ2n ≤ µ2n+1, λ2n+1 ≤ Λ2n+1 ≤ λ2n+2, (2) and µ2n ≤ ν2n+1 ≤ µ2n+1, λ2n+1 ≤ ν2n+2 ≤ λ2n+2. (3)

All solutions of Hill’ s equation are bounded in (−∞,+∞) for λ ∈ (λ2n, µ2n)∪(µ2n+1,λ2n+1), while all nontrivial solutions

∗ Corresponding author: [email protected] 19 / of Hill’ s equation are unbounded in (−∞,+∞) for λ ∈ (−∞,λ0) ∪ (µ2n, µ2n+1) ∪ (λ2n+1,λ2n+2).

A symmetric single well potential on [0,a] is deﬁned as symmetric for the midpoint a/2 and nonincreasing on [0,a/2]. For example, in a symmetric hydrogen bond that occurs in many biological structures such as DNA and water, the proton free energy landscape is a symmetric single well potential. Besides, especially in recent years, since quantum mechanic has gained importance, there are a lot of studies on eigenvalues of Hill’ s equation and Schrödinger’ s operator with symmetric single well potential such as anharmonic oscillator. The eigenvalues of these equations represent excitation energy and eigenfunctions are named as wavefunction in physics.

Instability intervals for Hill’ s equation with symmetric single well potential have been investigated by many authors. Some important studies of them are [2], [4], [5], [6], [7], [8]. Especially, we refer to [4] includes the length for Λ0 − λ0. In this study, the upper and lower bounds for the length between Dirichlet and the periodic eigenvalues are determined for Hill’ s equation when the potential is symmetric single well. We note that a symmetric single well potential on [0,a] means a continuous function q(x) on [0,a] which is symmetric about x = a/2 and non-increasing on [0,a/2], that is, q(x) = q(a − x) mathematically.

2. Main Results Firstly, we emphasize that q0 (x) exists since a monotone function on an interval I is differentiable almost everywhere on I [3]. Our analysis is based on the following theorem of [8]: The periodic eigenvalues of (1) satisfy, as n → ∞

a/2 Z 1/2 2(n + 1)π a 0 4(n + 1)π λ n+ = − q (t)sin t dt 2 1 a 8(n + 1)2π2 a 0  a/2 a/2  2 Z Z (4) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 64(n + 1)3π3 0 0 + on−3 and

a/2 Z 1/2 2(n + 1)π a 0 4(n + 1)π λ n+ = + q (t)sin t dt 2 2 a 8(n + 1)2π2 a 0  a/2 a/2  2 Z Z (5) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 64(n + 1)3π3 0 0 + on−3. The semi-periodic eigenvalues of (1) satisfy, as n → ∞

a/2 Z 1/2 (2n + 1)π a 0 2(2n + 1)π µ n = − q (t)sin t dt 2 a 2(2n + 1)2π2 a 0  a/2 a/2  2 Z Z (6) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 8(2n + 1)3π3 0 0 + on−3 and

a/2 Z 1/2 (2n + 1)π a 0 2(2n + 1)π µ n+ = + q (t)sin t dt 2 1 a 2(2n + 1)2π2 a 0  a/2 a/2  2 Z Z (7) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 8(2n + 1)3π3 0 0 + on−3. The purpose of this study is to prove the following theorem: 20 /

Theorem 2.1. Let q(x) be a symmetric single well potential on [0,a]. Then, the bounds for the length between Dirichlet and the periodic eigenvalues as n → ∞

a/2 2 Z (4n + 1)π 1 0 2(2n + 1)π Λ2n − λ2n ≥ − q (t)sin t dt a2 (2n + 1)π a 0 a/2 Z 1 0 4nπ − q (t)sin t dt 2nπ a 0  a/2 a/2  " # Z Z a a 2 0 0 − − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 4(2n + 1)2π2 16n2π2 0 0 + on−2 and

a/2 2 Z (4n + 1)π 1 0 2(2n + 1)π Λ2n − λ2n ≤ + q (t)sin t dt a2 (2n + 1)π a 0 a/2 Z 1 0 4nπ − q (t)sin t dt 2nπ a 0  a/2 a/2  " # Z Z a a 2 0 0 − − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 4(2n + 1)2π2 16n2π2 0 0 + on−2. Proof. Firstly, if we take the squares of the eigenvalues in (5)-(7), we ﬁnd that

a/2 2 2 Z 4(n + 1) π 1 0 4(n + 1)π λ2n+2 = + q (t)sin t dt a2 2(n + 1)π a 0  a/2 a/2  Z Z (8) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 16(n + 1)2π2 0 0 + on−2,

a/2 2 2 Z (2n + 1) π 1 0 2(2n + 1)π µ2n = − q (t)sin t dt a2 (2n + 1)π a 0  a/2 a/2  Z Z (9) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 4(2n + 1)2π2 0 0 + on−2 and

a/2 2 2 Z (2n + 1) π 1 0 2(2n + 1)π µ2n+1 = + q (t)sin t dt a2 (2n + 1)π a 0  a/2 a/2  Z Z (10) a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 4(2n + 1)2π2 0 0 + on−2.

By (2), we obtain the bounds for Λ2n − λ2n that

µ2n − λ2n ≤ Λ2n − λ2n ≤ µ2n+1 − λ2n. (11) 21 /

By rewriting λ2n+2 from (8) for 2n, we get that

a/2 2 2 Z 4n π 1 0 4nπ λ2n = + q (t)sin t dt a2 2nπ a 0  a/2 a/2  Z Z a 2 0 0 − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 16n2π2 0 0 + on−2. By using the last equation and (9)-(10), we calculate that

a/2 2 Z (4n + 1)π 1 0 2(2n + 1)π µ2n+1 − λ2n = + q (t)sin t dt a2 (2n + 1)π a 0 a/2 Z 1 0 4nπ − q (t)sin t dt 2nπ a 0  a/2 a/2  " # Z Z a a 2 0 0 − − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 4(2n + 1)2π2 16n2π2 0 0 + on−2 and

a/2 2 Z (4n + 1)π 1 0 2(2n + 1)π µ2n − λ2n = − q (t)sin t dt a2 (2n + 1)π a 0 a/2 Z 1 0 4nπ − q (t)sin t dt 2nπ a 0  a/2 a/2  " # Z Z a a 2 0 0 − − aq (a) + 2a q(t)q (t)dt − 4 tq(t)q (t)dt 4(2n + 1)2π2 16n2π2 0 0 + on−2. By using last equations and (11), we prove the theorem.

Example 2.2. Consider an eigenvalue problem is named as anharmonic oscillator

y00 (x) + [λ − q(x)]y(x) = 0, x ∈ [0,π]

1 π 4 1 π 2 where q(x) = 4 x − 2 + 2 x − 2 and extended by periodicity. Since q(x) has mean value zero in the reference theorem, q(x) can taken as follows:

1 π 4 1 π 2 π2 π4 q(x) = x − + x − − − . 4 2 2 2 24 320 In this case, by calculating integral terms in the theorem, we get as n → ∞

1 3 2 3 3 Λ2n − λ2n ≥(4n + 1) − 4π + 16π n + 4π + 16π n + π − 2π 16(2n + 1)4π 1 − 2π3 + 8πn2 − 3π 128n4π " # 1 1 π4 − − 112π4 + 1920π2 + 8960 4(2n + 1)2 16n2 6451200 + on−2 and

22 /

1 3 2 3 3 Λ2n − λ2n ≤(4n + 1) + 4π + 16π n + 4π + 16π n + π − 2π 16(2n + 1)4π 1 − 2π3 + 8πn2 − 3π 128n4π " # 1 1 π4 − − 112π4 + 1920π2 + 8960 4(2n + 1)2 16n2 6451200 + on−2.

3. Discussion and Conclusion

We obtain the following conclusion from the main theorem: Conclusion: If q(x) is a constant, the length between Dirichlet and the periodic eigenvalues as n → ∞

2 " # (4n + 1)π 1 1 2 2 Λ2n − λ2n = − − a q (a) a2 4(2n + 1)2π2 16n2π2 + on−2. This conclusion holds since q0 (x) = 0

Acknowledgment

The author is grateful to Prof. Dr. Haskız CO¸SKUNfor supervisor.

References

[1] Eastham, M. S. P. 1973. The spectral theory of periodic differential equations. Scottish Academic Press, Edinburgh and London, chapter 2 and 3. [2] Ashbaugh, M. and Benguria, R. 1989. Optimal lower bound for the gap between the first two eigenvalues of one- dimensional Schrödinger operators with symmetric single-well potentials and related results. Proc. Amer. Math. Soc., 105(1989), 419-424. [3] Haaser, N. B. and Sullivian, J. A. 1991. Analysis. Van Nostrand Reinhold Co., New York. [4] Huang, M.- J. 1997. The first instability interval for Hill equations with symmetric single-well potentials. Proc. Amer. Math. Soc., 125(1997), no.3, 775-778. [5] Horvath, M. 2002. On the first two eigenvalues of Sturm-Liouville Operators. Proc. Amer. Math. Soc., 131(2002), no.4, 1215-1224. [6] Huang, M.- J. and Tsai, T.- M. 2009. The eigenvalue gap for one-dimensional Schrödinger operators with symmetric potentials. Proc. Roy. Soc. Edinburg, 139(2009), 359-366. [7] Chen, D.- Y. and Huang, M.- J. 2011. Comparison theorems for the eigenvalue gap of Schrödinger operators on the real line. Ann. Henri Poincare, 13(2011), 85-101. [8] Co¸skun,H., Ba¸skaya,E. and Kabata¸s,A. 2019. Instability intervals for Hill’s equation with symmetric single well potential. Ukr. Mat. Zh., 71(2019), no.6, 858-864.

23 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERENCE PROCEEDING

1 On solving the (2+1)-dimensional Zoomeron equation via (G 0 )- expansion method

Hasan Bulut∗1, Gizem Aydin2 1Department of Mathematics, Firat University, Elazig, Turkey 2Department of Mathematics, Firat University, Elazig, Turkey

1 Keywords Abstract: In this article, ( G0 )-expansion method is applied on (2+1)-dimensional Zoomeron equation, Zoomeron equation to seek some new traveling wave solutions. Many operations have 1 ( G0 )-expansion method. been performed with the package program code for the accuracy of these solutions. The gained solutions in this paper are veriﬁed by plugging them back to the studied partial differential equation. The exact solutions of the suggested equation are obtained via using 1 the ( G0 )-expansion method. We are conﬁdent that the solutions will be useful for scientists to work in this area.

1. Introduction

Nonlinear partial differential equations (PDEs) have an important place in physics and mathematics. Such equations are mathematical models for processes in the aspects of physics, genetics, chemicals and science. Computer programs like Matlab, Maple, and Mathematica are used to gain the general solutions of PDEs. The general solutions of these equations provide researchers with information about the character of physical events. To understand the nature of physical phenomena, researchers have studied many methods, some of these methods can be listed as follows: an inverse spectral 1 transform technique[1], the Laplace perturbation method [2],( G0 )-expansion method [3], direct algebraic method [4], the hyperbolic tangent function expansion method [5], the generalized expansion method [6], sine-Gordon expansion method (SGEM) [7], the Hirota‘s bilinear transformation method [8], the Bernoulli sub-equation function method [9][10], sechp − tanhp functions methods [11], the modiﬁed simple equation method [12], generalized Kudryashov method G0 [13],( G )-expansion method [14], Tanh-Coth method [15], a multiplier method [16], and more [17-23].The nonlinear Zoomeron equation in (2+1)-dimension is given by [1] can be written as: u u xy − xy + 2u2 = 0, (1) u xy u xx xt here u(x,y,t) represent the amplitude of the relative wave mode.

1 2. ( G0 )- Expansion Method

1 In this portion, the general details of the ( G0 )-expansion method are provided. Let us consider a three-variable general form of the nonlinear PDE as O(u,ut ,uux,uuy,uxx,....) = 0, (2) and transform equation (2) with u(x,y,t) = u(ξ) = u,ξ = κ1x + κ2y − κ3t,in which κ1,κ2 and κ3 are non-zero constants. After conversion, we’ll have a nonlinear ordinary differential equation (ODE)

S(u,u0,uu00,u000,.....) = 0. (3)

The solution of (3) shall be taken in the following form

m 1 i u(ξ) = a0 + ∑ ai( 0 ) . (4) i=1 G

Here G = G(ξ) is a function that veriﬁes the following second-order ordinary differential equation

G00 + λG0 + µ = 0, (5)

∗ Corresponding author: hbulut@firat.edu.tr 24 / here a0,a1,a2,...,λ, µ are constants and m is a term of balance. The term balancing is a fixed number obtained between the linear term and the non-linear term in any nonlinear ODE that have highest order. This number is written in place of (4) 00 1 and then the derivatives required for the solution are obtained. These derivatives are taken as G = −λG − µ.( G0 ) is a polynomial and homogeneous equation. Here we gain a system of algebraic equations by equalizing the coefficients of 1 i ( Gi ) ,i = 1,2,3,.... terms to zero. This algebraic equation scheme is solved automatically or with the aid of a computer program depending on the degree of difficulty. These solutions (4) given by the solution function and moving wave transformation is written in place (2 ) given by the solution of the PDE is found [24]. The general solution of the ODE (5) could be reported as below. µξ G = G(ξ) = − − c e−λξ + c , (6) λ 1 2 here c1 and c2 are fixed. Depending on the variable the derivative of the (6) solution function is taken and necessary arrangements are made then 1 1 = , (7) G0 µ λξ − λ + c1e is obtained. When we transform the algebraic expression provided in (7) to a trigonometric function with c1 = A, then it may write as 1 λ = . (8) G0 −µλA(Cosh(ξλ) − Sinh(ξλ))

1 3. Implementation ( G0 )- Expansion Method

1 Now, we use the ( G0 ) -expansion method on (2 + 1) dimensional Zoomeron equation. Consider the (2 + 1) dimensional Zoomeron equation given by Eq. (1). If we apply

u(x,y,t) = U(ξ),ξ = κ1x + κ2y − κ3t, (9) using a wave transformation 00 00 00 00 U U 00 κ κ κ2 − κ3κ + 2κ κ u2 = 0, (10) 1 2 3 u 1 2 u 1 3 we acquire the nonlinear ODE. Now integrating the above equation two times with belong to ξ , we have

2 2 00 3 κ1κ2(κ3 − κ1 )U − κ1κ2U + εU = 0, (11) we get the equation. Here ε is the scalar of integration. Balancing between the U00 and U3 in Equation (11), one have m = 1. In this case, the equation (4) becomes 1 u(ξ) = a + a ( ), (12) 0 1 G0

1 i the solution (12) together with its derivative inserting into Eq. (11) and the necessary arrangements are made, Gi ) ,i = 1,2,3....,m coefﬁcients of the term polynomial terms equal to zero 1 ( )0 : εa − κ κ a3, G0 0 1 3 0 1 ( )1 : εa − κ3λ 2κ a + κ λ 2κ κ2a − 6κ κ a2a , G0 1 1 2 1 1 2 3 1 1 3 0 1 1 ( )2 : −3κ3λ µκ a + 3κ λ µκ κ2a − 6κ κ a a2, G0 1 2 1 1 2 3 1 1 3 0 1 1 ( )3 : −2κ3µ2κ a + 2κ µ2κ κ2a − 2κ κ a3, G0 1 2 1 1 2 3 1 1 3 1 (13) algebraic equation system is obtained. The system of equations (13) given by the computer package program is solved. √ √ 1 √ √ 1 √ 4 4 3 2 i ε λκ2 i 2 εµκ2 2ε+κ1 λ κ2 Case-1 When a0 = − , a1 = − , κ3 = − √ √ , one could gain the following √ 1 1 √ 1 1 κ1λ κ2 4 3 2 4 4 3 2 4 2κ1 (2ε+κ λ κ2) λκ1 (2ε+κ1 λ κ2) complex soliton solution

√ √ 1 √ √ 1 4 4 i ε λκ2 i 2 εµκ2 u(x,y,t) = − 1 − √ 1 , √ 1 1 µ 4 3 2 4 4 3 2 4 2κ1 (2ε + κ λ κ2) λκ1 (2ε + κ1 λ κ2) − λ +C1cosh[λξ] −C1sinh[λξ]

25 / where ξ = κ1x + κ2y − κ3t. √ √ 1 i ε λκ 4 u(x,t) = − 2 √ 1 1 4 3 2 4 2κ1 (2ε + κ λ κ2) √ √ 1 i 2 εµκ 4 − √ 2 √ √ 1 1 3 2 3 2 4 3 2 µ t 2ε+κ1 λ κ2 t 2ε+κ1 λ κ2 λκ (2ε + κ λ κ2) 4 − +C1cosh[λ(κ1x + κ2y − √ √ )] −C1sinh[λ(κ1x + κ2y − √ √ )] 1 1 λ κ1λ κ2 κ1λ κ2 providing ε > 0, κ1 > 0, λ > 0, κ2 > 0. √ √ 1 √ √ 1 √ 4 4 3 2 ε λκ2 2 εµκ2 2ε+κ1 λ κ2 Case-2 When a0 = − , a1 = − , κ3 = − √ √ , we can gain the following real √ 1 1 √ 1 1 κ1λ κ2 4 3 2 4 4 3 2 4 2κ1 (2ε+κ λ κ2) λκ1 (2ε+κ1 λ κ2) soliton solution √ √ 1 √ √ 1 4 4 ε λκ2 2 εµκ2 u(x,y,t) = − 1 − √ 1 , √ 1 1 µ 4 3 2 4 4 3 2 4 2κ1 (2ε + κ λ κ2) λκ1 (2ε + κ1 λ κ2) − λ +C1cosh[λξ] −C1sinh[λξ] where ξ = κ1x + κ2y − κ3t. √ √ 1 ε λκ 4 u(x,y,t) = − 2 √ 1 1 4 3 2 4 2κ1 (2ε + κ λ κ2) √ √ 1 2 εµκ 4 − √ 2 √ . √ 1 1 3 2 3 2 4 3 2 µ t 2ε+κ1 λ κ2 t 2ε+κ1 λ κ2 λκ (2ε + κ λ κ2) 4 − +C1cosh[λ(κ1x + κ2y − √ √ )] −C1sinh[λ(κ1x + κ2y − √ √ )] 1 1 λ κ1λ κ2 κ1λ κ2 √ √ 2 2 2 2 2 2 λ κ2(−κ1 +κ3 ) λ κ2(−κ1 +κ3 ) 1 2 2 2 Case-3 When a0 = √ , a1 = √ , ε = κ1λ κ2(−κ + κ ), we have the following real soliton 2 κ3 λ κ3 2 1 3 solution q q 2 2 2 2 2 2 λ κ2(−κ1 + κ3 ) λ κ2(−κ1 + κ3 ) u(x,y,t) = − √ − √ µ , 2 κ3 λ κ3(− λ +C1cosh[λξ] −C1sinh[λξ]) where ξ = κx + νy − ωt. q 2 2 2 2µ λ κ2(−κ1 + κ3 ) −1 + µ−C λcosh[λ(κ x+κ y−κ t)]−C λsinh[λκ x+κ y−κ t] u(x,y,t) = 1 √ 1 2 3 1 1 2 3 , 2 κ3

2 2 2 providing λ κ2(−κ + κ ) > 0,κ3 > 0. 1 3√ √ √ ε 2 εµ 2 √ √ √ √ √ ε Case-4 When a0 = − ,a1 = − κ λ κ , κ2 = 2 2 2 , 2 κ1 κ3 1 3 κ1λ (−κ1 −κ3 ) we have the following real soliton solution √ √ √ ε 2 εµ u(x,y,t) = −√ √ √ − √ √ µ , 2 κ1 κ3 κ1λ κ3(− λ +C1cosh[λξ] −C1sinh[λξ]) where ξ = κ1x + κ2y − κ3t.   √ 2µ ε −1 +  2ε 2ε µ−C1λcosh λ xκ1−tκ3+ +C1sinh λ xκ1−tκ3+ κ λ2(−κ2−κ2) κ λ2(−κ2−κ2) u(x,y,t) = 1√ √1 √3 1 1 3 , 2 κ1 κ3 providing κ1 > 0,κ3 > 0,ε > 0.

4. Conclusions

1 We have implemented the( G0 ) -expansion method successfully to the Zoomeron equation. The gained solutions are novel compared to other results presented in article research [25-27]. The ﬁndings of these studies would be useful in describing the physical deﬁnitions of certain non-linear models in research. This approach can also be extended to many other nonlinear equations. 26 /

References

[1] F. Calogero and A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform.-I, Nuovo Cim. B Ser. 11, 1976. [2] M. Yavuz, N. Ozdemir, and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, Eur. Phys. J. Plus, 2018. [3] K. K. Ali, H. Dutta, R. Yilmazer, and S. Noeiaghdam, On the new wave behaviors of the Gilson-Pickering equation, Front. Phys., vol. 8, p. 54, 2020. [4] A. R. Seadawy, Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma, Comput. Math. with Appl., 2014. [5] S. Liu, Z. Fu, S. Liu, and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. Sect. A Gen. At. Solid State Phys., 2001. [6] R. Sabry, W. M. Moslem, F. Haas, S. Ali, and P. K. Shukla, Nonlinear structures: Explosive, soliton, and shock in a quantum electron-positron-ion magnetoplasma, Phys. Plasmas, 2008. [7] G. Yel, H. M. Baskonus, and H. Bulut, Novel archetypes of new coupled Konno-Oono equation by using sine-Gordon expansion method, Opt. Quantum Electron., 2017. [8] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, vol. 85, no. 8-9, pp. 407-408, 1981. [9] H. H. Abdulkareem, H. F. Ismael, E. S. Panakhov, and H. Bulut, Some Novel Solutions of the Coupled Whitham- Broer-Kaup Equations, in International Conference on Computational Mathematics and Engineering Sciences, 2019, pp. 200-208. [10] H. F. Ismael and H. Bulut, On the Solitary Wave Solutions to the (2+ 1)-Dimensional Davey-Stewartson Equations, in International Conference on Computational Mathematics and Engineering Sciences, 2019, pp. 156-165. [11] M. Alquran and K. Al-Khaled, Mathematical methods for a reliable treatment of the (2+1)-dimensional Zoomeron equation, Math. Sci., 2013. [12] K. Khan and M. Ali Akbar, Traveling wave solutions of the (2 + 1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method, Ain Shams Eng. J., 2014. [13] E. Aksoy, A. C. Cevikel, and A. Bekir, Soliton solutions of (2+1)-dimensional time-fractional Zoomeron equation, Optik (Stuttg)., 2016. [14] R. Abazari, The solitary wave solutions of Zoomeron equation, Appl. Math. Sci, vol. 5, no. 59, pp. 2943-2949, 2011. [15] A. IRSHAD and S. T. MOHYUD-DIN, Solitary wave solutions for Zoomeron equation, Walailak J. Sci. Technol., vol. 10, no. 2, pp. 201-208, 2012. [16] T. Motsepa, C. Khalique, and M. Gandarias, Symmetry Analysis and Conservation Laws of the Zoomeron Equation, Symmetry (Basel)., 2017. [17] H. F. Ismael, H. Bulut, and H. M. Baskonus, Optical soliton solutions to the Fokas-Lenells equation via sine-Gordon G0 expansion method and (m+ ( G )) -expansion method, Pramana - J. Phys., vol. 94, no. 1, Dec. 2020. [18] H. Dutta, H. Gunerhan, K. K. Ali, and R. Yilmazer, Exact soliton solutions to the cubic-quartic nonlinear Schrodinger equation with conformable derivative, Front. Phys., vol. 8, p. 62, 2020. [19] H. F. Ismael, Carreau-Casson fluids flow and heat transfer over stretching plate with internal heat source/sink and radiation, Int. J. Adv. Appl. Sci. J., vol. 6, no. 2, pp. 81-86, 2017. [20] W. G. and H. F. I. and H. B. and H. M. Baskonus, Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media, Phys. Scr., 2019. [21] W. Gao, H. F. Ismael, A. M. Husien, H. Bulut, and H. M. Baskonus, Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schro¨dinger and Resonant Nonlinear Schrodinger Equation with the Parabolic Law, Appl. Sci., vol. 10, no. 1, p. 219, Dec. 2019. [22] C. K. Kuo and B. Ghanbari, Resonant multi-soliton solutions to new (3+1)-dimensional Jimbo-Miwa equations by applying the linear superposition principle, Nonlinear Dyn., 2019. [23] W. Gao, H. F. Ismael, S. A. Mohammed, H. M. Baskonus, and H. Bulut, Complex and real optical soliton properties of the paraxial nonlinear Schrodinger equation in Kerr media with M-fractional, Front. Phys., vol. 7, p. 197, 2019. [24] A. Yokus, An expansion method for finding traveling wave solutions to nonlinear pdes," 2015.

27 /

G0 [25] K. Khan, M. Ali Akbar, M. Abdus Salam, and M. Hamidul Islam, A note on enhanced ( G )-expansion method in nonlinear physics, Ain Shams Eng. J., 2014. [26] O. Guner and A. Bekir, Bright and dark soliton solutions for some nonlinear fractional differential equations, Chinese Phys. B, 2016. [27] A. Bekir, F. Tascan, and O¨. U¨ nsal, Exact solutions of the Zoomeron and Klein-Gordon-Zakharov equations, J. Assoc. Arab Univ. Basic Appl. Sci., 2015.

28 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

Some Comparisons of BLUPs under General Linear Random Effects Models

Melek ERI˙S¸ BÜYÜKKAYA∗1, Nesrin GÜLER2 1Karadeniz Technical University, Faculty of Science, Department of Statistics and Computer Sciences, TURKEY 2Sakarya University, Faculty of Political Sciences, Department of Econometrics, TURKEY

Keywords Abstract: In the study, covariance matrices’ comparison of predictors is considered in BLUP, the context of general linear random-effects models. Assume a general linear random- Covariance matrix, effects model with its sub-sample models having correlated random-effects. Our aim is to Inertia, give some comparisons for covariance matrices of the Best Linear Unbiased Predictors Linear random-effects model, (BLUPs) of unknown vectors under sub-sample models by using some well-known rank Rank and inertia formulas for matrices and their Moore-Penrose generalized inverses.

1. Introduction

A general linear random-effects model, deﬁned by

M : y = Xβ + ε, β = Aα + γ, (1)

nx nxp px where y ∈ R 1 is a vector of observable response variables, X ∈ R is a known matrix of arbitrary rank, β ∈ R 1 is pxk kx an unknown random vector, A ∈ R is a known matrix of arbitrary rank, α ∈ R 1 is a vector of ﬁxed but unknown px nx parameters, γ ∈ R 1 is a vector of unobservable random variables, and ε ∈ R 1 is an unobservable vector of random errors. We can divide y, X and ε in M as           y1 X1 ε1 X1 ε1  y2   X2   ε2   X2   ε2  :   =   +   =  (A + ) +   (2) M  .   . β  .   .  α γ  .   .   .   .   .   .  yn Xn εn Xn εn and then we obtain sub-sample models of general linear random-effects model of M as follows

M 1 : y1 = X1β + ε1 = X1 (Aα + γ) + ε1,

M 2 : y2 = X2β + ε2 = X2 (Aα + γ) + ε2, (3) . . M n : yn = Xnβ + εn = Xn (Aα + γ) + εn.

Models M i, i = 1,...,n are transformed version of model M . They are obtained from pre-multiplying the model M by the matrices T1 = In1 0 ··· 0 , T2 = 0 In2 ··· 0 ,..., Tn = 0 0 ··· Inn , (4) respectively. For the models M i, i = 1,...,n, we will assume the following general assumptions on the expectation vector and covariance matrix of random vectors       γ γ Σ00 Σ01 ··· Σ0n  ε1   ε1   Σ10 Σ11 ··· Σ1n  E  = 0, cov  =   := Σ, (5)  .   .   . . .. .   .   .   . . . .  εn εn Σn0 Σn1 ··· Σnn

∗ Corresponding author: [email protected] 29 /

(n+p)×(n+p) where Σ ∈ R is a positive semi-deﬁnite matrix and all the entries of Σ are known. Under the assumptions in (5), we can obtain the following formulas of the expectations and dispersion of yi

0 E(yi) = Xbiα and cov(yi,yi) = D(yi) = XeiΣXei := Vi, (6) where we use the following expressions for convenience of the representation of above formulas

Xbi = XiA and Xei = [Xi Ti], i = 1,...,n. (7)

We assume that the sub-sample models Mi are consistent, i.e., yi ∈ C [Xbi Vi] holds with probability 1; see, [17, p. 282]. It is known that if the general linear random-effects model M is consistent, then the sub-models are consistent; see, [24].

In order to establish some general results on simultaneous predictions of all unknown parameters under the models, we can consider the following general linear function of ﬁxed effects and random-effects

φ i = Fα + Gγ + HiTiε, i = 1,...,n, (8)

sxk sxp sxni with E(φ i) = Fα for given matrices F ∈ R , G ∈ R and Hi ∈ R . From the assumptions in (5),

0 0 D(φ i) = JiΣJi and cov(φ i,yi) = JiΣXei := Ci (9) is obtained, where Ji = [GHiTi], i = 1,...,n. The vector φ i in (8) is said to be predictable under M i if there exists an sxn Li ∈ R i such that D(Liyi − φ i) = min and E (Liyi − φ i) = 0, i = 1,...,n, (10) holds in the Löwner partial ordering, the linear statistic Li is deﬁned to be the best linear unbiased predictor (BLUP) of φ i under M i and is denoted by

Liyi = BLUPM i (φ i) = BLUPM i (Fα + Gγ + HiTiε). (11)

This is well-known deﬁnition of the BLUP of φ i which is originated from [3]. If G = 0 and Hi = 0, (11) reduces the best linear unbiased estimator (BLUE) of Fα under Mi.

Consideration of the models M and Mi separately or simultaneously for making prediction on unknown vectors is meaningful because they have the same random coefﬁcient vectors. Predictors of unknown vectors have different algebraic expressions and different properties under these models. Then it is natural to consider the comparison problem among covariance matrices of predictors because of having different performances under the models. Covariance matrices of BLUPs are usually used as a comparison criteria to determine optimal predictors among other types of unbiased predictors because of their minimum covariance requirement in the Löwner partial ordering. In this study, we give variety of inequalities and equalities for comparison of covariance matrices of BLUPs under two sub-sample models of M by using rank and inertia formulas. We also present results for comparisons of covariance matrices of BLUEs under considered models. As further reference for comparison of covariance matrix of predictors/estimators, we may mention [5, 6, 22, 23, 25]. For more details on inertias and ranks of symmetric matrices and relations between inertias and Löwner partial ordering of symmetric matrices; see, e.g., [16, 18, 19, 26]. For more related work on BLUP under linear mixed models; see, e.g., [2, 4, 7–14, 20, 21].

mxn 0 + Let R stand for the collection of all m × n real matrices, A , r (A), C (A), and A denote the transpose, the rank, mxn the column space, and the Moore-Penrose generalized inverse of a matrix A ∈ R , respectively, Im denote the identity + ⊥ + + matrix of order m. Furthermore, let PA = AA , EA = A = Im −AA , FA = In −A A stand for the orthogonal projectors. mxm The number of positive and negative eigenvalues of symmetric matrix A ∈ R counted with multiplicities are denoted by i+ (A) and i− (A), called the positive and negative inertia of A, respectively, and also for brief i± (A) denotes the both numbers. It is easy to see that r (A) = i+ (A)+i− (A). A 0, A ≺ 0, A 0 and A 0 mean that A is a symmetric positive definite, negative definite, positive semi-definite, and negative semi-definite matrix, respectively.

2. Preliminaries

In the study of the equality and inequality for covariance matrices of the BLUPs of unknown parameters under general linear random-effects model, it is important to simplify all kinds of matrix expressions involving Moore–Penrose generalized inverses of matrices. One of the powerful tool for simplifying complicated matrix expressions is rank formulas for partitioned matrices. The following rank equalities for partitioned matrices are well known; see [15].

m×n m×k l×n l×k Lemma 2.1. Let A ∈ R , B ∈ R , C ∈ R , and D ∈ R . Then,

r[A B] = r(A) + r(EAB) = r(B) + r(EBA), (12) 30 /

A r = r(A) + r(CF ) = r(C) + r(AF ), (13) C A C AB r = r(B) + r(C) + r(E AF ), (14) C 0 B C AB r = r(A) + r(D − CA+B) if (B) ⊆ (A) and (C0) ⊆ (A0). (15) CD C C C C The following lemmas were given by [18], which will be used to establish rank and inertias of matrices and to characterize equalities and inequalities of BLUPs’ covariances matrices. m×n m×m Lemma 2.2. Let A, B ∈ R , or, A, B ∈ R be symmetric matrices. Then, the following results hold. A = B ⇔ r(A − B) = 0, (16)

A B ⇔ i+(A − B) = m and A ≺ B ⇔ i−(A − B) = m, (17) A < B ⇔ i−(A − B) = 0 and A 4 B ⇔ i+(A − B) = 0. (18) m×m n×n m×n Lemma 2.3. Let A ∈ R and B ∈ R be symmetric matrices and Q ∈ R . Then, + i±(A ) = i±(A), i±(−A) = i∓(A), (19) AQ A −Q −AQ i = i = i , (20) ± Q0 B ± −Q0 B ∓ Q0 −B A 0 0 Q 0 Q i = i (A) + i (B), i = i = r(Q). (21) ± 0 B ± ± + Q0 0 − Q0 0 m×m n×n m×n Lemma 2.4. Let A ∈ R and D ∈ R be symmetric matrices and B ∈ R . Then, AB AB i = r(B) + i (E AE ) and r = 2r(B) + r(E AE ). (22) ± B0 0 ± B B B0 0 B B In particular, AA0 B AA0 B i = r[AB], i = r(B), (23) + B0 0 − B0 0 AA0 B AA0 BC r = r[AB] + r(B), r = r[ABC] + r(B), (24) B0 0 B0 0 0 AB i = i (A) + i (D − B0A+B) if (B) ⊆ (A). (25) ± B0 D ± ± C C For the following lemma, see [20].

Lemma 2.5. Let Mi be as given in (3), i = 1,...,n. The parameter vector φ i in (8) is predictable by yi under Mi if and only if 0 0 C (F ) ⊆ C (Xbi). (26)

In this case, a linear statistic Liyi is the BLUP for φ i if and only if the equation ⊥ ⊥ Li[Xbi ViXbi ] = [FCiXbi ] (27) is satisﬁed. Then, the general expression of Li and the corresponding BLUPMi (φ i) can be written as ⊥ ⊥ + ⊥ ⊥ BLUPMi (φ i) = Liyi = [FCiXbi ][Xbi ViXbi ] + Ui[Xbi ViXbi ] yi, (28) s×n where Ui ∈ R i is arbitrary. Furthermore, the following results hold. ⊥ ⊥ ⊥ a) r[Xbi ViXbi ] = r[Xbi Vi], C [Xbi ViXbi ] = C [Xbi Vi], and C (Xbi) ∩ C (ViXbi ) = {0},

b) Li is unique if and only if r[Xbi Vi] = ni,

c) BLUPMi (φ i) is unique if and only if yi ∈ C [Xbi Vi] holds with probability 1,

d) BLUPMi (φ i) satisﬁes + + 0 D[BLUPMi (φ i)] = (ZiWi )Vi(ZiWi ) , (29) + 0 cov(BLUPMi (φ i),φ i) = ZiWi Ci, (30) + + 0 D[φ i − BLUPMi (φ i)] = (ZiWi Xei − Ji)Σ(ZiWi Xei − Ji) , (31) ⊥ ⊥ where Wi = [Xbi ViXbi ], Zi = [FCiXbi ] and Ji = [GHiTi].

It is noteworthy that the requirement in (26) also means Fα is an estimable parametric function under Mi, which is a well-known deﬁnition for estimability of a parametric function, see, e.g., [1]. 31 /

3. Main Results

In this section, we give links between the covariance matrix of BLUP of φ i under two sub-sample models of general linear random-effects model M .

Theorem 3.1. Assume that φ k and φ l are predictable under sub-sample models M k and M l, respectively, k,l = 1,...,n, k 6= l, and let the BLUP equation be as given in (28). Also, denote

 0  Vk 0 Ck 0 Xbk  0   0 −Vl Cl −Xbl 0   0 0  N =  Ck −Cl JkΣJk − JlΣJl −FF .  0 0   0 −Xbl −F 0 0  0 0 Xbk 0 F 0 0 Then, i+ cov φ k − BLUP M k (φ k) − cov φ l − BLUP M l (φ l) = i+ (N) − r Xbk Vk − r Xbl , (32)

i− cov φ k − BLUP M k (φ k) − cov φ l − BLUP M l (φ l) = i− (N) − r Xbk − r Xbl Vl , (33)

r (cov[φ k − BLUP(φ k)] − cov[φ l − BLUP(φ l)]) (34) = r (N) − r Xbk Vk − r Xbl − r Xbk − r Xbl Vl . In consequence, the following results hold. a) cov φ l − BLUP M l (φ l) cov φ k − BLUP M k (φ k) ⇔ i− (N) = r Xbk + r Xbl Vl + s.

b) cov φ l − BLUP M l (φ l) ≺ cov φ k − BLUP M k (φ k) ⇔ i+ (N) = r Xbk Vk + r Xbl + s.

c) cov φ l − BLUP M l (φ l) cov φ k − BLUP M k (φ k) ⇔ i+ (N) = r Xbk Vk + r Xbl .

d) cov φ l − BLUP M l (φ l) cov φ k − BLUP M k (φ k) ⇔ i− (N) = r Xbk + r Xbl Vl . e) cov φ l − BLUP M l (φ l) = cov φ k − BLUP M k (φ k) ⇔ r (N) = r Xbk Vk + r Xbl + r Xbk + r Xbl Vl . Proof. From (31), cov φ k − BLUP M k (φ k) − cov φ l − BLUP M l (φ l) is written as 0 0 + + + + ZkWk Xek − Jk Σ ZkWk Xek − Jk − ZlWl Xel − Jl Σ ZlWl Xel − Jl . (35)

Then applying (25) to (35) and simplifying by Lemma 2.3 and Lemma 2.4, and congruence operations, we obtain i± cov φ k − BLUP M k (φ k) − cov φ l − BLUP M l (φ l)

0 + + = i± cov φ k − BLUP M k (φ k) − ZlWl Xel − Jl Σ ZlWl Xel − Jl

0  +  Σ Σ ZlWl Xel − Jl = i± − i± (Σ)  +  ZlWl Xel − Jl Σ cov φ k − BLUP M k (φ k)

" 0 0 + # Σ −ΣJl ΣXel 0 0 Wl XelΣ 0 = i± + 0 0 − i± (Σ) −JlΣ cov φ k − BLUP M k (φ k) 0 Zl Wl 0 0 Zl

  0 −Wl XelΣ 0 0 0  −Wl 0 0 Zl  0 Wl = i±  0 0  − i∓ 0 − i± (Σ)  ΣXe 0 Σ −ΣJ  Wl 0 l l 0 Zl −JlΣ cov φ k − BLUP M k (φ k)

32 /

 0 ⊥ 0  XelΣXel −Xbl −VlXbl 0 XelΣJl  −X0 0 0 0 F0   bl  i  ⊥ ⊥ 0  − r ( ) − i ( ) = ±  −Xbl Vl 0 0 0 Xbl Cl  W ± Σ  0 0   ΣXel 0 0 Σ −ΣJl  0 ⊥ 0 JlΣXel FClXbl 0 cov φ k − BLUP M k (φ k) − JlΣJl  0 ⊥ 0  XelΣXel −Xbl −VlXbl XelΣJl  −X0 0 0 F0  i  bl  − r ⊥ = ±  ⊥ ⊥ 0  Xbl VlXbl  −Xbl Vl 0 0 Xbl Cl  0 ⊥ 0 JlΣXel FClXbl cov φ k − BLUP M k (φ k) − JlΣJl  0  Vl −Xbl 0 Cl 0 0  −Xbl 0 0 F  ⊥ = i±  ⊥ ⊥  − r Xbl VlXbl  0 0 Xbl VlXbl 0  0 Cl F 0 cov φ k − BLUP M k (φ k) − JlΣJl  0  Vl Xbl Cl h i 0 0 ⊥ ⊥ = i∓  Xbl 0 F  + i± Xbl VlXbl − r Xbl Vl 0 Cl FJlΣJl − cov φ k − BLUP M k (φ k)  0  Vl Xbl Cl 0 0 h i  Xbl 0 F  ⊥ ⊥ = i∓  0  + i± Xbl VlXbl − r Xbl Vl  0 + +  Cl FJlΣJl − ZkWk Xek − Jk Σ ZkWk Xek − Jk  0  Vl Cl Xbl 0 h i  0 + +  ⊥ ⊥ = i∓  C J ΣJ − Z W X − J Σ Z W X − J F  + i± Xb VlXb − r Xbl Vl  l l l k k ek k k k ek k  l l 0 0 Xbl F 0  0     Vl Cl Xbl 0 0 0 + + = i∓  Cl JlΣJl F  −  Is  ZkWk Xek − Jk Σ ZkWk Xek − Jk 0 Is 0  0 0 Xbl F 0 0 h ⊥ ⊥i +i± Xbl VlXbl − r Xbl Vl

 0  Vk Ck Xbk     0    0 Vl Cl Xbl   0 0  = i±  Ck  Is JkΣJk 0 Is 0 −  Cl JlΣJl F  F   0 0 0   Xbl F 0  0 0 Xbk F 0 h ⊥ ⊥i h ⊥ ⊥i +i∓ Xbk VkXbk − r Xbk Vk + i± Xbl VlXbl − r Xbl Vl

 0  Vk 0 Ck 0 Xbk  − 0 −   0 Vl Cl Xbl 0  h i h i  0 0  ⊥ ⊥ ⊥ ⊥ = i±  Ck −Cl JkΣJk − JlΣJl −FF  + i∓ Xbk VkXbk − r Xbk Vk + i± Xbl VlXbl − r Xbl Vl ,  0 0   0 −Xbl −F 0 0  0 0 Xbk 0 F 0 0 that is, i+ cov φ k − BLUP M k (φ k) − cov φ l − BLUP M l (φ l) = i+ (N) − r Xbk Vk − r Xbl , i− cov φ k − BLUP M k (φ k) − cov φ l − BLUP M l (φ l) = i− (N) − r Xbk − r Xbl Vl . In consequence, from (22) and (23), the required results (32) and (33) are obtained. Adding these equalities yield (34). Applying Lemma 2.2 to (32)-(34) yields (a)-(e).

The following result is an immediate consequences of Theorem 3.1.

Corollary 3.2. Assume that the Fα is estimable under two sub-sample models M k and M l , k,l = 1,...,n, k 6= l. Denote

33 /

  Vk 0 0 0 Xbk    0 −Vl 0 −Xbl 0    N =  0 0 0 −FF .  0 0   0 −Xbl −F 0 0  0 0 Xbk 0 F 0 0 Then, i+ cov BLUE M k (Fα) − cov BLUE M l (Fα) = i+ (N) − r Xbk Vk − r Xbl , i− cov BLUE M k (Fα) − cov BLUE M l (Fα) = i− (N) − r Xbk − r Xbl Vl , r cov BLUE M k (Fα) − cov BLUE M l (Fα) = r (N) − r Xbk Vk − r Xbl − r Xbk − r Xbl Vl . In consequence, the following results hold. a) cov BLUE M l (Fα) cov BLUE M k (Fα) ⇔ i− (N) = r Xbk + r Xbl Vl + s.

b) cov BLUE M l (Fα) ≺ cov BLUE M k (Fα) ⇔ i+ (N) = r Xbk Vk + r Xbl + s.

c) cov BLUE M l (Fα) cov BLUE M k (Fα) ⇔ i+ (N) = r Xbk Vk + r Xbl .

d) cov BLUE M l (Fα) cov BLUE M k (Fα) ⇔ i− (N) = r Xbk + r Xbl Vl .

e) cov BLUE M l (Fα) = cov BLUE M k (Fα) ⇔ r (N) = r Xbk Vk + r Xbl + r Xbk + r Xbl Vl .

4. Conclusion

A general linear random-effects model that includes both ﬁxed and random-effects are considered with its sub-sample models without making any restrictions on correlation of random-effects and also any full rank assumptions. It is assumed that a general linear random-effects model have correlated random-effects. In this study, we establish a variety of inequalities and equalities for comparison of covariance matrices of BLUPs/BLUEs under sub-sample models of the general linear random-effects model. We use known rank and inertia formulas for deriving the formulas for calculating the comparison of BLUPs’ covariance matrices. The results obtained in this paper can present useful aspect for describing performances of BLUPs/BLUEs under sub-sample models.

References

[1] Alalouf, I. S. & Styan, G. P. H. (1979). Characterizations of estimability in the general linear model. Ann. Stat., 7, 194–200. [2] Arendacká, B. & Puntanen, S. (2015). Further remarks on the connection between ﬁxed linear model and mixed linear model. Stat. Papers, 56 (4), 1235–1247. [3] Goldberger, A. S. (1962). Best linear unbiased prediction in the generalized linear regression model. J. Amer. Statist. Assoc., 57, 369–375. [4] Güler, N. (2020). On relations between BLUPs under two transformed linear random-effects models. Communications in Statistics - Simulation and Computation, DOI: 10.1080/03610918.2020.1757709. [5] Güler, N. & Büyükkaya, M. E. (2019a). Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models. Communications in Statistics - Theory and Methods, DOI: 10.1080/03610926.2019.1599950. [6] Güler, N. & Büyükkaya, M. E. (2019b). Notes on Comparison of Covariance Matrices of BLUPs Under Linear Random-Effects Model with Its Two Sub-sample Models. Iranian J. Sci. Tech. Transactions A, 43 (6), 2993–3002. [7] Haslett, S. J., Isotalo, J., Liu, Y. & Puntanen, S. (2014). Equalities between OLSE, BLUE and BLUP in the linear model. Stat. Papers, 55, 543–561. [8] Haslett, S.J. & Puntanen, S. (2010a). A note on the equality of the BLUPs for new observations under two linear models. Acta Comment. Univ. Tartu. Math., 14, 27–33. [9] Haslett, S. J. & Puntanen, S. (2010b). Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat. Papers, 51 (2), 465–475. 34 /

[10] Haslett, S.J. & Puntanen, S. (2011). On the equality of the BLUPs under two linear mixed models. Metrika, 74, 381–395. [11] Haslett, S.J. & Puntanen, S. (2013). A review of conditions under which BLUEs and/or BLUPs in one linear mixed model are also BLUEs and/or BLUPs in another. Calcutta Statistical Association Bulletin, 65 (1-4), 25–42. [12] Haslett, S. J., Puntanen, S. & Arendacká, B. (2015). The link between the mixed and ﬁxed linear models revisited. Stat. Papers, 56, 849–861. [13] Isotalo, J. & Puntanen, S. (2006). Linear prediction sufﬁciency for new observations in the general Gauss–Markov model. Commun. Statist. A, 35, 1011–1023. [14] Liu, X. & Wang, Q. W. (2013). Equality of the BLUPs under the mixed linear model when random components and errors are correlated. J. Multivariate Anal., 116, 297–309. [15] Marsaglia, G. & Styan, G. P. H. (1974). Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra, 2, 269–292. [16] Puntanen, S., Styan, G. P. H. & Isotalo, J. (2011). Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer, Heidelberg. [17] Rao, C. R. (1973). Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J. Multivariate Anal., 3, 276–292. [18] Tian, Y. (2010). Equalities and inequalities for inertias of Hermitian matrices with applications. Linear Algebra Appl., 433, 263–296. [19] Tian, Y. (2012). Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method. Nonlinear Analysis, 75, 717–734. [20] Tian, Y. (2015a). A new derivation of BLUPs under random-effects model. Metrika, 78, 905–918. [21] Tian, Y. (2015b). A matrix handling of predictions under a general linear random-effects model with new observations. Elektron. J. Linear Algebra, 29, 30–45. [22] Tian, Y. (2017a). Some equalities and inequalities for covariance matrices of estimators under linear model. Stat. Papers, 58, 467–484. [23] Tian, Y. (2017b). Matrix rank and inertia formulas in the analysis of general linear models. Open Math., 15 (1), 126–150. [24] Tian, Y. (2017c). Transformation approaches of linear random-effects models. Stat. Methods Appl., 26 (4), 583–608. [25] Tian, Y. & Guo, W. (2016). On comparison of dispersion matrices of estimators under a constrained linear model. Stat. Methods Appl., 25 (4), 623–649. [26] Tian, Y. & Jiang, B. (2016). Matrix rank/inertia formulas for least-squares solutions with statistical applications. Spec. Matrices, 4, 130–140.

35 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

On generalized statistical convergence via ideal in cone metric spaces

Isıl¸ Açık Demirci∗1, Mehmet Gürdal2 1Mehmet Akif Ersoy University, Department of Mathematics Education, Turkey 2Suleyman University, Department of Mathematics, Turkey

Keywords Abstract: Every metric space is a cone metric space, and every cone metric space is a Cone metric, topological space. In this study, we introduce the concept of I -statistical convergence multiple sequence, for double and multiple index sequences in cone metric spaces over topological vector ideal convergence spaces. Consequently, we have generalized several results in cone metric spaces from metric space.

1. Introduction and preliminaries

The notion of statistical convergence of sequences in real spaces was introduced in 1951 by Fast in [6]. This concept has been studied by many mathematicians up to date (see, for example, [7, 15, 16]). The idea of statistical convergence was further extended to I -convergence in [11] using the notion of ideals of natural numbers, with many interesting consequences. More investigations in this direction and more applications of ideals can be found in [4, 8, 18, 23]. In another direction, a new type of convergence, called I -statistical convergence, was introduced in [5]. Later on, it was further investigated by Sava¸sand Das [21, 22] and Yamancı and Gürdal [25, 26]. In this paper, we shall study I -statistical convergence in cone metric spaces for double and multiple index sequences. Cone metric spaces have been actually defined many years ago by several authors and appeared in the literature under different names. Dj. Kurepa was the first who introduced such spaces in 1934 under the name “espaces pseudo-distenciés” [14]. Cone metric spaces are one of many generalizations of metric spaces, and play an important role in fixed point theory, computer science, and some other research areas as well as in general topology (see, for example [1–3, 9]). There have been a lot of papers dealing with the theory of cone metric spaces. In this paper, we shall introduce and investigate I -statistical convergence in cone metric spaces for double sequences and n-tuple sequence. The paper is organized so that introduction is followed by three sections. In Section 2, we familiarize the reader with the basic notions concerning ideal statistical convergence for double sequences in cone metric spaces and give some results of this convergence. In Section 3, we will introduce the concept of ideal statistical convergence for multiple index sequences in topological vector space (tvs) valued cone metric space. Throughout this paper, the set of positive integers is denoted by N. For undefined terms in the paper the readers can refer to [7, 11].

Deﬁnition 1.1. Let A ⊂ N, put An = {k ∈ A : k ≤ n}, ∀n ∈ N. Then |A | |A | δ (A) := lim inf n and δ (A) := lim sup n n→∞ n n→∞ n are called lower and upper asymptotic density of the set A, respectively. If δ (A) = δ (A), then

|A | δ (A) := lim n n→∞ n is called an asymptotic (or natural) density of the set A. All the three densities, if they exist, are in [0,1]. Utilizing above information, we say that a sequence (x ) is statistically convergent to x provided that for every > 0, k k∈N ε

δ ({k ∈ N : |xk − x| ≥ ε}) = 0. If (x ) is statistically convergent to x we write st-limx = x. k k∈N k

∗ Corresponding author: [email protected] 36 /

On the other hand, I -convergence in a metric space was introduced by Kostyrko et al. [11] and its definition is depending upon the definition of an ideal I in N. A family I ⊂ 2N is called an ideal if the following properties are held: (i) /0 ∈ I ; (ii) P ∪ R ∈ I for every P,R ∈ I ; (iii) R ∈ I for every P ∈ I and R ⊂ P. A non-empty family of sets F ⊂ 2N is a filter if and only if /0 ∈/ F , P ∩ R ∈ F for every P,R ∈ F , and R ∈ F for every P ∈ F and every R ⊃ P. An ideal I is said to be non-trivial if I 6= /0 and X ∈/ I . The I ⊂ 2X is a non-trivial ideal if X and only if F = F (I ) = {N\P : P ∈ I } is a filter on X. A non-trivial ideal I ⊂ 2 is called admissible if and only if I ⊃ {{x} : x ∈ X}. An admissible ideal ⊂ 2N is said to hold the property (AP) if for every family {P } with P ∩P = /0 (n 6= k), P ∈ I n n∈N n k n I ∞ [ (n ∈ ) there is a family {R } such that (P \R ) ∪ (R \P ) for all k ∈ and a limit set R = R ∈ ([11]). N n n∈N k n k k N k I k=1 Definition 1.2. ([11]) A sequence of reals {x } is said to be -convergent to L if, for each > 0, the set n n∈N I ε

A(ε) = {n ∈ N : |xn − L| ≥ ε} ∈ I .

For more information about I -convergent, see the references in [17, 18]. A nontrivial ideal I of N × N is called strongly admissible if {i} × N and N × {i} belong to I for each i ∈ N. It is evident that a strongly admissible ideal is admissible also. Let I0 = {A ⊂ N × N : (∃m(A) ∈ N)(i, j ≥ m(A) ⇒ (i, j) ∈/ A)}.Then I0 is a nontrivial strongly admissible ideal and clearly an ideal I is strongly admissible if and only if I0 ⊆ I . Definition 1.3. ([4]) Let (X,ρ) be a metric space. A double sequence x = xi j in X is said to be I -convergent to i, j∈N ξ ∈ X if for all ε > 0 we have A(ε) = (i, j) ∈ N × N : ρ (xi j,ξ) ≥ ε ∈ I and we write I − limi, j→∞ xi j = ξ. We now recall the following basic concepts from [10, 13] which will be needed throughout the paper. Let E be a Hausdorff topological vector space with the zero vector θ. P ⊂ E called a (convex) cone if (i) P 6= {θ},P is non-empty and closed; (ii) P + P ⊂ P and λP ⊂ P for all non-negative λ; (iii) P ∩ (−P) = {θ}. Given a cone P subset of E, we can define a partial ordering ≤ with respect to P by defining x ≤ y ⇐⇒ y − x ∈ P. We shall write x < y to indicate that x ≤ y but x 6= y, while x y will stand for y − x ∈ intP, where intP represent the set of the interior points of P. The sets of the form [x,y] are called order-intervals and are defined as the following:

[x,y] = {z ∈ E : x ≤ z ≤ y}.

It is seen that order-intervals are convex. If [x,y] ⊂ A whenever x,y ∈ A and x ≤ y, then A ⊂ E is called order-convex. If ordered topological vector space (E,P) has a neighborhoods’ base of θ which consists of order-convex sets then, it is order-convex. At this stage, the cone P is called a normal cone. Considering the normed space, this condition comes to mean that the unit ball is order-convex, it is equivalent to the condition that ∃k such that x,y ∈ E and θ ≤ x ≤ y ⇒ kxk ≤ kkyk. The smallest constant k is called the normal constant of P [10]. If each of the increasing sequence that is bounded above in P is convergent then, we call P as a regular cone. In other words, if there exists a sequence {xn} such that

x1 ≤ x2 ≤ ... ≤ xn ≤ ... ≤ y for some y ∈ E, then ∃x ∈ E such that lim kxn − xk = 0. Equivalently the cone P is regular if every decreasing sequence n→∞ which is bounded from below is convergent. It is well known that if P is a regular cone then it is a normal cone. Let E is a topological vector space, V ⊂ E is an absolutely convex and absorbent subset, the corresponding Minkowski functional qv : E → R is deﬁned x 7→ qv(x) = inf{λ > 0 : x ∈ λV}. It is a semi-norm on E. If V is an absolutely convex neighborhood of θ ∈ E, then qv is continuos and

{x ∈ E : qv(x) < 1} = intV ⊂ V ⊂ V = {x ∈ E : qv(x) ≤ 1}.

Let (E,P) be an ordered tvs and e ∈ intP. Then

[−e,e] = (P − e) ∩ (e − P) = {z ∈ E : −e z e} is an absolutely convex neighborhood of θ. We denote the corresponding Minkowski functional q[−e,e] by qe. It can be verified that int[−e,e] = (intP − e) ∩ (e − intP). If P is normal and solid, then the Minkowski functional qe is a norm 37 / on E. Furthermore, it is an increasing function on P. In fact, for θ x1 x2 the set {λ : x1 ∈ λ [−e,e]} is the subset of {λ : x2 ∈ λ [−e,e]} and it follows that qe(x1) ≤ qe(x2). Let X be a nonempty set. Suppose that the mapping d : X ×X → E satisfies (d1) θ < d(x,y) for all x,y ∈ X and d(x,y) = 0 iff x = y; (d2) d(x,y) = d(y,x) for all x,y ∈ X; (d3) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z ∈ X. Then d is said to be a cone metric on X, and (X,d) is said to be a cone metric space. It is obvious that the notion of cone metric spaces generalizes the notion of metric spaces. We now give an example of a cone metric space. Let E = R2, P = {x,y ∈ E : x,y ≥ 0} ⊂ R2, X = R and d : X × X → E is given by d(x,y) = (β |x − y|,α |x − y|), where α,β ≥ 0 are constants. Then it is easy to verify that (X,d) is a cone metric space. Let (X,d) is a cone metric space. {x } be a sequence in X and let x ∈ X. If for every c ∈ E with 0 c there is N ∈ n n∈N N such that for all n > N, d(x ,x) c, then {x } is said to be convergent to x and x is called the limit of the sequence n n n∈N {x } . n n∈N If for any c ∈ E with 0 c there is N ∈ such that for all n,m > N, d(x ,x ) c, then {x } is called a Cauchy N n m n n∈N sequence in X. If every Cauchy sequence in X is convergent in X then X is called a complete cone metric space [13]. It is known that [24] any cone metric space is a first countable Hausdorff topological space with the topology induced by the open balls defined naturally for each element z in X and for each element c in intP. So as in [12] we can show that I ∗-convergence always implies I -convergence but the converse is not true. The two concepts are equivalent if and only if the ideal I has condition (AP).

2. New results on double sequences

In [19] and [20], the authors introduced the concepts of statistical convergence and ideal convergence for a double sequence in cone metric space. In this section, we introduce the notion of I and I ∗-statistical convergence of double sequences in tvs-cone metric space. Deﬁnition 2.1. Let (X,d) be a tvs-cone metric space. A double sequence xi j in a tvs-cone metric space (X,d) is i, j∈N said to be I -statistically convergent to a point ξ ∈ X if for each c,γ ∈ E with c θ and γ θ, there is N ∈ N such that 1 (m,n) ∈ N × N : i ≤ m, j ≤ n : d (xi j,ξ) c γ ∈ I , mn or equivalently if for each c,δ ∈ E with c θ and γ θ,

|Amn(c,δ)| δ (A(c,δ)) = I − lim = 0, I m,n mn where Amn(c,δ) = i ≤ m, j ≤ n : d (xi j,ξ) c . If a double sequence xi j is I -statistically convergent to ξ in a tvs-cone metric space (X,d) then we write i, j∈N

I − st − lim xi j = ξ i, j→∞ where ξ is called I -statistical limit of the sequence xi j . i, j∈N Deﬁnition 2.2. A double sequence x = xi j in a tvs-cone metric space (X,d) is called I -statistical bounded if there i, j∈N exists C,γ ∈ E with C θ and γ θ and N ∈ N such that 1 (m,n) ∈ N × N : i ≤ m, j ≤ n : d (xi j,θ) C γ ∈/ I . mn

2 2 Let I be the ideal I0 of N × N, d be a cone metric defined as d : R × R → (E,P), where P is cone subset of the tvs E. If we define double sequence xi j by i, j∈N ( (i,1), if j = 2, i ∈ N x = , i j 1 , 1 , otherwise (i+ j)2 i j then xi j is unbounded but this sequence is I -statistically convergent. i, j∈N Definition 2.3. A double sequence x = xi j in a tvs-cone metric space (X,d) is said to be a I -statistically Cauchy i, j∈N sequence if each c θ and γ θ there exists (p,q) ∈ N × N such that 1 (m,n) ∈ N × N : i ≤ m, j ≤ n : d (xi j,xpq) c γ ∈ I . mn

38 /

Lemma 2.4. (X,d) be a tvs-cone metric space, e ∈ intP and qe be the Minkowski functional of [−e,e] and dq = qe ◦ d. Let xi j i, j∈ be a double sequence in X. Then N (i) xi j i, j∈ tvs-cone statistically convergent to ξ if and only if dq (xi j,ξ) → 0 (statistically) as i, j → ∞, N (ii) xi j is a statistically Cauchy sequence if and only if dq (xi j,xnm) → 0 (statistically) as (i, j,n,m → ∞) i, j∈N Proof. The proof is analogous to [20] and so omitted. Theorem 2.5. (X,d) be a tvs-cone metric space. If a double sequence xi j is a I -statistically convergent then it is i, j∈N I -statistically Cauchy sequence. Proof. Let xi j be I -statistically convergent to ξ. Then for every c θ and γ θ, the set i, j∈N 1 n c o (m,n) ∈ × : i ≤ m, j ≤ n : d (xi j,ξ) γ ∈ I . N N mn 2

Select S and R such that (S,R) ∈/ Bc. Then we write

d (xi j,xSR) ≤ d (xi j,ξ) + d (ξ,xSR).

Now let Ac = (i, j) : d (xi j,xSR) c , c Bc = (i, j) : d (xi j,ξ) 2 , c Cc = (i, j) : d (ξ,xSR) 2 . Then, Ac ⊆ Bc ∪Cc and hence xi j is I -statistically Cauchy. i, j∈N

Theorem 2.6. Suppose that (X,d) be a tvs-cone metric space, e ∈ intP, qe be the Minkowski functional of [−e,e] and dq = qe ◦ d. And let xi j i, j∈ and yi j i, j∈ be a two double sequences in X and xi j i, j∈ is I -statistically convergent N N N to ξ and yi j is I -statistically convergent to η. Then dq (xi j,yi j) is I -statistically convergent to dq (ξ,η) as i, j∈N i, j∈N i, j → ∞.

Proof. For every ε > 0 we have d (xi j,yi j) ≤ d (xi j,ξ) + d (ξ,η) + d (yi j,η). Hence d (xi j,yi j) − d (ξ,η) ≤ d (xi j,ξ) + d (yi j,η) and we have (i, j) ∈ N × N : qe (d (xi j,yi j) − d (ξ,η)) ε n ε o n ε o ⊂ (i, j) ∈ × : d (x ,ξ) ∪ (i, j) ∈ × : d (y ,η) . N N q i j 2 N N q i j 2 Thus 1 (m,n) ∈ × : i ≤ m, j ≤ n : qe (d (xi j,yi j) − d (ξ,η)) ε γ ∈ I . N N mn and the result follows. Theorem 2.7. (X,d) be a tvs-cone metric space. If xi j is a I -statistically convergent then its I -statistically limit i, j∈N point is unique.

Proof. Let c θ and γ θ. It sufﬁces to show that for any A1,A2 ∈ I , we have (N × N \ A1) ∩ (N × N \ A2) 6= θ, since the last two sets belong to the ﬁlter associated with I . If there are two limits ξ,η ∈ X, ξ 6= η, choose c and γ such that 1 1 θ < c < 2 d (ξ,η), θ < γ < 2 d (ξ,η) and put

1 A1 = (m,n) ∈ N × N : mn i ≤ m, j ≤ n : d (xi j,ξ) c γ ∈ I , 1 A2 = (m,n) ∈ N × N : mn i ≤ m, j ≤ n : d (xi j,η) c γ ∈ I .

Since the sets (N×N \ A1) and (N×N \ A2) are in the ﬁlter of I , the intersection of these two sets cannot be an empty set. So, this contradicts disjointness of the neighborhoods of A1 and A2.

∗ Theorem 2.8. Suppose that I be a strongly admissible ideal if I − st − limi, j→∞ xi j = ξ then I − st − limi, j→∞ xi j = ξ .

39 /

∗ Proof. Let c θ and γ θ. Since I − st − limi, j→∞ xi j = ξ, there is a set H ∈ I such that

1 (i, j) ∈ M : d (xi j,ξ) c i j exists for M = N × N \ H (M ∈ F(I )). Then 1 A(c) = (m,n) ∈ × : i ≤ m, j ≤ n : d (xi j,ξ) c γ N N mn

⊂ H ∪ (M ∩ (({1,2,...,(k0 − 1)} × N) ∪ (N × {1,2,...,(k0 − 1)}))).

This shows that A(c) ∈ I . Hence we get I − st − lim xi j = ξ. m,n→∞

3. Some results for multiple sequences

In this section, we introduce some results for n-tuple sequence in tvs cone metric space in analogy to Moricz [15]. The n notion and results can be extended to n-tuple sequences in the previous section. Let i ∈ N such that i := (i1,i2,...,in) with nonnegative integers for coordinates i j, where n is a ﬁxed positive integer. i and m : = (m1,m2,...,mn) are distinct n iff i j 6= m j for at least one j. N is partially ordered by agreeing that i ≤ m iff i j ≤ m j for each j = 1,...,n. We say that a n-tuple sequence {x } n is ideal statistically convergent to in cone metric space if for each c and , i i∈N ξ θ γ θ 1 m ∈ n : |{i ≤ m : d (x ,ξ) c}| γ ∈ I N |m| i

n where |m| := ∏m j. j=1 Furthermore, we say that {x } n is ideal statistically Cauchy if for each c, ∈ E, c 0 and there exists i i∈N γ γ θ n k := (k1,k2,...,kn) ∈ N such that 1 m,n ∈ n : |{i ≤ m, k ≤ n : d (x ,x ) c}| γ ∈ I N |mn| i k

n n n− n− Here, a nontrivial ideal I of N × N is called strongly admissible if {i} × N 1 and N 1 × {i} belong to I for each i ∈ N. Subsequently, we give the results for n-tuple sequences which are just generalization of the results for double sequences, without proofs.

Theorem 3.1. If an n-tuple sequence {x } n is ideal statistically convergent in tvs cone metric space (X,d), then {x } n i i∈N i i∈N is ideal statistically Cauchy. Theorem 3.2. (X,d) be a tvs cone metric space. Let {x } n and y be an n-tuple sequences in X and {x } n is i i∈N j j∈ n i i∈N N I -statistically convergent to ξ and yj n is I -statistically convergent to η. Then d xi,yj n n is I -statistically j∈N i∈N ,j∈N convergent to d (ξ,η).

Theorem 3.3. (X,d) be a tvs cone metric space. If an n-tuple sequence {x } n is a -statistically convergent then its i i∈N I I -statistically limit point is unique. ∗ Theorem 3.4. Let I be a strongly admissible ideal if I − st − limi j→∞ xi = ξ then I − st − limi j→∞ xi = ξ.

References

[1] Abbas, M., Rhoades, B.E., 2009. Fixed and periodic point results in cone metric space, Appl. Math. Lett., 22, 511-515. [2] Aliprantice, C.D., Tourky, R., 2007. Cones and duality, Amer. Math. Soc., 30, 3357-3366. [3] Chi, K.P., An, T.V., 2011. Dugungji’s theorem for cone metric spaces, Appl. Math. Lett., 24, 387-390. [4] Das, P., Kostyrko, P., Wilczyncki, W., Malik, P., 2008. I and I ∗-convergence of double sequence, Math. Slovaca, 58(5), 605-620. [5] Das, P., Sava¸sE., Ghosal, S., 2011. On generalized of certain summability methods using ideals, Appl. Math. Lett., 26, 1509-1514. [6] Fast, H., 1951. Sur la convergence statistique, Colloq. Math., 2, 241-244. [7] Fridy, J.A., 1985. On statistical convergence, Analysis (Munich), 5, 301-313. 40 /

[8] Gürdal, M., ¸Sahiner, A., 2008. Extremal I -Limit Points of Double Sequences, Appl. Math. E-Notes, 8, 131-137. [9] Huang, L.G., Zhang, X., 2007. Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2), 1467-1475. [10] Kadelburg, Z., Radenovic, S., Rakocevic, V., 2011. A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 24, 370-374. [11] Kostyrko, P., Salat, T., Wilczynski, W., 2000-2001. I -convergence, Real Anal. Exch., 26, 669-686. [12] Lahiri, B.K., Das, P., 2005. I and I ∗-convergence in topological spaces, Math. Bohemica, 130(2), 153-160. [13] Long-Guang, H., Xian, Z., 2007. Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332, 1468-1476. [14] Mamuzic, Z.P., 1963. Introduction to general topology, P. Noordhoff, Ltd., The Netherlands. [15] Moricz, F., 2003. Statistical convergence of multiple sequences, Arc. Math., 81, 82-89. [16] Mursaleen, M., Edely, O.H.H., 2003. Statistical convergence of double sequences, J. Math. Anal. Appl., 288, 223-231. [17] Mursaleen, M., Mohiuddine, S.A., Edely, O.H.H., 2010. On ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59, 603-611. [18] Nabiev, A., Pehlivan, S., Gürdal, M., 2007. On I -Cauchy sequences, Taiwanese J. Math., 12, 569-576. [19] ¸Sahiner, A., Yılmaz, N., 2014. Multiple sequences in cone metric spaces, TWMS J. App. Eng. Math., 4(2), 226-233. [20] ¸Sahiner, A., Yigit,˘ T., Yılmaz, N., 2014. I -convergence of multiple sequences in cone metric spaces, Contemporary Analy. Appl. Math., 2(1), 116-126. [21] Sava¸s,E., Das, E., 2011. A generalized statistical convergence via ideals, Appl. Math. Lett., 24, 826-830. [22] Sava¸s,E., Das, E., 2014. On I -statistically pre-Cauchy sequences, Taiwanese J. Math., 18(1), 115-126. [23] Pal, S.K., Sava¸s,E., Çakallı, H., 2013. I -convergence on cone metric spaces, Sarajevo J. Math., 9(21), 85-93. [24] Turkoglu, D., Abuloha, M., 2010. Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta Math. Sinica, 26, 489-496. [25] Yamancı, U., Gürdal, M., 2014. I -statistical convergence in 2-normed space, Arab J. Math. Sci., 20(1), 41-47. [26] Yamancı, U., Gürdal, M., 2014. I -statistically pre-Cauchy double sequences, Global J. Math. Anal., 2(4), 297-303.

41 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

I -statistical limit superior and I -statistical limit inferior of triple sequences

Mualla Birgül Huban∗1, Mehmet Gürdal2, Ekrem Savas¸3 1Isparta University of Applied Sciences, Turkey 2Süleyman Demirel University, Department of Mathematics, Turkey 3U¸sak University, Turkey

Keywords Abstract: In this work, we introduce the notions of I -statistical convergence, I - I -limit superior, statistical inferior and I -statistical superior for triple sequences. We also investigate I -limit inferior, some of their properties for the triple sequence. I -statistical convergence, triple sequence

1. Introduction and preliminaries

The definition of statistical convergence of sequences in real spaces has been firstly presented by Fast in [4]. Then Salat continued to study statistically convergent sequences of real numbers [18]. This concept has been studied by many mathematicians up to date (see, [5, 11, 12]). The idea is based on the notion of the natural density of subsets of N, the set of all positive integers which is defined as follows: The natural density of a subset A of N denoted as δ (A) is defined by 1 δ (A) = limn→∞ n |{k ≤ n : k ∈ A}|. Statistical limit points studied by Fridy [6]. The study of statistical convergence in a double sequence has been initiated by Tripathy [22], Mursaleen and Edely [12] and Moricz [11]. Also, ¸Sahineret al. [15] studied the statistical convergence for the triple sequence. As a natural consequence, Fridy and Orhan [7] introduced the characterization of statistical limit superior and limit inferior. The idea of I -convergence was introduced by Kostyrko et al. [9]. More investigations in this direction and more applications of ideals can be found in [8, 13, 14, 16]. Also, Demirci [3] presented the notions of I -limit superior and inferior of a real sequence and gave some properties. Later on, it was further investigated by Lahiri and Das [10]. In another direction, a new type of convergence, called I -statistical convergence, was introduced in [2]. Recently, various types of I -statistical convergence for sequences has been studied by many authors [19, 20, 23, 24]. In this paper, we shall study the concepts of I -statistical limit superior and I -statistical limit inferior for triple sequences. Throughout this paper, the set of positive integers is denoted by N. For undefined terms in the paper the readers can refer to [5, 9]. We say that a sequence (x ) is statistically convergent to x provided that for every > 0, k k∈N ε

δ ({k ∈ N : |xk − x| ≥ ε}) = 0. If (x ) is statistically convergent to x we write st-limx = x. k k∈N k On the other hand, I -convergence in a metric space was introduced by Kostyrko et al. [9] and its definition is depending upon the definition of an ideal I in N. A family I ⊂ 2N is called an ideal if the following properties are held: (i) /0 ∈ I ; (ii) P ∪ R ∈ I for every P,R ∈ I ; (iii) R ∈ I for every P ∈ I and R ⊂ P. A non-empty family of sets F ⊂ 2N is a filter if and only if /0 ∈/ F , P ∩ R ∈ F for every P,R ∈ F , and R ∈ F for every P ∈ F and every R ⊃ P. An ideal I is said to be non-trivial if I 6= /0 and X ∈/ I . The I ⊂ 2X is a non-trivial ideal if X and only if F = F (I ) = {N − P : P ∈ I } is a filter on X. A non-trivial ideal I ⊂ 2 is called admissible if and only if I ⊃ {{x} : x ∈ X}.

Deﬁnition 1.1. ([9]) Let I be an ideal on N. A sequence x = {xn} of real numbers is said to be I -convergent to L if for each ε > 0, A(ε) = {n ∈ N : |xn − L| ≥ ε} ∈ I . In this case write I -limx = L. For more information about I -convergent, see the references in [3, 13, 14, 16].

∗ Corresponding author: [email protected] 42 /

Deﬁnition 1.2. ([19]) A sequence {xn} is said to be I -statistically convergent to L if for each ε > 0 and γ > 0, 1 n ∈ : |{k ≤ n : |x − L| ≥ ε}| ≥ γ ∈ I . N n k

L is called I -statistical limit of the sequence {xn} and we write I -st − limn→∞ xn = L. We now recall the following basic concepts from [1, 15, 17, 21] which will be needed throughout the paper. A function x : N × N × N → R (or C) is called a real (complex) triple sequence. A triple sequence (xnkl) is said to be convergent to L in Pringsheim’s sense if for every ε > 0, there exists n0 (ε) ∈ N such that |xnkl − L| < ε whenever n,k,l ≥ n0. Deﬁnition 1.3. A subset K of N × N × N is said to have natural density δ(K) if |K | δ(K) = P − lim nkl n,k,l→∞ nkl exists, where the vertical bars denote the number of (n,k,l) in K such that p ≤ n, q ≤ k, r ≤ l. Then, a real triple sequence x = (xnkl) is said to be statistically convergent to L in Pringsheim’s sense if for every ε > 0,

δ ({(n,k,l) ∈ N × N × N : |xnkl − L| ≥ ε}) = 0. × × × Throughout the paper we consider the ideals of 2N by I ; the ideals of 2N N by I2 and the ideals of 2N N N by I3.

Deﬁnition 1.4. A real triple sequence (xnkl) is said to be I -convergent to L in Pringsheim’s sense if for every ε > 0,

{(n,k,l) ∈ N × N × N : |xnkl − L| ≥ ε} ∈ I3.

In this case, one writes I3-limxnkl = L.

2. Main Results

In this section, we study the concepts of I -statistical limit superior and I -statistical limit inferior for triple sequences. From now on, unless otherwise expressed we shall deal with an admissible ideal of 2N×N×N and the notations mentioned above. Deﬁnition 2.1. A triple sequence {x } is said to be -statistically convergent to L in Pringsheim’s sense if for nkl n,k,l∈N I each ε > 0 and γ > 0, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr − L ≥ ε ≥ γ ∈ I3. N N N nkl or equivalently if for each ε > 0 and γ > 0,

|Ankl(ε,γ)| δI (A(ε,γ)) = I − lim = 0, n,k,l nkl where Ankl(ε,γ) = p ≤ n,q ≤ k,r ≤ l : xpqr − L ≥ ε .

L is called I -statistical limit of the sequence {xnkl} and we wirte I3-st − limxnkl = L. We write 1 Mt = (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > t > γ N N N nkl and t 1 M = (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < t > γ N N N nkl for t ∈ R and for a triple sequence {xnkl}.

Deﬁnition 2.2. (i) If there is a t ∈ R such that Mt ∈/ I3, we put

I3 − limsupx = sup{t ∈ R : Mt ∈/ I3}.

If Mt ∈ I3 holds for each t ∈ R then we put I3 − limsupx = −∞. t (ii) If there is a t ∈ R such that M ∈/ I3, we put t I3 − liminfx = inf t ∈ R : M ∈/ I3 . t If M ∈ I3 holds for each t ∈ R then we put I3 − liminfx = +∞. 43 /

Theorem 2.3. (i) If β = I3-st − limsupx is finite, then for every positive number ε, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > β − ε > γ ∈/ I3 N N N nkl and 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > β + ε > γ ∈ I3. N N N nkl (ii) Similarly, if α = I -st − liminfx is finite, then for every positive number ε, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < α + ε > γ ∈/ I3 N N N nkl and 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < α − ε > γ ∈ I3. N N N nkl Proof. It follows from the definition.

Theorem 2.4. For every real triple number sequence (xnkl), I3-st − liminfxnkl ≤ I3-st − limsupxnkl.

Proof. If (xnkl) is any real triple sequence we have three possibilities: (1) The case I3-limsupxnkl = +∞ is clear. (2) If I3-limsupxnkl = −∞, then we have

t t ∈ R ⇒ Mt ∈ I3 and M ∈/ I3.

t t Hence, since M = R, I3-st − liminfxnkl = inf{t ∈ R : M ∈/ I3} = −∞ and I3-st − liminfxnkl ≤ I3-st − limsupxnkl (3) If −∞ < β = I3-st − limsupxnkl < ∞ and α = I3-st − liminfxnkl then for ε > 0, γ > 0, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > β + ε > γ ∈ I3. N N N nkl

This implies, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < β + ε > γ ∈ F (I3) N N N nkl i.e., 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < β + ε > γ ∈/ I3. N N N nkl t t So, β + ε ∈ {t ∈ R : M ∈/ I3}. Since ε was arbitrary and by deﬁnition α = inf{t ∈ R : M ∈/ I3}. Therefore, α < β + ε. This proves that α ≤ β.

Deﬁnition 2.5. A number ξ is said to be an I -statistical limit point of the triple sequence x = (xnkl) provided that there exists a set M = {n1 < n2 < ...} × {k1 < k2 < ...} × {l1 < l2 < ...} ⊂ N × N × N such that M ∈/ I3 and P − limxnik jlm = ξ for all i, j,m = 1,2,... .

Deﬁnition 2.6. The real triple sequence x = (xnkl) is said to be I3-st bounded if there is a number K such that 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > K > γ ∈ I3. N N N nkl

Remark 2.7. If a triple sequence is I3-st bounded then I3-st − limsup and I3-st − liminf of that sequence are ﬁnite.

Deﬁnition 2.8. A number ζ is called to be an I -statistical cluster point of the triple sequence x = (xnkl) if for each ε > 0 and γ > 0, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr − ζ ≥ ε < γ ∈/ I3. N N N nkl Theorem 2.9. If a I -statistically bounded real triple sequence has one cluster point then it is I -statistically convergent.

44 /

Proof. Let (xnkl) be a I -statistically bounded triple sequence which has one cluster point. Then 1 M = (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > K > γ ∈ I3. N N N nkl

0 0 So, there exists a set M = {n1 < n2 < ...}×{k1 < k2 < ...}×{l1 < l2 < ...} ⊂ N×N×N such that M ∈/ I3 and xnik jlm is a statistically bounded sequence. Now, since (xnkl) has only one cluster point and xnik jlm is a statistically bounded subsequence of (xnkl). So xnik jlm also has only one cluster point. Hence xnik jlm is statistically convergent. Let, st − lim xnik jlm = ξ, then for any ε > 0 and δ > 0 we have the inclusion, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr − ξ ≥ ε ≥ γ ⊆ M ∪ F ∈ I3 N N N nkl where F is a ﬁnite set, i.e., (xnkl) is I -statistical convergent to ξ.

Theorem 2.10. A real triple sequence x is I -statistical convergent if and only if I3-st − liminfx = I3-st − limsupx, provided x is I3-st bounded.

Proof. Let α = I3-st − liminfx and β = I3-st − limsupx and I3-st − limx = L. Then 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > L + ε ≥ γ ∈ I3 N N N nkl and 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < L − ε ≥ γ ∈ I3 N N N nkl which implies β ≤ L and L ≤ α. Therefore, β ≤ α. But we know that α ≤ β, i.e., α = β. To prove sufﬁciency let ε > 0, δ > 0 and L = I3-st − liminfx = I3-st − limsupx. Then 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr − L > ε > γ N N N nkl 1 n ε o ⊂ (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : x > L + > γ N N N nkl pqr 2 1 n ε o ∪ (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : x < L − > γ . N N N nkl pqr 2

Since 1 n ε o (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > L + > γ ∈ I N N N nkl 2 3 and 1 n ε o (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < L − > γ ∈ I , N N N nkl 2 3 we have 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr − L > ε > γ ∈ I3. N N N nkl

So, x = (xnkl) is I -statistical convergent.

Theorem 2.11. If x = (xnkl), y = (ynkl) are two I3-st bounded triple sequences in Pringsheim’s sense, then (i) I3-st − limsup(x + y) ≤ I3-st − limsupx + I3-st − limsupy (ii) I3-st − liminf(x + y) ≥ I3-st − liminfx + I3-st − liminfy.

Proof. Since the proof of (ii) is analogous we prove only (i). Let l1 = I3-st − limsupx, l2 = I3-st − limsupy, ε > 0 and γ > 0 be given. Let 1 B = c ∈ : (n,k,l) : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > c > γ ∈/ I3 . (x+y) R nkl

45 /

We can also assume that B(x+y) is not empty. We have 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > l1 + l2 + ε > γ N N N nkl 1 n ε o ⊂ (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : x > l + > γ N N N nkl pqr 1 2 1 n ε o ∪ (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : y > l + > γ . N N N nkl pqr 2 2

Since both sets on the right hand side belong to I3 we conclude 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > l1 + l2 + ε > γ ∈ I3. N N N nkl

If c ∈ B(x+y), then by deﬁnition 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > c > γ ∈/ I3. N N N nkl

We show that c < l1 + l2 + ε. Conversely assume that c ≥ l1 + l2 + ε. We would have 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > c > γ N N N nkl 1 ⊆ (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > l1 + l2 + ε > γ N N N nkl which means 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr + ypqr > c > γ ∈ I3, N N N nkl a contradiction. Hence, c < l1 + l2 + ε and we deduce I3-st − limsup(x + y) = supB(x+y) < l1 + l2 + ε. Since ε > 0 is arbitrary, this completes the proof. We need the following deﬁnition for the subsequent theorem.

Deﬁnition 2.12. A triple sequence x = (xnkl) is said to be I3-statistically convergent to +∞ (or −∞) in Pringsheim’s sense if for every real number G > 0, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr ≤ G > γ ∈ I3 N N N nkl

1 (or (n,k,l) : nkl p ≤ n,q ≤ k,r ≤ l : xpqr ≥ −G > γ ∈ I3).

Theorem 2.13. If I3 is an admissible ideal and I3-st − limsupx = `, then there exists a subsequence of x = (xnkl) that is I -statistically convergent to ` in Pringsheim’s sense.

Proof. Since /0 ∈ I3 and I3 is admissible ideal, we can assume that x is a non-constant triple sequence having inﬁnite number of distinct elements. Case-I: If ` = −∞, then {t ∈ R : Mt ∈/ I3} = /0. So, if G > 0 and γ > 0, then 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr ≥ −G > γ ∈ I3, N N N nkl i.e., I3-st − limx = −∞. Case-II: If ` = +∞, then {t ∈ R : Mt ∈/ I3} = R. So for any t ∈ R and γ > 0, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > t > γ ∈/ I3. N N N nkl

Let, xn1k1l1 be arbitrary term of x = (xnkl) and so, 1 Ax = (n,k,l) ∈ N × N × N : p ≤ n,q ≤ k,r ≤ l : xpqr > xn k l + 1 > γ ∈/ I3. n1k1l1 nkl 1 1 1

46 / Since I3 is an admissible ideal, so Axn k l must be an inﬁnite set. That is δ p ≤ n,q ≤ k,r ≤ l : xpqr > xn1k1l1 + 1 6= 0. 1 1 1 We claim that there is at least (p,q,r) ∈ p ≤ n,q ≤ k,r ≤ l : xpqr > xn1k1l1 + 1 such that p > n1 +1, q > k1 +1, r > l1 +1. For otherwise p ≤ n,q ≤ k,r ≤ l : xpqr > xn1k1l1 + 1 ⊆ {(1,1,1),(2,2,2),...,(n1 + 1,n1 + 1,n1 + 1)}, i.e., δ p ≤ n,q ≤ k,r ≤ l : xpqr > xn1k1l1 + 1 ⊆ δ ({(1,1,1),(2,2,2),...,(n1 + 1,n1 + 1,n1 + 1)}) = 0, which is a contradiction. We call this (n,k,l) as (n2,k2,l2). Thus xn k l > xn k l + 1. Proceeding in this way, we obtain a 2 2 2 1 1 1 subsequence xnikili of x with xnikili > xni−1ki−1li−1 + 1 for all i. Since for any G > 0, 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr < G > γ ∈ I3, N N N nkl because I3 is admissible ideal, so,I3-st − limxnk = +∞. Case-III: Let −∞ < ` < +∞. So, 1 1 (n,k,l) ∈ N × N × N : p ≤ n,q ≤ k,r ≤ l : xpqr > ` + > γ ∈ I3, nkl 2 and 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > ` − 1 > γ ∈/ I3. N N N nkl

So, there must be a (n1,k1,l1) in this set for which

1 p ≤ n1,q ≤ k1,r ≤ l1 : xpqr > ` − 1 > γ n1k1l1 and 1 1 p ≤ n1,q ≤ k1,r ≤ l1 : xpqr ≤ ` + > δ. n1k1l1 2 For otherwise 1 (n,k,l) ∈ × × : p ≤ n,q ≤ k,r ≤ l : xpqr > ` − 1 > γ N N N nkl 1 1 ⊂ (n,k,l) ∈ N × N × N : p ≤ n,q ≤ k,r ≤ l : xpqr ≤ ` + > δ ∈ I3, nkl 2 which is a contradiction. Hence we have 1 ` − 1 < x ≤ ` + . n1k1l1 2 1 Next we proceed to choose an element xn2k2l2 from x, n2 > n1, k2 > k1, l2 > l1such that ` − 2 < xn2k2l2 . Now 1 1 (n,k,l) ∈ N × N × N : p ≤ n,q ≤ k,r ≤ l : xpqr > ` − > δ nkl 2

1 is an inﬁnite set. So, δ p ≤ n,q ≤ k,r ≤ l : xpqr > ` − 2 6= 0. We observe that there is at least one p > n1, q > k1, 1 r > l1 for which xpqr > ` − 2 , for otherwise 1 δ p ≤ n,q ≤ k,r ≤ l : x > ` − ≤ δ ({(1,1,1),(2,2,2),...,(n ,n ,n )}) = 0 pqr 2 1 1 1 which is a contradiction. Let 1 E = p ≤ n,q ≤ k,r ≤ l : n ≤ p, k < q, l < r and x > ` − 6= /0. n1k1l1 1 1 1 pqr 2

1 If (p,q,r) ∈ En1k1l1 always implies xpqr ≥ ` + 2 , then, 1 E ⊆ p ≤ n,q ≤ k,r ≤ l : x > ` + , n1k1l1 pqr 2 47 /

1 i.e., δ En1k1l1 ≤ δ p ≤ n,q ≤ k,r ≤ l : xpqr > ` + 2 = 0. Since, 1 1 (n,k,l) ∈ N × N × N : p ≤ n,q ≤ k,r ≤ l : xpqr > ` + < δ ∈ F (I3). nkl 2 Thus, 1 p ≤ n,q ≤ k,r ≤ l : x > ` + pqr 2

⊆ {(1,1,1),(2,2,2),...,(n1,n1,n1)} ∪ En1k1l1 . So, 1 δ p ≤ n,q ≤ k,r ≤ l : x > ` + pqr 2 ≤ δ ({(1,1,1),(2,2,2),...,(n1,n1,n1)}) + δ En1k1l1 ≤ 0,

1 1 which is a contradiction. This shows that there is a n2 > n1, k2 > k1, l2 > l1 such that ` − 2 < xn2k2l2 < ` + 2 . Proceeding 1 1 in this way we obtain a subsequence xn k l of x, ni > ni−1, ki > ki−1, li > li−1 such that ` − < xn k l < ` + for each i. i i i i i i i i This subsequence xnikili converges to ` in Pringsheim’s sense and thus I -statistically convergent to ` in Pringsheim’s sense. This proves the theorem. The following results are an immediate corollaries of Theorem 2.13 and Remark 2.7.

Theorem 2.14. If I3-st −liminfx = `, then there exists a subsequence of x = (xnkl) which is I -statistically convergent to ` in Pringsheim’s sense.

Theorem 2.15. Every I3-st bounded triple sequence x = (xnkl) in Pringsheim’s sense has a subsequence which is I -statistically convergent to a ﬁnite real number in Pringsheim’s sense.

References

[1] Esi, A., Sava¸s,E., 2015. On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. Inf.Sci., 9(5), 2529-2534. [2] Das, P., Sava¸sE., Ghosal, S., 2011. On generalized of certain summability methods using ideals, Appl. Math. Lett., 26, 1509-1514. [3] Demirci, K., 2001. I -limit superior and limit inferior, Math. Commun., 6, 165-172. [4] Fast, H., 1951. Sur la convergence statistique, Colloq. Math., 2, 241-244. [5] Fridy, J.A., 1985. On statistical convergence, Analysis (Munich), 5, 301-313. [6] Fridy, J.A., 1993. Statistical limit points, Proc. Amer. Math. Soc. 118, 1187-1192. [7] Fridy, J.A., Orhan, C., 1997. Statistical limit superior and limit inferior, Proc. Amer. Math. Soc., 125, 3625-3631. [8] Gürdal, M., ¸Sahiner, A., 2008. Extremal I -Limit Points of Double Sequences, Appl. Math. E-Notes, 8, 131-137. [9] Kostyrko, P., Salat, T., Wilczynski, W., 2000/2001. I -convergence, Real Anal. Exch., 26, 669-686. [10] Lahiri, B.K., Das, P., 2003. Further results on I -limit superior and limit inferior, Math. Commun. 8, 151-156. [11] Moricz, F., 2003. Statistical convergence of multiple sequences, Arc. Math., 81, 82-89. [12] Mursaleen, M., Edely, O.H.H., 2003. Statistical convergence of double sequences, J. Math. Anal. Appl., 288, 223-231. [13] Mursaleen, M., Mohiuddine, S.A., Edely, O.H.H., 2010. On ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59, 603-611. [14] Nabiev, A., Pehlivan, S., Gürdal, M., 2007. On I -Cauchy sequences, Taiwanese J. Math., 12, 569-576. [15] ¸Sahiner, A., Gürdal, M., Düden, F.K., 2007. Triple sequences and their statistical convergence, Selçuk J. Appl. Math., 8(2), 49-55. [16] ¸Sahiner, A., Gürdal, M., Saltan, S., Gunawan, H., 2007. Ideal Convergence in 2-normed spaces, Taiwanese J. Math., 11(4), 1477-1484. [17] ¸Sahiner, A., Tripathy, B.C., 2008. Some I -related properties of triple sequences, Selçuk J. Appl. Math., 9(2), 9-18. [18] Šalát, T., 1980. On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2), 139–150. [19] Sava¸s,E., Das, E., 2011. A generalized statistical convergence via ideals, Appl. Math. Lett., 24, 826-830. 48 /

[20] Sava¸s,E., Das, E., 2014. On I -statistically pre-Cauchy sequences, Taiwanese J. Math., 18(1), 115-126. [21] Subramanian, N., Esi, A., Esi, A., 2020. Rough I -convergence on triple Bernstein operator sequences, Southeast Asian Bull., 44, 417-432. [22] Tripathy, B.C., 2003. Statistically Convergent Double Sequences, Tamkang J. Math., 34(3), 231-237. [23] Yamancı, U., Gürdal, M., 2014. I -statistical convergence in 2-normed space, Arab J. Math. Sci., 20(1), 41-47. [24] Yamancı, U., Gürdal, M., 2014. I -statistically pre-Cauchy double sequences, Global J. Math. Anal., 2(4), 297-303.

49 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

I -statistically localized sequences of weighted g via modulus functions in 2-normed spaces

Mualla Birgül Huban∗1, Mehmet Gürdal2, Ekrem Savas¸3 1Isparta University of Applied Sciences, Turkey 2Süleyman Demirel University, Department of Mathematics, Turkey 3U¸sak University, Turkey

Keywords Abstract: In this work, by using the concept of ideal and weighted density via modulus Ideal convergence, functions, I -statistically localized sequences are deﬁned and some basic properties of 2-normed spaces, I -statistically localized sequences are given. Also, it is proved that a sequence is I - localized sequence, statistically Cauchy of weighted g via modulus functions if and only if its I -statistical weight function, barrier is equal to zero. modulus function

1. Introduction and background

The notion of statistical convergence of sequences of real numbers was introduced by Fast in [5] and is based on the notion of asymptotic density of a set A ⊂ N. However, the first idea of statistical convergence appeared (under the name almost convergence) in the first edition (Warsaw, 1935) of the celebrated monograph [27] of Zygmund. It should be also mentioned that the notion of statistical convergence has been considered, in other contexts [7, 9, 11]. Statistical convergence has several applications in different fields of mathematics: summability theory [4, 23], trigonometric series [27], measure theory [16], and approximation theory [13]. The idea of I -convergence for sequences, was inspired by the concept of statistical convergence introduced in [14], see Kostyrko, Salát, and Wilczynski´ [14] for a comprehensive bibliography. All the results of [14] apply to sequences of functions with domains being singletons. Recently, in [4, 22], Sava¸set al. studied the I -statistically convergence of sequences and obtained some results of this concept. Particularly, in [9], Gürdal introduced the concept of ideal convergent sequences in 2-normed spaces. Gürdal and Açık [10] presented I -Cauchy sequences in 2-normed spaces. For more informations about I -convergence and I -statistically convergence, see [3, 10, 17, 21, 24]. In this paper, we are concerned with I -statistically localized sequence by using the concept of ideal and weighted density via modulus functions, and some important results are established. Now we recall some definitions and notations that will be used in paper. The notion of a statistically convergent sequence can be defined using the asymptotic density of subsets of the set of positive integers N ={1,2,...}. For any K ⊆ N and n ∈ N we denote K (n) := cardK ∩ {1,2,...,n} and we define lower and upper asymptotic density of the set K by the formulas

K (n) K (n) ρ (K) := liminf ; ρ (K) := limsup . n→∞ n n→∞ n

If ρ(K) = ρ(K) =: ρ(K), then the common value ρ(K) is called the asymptotic density of the set K and

K (n) ρ (K) = lim . n→∞ n Obviously all three densities ρ(K), ρ(K) and ρ(K) (if they exist) lie in the unit interval [0,1].

1 1 n ρ (K) = lim |Kn| = lim χK (k), n n ∑ n n k=1 if it exists, where χK is the characteristic function of the set K. Utilizing above information, we say that a number sequence x = (x ) statistically converges to a point L if for each k k∈N ε > 0 we have ρ (K (ε)) = 0, where K (ε) = {k ∈ N : |xk − L| ≥ ε} and in such situation we will write L = st-limxk.

∗ Corresponding author: [email protected] 50 /

On the other hand, I -convergence in a metric space was introduced by Kostyrko et al. [14] and its deﬁnition is depending upon the deﬁnition of an ideal I in N. If Y is a non-empty set, then a family of subsets of Y is called an ideal in Y iff (i) /0 ∈ I ; (ii) A,B ∈ I implies A ∪ B ∈ I ; (iii) for each A ∈ I and B ⊂ A we have imply B ∈ I . I is called a nontrivial ideal if Y ∈ I 6= /0and P (Y) is the power set of Y. A nontrivial ideal I in Y is called an admissible ideal if it is different from P (N) and contains all singletons, i.e., {x} ∈ I for each x ∈ Y. Let I ⊂ P (Y) be a nontrivial ideal. Then a class

F (I ) = {M ⊂ N : M = Y\A, for some A ∈ I } is a filter on Y, called the filter associated with the ideal I . An admissible ideal ⊂ ( ) is said to satisfy the condition (AP) if for every sequence (A ) of pairwise disjoint sets I P N n n∈N from I there are sets Bn ⊂ N, n ∈ N such that the symmetric difference An∆Bn is a finite set for every n and ∪n∈NBn ∈ I . Definition 1.1. ([14]) A sequence of reals {x } is said to be -convergent to L if, for each > 0, the set n n∈N I ε

K (ε) = {n ∈ N : |xn − L| ≥ ε} ∈ I . In the recent times in [1], using the natural density of weight g where g : N →[0,∞) is a function with property that n lim g(n) = ∞ and 9 0 as n → ∞, the concept of natural density was extended as follows: The upper density of n→∞ g(n) weight g was defined by K (1,n) ρ (K) = lim sup g n→∞ g(n) for K ⊂ N, where K (1,n) denotes the number of elements in K ∩ [1,n]. The lower density of weight g is defined in a similar manner. Then, the family n o Ig = K ⊂ N : ρg (K) = 0 n creates an ideal. It was seen in [1] that ∈ Ig iff → 0 as n → ∞. Furthermore, we suppose that n/g(n) 0 as N g(n) 9 n → ∞ so that N ∈/ Ig and Ig is a proper admissible ideal of N. We denote by G the collection of such weight functions g satisfying the above properties. Very recently in [2], a new kind of density was defined by using the modulus function and and the weighted function as + + follows: The modulus functions are defined as functions f : R ∪ {0} → R ∪ {0} satisfying the following properties. (i) f is increasing + (ii) f (x + y) ≤ f (x) + f (y) for all x,y ∈ R (iii) f (x) = 0 ⇔ x = 0 (iv) f is right continuous at 0. f Then the ρg (K) was defined in [2] by f f (K (1,n)) ρg (K) = lim . n→∞ f (g(n)) f (n) All the conditions to be a density function are satisfied by this new density function ρ f (K) apart from that as g f (g(n)) n → ∞ might not be equal to 1. Also, the modulus function f applied to have the generalized density function as above f (K (1,n)) M must be unbounded. Otherwise, take | f (x)| ≤ M for all x and some M > 0. Then for any K ⊆ , ≤ < N f (g(n)) f (g(n)) M → 0 which is of no interest again. Hence, we suppose that the modulus function to be unbounded. Moreover, the g(n) family g f I ( f ) = K ⊂ N : ρg (K) = 0 constructs an ideal. As a natural result, we can define the following definition. f f Definition 1.2. A sequence (xn) of real numbers is called the ρg -statistically convergent to x if for any ε > 0, ρg (K (ε)) = 0, where K (ε) is as in Definition 1. The following 2-normed space was given by Gähler [8].

Definition 1.3. Let X be a real vector space of dimension d, where 2 ≤ d < ∞. A 2-norm on X is a function k.,.k : X ×X → R which satisfies (i) kx,yk = 0 if and only if x and y are linearly dependent; (ii) kx,yk = ky,xk; (iii) kαx,yk = |α|kx,yk, α ∈ R; (iv) kx,y + zk ≤ kx,yk + kx,zk. The pair (X,k.,.k) is then called a 2-normed space. After this definition, many authors studied statistical convergence, I -convergence, I -Cauchy sequence, I ∗-convergent and I ∗-Cauchy sequence on this space (see [9, 10, 21, 24]). Depending upon the I -convergence and statistical convergence, Das et al. [4] introduced the I -statistical convergence as follows: 51 /

Deﬁnition 1.4. A sequence (xn) is said to be I -statistically convergent to L ,if for each ε > 0 and δ > 0 1 n ∈ : |{k ≤ n : kx − L k ≥ ε}| ≥ δ ∈ I . N n k

Later on, the concepts of I -statistically convergent and I -statistically Cauchy were given in 2-normed spaces and some results were obtained in [24].

Deﬁnition 1.5. Let I ⊂ 2N be a nontrivial ideal in N. The sequence (xn) of X is said to be I -statistically convergent to L ,if for each ε > 0, δ > 0 and nonzero z in X the set 1 n ∈ : |{k ≤ n : kx − L ,zk ≥ ε}| ≥ δ ∈ I N n k or equivalently if for each ε > 0 |An (ε)| ρ (An (ε)) = I -lim = 0 I n where An (ε) = {k ≤ n : kxk − L ,zk ≥ ε}.

If (xk) is I -statistically convergent to L then we write I -st − lim kxk − L ,zk = 0 or I -st − lim kxk,zk = kL ,zk. k→∞ k→∞

Definition 1.6. A sequence {xn} in 2-normed space (X,k.,.k) is said to be I -statistically Cauchy sequence in X if for every ε > 0, δ > 0 and every nonzero z ∈ X there exists a number N such that 1 ρ n ∈ : |{k ≤ n : kx − x ,zk ≥ ε}| ≥ δ = 0, I N n k N i.e., for every nonzero z ∈ X, 1 n ∈ : |{k ≤ n : kx − xN,zk ≥ ε}| ≥ δ ∈ I . N n k ∗ Definition 1.7. ([26]) A sequence {xn} in 2-normed space (X,k.,.k) is said to be I -statistically convergent to µ ∈ X if and only if there exists a set B ∈ F (I ) such that st- lim kxm − µ,zk = 0 and B = {b1 < b2 < ... < bn < ...} ⊂ . n→∞ n N Definition 1.8. ([26]) A sequence (x ) in 2-normed space (X,k.,.k) is said to be ∗-statistically Cauchy sequence if n n∈N I and only if there is a set B = {b1 < b2 < ... < bk} such that

st- lim xm − xmp ,z = 0. k,p→∞ k

A sequence (xn) in a metric space X is said to be localized in some subset M ⊂ X if the number sequence d (xn,x) converges for all x ∈ M (see [15]). This deﬁnition has been extended to statistical localized and I -localized in metric space [18, 19] and 2-normed spaces [25, 26], and they obtained interesting results in this concept. In this paper, by using the concept of ideal, the I -statistically localized sequences of weight g via modulus function f are deﬁned and some basic properties of I -statistically localized sequences of weight g via modulus function f are given. Also, it is proved that a sequence is I -statistically Cauchy of weight g via modulus function f if and only if its I -statistical barrier of weight g via modulus function f is equal to zero.

2. Main Results

Our main deﬁnitions and notations are as following: Deﬁnition 2.1. (a) A sequence (x ) in 2-normed space (X,k.,.k) is called as the -statistically localized of weight g n n∈N I via modulus function f in the subset M ⊂ X if and only if f (|{k ≤ n : kxk − x,zk ≥ ε}|) n ∈ : ≥ δ ∈ I N f (g(n)) exists for every x,z ∈ M. (b) the maximal set on which a sequence (xn) is I -statistically localized of weight g via modulus function f is said to be -statistically locator of weight g via modulus function f of (xn) and it is denoted by loc g (xn). I Ist ( f ) (c) A sequence (xn) in 2-normed space (X,k.,.k) is said to be I -statistically localized of weight g via modulus function f everywhere if (xn) is I -statistically locator of weight g via modulus function f of (xn) coincides with X. (d) A sequence (xn) in 2-normed space (X,k.,.k) is called as the I -statistically localized of weight g via modulus function f in itself if n o n ∈ : xn ∈/ loc g (xn) ⊂ . N Ist ( f ) I

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From the above deﬁnition, if (xn) is an I -statistically Cauchy sequence of weight g via modulus function f , then it is I -statistically localized of weight g via modulus function f everywhere. Actually, owing to f n ≤ k : f kxn − x,zk − xn0 − x,z > ε f n ≤ k : xn − xn0 ,z > ε f (g(k)) 6 f (g(k)) we have ( ) f n ≤ k : kxn − x,zk − xn0 − x,z > ε k ∈ : ≥ δ N f (g(k)) ( ) f n ≤ k : xn − xn0 ,z > ε ⊂ k ∈ : ≥ δ . N f (g(k))

So, the sequence is I -statistically localized of weight g via modulus function f if it is I -statistically Cauchy sequence of weight g via modulus function f . Also, we are able to say that each I -statistically convergence sequence of weight g via modulus function f is I -statistically localized of weight g via modulus function f . Note that if I is an admissible ideal, then every statistically localized sequence of weight g via modulus function f in 2-normed space (X,k.,.k) is I -statistically localized sequence of weight g via modulus function f in (X,k.,.k).

∗ Deﬁnition 2.2. We say the sequence (xn) to be I -statistically localized of weight g via modulus function f in 2-normed ∗ space (X,k.,.k) if and only if the number sequence kxn − x,zk is I -statistically convergent of weight g via modulus function f for every x,z ∈ X.

From the above deﬁnition, one sees that every I ∗-statistically Cauchy sequence or I ∗-statistically convergent of weight g via modulus function f in 2-normed space (X,k.,.k) is I ∗-statistically localized of weight g via modulus function f in (X,k.,.k). Note that for an admissible ideal, I ∗-statistically convergence and I ∗-statistically Cauchy of weight g via modulus function f criteria imply I -statistically convergence and I -statistically Cauchy criteria of weight g via modulus function f , respectively.

∗ Lemma 2.3. Let X be a 2-linear normed space and I be an admissible ideal on N. If a sequence (xn) ⊂ X is I - statistically localized of weight g via modulus function f on the set M ⊂ X, then (xn) is I -statistically localized of weight g via modulus function f on the set M and loc ∗g (xn) ⊂ loc g (xn). Ist ( f ) Ist ( f ) ∗ Proof. Assume that (xn) is I -statistically localized of weight g via modulus function f on M. Then, there is a set P ∈ I such that f j ∈ N : x j − x,z ε lim > j→∞ f (g( j)) C ∗ exists for each x,z ∈ M and P = N\P = p1 < p2 < ... < p j . Then, the sequence kxn − x,zk is an I -statistically Cauchy sequence of weight g via modulus function f , which means that f (|{k ≤ n : kxk − xN,zk ≥ ε}|) n ∈ : ≥ δ ∈ I . N f (g(n))

Therefore, the number sequence kxn − x,zk is I -statistically convergent of weight g via modulus function f , which gives that (xn) is I -statistically localized of weight g via modulus function f on the set M. Now we are ready to give our main results about I -statistically localized sequences.

Proposition 2.4. Let (xn) be an I -statistically localized sequence of weight g via modulus function f in a linear 2-normed space (X,k.,.k). Then (xn) is I -statistically bounded of weight g via modulus function f .

Proof. Suppose that (xn) is I -statistically localized of weight g via modulus function f . Then, the number sequence kxn − x,zk is I -statistically convergent of weight g via modulus function f for some x,z ∈ X. This means that f (|{n ≤ k : kxn − x,zk > K}|) k ∈ : > δ ∈ I N f (g(k)) for some K > 0 and δ > 0. As a result, the sequence (xn) is I -statistically bounded of weight g via modulus function f .

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Proposition 2.5. Let be an admissible ideal satisfying the property (AP) and M = loc g (xn). Also, a point y ∈ X be I Ist ( f ) such that there exists x ∈ M for any ε > 0, δ > 0 and every nonzero z ∈ M such that f (|{n ≤ k : |kx − xn,zk − ky − xn,zk| > ε}|) k ∈ : > δ ∈ I . (1) N f (g(k)) Then y ∈ M.

Proof. To show that the number sequence kxn − y,zk is an I -statistically Cauchy sequence of weight g via modulus function f is enough. Let be ε > 0 and x ∈ M = loc g (xn) is a point satisfying the property (1). From the (AP) property Ist ( f ) of I , we get f (|{n ≤ k : |kx − x ,zk − ky − x ,zk| ε}|) kn kn > → 0 f (g(k)) and f (|{(n,m) : |kx − x,zk − kx − x,zk| ε, n,m ≤ k}|) kn km > → 0 f (g(k)) as m,n → ∞, where K = {k1 < k2 < ... < kn < ...} ∈ F (I ). Therefore, there is n0 ∈ N for any ε > 0, δ > 0 and every nonzero z ∈ M such that ε f n ≤ k : |kx − xk ,zk − ky − xk ,zk| δ n n > 3 < (2) f (g(k)) 3 ε f (n,m) : |kx − xk ,zk − kx − xk ,zk| , n,m ≤ k δ n m > 3 < . (3) f (g(k)) 3 for all n ≥ n0, m ≥ m0. Since

f (|{(n,m) : |ky − xkn ,zk − ky − xkm ,zk| > ε, n,m ≤ k}|) f (g(k)) ε f n ≤ k : |ky − xk ,zk − kx − xk ,zk| ≤ n n > 3 f (g(k)) ε f n ≤ k : |kx − xk ,zk − kx − xk ,zk| + n m > 3 f (g(k)) ε f n ≤ k : |kx − xk ,zk − ky − xk ,zk| + m n > 3 , f (g(k)) we obtain by using (2) and (3) together with above inequality f (|{(n,m) : |ky − x ,zk − ky − x ,zk| ε, n,m ≤ k}|) kn km > < δ f (g(k)) for all n ≥ n0, m ≥ n0. So, f (|{(n,m) : |ky − x ,zk − ky − x ,zk| ε, n,m ≤ k}|) kn km > → 0 as m,n → ∞ f (g(k)) for the K = (kn) ⊂ N and K ∈ F (I ). Hence ky − xn,zk is an I -statistically Cauchy sequence of weight g via modulus function f , which ﬁnishes the proof. Deﬁnition 2.6. ([20])A point a in a 2-normed space (X,k.,.k) is called a limit point of a set M in X if for an arbitrary Σ = {(b1,ε1),...,(bn,εn)}, there is a point aΣ ∈ M, aΣ 6= a such that aΣ ∈ WΣ (a). Moreover, a subset Y ⊂ X is called a closed subset of X if Y contains every its limit point. If Y 0 is the set of all limit points of a subset Y ⊂ X, then the set Y = Y ∪Y 0 is called the closure of the set Y. Proposition 2.7. I -statistical locator of weight g via modulus function f of any sequence is a closed subset of the 2-normed space (X,k.,.k).

Proof. Let y ∈ loc g (xn). Then, there is a point x ∈ loc (xn) for arbitrary Σ = {(b ,ε ),...,(bn,εn)} such that x 6= y Ist ( f ) Ist 1 1 and x ∈ W (y). Thus, for any ε > 0, δ > 0 and every z ∈ loc g (xn) Σ Ist ( f ) f (|{n ≤ k : |kx − xn,zk − ky − xn,zk| > ε}|) k ∈ : > δ ∈ I N f (g(k)) due to f (|{n ≤ k : |kx − x ,zk − ky − x ,zk| ε}|) f (|{n ≤ k : ky − x ,zk ε}|) n n > ≤ n > < δ f (g(k)) f (g(k)) for every n ∈ . In conclusion, the hypothesis of Proposition 2.5 is satisﬁed, and then we reach that y ∈ loc g (xn), that N Ist ( f ) is, loc g (xn) is closed. Ist ( f ) 54 /

Deﬁnition 2.8. A point y is an I -statistical limit point of weight g via modulus function f of the sequence (xn) in 2-normed space (X,k.,.k) if there is a set K = {k1 < k2 < ... < kn} ⊂ N such that K ∈/ I and f (|{n ∈ : kx − y,zk ε}|) lim N kn > = 0. n→∞ f (g(n))

A point ξ is said to be an I -statistical cluster point of weight g via modulus function f of the sequence (xn) if for each ε > 0, δ > 0 and every z ∈ X f (|{n ≤ k : kxn − ξ,zk > ε}|) k ∈ : < δ ∈/ I . N f (g(k)) We can have the following result owing to

f (|{n ≤ k : |kx − y,zk − kx − y,zk| ε}|) f (|{n ≤ k : kx − x,zk ε}|) n > ≤ n > . f (g(k)) f (g(k))

Proposition 2.9. Let y ∈ X be an I -statistical limit point (an I -statistical cluster point) of weight g via modulus function f of a sequence (xn) in 2-normed space (X,k.,.k). Then, the number ky − x,zk is an I -statistical limit point (an I - statistical cluster point) of weight g via modulus function f of the sequence {kxn − x,zk} for each x ∈ X and every nonzero z ∈ X.

Deﬁnition 2.10. Let (xn) be the I -statistical localized sequence of weight g via modulus function f with the I -statistical locator L = loc g (xn). The number Ist ( f )

g σ = inf I ( f )-st- lim kx − xn,zk x∈L n→∞ is called as the I -statistical barrier of weight g via modulus function f of (xn). Theorem 2.11. Let (X,k.,.k) be a 2-normed space and let I ⊂ 2N be an ideal satisfying the (AP) property. Then, an I -statistically localized sequence of weight g via modulus function f is I -statistically Cauchy sequence of weight g via modulus function f if and only if σ = 0.

Proof. Assume that (xn) is an I -statistically Cauchy sequence of weight g via modulus function f in 2-normed space

(X,k.,.k). Then, there is a set R = {r1 < r2 < ... < rn} ⊂ N such that R ∈ F (I ) and st-limn,m→∞ kxrn − xrm ,zk = 0. As a consequence, there is a n0 ∈ N for every ε > 0, δ > 0 and every z ∈ X such that f (|{n ≤ k : kx − ξ,zk ε}|) n > < δ f (g(k))

g for all n ≥ n0. Because (xn) is I -statistically localized sequence of weight g via modulus function f , I ( f )-st- limn→∞ xn − xr ,z exists and we get n0

g I ( f )-st- lim xn − xr ,z ≤ δ. n→∞ n0

Therefore, σ ≤ δ. Because δ > 0 is arbitrary, we obtain σ = 0.

Now assume that σ = 0. Then, there is a x ∈ locIst (xn) for every ε > 0, δ > 0 and every nonzero z ∈ X such that

g δ kx,zk = I ( f )-st- lim kx − xn,zk < . n→∞ 2 Then f (|{n ≤ k : |kx,zk − kx − xn,zk| > ε}|) δ k ∈ : > − kx,zk ∈ I . N f (g(k)) 2 Hence, we have f (|{n ≤ k : kx − xn,zk > ε}|) δ k ∈ : > ∈ I . N f (g(k)) 2 g So, I ( f )-st-limn→∞ kx − xn,zk = 0, which shows that (xn) is I -statistically Cauchy sequence of weight g via modulus function f .

Deﬁnition 2.12. A sequence (xn) in 2-normed space (X,k.,.k) is called as the uniformly I -statistically localized of weight g via modulus function f on a subset L ⊂ X if the sequence {kx − xn,zk} is uniformly I -statistically converges of weight g via modulus function f for all x,z ∈ L.

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Proposition 2.13. Let the sequence (xn) be uniformly I -statistically localized of weight g via modulus function f on the set L ⊂ X and v ∈ Y is such that for every ε > 0, δ > 0 and every nonzero z in L there is y ∈ L such that f (|{n ≤ k : |kv − xn,zk − ky − xn,zk| > ε}|) k ∈ : > δ ∈ I N f (g(k))

Then v ∈ loc g (xn) and (xn) is uniformly -statistically localized of weight g via modulus function f on the set of such Ist ( f ) I points v. Since the proof of Proposition 2.13 is analog to Proposition 2.5, we omit it. References [1] Balcerzak, M., Das, P., Filipczak, M., Swaczyna, J., 2015. Generalized kinds of density and the associated ideals, Acta Math. Hungar., 147, 97-115. [2] Bose, K., Das, P., Kwela, A., 2018. Generating new ideals using weighted density via modulus functions, Indag. Math. 29(5), 1196–1209. [3] Das, P., Sava¸s,E., 2014. On I -statistically pre-Cauchy sequences, Taiwanese J. Math., 18(1), 115-126. [4] Das, P., Sava¸sE., Ghosal, S., 2011. On generalized of certain summability methods using ideals, Appl. Math. Lett., 26, 1509-1514. [5] Fast, H., 1951. Sur la convergence statistique, Colloq. Math., 2, 241-244. [6] Freedman, A.R., Sember, J.J., 1981. Densities and summability, Pacitiﬁc J. Math. 95, 10-11. [7] Fridy, J.A., 1985. On statistical convergence, Analysis (Munich), 5, 301-313. [8] Gähler, S., 1963. 2-metrische Räume und ihre topologische Struktur. Math. Nachr., 26, 115-148. [9] Gürdal, M., 2006. On ideal convergent sequences in 2-normed spaces. Thai Journal of Mathematics, 4(1), 85-91. [10] Gürdal, M., Açık, I., 2008. On I -Cauchy sequences in 2-normed spaces. Math. Inequal. Appl., 2(1), 349-354. [11] Gürdal, M., Pehlivan, S., 2004. The statistical convergence in 2-Banach spaces, Thai. J. Math., 2(1), 107-113. [12] Gürdal, M., ¸Sahiner, A., 2008. Extremal I -Limit Points of Double Sequences, Appl. Math. E-Notes, 8, 131-137. [13] Gürdal, M., ¸Sahiner, A., Açık, I., 2009. Approximation theory in 2-Banach spaces, Nonlinear Analysis, 71(5-6), 1654-1661. [14] Kostyrko, P., Salat, T., Wilczynski, W., 2000/2001. I -convergence, Real Anal. Exch., 26, 669-686. [15] Krivonosov, L.N., 1974. Localized sequences in metric spaces, Izv. Vyssh. Uchebn. Zaved. Mat., 4, 45-54; Soviet Math. (Iz. VUZ), 18(4), 37-44. [16] Miller, H.I., 1995. A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347, 1811-1819. [17] Nabiev, A., Pehlivan, S., Gürdal, M., 2007. On I -Cauchy sequences, Taiwanese J. Math., 12, 569-576. [18] Nabiev, A.A., Sava¸s,E., Gürdal, M., 2019. Statistically localized sequences in metric spaces. J. App. Anal. Comp., 9(2), 739-746. [19] Nabiev, A.A., Sava¸s,E., Gürdal, M., 2020. I -localized sequences in metric spaces, Facta Universitatis’Series Mathematics and Informatics, 35(2), 459-469. [20] Raymond, W.F., Cho, Ye.J., 2001. Geometry of linear 2-normed spaces, Huntington, N.Y. Nova Science Publishers. [21] ¸Sahiner, A., Gürdal, M., Saltan, S., Gunawan, H., 2007. Ideal Convergence in 2-normed spaces, Taiwanese J. Math., 11(4), 1477-1484. [22] Sava¸s,E., Das, E., 2011. A generalized statistical convergence via ideals, Appl. Math. Lett., 24, 826-830. [23] Sava¸s,E., Gürdal, M., 2014. Certain summability methods in intuitionistic fuzzy normed spaces, Journal of Intelligent and Fuzzy Systems, 27(4), 1621-1629. [24] Yamancı, U., Gürdal, M., 2014. I -statistical convergence in 2-normed space, Arab J. Math. Sci., 20(1), 41-47. [25] Yamancı, U., Nabiev, A.A., Gürdal, M., 2020. Statistically localized sequences in 2-normed spaces, Honam Mathematical J., 42(1), 161-173. [26] Yamancı, U., Gürdal, M., 2020. I -statistically localized sequence in 2-normed spaces, Lecture Notes on Data Engineering and Communications Technologies, vol 43. Springer, Cham, pp. 1039-1046. [27] Zygmund, A, 1979. Trigonometric Series, second ed., Cambridge Univ. Press, Cambridge.

56 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

Ay¸se Kabata¸s∗1 1Karadeniz Technical University, Science Faculty, Department of Mathematics, Turkey

Keywords Abstract: In this paper, asymptotic formulae are investigated for the solutions of Hill’s Green’s function, equation when the potential is symmetric single well. For this reason, the obtained results Hill’s equation, for the solutions of Hill’s equation by Eastham (1973) are improved. Here, the potential Symmetric single well function q(x) is a real-valued, absolutely continuous and periodic function with period a. potential, By using these estimates, Green’s function is derived for the Hill’s equation under periodic Eigenfunctions, boundary conditions. The method developed by Fulton (1977) is followed. Because of Asymptotics this, asymptotic approximations for the derivatives of solutions are also calculated with different types of restrictions on the potential.

Green’s Function for Hill’s Equation with Symmetric Single Well Potential

1. Introduction

Consider the Hill’s equation y00(x) + [λ − q(x)]y(x) = 0, x ∈ [0,a] (1) where λ is a real parameter and the potential q(x) is a real-valued, absolutely continuous and periodic function with period a. In particular, the equation 1 coupled with periodic boundary conditions y(0) = y(a),y0(0) = y0(a) is a self-adjoint problem.

The Hill’s equation with various types of restrictions on the potential has been widely studied in the literature ([7, 10, 13]). [6, 12] are specially referred that q(x) is a symmetric single well potential. Important results about the eigenvalues and instability intervals were obtained in [2, 6, 12]. The properties of the Green’s functions and some criteria for the maximum and anti-maximum principles were investigated in [4, 5]. In addition, there are many studies on the inverse problem ([3, 8]).

In this paper, asymptotic formulae are derived for the solutions of the Hill’s equation when the potential q(x) is a symmetric single well with mean value zero. By a symmetric single well potential on [0,a], we mean a continuous function q(x) on a a [0,a] which is symmetric about x = 2 and non-increasing on [0, 2 ].

The following results obtained in [10] are improved to determine the solutions.

Let φ1(x,λ) and φ2(x,λ) be the linearly independent solutions of (1) which satisfy the initial conditions

0 0 φ1(0,λ) = 1, φ1(0,λ) = 0, φ2(0,λ) = 0, φ2(0,λ) = 1. (2)

Theorem 1.1. [10, §4.3] Let q(x) be absolutely continuous. Then, as λ → ∞,

√ √ √ 1 − 1 1 −1 1 2 −1 φ (x,λ) = cosx λ + λ 2 Q(x)sinx λ + λ q(x) − q(0) − Q (x) cosx λ + o(λ ), (3) 1 2 4 2

√ √ √ − 1 1 −1 1 − 3 1 2 − 3 φ (x,λ) = λ 2 sinx λ − λ Q(x)cosx λ + λ 2 q(x) + q(0) − Q (x) sinx λ + o(λ 2 ) (4) 2 2 4 2 where Z x Q(x) = q(t)dt. (5) 0

∗ Corresponding author: [email protected] 57 Theorem 1.2. [10, §4.3] Let q(x) be piecewise continuous. Then, as λ → ∞,

√ Z x √ √ − 1 φ1(x,λ) = cosx λ + λ 2 sin{(x −t) λ}q(t)cost λdt 0 Z x √ Z t √ √ −1 − 3 + λ sin{(x −t) λ}q(t)dt sin{(t − u) λ}q(u)cosu λdu + O(λ 2 ), (6) 0 0

√ Z x √ √ − 1 −1 φ2(x,λ) = λ 2 sinx λ + λ sin{(x −t) λ}q(t)sint λdt 0 Z x √ Z t √ √ − 3 −2 + λ 2 sin{(x −t) λ}q(t)dt sin{(t − u) λ}q(u)sinu λdu + O(λ ). (7) 0 0

The second goal of this paper is to determine the Green’s function asymptotics for the periodic Hill’s equation using the estimates on the solutions. One of the uses of Green’s function is to derive sampling representations for transforms associated with eigenvalue problems ([1]). The method developed by Fulton [11] is followed. Since the derivatives of the solutions are needed in this method, the asymptotic approximations for the derivatives of φ1(x,λ) and φ2(x,λ) are also calculated with different types of restrictions on the potential q(x).

2. Approximations for the Eigenfunctions

In this section, approximations are obtained for the solutions φ1(x,λ) and φ2(x,λ) of the Hill’s equation with the initial conditions (2).

Before, the following lemma is given for q(x) being of a symmetric single well potential.

Lemma 2.1. If q(x) is a symmetric single well potential on [0,a], then Z x Z a Z x q(t)dt = xq(x) + a{q(a) − q(a/2)} − tq0(t)dt − tq0(t)dt. (8) 0 a/2 a/2 Proof. Using integration by parts gives

Z x x Z x q(t)dt = tq(t) − tq0(t)dt 0 t=0 0 Z a/2 Z x = xq(x) − tq0(t)dt + tq0(t)dt 0 a/2 Z a/2 Z x = xq(x) − − tq0(a −t)dt + tq0(t)dt 0 a/2 Z a/2 Z x = xq(x) − (a −t)q0(t)dt − tq0(t)dt a a/2 Z a Z x = xq(x) + (a −t)q0(t)dt − tq0(t)dt. a/2 a/2 Z a Z x = xq(x) + a{q(a) − q(a/2)} − tq0(t)dt − tq0(t)dt. a/2 a/2

Theorem 2.2. Let q(x) be a symmetric single well potential on [0,a]. Then, as λ → ∞

√ Z a Z x √ 1 − 1 0 0 φ1(x,λ) = cosx λ + λ 2 xq(x) + a{q(a) − q(a/2)} − tq (t)dt − tq (t)dt sinx λ 2 a/2 a/2 ( ) 1 1 Z a Z x 2 √ + λ −1 q(x) − q(0) − xq(x) + a{q(a) − q(a/2)} − tq0(t)dt − tq0(t)dt cosx λ + o(λ −1), 4 2 a/2 a/2 (9)

58 √ Z a Z x √ − 1 1 −1 0 0 φ2(x,λ) = λ 2 sinx λ − λ xq(x) + a{q(a) − q(a/2)} − tq (t)dt − tq (t)dt cosx λ 2 a/2 a/2 ( Z a Z x 2) √ 1 − 3 1 0 0 − 3 + λ 2 q(x) + q(0) − xq(x) + a{q(a) − q(a/2)} − tq (t)dt − tq (t)dt sinx λ + o(λ 2 ). 4 2 a/2 a/2 (10)

Proof. If we use Theorem 1.1 and substitute (8) in (5), the proof is done.

We have also some approximations for the derivatives of φ1(x,λ) and φ2(x,λ). They will be used in calculation of the Green’s function.

Lemma 2.3. If q(x) is a piecewise continuous function, then the derivative of (6) is, as λ → ∞,

√ Z x √ √ 0 1 φ1(x,λ) = −λ 2 sinx λ + cos{(x −t) λ}q(t)cost λdt 0 Z x √ Z t √ √ − 1 −1 + λ 2 cos{(x −t) λ}q(t)dt sin{(t − u) λ}q(u)cosu λdu + O(λ ). (11) 0 0 Proof. The usual variation of constants formula [9, §2.5] gives √ Z x √ − 1 φ1(x,λ) = cosx λ + λ 2 sin{(x −t) λ}q(t)φ1(t,λ)dt. (12) 0 If we arrange this formula, one can write

√ √ Z x √ √ Z x √ − 1 φ1(x,λ) = cosx λ + λ 2 sinx λ cost λq(t)φ1(t,λ)dt − cosx λ sint λq(t)φ1(t,λ)dt . (13) 0 0

It is obtained by differentiating (13) with respect to x and substituting φ1(t,λ) from (6) in the integral that

√ √ Z x √ √ Z x √ 0 1 − 1 1 1 φ1(x,λ) = −λ 2 sinx λ + λ 2 λ 2 cosx λ cost λq(t)φ1(t,λ)dt + λ 2 sinx λ sint λq(t)φ1(t,λ)dt 0 0 1 √ Z x √ = −λ 2 sinx λ + cos{(x −t) λ}q(t)φ1(t,λ)dt 0 1 √ Z x √ √ = −λ 2 sinx λ + cos{(x −t) λ}q(t)cost λdt 0 Z x √ Z t √ √ − 1 −1 + λ 2 cos{(x −t) λ}q(t)dt sin{(t − u) λ}q(u)cosu λdu + O(λ ). 0 0

In [10, §4.3], it is determined similarly √ Z x √ √ 0 − 1 φ2(x,λ) = cosx λ + λ 2 cos{(x −t) λ}q(t)sint λdt 0 Z x √ Z t √ √ −1 − 3 + λ cos{(x −t) λ}q(t)dt sin{(t − u) λ}q(u)sinu λdu + O(λ 2 ). (14) 0 0

Lemma 2.4. Let q(x) be absolutely continuous. Then, the derivatives of (3) and (4) are, as λ → ∞,

√ √ √ 0 1 1 1 − 1 1 2 − 1 φ (x,λ) = −λ 2 sinx λ + Q(x)cosx λ + λ 2 q(x) + q(0) + Q (x) sinx λ + o(λ 2 ), (15) 1 2 4 2

√ √ √ 0 1 − 1 1 −1 1 2 −1 φ (x,λ) = cosx λ + λ 2 Q(x)sinx λ − λ q(x) − q(0) + Q (x) cosx λ + o(λ ). (16) 2 2 4 2

59 Proof. If q(x) is absolutely continuous, this implies that q0(x) exists p.p. and is integrable. Under these conditions, let consider the second term on the right of (11). We have

Z x √ √ 1 Z x h √ √ i cos{(x −t) λ}q(t)cost λdt = cosx λ + cos{(x − 2t) λ} q(t)dt 0 2 0 1 √ 1 Z x √ = Q(x)cosx λ + cos{(x − 2t) λ}q(t)dt 2 2 0 √ √ x 1 1 1 − 1 = Q(x)cosx λ + [− λ 2 q(t)sin{(x − 2t) λ} 2 2 2 t=0 Z x √ 1 − 1 0 + λ 2 q (t)sin{(x − 2t) λ}dt] 2 0 √ √ 1 1 − 1 = Q(x)cosx λ + λ 2 [q(x) + q(0)]sinx λ 2 4 Z x √ 1 − 1 0 + λ 2 q (t)sin{(x − 2t) λ}dt. 4 0 The last integral on the right here is o(1) as λ → ∞ by the Riemann-Lebesgue Lemma. So,

Z x √ √ √ √ 1 1 − 1 − 1 cos{(x −t) λ}q(t)cost λdt = Q(x)cosx λ + λ 2 [q(x) + q(0)]sinx λ + o(λ 2 ). (17) 0 2 4 Also, from [10, §4.3] Z x √ √ √ √ 1 1 − 1 − 1 sin{(x −t) λ}q(t)cost λdt = Q(x)sinx λ + λ 2 [q(x) − q(0)]cosx λ + o(λ 2 ). (18) 0 2 4 For the third term on the right of (11), together with (18) we ﬁnd √ √ √ − 1 R x R t λ 2 0 cos{(x −t) λ}q(t)dt 0 sin{(t − u) λ}q(u)cosu λdu Z x √ √ 1 − 1 −1 = λ 2 cos{(x −t) λ}q(t)Q(t)sint λdt + O(λ ) 2 0 Z x √ √ 1 − 1 h i −1 = λ 2 sinx λ − sin{(x − 2t) λ} q(t)Q(t)dt + O(λ ) 4 0 √ 2 x 1 − 1 Q (t) − 1 = λ 2 sinx λ + o(λ 2 ) 4 2 t=0 √ 1 − 1 2 − 1 = λ 2 Q (x)sinx λ + o(λ 2 ), (19) 8 again by using the Riemann-Lebesgue Lemma. (17) and (19) prove (15). The proof of (16) is similar.

Lemma 2.5. If q(x) is a symmetric single well potential on [0,a], then as λ → ∞

√ Z a Z x √ 0 1 1 0 0 φ1(x,λ) = −λ 2 sinx λ + xq(x) + a{q(a) − q(a/2)} − tq (t)dt − tq (t)dt cosx λ 2 a/2 a/2 ( Z a Z x 2) √ 1 − 1 1 0 0 − 1 + λ 2 q(x) + q(0) + xq(x) + a{q(a) − q(a/2)} − tq (t)dt − tq (t)dt sinx λ + o(λ 2 ), 4 2 a/2 a/2 (20)

√ Z a Z x √ 0 1 − 1 0 0 φ2(x,λ) = cosx λ + λ 2 xq(x) + a{q(a) − q(a/2)} − tq (t)dt − tq (t)dt sinx λ 2 a/2 a/2 ( ) 1 1 Z a Z x 2 √ − λ −1 q(x) − q(0) + xq(x) + a{q(a) − q(a/2)} − tq0(t)dt − tq0(t)dt cosx λ + o(λ −1). 4 2 a/2 a/2 (21)

Proof. Using (8) in (5) and substituting this in (15) and (16) prove the lemma.

60 3. Approximations for Green’s Function In this section, we aim to improve asymptotic formulae for Green’s function of the periodic Hill’s equation with symmetric single well potential. Firstly, we deﬁne w(λ) as follows 0 0 w(λ) := φ1(x,λ)φ2(x,λ) − φ1(x,λ)φ2(x,λ). (22)

It is known as the Wronskian function of φ1(x,λ) and φ2(x,λ). So, the Green’s function of the problem is ( φ (ξ,λ)φ (x,λ) 1 2 , 0 ≤ ξ ≤ x ≤ a G(x,ξ,λ) = w(λ) (23) φ1(x,λ)φ2(ξ,λ) w(λ) , 0 ≤ x ≤ ξ ≤ a. Theorem 3.1. For 0 ≤ ξ ≤ x ≤ a, as λ → ∞

√ √ √ √ √ √ − 1 1 −1 h i G(x,ξ,λ) = λ 2 cosξ λ sinx λ − λ A(x)cosξ λ cosx λ − A(ξ)sinξ λ sinx λ 2 √ √ 1 − 3 1 2 2 + λ 2 { q(ξ) + q(x) − A (ξ) + A (x) cosξ λ sinx λ 4 2 √ √ − 3 − A(ξ)A(x)sinξ λ cosx λ} + o(λ 2 ) (24) where Z a Z x A(x) := xq(x) + a{q(a) − q(a/2)} − tq0(t)dt − tq0(t)dt. (25) a/2 a/2 Similar result holds for 0 ≤ x ≤ ξ ≤ a changing the role of ξ and x. Proof. We begin to the proof by evaluating the Wronskian function w(λ). For this, we substitute (9), (10), (20) and (21) into (22). Hence, 1 1 √ 1 1 √ w(λ) = 1 − λ −1 q(x) − q(0) + A2(x) cos2 x λ + λ −1 q(x) + q(0) − A2(x) sin2 x λ 4 2 4 2 1 1 1 √ + λ −1A2(x) + λ −1 q(x) − q(0) − A2(x) cos2 x λ 4 4 2 1 1 √ − λ −1 q(x) + q(0) + A2(x) sin2 x λ + o(λ −1) 4 2 1 1 = 1 − λ −1A2(x) + λ −1A2(x) + o(λ −1) 4 4 = 1 + o(λ −1). (26) From that, we can write 1 1 = = 1 + o(λ −1). (27) w(λ) 1 + o(λ −1) Finally, using (9), (10), (27) in (23) we ﬁnd

√ √ √ φ1(ξ,λ)φ2(x,λ) 1 − 1 1 −1 1 2 −1 = cosξ λ + λ 2 A(ξ)sinξ λ + λ q(ξ) − q(0) − A (ξ) cosξ λ + o(λ ) w(λ) 2 4 2 √ √ √ − 1 1 −1 1 − 3 1 2 − 3 × λ 2 sinx λ − λ A(x)cosx λ + λ 2 q(x) + q(0) − A (x) sinx λ + o(λ 2 ) 2 4 2 × 1 + o(λ −1) √ √ √ √ − 1 1 −1 1 − 3 1 2 = {λ 2 cosξ λ sinx λ − λ A(x)cosξ λ cosx λ + λ 2 q(x) + q(0) − A (x) 2 4 2 √ √ √ √ √ √ 1 −1 1 − 3 × cosξ λ sinx λ + λ A(ξ)sinξ λ sinx λ − λ 2 A(ξ)A(x)sinξ λ cosx λ 2 4 √ √ 1 − 3 1 2 − 3 −1 + λ 2 q(ξ) − q(0) − A (ξ) cosξ λ sinx λ + o(λ 2 )} × 1 + o(λ ) 4 2 √ √ √ √ √ √ − 1 1 −1 h i = λ 2 cosξ λ sinx λ − λ A(x)cosξ λ cosx λ − A(ξ)sinξ λ sinx λ 2 √ √ 1 − 3 1 2 2 + λ 2 { q(ξ) + q(x) − A (ξ) + A (x) cosξ λ sinx λ 4 2 √ √ − 3 − A(ξ)A(x)sinξ λ cosx λ} + o(λ 2 ). Thus, the proof is completed. 61 Theorem 3.2. Green’s function of the periodic Hill’s equation satisfy, as n → ∞ (n = 0,1,2,...)

a 2(n + 1)πξ 2(n + 1)πx a2 2(n + 1)πξ 2(n + 1)πx G(x,ξ,n) = cos sin − [A(x)cos cos 2(n + 1)π a a 8(n + 1)2π2 a a 2(n + 1)πξ 2(n + 1)πx a3 1 − A(ξ)sin sin ] + { q(ξ) + q(x) − A2(ξ) + A2(x) a a 32(n + 1)3π3 2 2(n + 1)πξ 2(n + 1)πx 2(n + 1)πξ 2(n + 1)πx × cos sin − A(ξ)A(x)sin cos } + o(n−3) a a a a for 0 ≤ ξ ≤ x ≤ a. Similar result holds for 0 ≤ x ≤ ξ ≤ a changing the role of ξ and x. Proof. The periodic eigenvalues obtained in [6] are substituted in Theorem 3.1.

References

[1] Annaby, M. H., Tharwat, M. M. 2006. On Sampling Theory and Eigenvalue Problems with an Eigenparameter in the Boundary Conditions. Science University of Tokyo Journal of Mathematics, Vol. 42, Number 2 (2006),157-176. [2] Ba¸skaya,E. 2020. Periodic and Semi-Periodic Eigenvalues of Hill’s Equation with Symmetric Double Well Potential, TWMS J. App. Eng. Math., Vol. 10, No. 2 (2020), 346-352. [3] Borg, G. 1946. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung- durch die Eigenwerte, Acta Math. 78 (1946), 196. MR 7:382d [4] Cabada, A., Cid, J. A. 2012. On Comparison Principles for the Periodic Hill’s Equation, J. London Math. Soc. (2) 86 (2012), 272-290. [5] Cabada, A., Cid J. A., Lopez-Somoza, L. 2016. Green’s Functions and Spectral Theory for the Hill’s Equation, Applied Mathematics and Computation 286 (2016), 88-105. [6] Co¸skun,H., Ba¸skaya,E., Kabata¸s,A. 2019. Instability Intervals for Hill’s Equation with Symmetric Single Well Potential, Ukrainian Mathematical Journal (2019), Vol.71, No. 6, 977-983. [7] Co¸skun,H. 2003. On the Spectrum of a Second Order Periodic Differential Equation, Rocky Mountain J. Math. 33 (2003), 1261-1277. [8] Co¸skun,H. 2002. Some Inverse Results for Hill’s Equation, J. Math. Anal. Appl. 276 (2002), 833-844. [9] Eastham, M. S. P. 1970. Theory of Ordinary Differential Equations, Van Nostrand Reinhold. [10] Eastham, M. S. P. 1973. The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh and London. [11] Fulton, C. T. 1977. Two-Point Boundary Value Problems with Eigenvalue Parameter Contained in the Boundary Conditions, Proceedings of the Royal Society of Edinburgh, 77A (1977), 293-308. [12] Huang, M. J. 1997. The First Instability Interval for Hill Equations with Symmetric Single Well Potentials, Proc. Amer. Math. Soc. 125 (1997), 775-778. [13] Magnus, W., Winkler, S. 1979. Hill’s Equation, Dover Publications, New York.

62 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

N-dimensional Bound State Solutions of the Hyperbolic Potential Function in Approximate Analytic Form

Aysel Özﬁdan∗1, Aysen Durmus2 1Tarsus University,Faculty of Engineering , Department of Natural and Mathematical Sciences, Turkey 2Erciyes University, Faculty of Science, Department of Physics, Turkey

Keywords Abstract: Approximate bound state solutions of the Schrödinger equation for a quantum Hyperbolic potential function, system interacting with the hyperbolic potential function have been investigated within Asymptotic iteration method, the framework of Greene-Aldrich approximation and asymptotic iteration formalism in N-dimensions hyperspherical coordinates. We have obtained the N-dimensional energy eigenvalues and the corresponding wave functions in the context of non-relativistic theory. Our present results provide also an appropriate test of the accuracy of asymptotic iteration method based on solving second order homogeneous linear differential equations.

1. Introduction

Constructing the solutions to the Schrödinger equation for any lN−1–state has been a major preoccupation of quantum mechanics. However, N–dimensional Schrödinger equation for central potentials containing centrifugal term cannot be solved in an analytical form. In this connection, approximate expressions play a leading role in calculating the non- relativistic energies and wave functions for the exponential-type potential models. To deal with the centrifugal term, we have to apply the proper approximation scheme which is called the Greene-Aldrich approximation [1]. On determining the N–dimensional energy spectrum and wave functions for the hyperbolic potential function, we utilize the asymptotic iteration method (AIM) in this work. Asymptotic iteration approach has been developed by Ciftci et al.[2–4] to solve the second-order homogeneous linear differential equation.

Approximate lN−1–state solutions of the N–dimensional non-relativistic and relativistic equations are one of the most prior- ity problems in the ﬁeld of quantum mechanics. Hyperspherical coordinates provide a suitable framework for understanding the problem. It is also possible to achieve systematically 2D and 3D results from those obtained in N–dimensions. It should be noted that Louck and Shaffer[5], Louck[6, 7], and Chatterjee[8] have comprehensively examined the mathematical tools for generalization of the orbital angular momentum in hyperspherical coordinates.

In hyperspherical coordinates, hyperbolic potential function[9] can be expressed as

2 2 V(r) = V1tanh (αr) −V2sech (αr) (1) in which r ∈ [0, ∞], the parameters V1 and V2 deﬁne the property of the potential well the range of potential, is the potential range parameter. Because of describing the vibrations, interactions of diatomic and polyatomic molecules, hyperbolic potential is one of the convenient empirical mathematical models. The work of hyperbolic potential function has important applications in various ﬁelds of quantum chemistry and molecular physics. For this reason, this potential has been studied by many researchers[10–13]. For example, in spherical coordinates, partial-wave[12], supersymmetric quantum mechanics[13] are just a few methods in the literature. However, the non-relativistic solutions of the hyperbolic potential function in approximate analytic form have not been investigated so far in hyperspherical coordinates. In this work, we attempt to probe the N-dimensional bound state solutions for this potential via the asymptotic iteration approach.

This paper is organized as follows: in the following section, we review brieﬂy asymptotic iteration approach. N-dimensional solutions of the Schrödinger particle interacting with the hyperbolic potential function within the framework of Greene- Aldrich approximation and AIM are introduced in section 3. Afterall, concluding remarks are given in section 4.

∗ Corresponding author: ayselozﬁ[email protected] 63 /

2. A General View of Asymptotic Iteration Approach

In this section, we present brieﬂy the asymptotic iteration method which is the procedure for solving the Schrödinger equation in hyperspherical coordinates. A detailed description of this method is given in Refs.[2–4]. AIM can be used to solve the second-order differential equation of the form

00 0 y (r) = λ0 (r)y (r) + s0 (r)y(r) (2) in which λ0(r) and s0(r) functions in C∞(a,b) are sufﬁciently differentiable. The general solution of Equation (2) can be obtained in the following form

r 0 0 h r r0 0i y(r) = exp(−∫ α(r )dr ) C2 +C1∫ exp ∫ [λ0(τ) + 2α(τ)]dτ dr (3)

For sufficiently large k, s (r) s (r) k = k−1 = α(r) (4) λk(r) λk−1(r) in which 0 λk (r) = λk−1 (r) + sk−1 (r) + λ0(r)λk−1 (r) 0 sk (r) = sk−1 (r) + s0 (r)λk−1 (r) (5) If the eigenvalue problem has exact analytical solutions, the termination condition Equation (4), or equivalently δk (r) = λk (r)sk−1 (r) − λk−1 (r)sk (r) = 0 (6) produces, at each iteration, an expression that is independent of r. It is noted that k displays the iteration number. Physically meaningful solution of Equation (2) is provided by the first term of Equation (3) not the second term, so we can use the first term as the wavefunction generator  0  Z r sk r 0 y(r) = C2exp− 0 dr  (7) λk r in which C2 denotes the integrant constant which can be determined by normalization.

3. N-dimensional Solutions for Hyperbolic Potential Function with Centrifugal Term

We briefly outline of seperating variables of the Schrödinger equation in hyperspherical coordinates. The motion of a particle in the spherically symmetric potential field in N-dimensions is described in the context of non-relativistic theory as follows 2 } 2 −→ −→ − ∇ +VN (r) ψ ( r ) = Eψ ( r ) (8) 2µ N where E and µ are non-relativistic energy and reduced mass, VN (r) is the N-dimensional central potential, } is Planck constant. The N-dimensional Laplacian is defined with respect to Cartesian coordinates x1,x2,x3 ...xN as N 2 2 ∂ ∇N = ∑ 2 (9) j=1 ∂x j In accordance with the works of Louck [6] and Chatterjee [7], we introduce the hyperspherical coordinates in N- dimensional space as follows x1 = rcosθ1 sinθ2 sinθ3 ...sinθN−1

x2 = rsinθ1 sinθ2 sinθ3 ...sinθN−1

x3 = rcosθ2 sinθ3 sinθ4 ...sinθN−1

x4 = rcosθ3 sinθ4 sinθ5 ...sinθN−1 . .

x j = rcosθ j−1 sinθ j sinθ j+1 ...sinθN−1 ,3 ≤ j ≤ N − 1 (10)

xN = rcosθN−1 for N=3,4,5. . . where the range of variable is 0 ≤ r ≤ ∞ , 0 ≤ θ1 ≤ 2π , 0 ≤ θ j ≤ π for j = 2, 3,...N − 1 and r is the radius of an N-dimensional sphere. In connection with the hyperspherical coordinates, the Laplacian has the form

64 /

N−2 2 1 ∂ N−1 ∂ 1 1 1 ∂ k−1 ∂ ∇N = N−1 r + 2 ∑ 2 2 2 k−1 sin θk r ∂r ∂r r k=1 sin θk+1sin θk+2 ...sin θN−1 sin θk ∂k ∂θk 1 1 ∂ N−2 ∂ + 2 N−2 sin θN−1 (11) r sin θN−1 ∂θN−1 ∂θN−1 By taking the hyperspherical total wavefunction as

N−1 − 2 ψ (r,θ1,θ2,...,θN−1) = r R(r)YlN−1,lN−2,...l2,l1 (θ1,θ2,...θN−1) (12) in which R(r) and YlN−1,lN−2,...l2,l1 (θ1,θ2,...θN−1) are the radial wavefunction, the generalized spherical harmonics, respectively and employing the way of separating variables, we obtain the non-relativistic wave equation in hyperspherical coordinates for the function R(r) d2R(r) 2µ (N − 1)(N − 3) l (l + N − 2) + (E −V(r)) − − N−1 N−1 R(r) = 0 (13) dr2 }2 4r2 r2 It should be mentioned that the more detailed information about the generalized angular momentum operators can be found in [6, 7]. Substituting the hyperbolic potential function Equation (1) into Equation (13), we have

2 d R(r) 2µ 2µ 2 2 (N − 1)(N − 3) lN−1 (lN−1 + N − 2) + E − V1tanh (αr) −V2sech (αr) − − R(r) = 0 (14) dr2 }2 }2 4r2 r2

Equation (14) cannot be solved analytically for any lN−1-state on account of the centrifugal term. For this reason, in order to solve this equation, we have to use an approximation in the following form

1 4α2e−2αr α2 ≈ = (15) r2 (1 − e−2αr)2 sinh2 (αr)

It is remarked that this approximation becomes valid for small r. If we apply this approximation and transformation y = tanh2 (αr) to Equation (15), we rewrite the hyperradial wave equation in non-relativistic theory

d2R(y) (1 − 3y) dR(y) 1 (κ + 1) + + γ − Λ1y + Λ2 (1 − y) − (1 − y) R(y) = 0 (16) dy2 2y(1 − y) dy y(1 − y)2 4y where we take the following notations µE µV1 µV2 ε = 2 ,Λ1 = 2 2 ,Λ2 = 2 2 (17) 2α2} 2α } 2α } To solve Equation (17) via asymptotic iteration approach, the following acceptable physical wavefunction is proposed R(y) = yρ (1 − y)β f (y) (18) where f (y) is a function to be determined, ρ and β are deﬁned as

(κ + 1) p ρ = ,β = Λ − ε (19) 2 1 If we insert Equation (18) into Equation (16), we obtain the second-order homogeneous linear differential equation as follows " # " 2 # d2 f (y) 2ρ + 2β + 3 y − 2ρ + 1 d f (y) ρ + β + 1 − δ 2 = 2 2 + 4 f (y) (20) dy2 y(1 − y) dy y(1 − y) with r 1 δ = Λ + Λ + (21) 1 2 16 It is noticed that Equation (20) is convenient to AIM solutions. By comparing Equation (20) with Equation (2), we can determine the values of λ0 and s0. With Equation (5), it is then easy to obtain the values of λn(y) and sn(y) in the following forms 2ρ + 2β + 3 y − 2ρ + 1 λ = 2 2 0 y(1 − y) 2 ρ + β + 1 − δ 2 s = 4 0 y(1 − y)

65 /

3 3 1 3 1 1 2 2 2ρ + 2β + 2 2ρ + 2β + 2 y − 2ρ + 2 2ρ + 2β + 2 y − 2ρ + 2 ρ + β + 4 − δ λ1 = − + + y(1 − y) y2 (1 − y) y(1 − y)2 y(1 − y) 2 2ρ + 2β + 3 y − 2ρ + 1 + 2 2 y2(1 − y)2

1 2 3 1 1 2 2 1 2 2 ρ + β + − δ 2 2ρ + 2β + y − 2ρ + ρ + β + 4 − δ ρ + β + 4 − δ 4 2 2 s1 = − + + y2 (1 − y) y(1 − y)2 y2(1 − y)2 . . (22) In order to calculate the radial energy eigenvalues in hyperspherical coordinates, the termination condition given by Equation (4) is used. Hence, these energy eigenvalues are obtained as follows s0 s1 1 = ⇒ β0 = − − ρ ± δ λ0 λ1 4

s1 s2 5 = ⇒ β1 = − − ρ ± δ λ1 λ2 4

s2 s3 9 = ⇒ β2 = − − ρ ± δ λ2 λ3 4 . . (23) Thus, based on Equation (23), we are able to generalize the eigenvalues as follows r 1 1 1 β = − n + − ρ ± δ = − n + − ρ ± Λ + Λ + (24) n 4 4 1 2 16

From Equation (24), we have selected the positif root of βn. This root becomes valid for β0, β1, β2 etc. If we substitute the used abbrevations into Equation (24), we obtain the rovibrational energy spectrum for the hyperbolic potential function in hyperspherical coordinates

2 2 2 " r # } α N 2µ 1 E = V1 − 2n + lN−1 + − (V1 +V2) + (25) 2µ 2 }2α2 4

In spherical coordinates, this energy spectrum coincides with the ones obtained previously in [13, 14]. We can also construct the corresponding wavefunctions of the radial Schrödinger equation with this potential Equation (1) by using the wavefunction generator given by Equation (7).

1 f (y) =C = C F 0, 2δ, 2ρ + ,y 0 2 22 1 2 1 f (y) =C 2ρ + − (2δ − 1)y 1 2 2 1 1 =C 2ρ + F −1, 2δ − 1, 2ρ + ,y 2 2 2 1 2 2 3 2 2 f2 (y) =C2 4ρ + 4ρ + − (8ρ − 6δ − 8δρ + 6)y + 4δ − 6δ + 2 y 4 (26) 1 3 1 =C 2ρ + 2ρ + F −2, 2δ − 2, 2ρ + ,y 2 2 2 2 1 2 3 2 3 f3 (y) =C2 64δ + 176δ − 192δ − 48 y + 480δρ − 288ρ + 600δ − 240δ 2 − 192δ 2ρ − 360y2 −384δρ + 192δρ2 − 576ρ + 180δ − 288ρ2 − 270y − 64ρ3 − 144ρ2 − 92ρ − 15 1 3 5 1 =C 2ρ + 2ρ + 2ρ + F −3, 2δ − 3, 2ρ + ,y 2 2 2 2 2 1 2

Thus, the wavefunction f (y) can be written as follows

1 1 fn (y) = C2 2ρ + 2F1 −n, 2δ − n, 2ρ + ,y (27) 2 n 2 66 /

Based on Equation (21), we are able to obtain the following form 1 1 1 fn (y) = C2 2ρ + 2F1 −n, 2ρ + 2β + + n, 2ρ + ,y (28) 2 n 2 2 in which 2F1 represents the Gauss hypergeometric function being deﬁned as n k (−n)k(b)kx 2F1 (−n, b,c,x) = ∑ k=0 (c)kk! Γ(α+k) the Pochhammer symbol (α)k is deﬁned by (α)0 = 1 and (α)k = α (α + 1)(α + 2)...(α + k − 1) = Γ(α) for k = 1,2,3... Hence, the unnormalized hyperradial wavefunction for the hyperbolic potential function is β ρ 1 1 R(y) = C2 (1 − y) y 2ρ + 2F1 −n, 2ρ + 2β + + n, 2ρ + 1,y (29) 2 n 2 By substituting the y = tanh2 (αr) into Equation (29), we can write down 2 β 2 ρ 1 R(r) = C2 (sech (αr)) tanh (αr) 2ρ + 2 n 1 × F −n, 2ρ + 2β + + n, 2ρ + 1, tanh2 (αr) (30) 2 1 2 where C2 denotes the integration constant.

4. Discussion and Conclusion

In this work, we have studied the N-dimensional Schrödinger equation with the hyperbolic potential function model for any lN−1-state from the point of view the asymptotic iteration approach. To overcome the centrifugal term, we have applied a proper approximation scheme which is called the Greene-Aldrich approximation. Thus, the approximate analytic expressions for the energy eigenvalues and the corresponding wavefunctions of quantum system with consideration potential have been built in hyperspherical coordinates. It should be emphasized that asymptotic iteration formalism provides a systematic and elegant method for the problems of solving the N-dimensional non-relativistic wave equation. We remark that N-dimensional results allow us to the comprehension of a general treatment of this problem. Moreover, the energy equation in spherical coordinates can also obtain directly from the N-dimensional results. Indeed, we show that the approximate analytical solutions obtained in this work match with the previous works in literature.

References

[1] Greene, R. L., Aldrich, C. 1976. Variational wave functions for a screened Coulomb potential. Phys. Rev. A, 14, 2363-2366. [2] Ciftci, H., Hall, R.L., Saad, N. 2003. Asymptotic iteration method for eigenvalue problems. J. Phys. A Math. Gen., 36, 11807-11816. [3] Ciftci, H., Hall, R.L., Saad, N. 2005. Construction of exact solutions to eigenvalue problems by the asymptotic iteration method. J. Phys. A Math. Gen., 38, 1147-1155. [4] Ciftci, H., Hall, R.L., Saad, N. 2005. Iterative solutions to the Dirac equation. Phys.Rev.A,72, 022101-7. [5] Louck, J.D., Shaffer, W.H. 1960. Generalized orbital angular momentum and the n-fold degenerate quantum mechanical oscillator: Part I the twofold degenerate oscillator. J. Mol. Spect., 4, 285-297. [6] Louck, J.D. 1960. Generalized orbital angular momentum and the n-fold degenerate quantum mechanical oscillator: Part II the n-fold degenerate oscillator. J. Mol. Spect., 4, 298-333. [7] Louck, J.D. 1960. Generalized orbital angular momentum and the n-fold degenerate quantum mechanical oscillator: Part III radial integrals. J. Mol. Spect., 4, 334-341. [8] Chatterjee, A. 1990. Large-N expansions in quantum mechanics, atomic physics and some O(N) invariant systems. Phys. Rep., 186, 249-370. [9] Büyükkilic, F., Egrifes, E., Demirhan, D. 1997. Solution of the Schrödinger equation for two different molecular potentials by the Nikiforov–Uvarov method. Theor. Chem. Acc., 98, 192-196. [10] Yang, Q.B. 2003. Deformed symmetrical double-well potential. Acta Photon. Sin., 32, 882-884. [11] Arai, A. 1991. Exactly solvable supersymmetric quantum mechanics. J. Math. Anal. Appl., 158, 63-79. [12] Wei, G.F., Chen, W.L., Dong, S.H. 2014. The arbitrary l continuum states of the hyperbolic molecular potential. Phys. Lett. A, 378, 2367-2370. 67 /

[13] Wei, G.F., Chen, W.L., 2014. Arbitrary l-wave bound states of the Schrödinger equation for the hyperbolical molecular potential. Int. J. Quantum Chem. 114, 1602-1606. [14] Durmus, A. 2018. Approximate treatment of the Dirac equation with hyperbolic potential function. Few-Body Syst., 59, 1-13.

68 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

Almost Lacunary p-bounded Variation and Matrix Transformations

RabIa˙ SAVAS¸1 1Department of Mathematics, Sakarya University, Sakarya, Turkey

Keywords Abstract: In this article, we present a topological result by considering the generalization Lacunary sequences, of lacunary almost p-bounded variation sequence spaces. Additionally, some matrix almost bounded variation, transformations have been discussed. matrix transformation,

1. Introduction

Let ω be the set of all sequences real and complex valued and `∞, c and c0 respectively be the Banach spaces of bounded, convergence and null sequences x = (xn) normed by ||x|| = supn |xn|. Let D be the shift operator on ω.

∞ 2 ∞ Dx = {xn}n=1 and D x = {xn}3

k and so on. It is evidence that D is bounded linear operator on `∞ onto itself and D = 1 for every k. It may be recalled that Banach limit L is a non-negative linear functional on `∞ such that L is invariant under the shift operator, that is L(Dx) = L(x) ∀x ∈ `∞ and that L(e) = 1 where e = (1,1,1,...). (see [1]). A sequence x ∈ `∞ is called almost convergence (see [3]) if all Banach limits of x coincide. Let cbdenote the set of all almost convergent sequences. Lorentz [3] prove that n o c = x : limdmn (x) exists uniformly in n b m where x + x + ··· + x d (x) = n n+1 n+m . mn m + 1 ∞ By a lacunary sequence θ = {kr}r=0 where k0 = 0, we shall mean an increasing sequence of non-negative integers with kr kr − kr−1 → ∞ as r → ∞. The intervals determined by θ are denoted by Ir = (kr−1,kr] and hr = kr − kr−1. The ratio kr−1 will usually be denoted by qr (see [12]). Recently, Das and Mishra [8] deﬁned Mθ , the set of almost lacunary convergent sequences as follows: ( ) 1 Mθ = x : lim (xk+i − l) = 0 uniformly in i and for some l . r→∞ h ∑ r k∈Ir

∧ In [10] it is shown that the lacunary almost convergence is related to the space BVθ , the set of all sequences of the lacunary ∧ almost bounded variation, in the same manner as the almost convergence is related to the space BV, the set of all sequences ∧ ∧ of almost bounded variation, was introduced by Nanda and Nayak [16]. Also the space BVθ extended to BVθ (p), the set ∧ ∧ of all sequences of lacunary almost p−bounded variation, just as BV was extended to BV(p), the set of all sequences of almost p−bounded variation (see [11]). In 2011, Karakaya and Savas [20] presented the following sequence spaces as following:

∧ pr BVθ (p) = x : ∑|ϕrn (x)| converges uniformly in n r ∧ ∧ pr BVθ (p) = x : sup∑|ϕrn (x)| < ∞ , n r

∗ 69 / where h 1 r ϕr,n (x) = ϕr,n = ∑ u xkr−1+u+1+n − xkr−1+u+n . hr (hr + 1) u=1 ∧ ∧ ∧ ∧ ∧ ∧ It is clear that BVθ (p) = BVθ , BVθ (p) = BVθ if pr = 1 for all r ∈ N. The main goal of this paper is to examine some inclusions results by considering above sequence spaces and also is to present some matrix transformations.

2. Main Results

In this section we will present some interesting results. ∧ ∧ Let E ⊂ BVθ (p) be a compact subset. Then the following hold: ∧ ∧ (i) If fλ : BVθ (p) → C is given fλ (x) = ∑λnxn n ∧ ∧ for all x ∈ BVθ (p), then fλ (E) is compact for all λ = (λn) such that

p0/M ∑|λn| < ∞. n

(ii) Given ε > 0, there is some j0 = j0 (ε) such that for all n,

1/M ∞ ! pr ∑ |ϕr,n(x)| < ε for all x ∈ E, j ≥ j0. r= j+1

∧ In particular, these results hold for a compact subset of BVθ (p). ∧ ∧ Proof. (i) For x ∈ BVθ (p), we obtain the following

p0/M p0/M

∑λnxn = ∑λnϕ0,n (x) n n p0/M p0/M ≤ ∑|λn| |ϕ0,n (x)| n 1/M p0/M pr ≤ ∑|λn| ∑|ϕr,n (x)| n r p0/M ≤ g(x)∑|λn| < ∞. n

∧ ∧ Therefore, fλ is a well-deﬁned function. Also, if x,y ∈ BVθ (p), then

p0/M p0/M | fλ (x) − fλ (y)| = ∑λn (xn − yn) n p0/M ≤ g(x − y)∑|λn| . n

Hence, fλ is continuous and therefore fλ (E) is compact. (ii) Let ε > 0 be given and for every x ∈ E, let

 ∧  ε  ∧ ε  U x, = y ∈ BVθ (p) : g(x − y) < . 2  2 

ε 1 2 N Then U x, 2 is an open cover for E. Since E is compact; there exist x ,x ,··· ,x ∈ E such that N ε E ⊂ ∪ U xi, . i=1 2 70 /

For each i, there is ji such that 1/M ∞ ! i pr ε ∑ ϕr,n(x ) < r= j+1 2 for j ≥ ji. Let j0 = max{ j1, j2,··· , jN}. Then 1/M ∞ ! i pr ε ∑ ϕr,n(x ) < r= j+1 2 for j ≥ j0. Now for any x ∈ E, there is i0 (1 ≤ i0 ≤ N) such that 1/M ∞ ! i pr ε ∑ ϕr,n(x − x 0 ) < . r= j+1 2

Now let j ≥ j0. Then 1/M 1/M ∞ ! ∞ ! pr i pr ∑ |ϕr,n(x)| ≤ ∑ ϕr,n(x − x 0 ) r= j+1 r= j+1 1/M ∞ ! i pr + ∑ ϕr,n(x 0 ) r= j+1 1/M ∞ 1/M ∞ ! i pr i pr ≤ ∑ ϕr,n(x − x 0 ) + ∑ ϕr,n(x 0 ) r r= j+1 ε ε ≤ + = ε. 2 2 This completes the proof. ∧ ∧ ∧ ∧ Suppose that for all r, qr ≤ pr. Then BVθ (q) ⊂ BVθ (p). ∧ ∧ Proof. Suppose that a ∈ BVθ (q). Then there is an integer M such that for all n ∞ qr ∑ |ϕr,n| ≤ 1. r=M

pr qr pr Hence, for r ≥ M and all n, |ϕr,n| ≤ 1, so that |ϕr,n| ≤ |ϕr,n| . The uniform convergence of ∑|ϕr,n| hence follows from qr |ϕr,n| . Let A = an,k be an infinite matrix of complex numbers. Let X and Y be any two subsets of the space of all sequences of complex numbers. We write Ax = (An (x)) if An (x) = ∑an,kxk converges for each n. If x ∈ X implies that Ax ∈ Y, then we k say that A defines a matrix transformation from X into Y and we denote it by A : X → Y. By (X,Y) we mean the class of matrices A such that A : X → Y. If in X and Y there is some notion of limit or sum, then we write (X,Y,P) to denote the subset of (X,Y) which preserves the limit or sum. We note that if Ax is defined, then for all integer r,n ≥ 0 ∞ ϕr,n(Ax) = ∑a(n,k,r)xk k=0 where

 h  1 r h (h +1) ∑ u(akr−1+u+1+n,k − akr−1+u+n,k), r ≥ 1; b(n,k,r) = r r u=0  a(n,k), r = 0.

The notation a(n,k) denotes the element an,k of the matrix A. We now have  ∧  ∧ Let 1 ≤ p < ∞. Then A ∈ `∞,BVθ (p) if and only if

!p ∑ ∑|b(n,k,r)| < ∞ (1) r k 71 / uniformly in n. Proof. Sufﬁciency. Suppose that (1) holds and a ∈ `∞. Then !p p ∑|ϕr,n (Aa)| ≤ ∑ ∑|b(n,k,r)ak| r r k !p p ≤ ||a||∞ ∑ ∑|b(n,k,r)| . r k

 ∧  ∧ p Therefore, ∑|ϕr,n (Aa)| converges uniformly in n and so A ∈ `∞,BVθ (p). m  ∧  ∧ Necessity. Suppose that A ∈ `∞,BVθ (p) and a ∈ `∞. Hence,

1/p p qn (a) = ∑|ϕr,n (Aa)| r exists uniformly in n. Now {qn}n is a sequence of continuous seminorms on `∞ and supqn (a) < ∞. From Banach- Steinhaus n theorem, we obtain !p ∑ ∑|b(n,k,r)| < ∞. r k

 ∧  ∧ p Let p ≥ 1. Then A ∈ l,BVθ (p) if and only if ∑|b(n,k,r)| is bounded for all n,k. r This can be proved as in Maddox ([19], 167).

References

[1] S. Banach, Theorie des operations lineaires, Warszawa, 1932. [2] G. G. Lorentz, A contribution to the theory of divergent series, Acta Math. 80 (1948), 167-190. [3] A. R. Freedman, J. J. Sember and M. Rapheal, Some Cesáro-type summability spaces, Proc. Lond. Math. Soc. 37 (3) (1973), 508-520. [4] G. Das and S. K. Mishra, Banach limits and lacunary strong almost convergence, J. Orissa Math. Soc., 2 (2) (1983), 61-70. [5] E. Sava¸sand V. Karakaya, Some new sequence spaces deﬁned by lacunary sequences, Math. Slovaca, 57 (4) (2007), 1-7. [6] E. Sava¸s, Some sequence spaces and almost convergence, An. Univ. Timi¸soaraSer. Math. Inform. (2-3) (1992), 303-309. [7] S. Nanda and K. C. Nayak, Some new sequence spaces, Indian J. Pure. Appl. Math. 9 (8) (1978), 836-846. [8] I. J. Maddox, Elements of Functional Analysis, second ed., Cambridge Univ. Press. Cambridge, UK, 1988. [9] V. Karakaya and E. Savas, On almost p−bounded variation of lacunary sequences, Computers and Mathematics with Applications, 61 (2011), 1502-1506.

72 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

A new sequence spaces deﬁned by bounded variation in n- normed spaces

E. Savas¸∗ Department of Mathematics, U¸sak University, U¸sak- Turkey ,[email protected]

Keywords Abstract: In this paper we introduce a new sequence spaces by using almost lacunary Sequence space, bounded variation and Orlicz function in n- normed spaces. Also various algebraic and Orlicz function, topological properties and certain inclusion relations involving this space have been paranorm discussed.

1. Introduction

Let w denote the set of all real and complex sequences x = (xk). By l∞ and c, we denote the Banach spaces of bounded and convergent sequences x = (xk) normed by ||x|| = supk |xk|, respectively. A linear functional L on l∞ is said to be a Banach limit [1] if it has the following properties:

1. L(x) ≥ 0 if n ≥ 0 (i.e. xn ≥ 0 for all n), 2. L(e) = 1 where e = (1,1,...),

3. L(Dx) = L(x), where D denotes the sift operator on `∞, that is D : `∞ → `∞ deﬁned by D(x) = D(xn) = {xn+1}.

Let B be the set of all Banach limits on l∞. A sequence x ∈ `∞ is said to be almost convergent if all Banach limits of x coincide. Let cˆ denote the space of the almost convergent sequences . Consider the sequences of bounded linear transformations dmn (x) : l∞ → l∞ deﬁned by

m 1 i dmn (x) = ∑ D xn m + 1 i=0 with D0 = 1. It is evident that

0 d0n (x) = xn = D xn (1)

Lorentz [12] proved that n o cˆ = x : limdmn (x) exists uniformly in n m Now deﬁne

−1 d−1n (x) = xn−1 = D xn (2) and then write for m,n ≥ 0,

tmn (x) = dmn (x) − dm−1,n (x) (3)

So that (1), (2) and (3), we write

0 −1 t0n (x) = D xn − D xn = xn − xn−1. (4)

When m ≥ 1 a straightforward calculation shows that

1 m tmn (x) = ∑ v(xn+v − xn+v−1) m(m + 1) v=1

∗ 73 /

Nanda and Nayak [13] deﬁne the following sequence spaces. BVc = x : ∑|tmn(x)| converges uniformly in n m and BVc = x : sup∑|tmn(x)| < ∞ . n m

A lacunary sequence is an increasing integer sequence θ = {kr}r∈N∪{0} such that k0 = 0 and hr = kr −kr−1 → ∞, as r → ∞. kr Let Ir = (kr−1,kr] and qr = . kr−1 Das and Mishra [3] observed almost lacunary convergence as follows: ( ) 1 Mθ = x : there exists L such that uniformly in i ≥ 0 lim (xk+i − L) = 0 . r→∞ h ∑ r k∈Ir

It is natural question to expect that lacunary almost convergence must be related to some concept BVdθ in the same vein as almost convergence is related to the concept of BVc . A sequence in BVdθ will mean a sequence of lacunary almost bounded variation. Sava¸sand Karakaya [25] deﬁned the following sequence spaces by using almost lacunary convergence, BVc θ = x : ∑|ϕrn (x)| converges uniformly in n r and c BVc θ = x : sup∑|ϕrn (x)| < ∞ . n r where h 1 r ϕrn (x) = ϕrn = ∑ u xkr−1+u+1+n − xkr−1+u+n . hr (hr + 1) u=1 Furthermore the following theorem was proved in [25]:

Theorem 1.1. BVc θ ⊂ BVc θ for every θ.

But we do not know whether BVc θ ⊂ BVc θ , that is BVc θ = BVc θ . It is still open problem.

2. New Sequence Spaces

The theory of 2-normed spaces was first introduced by Ga¨hler[4] in 1964. Later on it was extended to n-normed spaces by Misiak [14]. Since then many mathematicians have worked in this field and obtained many interesting results for instance n see Gunawan([5], [6]), Gunawan and Mashadi[7] and so on. Let n ∈ N and X be a linear metric space over the field K of real or complex numbers of dimension d, where d ≥ n ≥ 2. Definition 2.1. [18]. A real valued function ||·,....,·||on Xn satisfying the following four conditions:

1. ||x1,x2,...,xn|| = 0 if and only if x1,x2,...xn are linearly dependent;

2. ||x1,x2,...xn|| is invariant under permutation;

3. ||αx1,x2,...,xn|| = |α| ||x1,x2,...,xn|| for any α ∈ K; and 0 0 4. ||x + x ,x2,...,xn|| ≤ ||x,x2,...,xn|| +||x ,x2,...,xn||.

is called an n-norm on X and the pair (X,k·,...,·k) is called an n-normed space over the ﬁeld K. The standard n-norm on X, is deﬁned as:

1 2 x1,x1 ... x1,xn

.....

kx1,x2,...xnkE = ..... ,

.....

x1,x1 ... x1,x1

74 /

n .,. denotes the inner product on X. If X = R , then this n-norm is exactly the same as the Euclidean n-norm 1 2 kx1,x2,...,xnkE mentioned earlier. For n = 1 this n-norm is the usual norm kxk = x1,x1 . A sequence (xk) in an n-normed space (X,k·,...,·k) is said to converge to same L ∈ K if lim kxk − L,z1,...zn−1k = 0 for k→∞ every z1,z2,...zn−1 ∈ X.

Recall in [10] that an Orlicz function M : [0,∞) → [0,∞) is continuous, convex, non-decreasing function such that M(0) = 0 and M(x) > 0 for x > 0, and M(x) → ∞ as x → ∞. Subsequently Orlicz function was used to deﬁne sequence spaces by Parashar and Choudhary [15] and others ([11], [26], [27] ). If convexity of Orlicz function M is replaced by M(x + y) ≤ M(x) + M(y) then this function is called Modulus function, which was presented and discussed by Ruckle [17]. It should be mentioned that notable works involving Orlicz function, modulus function were done in ([2], [17], [19] -[24]). Let E be a sequence space. Then E is called solid (or normal), if (αnxn) ∈ E, whenever (xn) ∈ E for all sequences of scalar (αn) with |αn| ≤ 1 for all k ∈ N. Lemma 2.2. [8]. A sequence space E is solid implies E is monotone.

Let M be an Orlicz function, (X,k.,...,.k) be a n-normed space and p = (pr) be any sequence of strictly positive real numbers. Now we deﬁne the following sequence spaces,

∞ pr φr,n (x) BVc θ (M, p,k.,...,.k) = {x = (xk) : ∑ M ,z1,z2,...,zn−2,zn−1 < ∞ uniformly in n r=1 ρ and for some ρ > 0, and each z1,z2,...,zn−2,zn−1 ∈ X}.

Theorem 2.3. The sequence space BVˆ θ (M, p,k.,...,.k) is a linear space over the ﬁeld C of complex numbers.

Proof. Let x,y ∈ BVc θ (M, p,k.,...,.k) and α,β ∈ C. Then there exist positive numbers ρ1 and ρ2 such that ∞ pr φr,n (x) ∑ M ,z1,z2,...,zn−2,zn−1 < ∞ r=1 ρ1 and ∞ pr φr,n (x) ∑ M ,z1,z2,...,zn−2,zn−1 < ∞ r=1 ρ2 uniformly in n. Deﬁne ρ3 = max(2|α|ρ1,2|β|ρ2). Since M is non-decreasing and convex we have

∞ pr αφr,n (x) + βφr,n (y) ∑ M ,z1,z2,...,zn−2,zn−1 r=1 ρ3 ∞ pr αφr,n (x) βφr,n (y) ≤ ∑ M ,z1,z2,...,zn−2,zn−1 + ,z1,z2,...,zn−2,zn−1 r=1 ρ3 ρ3 ∞ pr 1 αφr,n (x) ≤ ∑ M ,z1,z2,...,zn−2,zn−1 r=1 2 ρ1 pr βφr,n (y) + M ,z1,z2,...,zn−2,zn−1 < ∞ ρ2 uniformly in n. This proves that BVc θ (M, p,k.,...,.k) is linear space over the ﬁeld C of complex numbers.

Theorem 2.4. For any Orlicz function M and a bounded sequence p = (pr) of strictly positive real numbers, BVc θ (M, p,k.,...,.k) is a paranormed space with  !1/K   ∞ (x) pr  pn/k φr,n g(x) = inf ρ : M ,z1,z2,...,zn−2,zn−1 ≤ 1, n≥ ∑ 1  r=1 ρ  uniformly in n, where K = max(1,sup pr). The proof of the theorem is straightforward. So we omit it. M(u/ρ) Theorem 2.5. For Orlicz function M, if limu (u/ρ) > 0 for some ρ > 0 then BVc θ (M, p,k.,...,.k) ⊆ BVc θ (p,k.,...,.k).

M(u/ρ) Proof. If limu (u/ρ) > 0 then there exists a number α > 0 such that M (u/ρ) ≥ α (u/ρ) for all u > 0 and some ρ > 0. Let x ∈ BVc θ (M, p,k,.,k) so that ∞ pr φr,n (x) ∑ M ,z1,z2,...,zn−2,zn−1 < ∞ r=1 ρ 75 / for some ρ > 0. Then ∞ pr φr,n (x) ∑ M ,z1,z2,...,zn−2,zn−1 ≥ r=1 ρ H ! ∞ α pr max 1, ∑ [(kφr,n (x),z1,z2,...,zn−2,zn−1k)] . ρ r=1

Hence x ∈ BVc θ (p,k.,...,.k).

Theorem 2.6. If p = (pr) and t = (tr) are bounded sequences of positive real numbers with 0 < pr ≤ tr < ∞ for each r ∈ N,then for any Orlicz function M

BVc θ (M, p,k.,...,.k) ⊆ BVc θ (M,t,k.,...,.k).

h ipr Proof. x ∈ BV (M, p,k,.,k). > ∞ M φr,n(x) ,z ,z ,...,z ,z < Suppose that c θ Then there exists some ρ 0 such that ∑r=1 ρ 1 2 n−2 n−1 ∞. This implies that pr φi,n (x) M ,z1,z2,...,zn−2,zn−1 ≤ 1 ρ for sufﬁciently large values of i, say that i ≥ r0 for some ﬁxed r0 ∈ N. Since M is non-decreasing, we have

∞ tr φr,n (x) M ,z1,z2,...,zn−2,zn−1 ≤ ∑ ρ r=r0

∞ pr φr,n (x) M ,z1,z2,...,zn−2,zn−1 < ∞. ∑ ρ r=r0

Hence x ∈ BVc θ (M,t,k.,...,.k).

Theorem 2.7. The sequence space BVˆ θ (M, p,k,.,k) is solid.

Proof. Let x ∈ BVc θ (M, p,k.,...,.k). This implies that

∞ pr φr,n (x) ∑ M ,z1,z2,...,zn−2,zn−1 < ∞. r=1 ρ

Let α = (αr) be sequence of scalars such that |αr| ≤ 1 for all r ∈ N. Then the result follows from the following inequality

∞ pr αrφr,n (x) ∑ M z1,z2,...,zn−2,zn−1 ≤ r=1 ρ

∞ pr φr,n (x) ∑ M ,z1,z2,...,zn−2,zn−1 < ∞. r=1 ρ

Hence αx ∈ BVˆ θ (M, p,k.,...,.k) for all sequence of scalar (αr) with |αr| ≤ 1 for all r ∈ N, whenever x ∈ BVc θ (M,t,k.,...,.k).

From Theorem 2.7 and Lemma we have:

Corollary 2.8. The sequence space BVc θ (M, p,k,.,k) is monotone.

References

[1] S. Banach, Theorie des Operations linearies, Warszawa,1932. [2] R. Colak, M. Et and E. Malkowsky, Strongly almost (w,λ)-summable sequences deﬁned by Orlicz functions, Hokkaido Math. J., 34(2) (2005), 265-276. [3] G. Das and S. K. Mishra, Banach limits and lacunary strong almost convergence, J. Orissa Math. Soc. 2(2)(1983), 61-70. [4] S. Gahler, Linear 2-normietre Rame, Math. Nachr., 28,(1965) 1-43. [5] H. Gunawan, The space of p-summable sequence and its natural n-norm, Bull. Aust. Math. Soc., 64,(2001)137-147. [6] H. Gunawan, n-inner product, n-norms and the Cauchy-Schwartz inequality, Sci.Math.Jpn., 5,(2001) 47-54. [7] H. Gunawan and Mashadi, On n-normed spaces, Int. J. Math. Sci.,27,(2001)631-639. 76 /

[8] P. K. Kamthan, M. Gupta, Sequence spaces and Series, (Marcel Dekker, 1980). [9] V. Karakaya and E. Sava¸s, On almost p-bounded variation of lacunary sequences , Computer and Math. with Appl., 61(2011), 1502-1506. [10] M. A. Krasnoselskii, Y. B. Rutitsky, Converx functions and Orlicz functions, (Groningen, Netharland 1961). [11] J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 101 (1971) 379-390. [12] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta. Math. 80(1948),167-190. [13] S. Nanda and K.C. Nayak, Some new sequence spaces , Indian J. Pure. Appl. Math. 9 (8)(1978) 836-846. [14] A. Misiak, n- inner product spaces, Math. Nachr., 140 (1989), 299 − 319. [15] S. D. Parashar, B. Choudhury, Sequence space defined by Orlicz functions, Indian J. Pure Appl. Math., 25(14) (1994)419-428. [16] E. Sava¸sand Richard F. Patterson, An Orlicz extension of some new sequence spaces. Rend. Istit. Mat. Univ. Trieste 37 (2005), no. 1-2, 145-154 (2006). [17] W. H. Ruckle, FK-spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25(1973) 973-978. [18] A.¸Sahinerand M. Gurdal, Ideal convergence in n-normed spaces and some new sequence spaces via n-norm, Journal of Fundamental Sciences, 4, (2008)233-244. [19] E. Sava¸sand R. Sava¸s, Some sequence spaces defined by Orlicz functions. Arch. Math. (Brno) 40 (2004), no. 1, 33-40. [20] E. Sava¸s,and R. F. Patterson, Some double lacunary sequence spaces defined by Orlicz functions, Southeast Asian Bull. Math., 35(1)(2011), 103-110. [21] E. Sava¸s, On some new double lacunary sequences spaces via Orlicz function, J. Comput. Anal. Appl., 11(3)(2009), 423-430. [22] E. Sava¸s,and Richard F. Patterson, . Some σ -double sequence spaces defined by Orlicz function. J. Math. Anal. Appl. 324 (2006), no. 1, 525-531. [23] E. Sava¸s, (A,σ) -double sequence spaces defined by Orlicz function and double statistical convergence, Comput. Math. Appl. 55 (2008), no. 6, 1293-1301.

[24] E. Sava¸s,E.; R. F.Patterson, (Aσ)∆ -double sequence spaces via Orlicz functions and double statistical convergence, Iran. J. Sci. Technol. Trans. A Sci. 31 (2007), no. 4, 357-367 [25] E. Sava¸s,and V. Karakaya, Some new sequence spaces deﬁned by lacunary sequences ,Math. Slovaca, 57(2007), 393-399. [26] E. Sava¸s, Some sequence spaces and almost converegnce, Analele Universitatii din Timi¸soaraVol.XXX, No.2-3, (1992) ,303-309. [27] B. C. Tripathy, M. Et and Y. Altin, Generalized Difference Sequence Spaces Deﬁned by Orlicz Function in a Locally Convex Space, J. Anal. Appl., 1(3)(2003), 175 − 192.

77 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

Hankel matrices for blind image deconvolution

Belhaj Skander∗1, Alsulami Abdulrahman2 1University of Jeddah, Faculty of Science, Department of Mathematics, Saudi Arabia 2University of Jeddah, Faculty of Science, Department of Mathematics, Saudi Arabia

Keywords Abstract: This paper introduces a fast algorithm for computing the univariate Greater Hankel matrix, Common Divisor (GCD) of several polynomials (not pairwise) by the use of the gen- Barnett’s method, eralized Hankel matrix. Focuses on the Barnett’s method, the proposed algorithm is Greatest common divisor, able to solve the problem of Blind Image Deconvolution (BID), even in cases where the Blind image deconvolution process of identifying both the true image and the blurring function comes from several blurred images in the same scene. The algorithm has been implemented in Matlab and can reconstruct initial images from several blurred images in a few seconds.

1. Introduction

The Blind Image Deconvolution (BID) problem is the concept of identifying both the true image and the blurring function from the degraded image. We represent a gray level image by a matrix whose dimension is equal to the size of the image and the intensities are whose entries; and a color image can be represented by three matrices which represent respectively the intensities of the red, green and blue. The relation between the original image matrix P and the distorted image matrix F is given by F = P ∗U + N, where U is the blurring matrix, N is the additive noise matrix and P ∗U is the convolution of P and U. In 1999 [8] Pillai et al. thoroughly transformed the problem of Blind Image Deconvolution to compute approximate GCDs of polynomials, when multiple blurred versions of the same scene are available. Their algorithm is based on the Sylvester matrix with a complexity of O(n4) operations. In 2010 [7] Li et al. have introduced an algorithm using the Bézout matrix and Fast Fourier transform (FFT) with a saving complexity which is O(n2 log(n)) operations. In 2018 [1] Belhaj et al. have proposed an algorithm based on the Hankel matrix and the FFT technique with a more reduced complexity about half of the complexity of [7]. m×n Suppose now that we have more than two blurred images of the same original image P ∈ C ,

Pj = P ∗ D j + Nj for j = 1,...,s,

p j(x,y) = p(x,y)d j(x,y) + n j(x,y) for j = 1,...,s. with gcd(d1(x,y),...,ds(x,y)) = 1. To ﬁnd the original image associated to p(x,y), we compute the bivariate gcd(p1(x,y),..., ps(x,y)) by sampling the −2kπi polynomials p1,..., ps in each variable at the DFT (Discrete Fourier transform) points on the unit circle xk = e m , k = −2lπi 0,1,...,m − 1 and yl = e n , l = 0,1,...,n − 1. Thus, by using a generalized Hankel matrix, we recover the GCD of the resulting univariate polynomials. Our new algorithm, based on the calculation of cofactors, is very cheap for polynomials arising from blurred images because the cofactors deg(d j), for j = 1,...,s are very low compared with the approximate GCD p(x,y) of high degree. The paper is organized as follows: Two algorithms for computing the approximate cofactors and the approximate GCD of several univariate polynomials (not pairwise) are introduced in Section 2. Section 3 presents an algorithm for computing the bivariate polynomials GCD for several polynomials. Experiments are done in Section 4. Finally, concluding remarks are given in Section 5 to complete the paper.

∗ Corresponding author: [email protected] 78 /

2. Approximate cofactors and GCD of univariate polynomials

We begin this section by introducing the following classical results about Hankel matrices:

Definition 2.1. H = (hi, j) is a Hankel matrix if hi, j = hi−k, j+k for all positive k, that is, if the entries of H are invariant under a shift along the anti-diagonal direction. A Hankel matrix is completely defined by its first row and last column. n m i i Definition 2.2. Let u(x) = ∑ uix and v(x) = ∑ vix be two polynomials in K[x](K = R or C) of degree n and m, i=0 i=0 respectively, where m ≤ n. The power series expansion of the function R(x) = v(x)/u(x) at the infinity :

∞ −k R(x) = ∑ hkx k=0 deﬁnes the n × n Hankel matrix, H = H(u,v). In addition, every nonsingular Hankel matrix can be viewed as a Hankel matrix associated to two polynomials.

2.1. Approximate univariate gcd via Hankel

Suppose that we have a family of polynomials p(x),q1(x),...,qt (x) in K[x] with m j = deg(q j) ≤ n for every j ∈ {1,...,t}. Then, the Generalized Hankel matrix associated to p(x),q1(x),...,qt (x) for every j ∈ {1,...,t} is deﬁned as follows. H(p,q1) . Hp(q1,...,qt ) = . . H(p,qt )

Then, we can relate the gcd of p(x),q1(x),...,qt (x) to the matrix Hp(q1,...,qt ) as follows. Theorem 2.3. (Barnett’s method via Hankel)

1. The degree of the greatest common divisor of p,q1,...,qt veriﬁes the equality

deg(gcd(p,q1,...,qt )) = n − rank(Hp(q1,...,qt )).

2. If k = rank(Hp(q1,...,qt )), then the ﬁrst k columns of Hp(q1,...,qt ) are linearly independent and therefore, the last n − k columns can be written as a linear combination of the ﬁrst k columns.

3. If C1,...,Cn are the columns of the matrix Hp(q1,...,qt ),

k ( j) Ck+i = ∑ hi Cj, i = 1,...,n − k j=1

and   d p 1 0 0 (k) d p p  h  1 1 0  1  d p p p  (k)  p0 2 = d0 2 1 0  h2  . . . . .  .  ......  .  . . . .  .  dn−k pn−k pn−k−1 pn−k−2 ··· p0 (k) hn−k then n−k n−k−1 G (x) = d0x + d1x + ... + dn−k−1x + dn−k

is a greatest common divisor for the polynomials p(x),q1(x),...,qt (x).

Proof. See [4].

79 /

Next, we present an algorithm for computing the approximate GCD of several polynomials by using Hankel matrices. Algorithm 2.4. Barnett’s algorithm via Hankel Input:P(x),Q1(x),...,Qt (x), deg(P(x)) = n and deg(Qi(x)) = mi, n ≥ mi for 1 ≤ i ≤ t, and a value ε. Output: An approximate gcd of P(x),Q1(x),...,Qt (x).

Step 1: Compute the generalized Hankel matrix Hp(q1,...,qt ).

Step 2: Estimate k the approximate rank of Hp(q1,...,qt ) by Remark 2.7. Step 3: Solve the least-squares system of equations

(cn−k+1,...,cn)X = (c1,...,cn−k)

where ci denote the i–th column of HP(Q1,...,Qt ). Step 4: Compute G(x) of degree n − k by recovering all its coefﬁcients form d p 0 0 d1 p1 p0 1 p0 . = d0 ...... X dn−k pk pk−1 ··· p0

Now, we present our main result which provides the approximate cofactors of several polynomials in terms of the Hankel matrix.

2.2. The computation of cofactors via Hankel

We extend the result of the Proposition 3.3 in [2] for several polynomials. n n−1 If p(x) = p0x + p1y + ... + pn−1x + pn, recall that the Horner basis associated to p(x), denoted by Ho, is deﬁned by n−i Ho = {α1(x),...,αn(x)} with αi(x) = p0x + ... + pn−i−1x + pn−i. Theorem 2.5.

Let t ΦP : ¶n−1 → ¶n−1 (1) u(x) 7→ rem q1(x)u(x), p(x) ,...,rem qt (x)u(x), p(x) ,

n−1 a linear map, deﬁned by the matrix Hp(q1,...,qt ) when considering the monomial basis {1,x,...,x } in the initial vector space and the Horner basis in the ﬁnal one, where ¶n−1 is the vector space of polynomials of degree ≤ n − 1. Then,

(a) The cofactor f (x) ∈ N (ΦP) where N (ΦP) is the null space of the linear map ΦP.

(b) The last vector of a triangular basis in decreasing order of the null space of HP(Q1,...,Qt ) provides the coefﬁcients of f (x) in the monomial basis.

Proof. We denote G (x) the GCD of p,q1,...,qt . Since p(x) = G (x) f (x) then we show rem(qi f , p) = 0, for 1 ≤ i ≤ t which satisfy the relation (a). Observe that the degree of f (x) is equal to the rank of the matrix Hp(q1,...,qt ), equal to k following the notation of the above section. Since the Horner basis is degree-graded in decreasing degree, the coordinates of f (x) will be (0,...,0, f0,..., fk) such that | {z } n−k−1

t t Hp(q1,...,qt )(0,...,0, f0,..., fk) = (0,...,0) . (2)

Hence, if we have n ( j) Cn−k = ∑ hn−kCj, i = 0,...,,n − k − 1 j=n−k+1

(n−k−1) (n) then one solution of the homogeneous linear system (2) is given by ( f0,..., fk) = (1,−hn−k ,...,−hn−k). As a consequence, if we obtain a triangular basis in decreasing order of the null space of HP(Q1,...,Qt ), then the last vector will provide the coefﬁcients of F(x) in the monomial basis. For more details see [4].

80 /

Next, we present an algorithm for computing the approximate cofactors and GCD for many polynomials. Algorithm 2.6. Cofactor’s algorithm via Hankel.

Input: p(x),q1(x),q2(x),...,qt (x) ∈ K[x], with n = deg(p), m j = deg(q j) ≤ n for every j ∈ {1,...,t} and ε > 0 the given tolerance.

Output: G (x) ∈ K[x] : an approximate GCD of p(x),q1(x),q2(x),...,qt (x) and f ∈ K[x] : an approximate cofactor corresponding to p.

Step 1: Form the generalized Hankel matrix Hp(q1,...,qt ).

Step 2: Estimate the rank k of Hp(q1,...,qt ) for the given tolerance ε by computing the SVD of HP(q1,...,qt ) and by applying Remark 2.7.

T Step 3: Let C = V(k + 1 : n,:) ; the rows of C covers the null space of Hp(q1,...,qt ). Step 4: Compute the QR decomposition of C. The last row of the triangular factor R is associated to f (x).

Step 5: Compute G (x) by the use of the fast polynomial division based on FFT to p(x) and f (x).

Remark 2.7. In order to compute the effective rank of the generalized Hankel matrix Hp(q1,...,qt ), we decide on a tolerance ε and count the singular values above it. Thus, according to Theorem 2.3, we can estimate efﬁciently the degree r = deg(gcd(p,q1,...,qt )). Remark 2.8. It is possible to recover all the approximate cofactors f , f1, f2,..., ft ∈ K[x] corresponding to p,q1,q2,...,qt , respectively. Simply by applying fast polynomial divisions in pairs based on FFT.

3. A bivariate polynomial GCD algorithm

We review firstly basic results from [1] and [3]. We let K = R or C. N−1 N Definition 3.1. Let q(z) = q0 + ··· + qN−1z ∈ K[z] and let Q = (q0,q1,...,qN−1) ∈ K . Then, the polynomial q(z) is called the z-transform of Q and denoted by ZT(Q), and Q is called the coefficient vector of q(z).

N M Deﬁnition 3.2. Let F = (F0,F1,...,FN−1) ∈ K and G = (G0,G1,...,GM−1) ∈ K . The convolution of F and G is the N+M− vector of K 1 denoted by F ∗ G given element-wise by

min(i,N−1) (F ∗ G)i = ∑ FjGi− j, 0 ≤ i ≤ N + M − 2. j=max(0,i−M+1)

Lemma 3.3. (1-D z-transform) N M Let F ∈ K and G ∈ K . Then the z-transform of the convolution of F and G is the product of their z-transforms: ZT(F ∗ G) = ZT(F)ZT(G).

Thus, convolution and polynomial multiplication are equivalent. Deﬁnition 3.4. (2-D z-transform) A two-dimensional z-transform maps an M × N matrix Q to the bivariate polynomial

M−1 N−1 T i j q(x,y) = XQY = ∑ ∑ qi, jx y ∈ K[x,y], i=0 j=0 where X = (1,x,x2,...,xM−1), Y = (1,y,y2,...,yN−1). Then the matrix Q can be considered as the coefﬁcient matrix of q(x,y), and q(x,y) is the two-dimensional z-transform of Q.

3.1. The algorithm

m×n Suppose that we have several blurred images of the same original image P ∈ C ,

Pj = P ∗ D j + Nj for j = 1,...,s,

0 0 where the noise matrices are the Njs, the blurring functions are the D js and the z-transforms of the blurring functions are setwise co–prime. Then, by two-dimensional z-transforms, we will have p j(x,y) = p(x,y)d j(x,y) + n j(x,y) for j = 1,...,s with gcd(d1(x,y),...,ds(x,y)) = 1. 81 /

To ﬁnd the original image associated to p(x,y), we compute the bivariate gcd(p1(x,y),..., ps(x,y)) by sampling the −2kπi polynomials p1,..., ps in each variable at the DFT (Discrete Fourier transform) points on the unit circle xk = e m , k = −2lπi 0,1,...,m − 1 and yl = e n , l = 0,1,...,n − 1. The blurring function always has very low degree when polynomials arise from blurred images. Thus, it is much better if we interpolate cofactor deg(d1) by the above technique, and then compute the GCD p(x,y) by the use of the fast polynomial division to p1(x,y). By applying the fast algorithm based on FFT (Fast Fourier Transform), we conclude the division of two bivariate polynomials. Then, the algorithm concerns the computation of the approximate bivariate polynomial GCD is given as follows: Algorithm 3.5. Approximate bivariate polynomial GCD algorithm.

Input: p1(x,y), p2(x,y),..., ps(x,y) ∈ C[x,y] , with degx(p1) = degx(p2) = ··· = degx(ps) = m and degy(p1) = degy(p2) = ··· = degy(ps) = n and ε > 0 the given tolerance. Moreover we suppose that degx(p1(x,yl)) = m for 0 ≤ l ≤ n − 1, and degy(p1(xt ,y)) = n for 0 ≤ t ≤ m − 1.

Output: p(x,y) ∈ C[x,y] : an approximate GCD of p1, p2,..., ps and di(x,y) ∈ C[x,y] : approximate cofactors of pi for i ∈ {1,...,s}.

Step 1: Compute the matrices Hp1 (p2,..., ps)(y1) and Hp1 (p2,..., ps)(x1). Estimate the ranks of both, k1 = rank(Hp1 (p2,..., ps)(y1)) and k2 = rank(Hp1 (p2,..., ps)(x1)) by Remark 2.7. Step 2:

Case 1: k1 + k2 m + n.

−2tπi (m−r+1)×(n−t+1) k +1 Step 2a-1: Apply the Cofactor’s algorithm via Hankel to compute [d1(xt ,yl)] ∈ C , where xt = e 1 , −2lπi k +1 0 ≤ t ≤ k1, yl = e 2 , 0 ≤ l ≤ k2.

Step 2b-1: Apply inverse FFT to [d1(xt ,yl)] to compute d1(x,y).

Step 2c-1: Compute p(x,y) by the use of the fast polynomial division to p1(x,y) and d1(x,y).

Step 2d-1: Compute di(x,y) for i ∈ {2,...,s} by applying fast polynomial divisions in pairs.

Case 2: k1 + k2 m + n does not hold.

−2tπi (m−k +1)×(n−k +1) m−k +1 Step 2a-2 Apply the Barnett’s algorithm via Hankel to compute [p(xt ,yl)] ∈ C 1 2 , where xt = e 1 , −2lπi n−k +1 0 ≤ t ≤ m − k1, yl = e 2 , 0 ≤ l ≤ n − k2.

Step 2b-2 Apply inverse FFT to [p(xt ,yl)] to compute p(x,y).

Step 2c-2: Compute d1(x,y) by the use of the fast polynomial division to p1(x,y) and p(x,y).

Step 2d-2: Compute di(x,y) for i ∈ {2,...,s} by applying fast polynomial divisions in pairs.

Remark 3.6. For s blurred images of size n×n, our algorithm requires O(sn2 log(n)) ﬂops when the degrees of the blurring functions are O(log2(n)), which corresponds to Case 1. 4. Experimental results

Performance tests for the algorithms proposed in this paper were implemented using Matlab 9.4.0.813654 (R2018a) and run along an Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz Laptop with 8.00 GB of RAM and 1.99GHz of processor. The two images employed here come from the literature on image deconvolution [5–8]. Restoration performances are quantitatively measured by the peak signal-to-noise ratio (PSNR) and the Structural SIMilarity (SSIM). Our algorithm can successfully reconstruct original images from blurred images in few seconds.

Example 1. (Reconstructing Gray level Image From Five Blurred Images) In Figure 1, Figure 1.(a) is an image of size 256 × 256. Figure 1.(b) includes five distorted images built by convolving Figure 1.(a) with five 3 × 3 coprime distortion random filters. Figure 1.(c) is the image reconstructed by running Algorithm 3.5 (Case 1) in about 0.68 seconds and restoration performances PSNR = +23.49dB and SSIM = 0.945220435384127. Example 2. (Reconstructing RGB Image From Five Blurred Images) In Figure 2, Figure 2.(a) is an image of size 128×170 scanned from [5]. Figure 2.(b) includes five distorted images built by convolving Figure 2.(a) with five 7 × 7 coprime distortion Gaussian filters. Figure 2.(c) is the image reconstructed by running Algorithm 3.5 (Case 1) in about 1.56 seconds and restoration performances PSNR = +19.15dB and SSIM = 0.998633476903645.

82 /

(a) Original image (b) Five distorted image (c) Reconstructed image Figure 1. Experiment with ﬁve Blurred Gray level images

(a) Original image (b) Five distorted image (c) Reconstructed image Figure 2. Experiment with ﬁve blurred RGB images

5. Conclusion

In this paper, we have designed a specialized algorithm via generalized Hankel based method for computing the GCD of bivariate polynomials of several blurred images in a compact way. Numerical experiments performed with a wide variety of images show the effectiveness of our algorithm in terms of computational time, the peak signal-to-noise ratio (PSNR) and the Structural SIMilarity (SSIM). However, since the Hankel matrix coefficients may increase exponentially, it is natural that our algorithm finds difficulties to recover initial images from distorted ones.

Acknowledgements

This work was funded by the University of Jeddah Saudi Arabia under grant No. UJ-02-012-DR. The authors therefore acknowledge with thanks the University technical and ﬁnancial support.

References

[1] S. Belhaj, H. Ben Kahla, M. Dridi, and M. Moakher. Blind image deconvolution via hankel based method for computing the gcd of polynomials. Mathematics and Computers in Simulation, 144:138-152, 2018. [2] D. Bini and L. Gemignani. Fast parallel computation of the polynomial remainder sequence via Bezout and Hankel matrices. SIAM J. Comput., 24(1):63-77, February 1995. [3] G. M. Diaz-Toca and S. Belhaj. Blind image deconvolution through bezoutians. Journal of Computational and Applied Mathematics, 315:98-106, 2017. [4] G.M. Diaz-Toca and L. Gonzalez-Vega. Barnett’s theorems about the greatest common divisor of several univariate polynomials through Bezout-like matrices. Journal of Symbolic Computation, 34(1):59-81, 2002. [5] D.C. Ghiglia, L. A. Romero, and G. A. Mastin. Systematic approach to two-dimensional blind deconvolution by zero-sheet separation. J. Opt. Soc. Am. A, 10(5):1024-1036, 1993. [6] E. Kaltofen, Z. Yang, and L. Zhi. Approximate greatest common divisors of several polynomials with linearly constrained coefﬁcients and singular polynomials. In Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC ’06, pages 169-176, 2006. [7] Z. Li, Z. Yang, and L. Zhi. Blind image deconvolution via fast approximate GCD. In Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC ’10, pages 155-162, New York, NY, USA, 2010. ACM. [8] S.U. Pillai and B. Liang. Blind image deconvolution using a robust GCD approach. Trans. Img. Proc., 8(2):295-301, 1999.

83 7Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2020) September 25-28 2020 at Bodrum Isis Hotel in Mugla, TURKEY ICRAPAM CONFERANCE PROCEEDING

Asymptotically Lacunary Statistical Equivalent Functions on Time Scales

Bayram Sozbir∗1, Selma Altundag1 1Sakarya University, Science and Art Faculty, Department of Mathematics, Turkey

Keywords Abstract: In this paper, we introduce asymptotically statistical equivalence for non- Asymptotical equivalent negative two delta measurable real-valued functions deﬁned on time scales, and also deﬁne functions, the concepts of asymptotically lacunary statistical equivalence and strong asymptotically Statistical convergence, lacunary equivalence on time scales. Furthermore, the relationships between these new Lacunary statistical concepts are investigated. Finally, we present some equivalence results for asymptotically convergence, lacunary statistical equivalent functions with respect to the partial order on time scales. Delta measure on time scales, Time scales

1. Introduction

The idea of statistical convergence was introduced by Fast [1] and also independently Steinhaus [2]. This concept is a generalization of the concept of ordinary convergence and has a close relationship with the notion of density of the subset of natural numbers N. Over the course of many years and under different names it has been discussed by various researchers −1 [3–9]. Let K ⊆ N and Kn = {k ≤ n : k ∈ K}. Then the natural density of K is defined by δ (K) = limnn |Kn| if the limit exists, where |Kn| indicates the cardinality of Kn. A sequence x = (xk) is said to be statistically convergent to L if for every ε > 0, the set Kε := {k ∈ N : |xk − L| ≥ ε} has natural density zero, i.e., for each ε > 0, 1 lim |{k ≤ n : |xk − L| ≥ ε}| = 0, n n and written as st − limx = L. In [10], Pobyvanets firstly introduced the concept of asymptotically regular matrices which preserve the asymptotic equivalence of two nonnegative number sequences. Furthermore, Fridy [11] investigated new ways of comparing convergence rate of nonnegative sequences. After this work of Fridy, Marouf [12] presented some main definitions for asymptotic regular matrices. Later on, Patterson [13] extended these concepts by using statistical limit and defined the concept of asymptotically statistical equivalence. In the following years, many researchers have studied generalizations and applications of this notion [14–19]. The main goal of our study is to move the concepts of asymptotically statistical equivalence and asymptotically lacunary statistical equivalence to an arbitrary time scale. A time scale is an arbitrary closed subset of the real numbers R in the usual topology, denoted by T. The calculus of time scales has been constructed by Hilger [20]. This theory is an efficient tool to unify continuous and discrete analysis in one theory as it allows integration and differentiation of independent the domain used. The time scale calculus, due to its applications in different branches of science and engineering, has received much attention and has become the object of many research works [21–23]. Moreover, the idea of statistical convergence has been studied on time scales in [24] and [25], independently. Following these studies, statistical convergence has been applied to time scales by various researchers [26–33].

2. Preliminaries

In this section, we remind some basic deﬁnitions and notations that are used in the sequel.

Deﬁnition 2.1. [12] Two non-negative sequences x = (xk) and y = (yk) are said to be asymptotically equivalent, denoted by x ∼ y, if lim xk = 1. k yk

∗ Corresponding author: [email protected] 84 /

Deﬁnition 2.2. [13] Two non-negative sequences x = (xk) and y = (yk) are said to be asymptotically statistical equivalent of multiple L provided that for every ε > 0 1 xk lim k ≤ n : − L ≥ ε = 0. n n yk

S S We denote it as x∼L y. It is also called simply asymptotically statistical equivalent if L = 1, denoted by x∼y.

By a lacunary sequence, we mean an increasing non-negative integer sequence θ = (kr) such that k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir := (kr−1,kr] [5].

Deﬁnition 2.3. [14] Let θ = (kr) be a lacunary sequence. Two non-negative sequences x = (xk) and y = (yk) are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ε > 0 1 xk lim k ∈ Ir : − L ≥ ε = 0. r hr yk

SL S We denote it as x∼θ y. It is also called simply asymptotically lacunary statistical equivalent if L = 1, denoted by x∼θ y.

Deﬁnition 2.4. [14] Let θ = (kr) be a lacunary sequence. Two non-negative sequences x = (xk) and y = (yk) are said to be strong asymptotically lacunary equivalent of multiple L provided that

1 xk lim − L = 0, r h ∑ y r k∈Ir k

NL denoted by x ∼θ y and simply strong asymptotically lacunary equivalent if L = 1.

Here we give a very short introduction to summability theory on time scales calculus. The forward jump operator σ : T → T is defined by σ (t) = inf{s ∈ T : s > t} for t ∈ T, and also the graininess function µ : T → [0,∞) is defined by µ (t) = σ (t)−t. A closed interval on a time scale T is given by [a,b] = {t ∈ : a ≤ t ≤ b}. Open intervals or half-open intervals are defined accordingly. T T In this paper, we use the Lebesgue ∆-measure by µ∆ introduced by Guseinov [22]. In this case, it is known that if a ∈ T\{maxT}, then the single point set {a} is ∆-measurable and µ∆ ({a}) = σ (a) − a. If a,b ∈ T and a ≤ b, then ([a,b) ) = b − a and ((a,b) ) = b − (a); if a,b ∈ \{max } and a ≤ b, then ((a,b] ) = (b) − (a) and µ∆ T µ∆ T σ T T µ∆ T σ σ ([a,b] ) = (b) − a [22]. µ∆ T σ Throughout this paper, we consider that T is a time scale satisfying infT = t0 > 0 and supT = ∞. Definition 2.5. [25]A ∆-measurable function f : T → R is statistically convergent to a number L on T if for every ε > 0, µ ({s ∈ [t ,t] : | f (s) − L| ≥ ε}) lim ∆ 0 T = 0 t→∞ ([t ,t] ) µ∆ 0 T which is denoted by st − lim f (t) = L. T t→∞

Let θ = (kr) is an increasing sequence of non-negative integers with k0 = 0 and σ (kr) − σ (kr−1) → ∞ as r → ∞. Then θ is called a lacunary sequence with respect to T [26]. Deﬁnition 2.6. [26] Let θ be a lacunary sequence on T.A ∆-measurable function f : T → R is said to be lacunary statistically convergent to L on T if for every ε > 0, µ ({s ∈ (k ,k ] : | f (s) − L| ≥ ε}) lim ∆ r−1 r T = 0, r→∞ ((k ,k ] ) µ∆ r−1 r T which is denoted by stT−θ − lim f (t) = L. Deﬁnition 2.7. [26] Let θ be a lacunary sequence on T and f : T → R be a ∆-measurable function. Then f is strongly lacunary Cesàro summable to L on T, if there exists some L ∈ R such that 1 Z lim | f (s) − L|∆s = 0. r→∞ µ∆ ((kr−1,kr] ) T (k ,k ] r−1 r T

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3. Main Results

This section contains our main definitions. Here, first of all, we introduce the concepts of asymptotically statistical equivalence, asymptotically lacunary statistical equivalence and strong asymptotically lacunary equivalence by taking non-negative two ∆-measurable real-valued functions defined on arbitrary time scales instead of sequences. We then examine the relationship between these new concepts.

Deﬁnition 3.1. Let f ,g : T → R be two non-negative ∆-measurable functions. If for every ε > 0, 1 f (s) lim µ∆ s ∈ [t0,t] : − L ≥ ε = 0, t→∞ ([t ,t] ) T g(s) µ∆ 0 T

SL T then we say that the functions f and g are asymptotically statistical equivalent of multiple L on T, and we denote it as f ∼ g. S It is also called simply asymptotically statistical equivalent if L = 1, denoted by f ∼T g. Furthermore, SL denotes the set of T SL f and g such that f ∼T g.

Remark 3.2. i) If g(t) = 1 for all t ∈ T, then asymptotically statistical equivalence on T is reduced to statistical convergence on T introduced in [25]. ii) If we choose T = N, then asymptotically statistical equivalence on T turns out to be the concept of asymptotically statistical equivalence introduced in [13].

Deﬁnition 3.3. Let θ = (kr) be a lacunary sequence on T and f ,g : T → R be two non-negative ∆-measurable functions. If for every ε > 0, 1 f (s) lim µ∆ s ∈ (kr−1,kr] : − L ≥ ε = 0, r→∞ ((k ,k ] ) T g(s) µ∆ r−1 r T then we say that the functions f and g are asymptotically lacunary statistical equivalent of multiple L on T, and we denote it SL S as f θ∼−T g. It is also called simply asymptotically lacunary statistical equivalent if L = 1, denoted by f θ∼−T g. Furthermore, SL SL denotes the set of f and g such that f θ∼−T g. θ−T Remark 3.4. i) If g(t) = 1 for all t ∈ T, then asymptotically lacunary statistical equivalence on T is reduced to lacunary statistical convergence on T introduced in [26]. ii) If we choose T = N, then asymptotically lacunary statistical equivalence on T turns out to be the concept of asymptotically lacunary statistical equivalence introduced in [14].

Deﬁnition 3.5. Let θ = (kr) be a lacunary sequence on T. Then the two non-negative ∆-measurable functions f : T → R and g : T → R are said to be strong asymptotically lacunary equivalent of multiple L on T provided that

1 Z f (s) lim − L ∆s = 0. r→∞ µ∆ ((kr−1,kr] ) g(s) T (k ,k ] r−1 r T

NL In this situation, we write f θ∼−T g. It is also called simply strong asymptotically lacunary equivalent if L = 1, denoted by N NL f θ∼−T g. Furthermore, NL denotes the set of f and g such that f θ∼−T g. θ−T The next theorem gives us the relation between the concepts of asymptotically lacunary statistical equivalence and strong asymptotically lacunary equivalence on time scales.

Theorem 3.6. Let θ = (kr) be a lacunary sequence on T. Then, we have the following: NL SL i) (a) If f θ∼−T g, then f θ∼−T g. (b) NL is proper subset of SL . θ−T θ−T SL NL θ−T θ−T ii) If f ,g ∈ Cb (T) and f ∼ g, then f ∼ g. iii) SL ∩C ( ) = NL ∩C ( ), θ−T b T θ−T b T where Cb (T) denotes the set of all bounded real valued and ∆-measurable functions on T. Proof. This can be proved by using the ideas of Theorem 1 and Theorem 2 in [26].

In the following example, we show that there exists two functions which are asymptotically lacunary statistical equivalent, but not strong asymptotically lacunary equivalent.

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Example 3.7. Let the functions f and g be deﬁned in each intervals (k , (k )] by r−1 σ r T  2, if t ∈ (k ,σ (k ) + 1) ,  r−1 r−1 T  3, if t ∈ [σ (k ) + 1,σ (k ) + 2) ,  r−1 r−1 T  . f (t) = .  √ √ √  ur + 1, if t ∈ σ (kr−1) + ur − 1,σ (kr−1) + ur ,  T  1, otherwise, and  1/2, if t ∈ (k ,σ (k ) + 1) ,  r−1 r−1 T  1/3, if t ∈ [σ (k ) + 1,σ (k ) + 2) ,  r−1 r−1 T  . g(t) = .  √ √ √  1/ ur + 1 , if t ∈ σ (kr−1) + ur − 1,σ (kr−1) + ur ,  T  1, otherwise, where ur = σ (kr) − σ (kr−1) and [x] denotes the largest integer not exceeding x. Then, one can writes

 4, if t ∈ (k ,σ (k ) + 1) ,  r−1 r−1 T  9, if t ∈ [ (k ) + 1, (k ) + 2) ,  σ r−1 σ r−1 T f (t)  . = . g(t) .  √ 2 √ √  ur + 1 , if t ∈ σ (kr−1) + ur − 1,σ (kr−1) + ur ,  T  1, otherwise. Then, we have √ 1 f (s) µ kr−1,σ (kr−1) + ur ∆ T µ∆ s ∈ (kr−1,kr] : − 1 ≥ ε ≤ ((k ,k ] ) T g(s) u µ∆ r−1 r T r √ ur = → 0 (as r → ∞). ur

S Hence, we get that f θ∼−T g. On the other hand,

1 Z f (s) − 1 ∆s µ∆ ((kr−1,kr] ) g(s) T (k ,k ] r−1 r T √ [ ur]+1 3 1 = ((k , (k ) + 1) ) + m2 − 1 ([ (k ) + m − 2, (k ) + m − 1) ) µ∆ r−1 σ r−1 T ∑ µ∆ σ r−1 σ r−1 T ur ur m=3 √ [ ur]+1 1 = ∑ m2 − 1 ur m=2 √ √ √ ([ ur]+1)([ ur]+2)(2[ ur]+3) √ − ur − 1 = 6 → ∞ (as r → ∞). ur

Nθ−T Hence, we obtain that f 6∼ g.

The relationship between the concepts of asymptotically statistical equivalence and asymptotically lacunary statistical equivalence on time scales is examined in the following two theorems.

SL SL σ(kr) T θ−T Theorem 3.8. Let θ = (kr) be a lacunary sequence on with liminf > 1, then f ∼ g implies f ∼ g. T r→∞ σ(kr−1) Proof. The proof can be done easily in a similar way to the Lemma 3.1 of [27]. Hence, we omit it.

SL SL σ(kr) θ∼−T ∼T Theorem 3.9. Let θ = (kr) be a lacunary sequence on T with limsup σ(k ) < ∞, then f g implies f g. r→∞ r−1 Proof. The proof can be done easily in a similar way to the Lemma 3.2 of [27]. Hence, we omit it.

Finally, via the partial order on time scales, we give some equivalence results for asymptotically lacunary statistical equivalent functions. Before doing this, we give the following notation. Let f ,g : T → R be two non-negative ∆-measurable functions. We use the notation f ≺ g if f (t) ≤ g(t) holds for all t ∈ T.

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0 (L−L0) SL SL S θ−T θ−T θ−T Theorem 3.10. Let θ = (kr) be a lacunary sequence on T. If h ≺ f and ( f − h) ∼ g, then f ∼ g implies h ∼ g. 0 SL Proof. Suppose that h ≺ f and ( f − h) θ∼−T g. Then, because of h ≺ f , the inequality

h(s) 0 f (s) f (s) − h(s) 0 − L − L ≤ − L + − L g(s) g(s) g(s) holds for all s ∈ T. Then, for a given ε > 0, since µ∆ is a measure, we have

1 n h(s) 0 o µ s ∈ (kr− ,kr] : − (L − L ) ≥ ε µ (k ,k ] ∆ 1 T g(s) ∆( r−1 r T)

1 n f (s) ε o ≤ µ s ∈ (kr− ,kr] : − L ≥ µ (k ,k ] ∆ 1 T g(s) 2 ∆( r−1 r T)

1 n f (s)−h(s) 0 ε o + µ s ∈ (kr− ,kr] : − L ≥ . µ (k ,k ] ∆ 1 T g(s) 2 ∆( r−1 r T)

(L−L0) S − Taking limit as r → ∞ and also using the hypothesis, we obtain that h θ ∼T g which is desired result.

0 /L00 SL SL S1 θ−T θ−T θ−T Theorem 3.11. Let θ = (kr) be a lacunary sequence on T. If g ≺ h and f ∼ (h − g), then f ∼ g implies f ∼ h, where 00 1 1 L := L + L0 . 0 SL θ−T Proof. Suppose that g ≺ h and f ∼ (h − g). Since g ≺ h, then g(s) ≤ h(s) for all s ∈ T. It follows that the function h − g = (h − g)(s) = h(s) − g(s) is a ∆-measurable non-negative real valued function on T. We also have

h(s) 00 h(s) − g(s) 1 g(s) 1 − L ≤ − + − f (s) f (s) L0 f (s) L for all s ∈ T. Then, for any given ε > 0, since µ∆ is a measure, we may write

1 n h(s) 00 o µ s ∈ (kr− ,kr] : − L ≥ ε µ (k ,k ] ∆ 1 T f (s) ∆( r−1 r T)

1 n h(s)−g(s) 1 ε o ≤ µ s ∈ (kr− ,kr] : − 0 ≥ µ (k ,k ] ∆ 1 T f (s) L 2 ∆( r−1 r T)

1 n g(s) 1 ε o + µ s ∈ (kr− ,kr] : − ≥ . µ (k ,k ] ∆ 1 T f (s) L 2 ∆( r−1 r T)

L0 L S − S − Because of f θ∼T (h − g), if f θ∼T g, the right side of last inequality tends to 0 as r → ∞, and consequently the right hand 00 S1/L side also tends to 0. Hence f θ∼−T h. This completes the proof.

Definition 3.12. If f : T → R is a function such that f (t) satisfies a property P for all t except a set which has zero lacunary density on time scale, then it is said that f (t) has the property P lacunary almost all t and we abbreviate this by “l∆.a.a.t". 1 Recall, the lacunary density of A on time scale is defined by δ (A) = lim µ ({s ∈ (kr− ,kr] : s ∈ A}), if θ−T µ (k ,k ] ∆ 1 T r→∞ ∆( r−1 r T) this limit exists, where A is a ∆-measurable set.

SL SL θ−T θ−T Theorem 3.13. Let θ = (kr) be a lacunary sequence on T. If h = f (l∆.a.a.t) and f ∼ g, then h ∼ g. Proof. Let A := {s ∈ T : h(s) 6= f (s)}. From the assumption, we have δθ−T (A) = 0. Then, for every ε > 0, the following inclusion n o s ∈ (k ,k ] : h(s) − L ≥ ε r−1 r T g(s) n o = s ∈ (k ,k ] : h(s) − L ≥ ε ∩ (Ac ∪ A) r−1 r T g(s) n o n o ⊆ s ∈ (k ,k ] : h(s) − L ≥ ε ∩ Ac ∪ s ∈ (k ,k ] : h(s) − L ≥ ε ∩ A r−1 r T g(s) r−1 r T g(s) n o ⊆ s ∈ (k ,k ] : f (s) − L ≥ ε ∪ {s ∈ (k ,k ] : s ∈ A} r−1 r T g(s) r−1 r T 88 / and so the inequality 1 n h(s) o µ s ∈ (kr− ,kr] : − L ≥ ε µ (k ,k ] ∆ 1 T g(s) ∆( r−1 r T)

1 n f (s) o ≤ µ s ∈ (kr− ,kr] : − L ≥ ε µ (k ,k ] ∆ 1 T g(s) ∆( r−1 r T)

1 + µ ({s ∈ (kr− ,kr] : s ∈ A}) µ (k ,k ] ∆ 1 T ∆( r−1 r T) L S − holds. By taking limit as r → ∞ on both sides of last inequality, we get h θ∼T g. This proves our assertion.

Acknowledgements

The first author would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their financial supports during his doctorate studies. The authors would also like to thank the reviewers for their valuable comments which are improved the paper.

References

[1] Fast, H. 1951. Sur la convergence statistique. Colloq. Math., 2(1951), 241–244. [2] Steinhaus, H. 1951. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math., 2(1)(1951), 73–74. [3] Schoenberg, I. J. 1959. The integrability of certain functions and related summability methods. Amer. Math. Monthly, 66(1959), 361–375. [4] Fridy, J. A. 1985. On statistical convergence. Analysis, 5(1985), 301–313. [5] Fridy, J. A., Orhan, C. 1993. Lacunary statistical convergence. Pacific J. Math., 160(1993), 43–51. [6] Connor, J. S. 1988. The statistical and strong p-Cesàro convergence of sequences. Analysis, 8(1988), 47–63. [7] Mursaleen, M. 2000. λ-statistical convergence. Math. Slovaca, 50(1)(2000), 111–115. [8] Móricz, F. 2004. Statistical limits of measurable functions. Analysis, 24(1)(2004), 1–18. [9] Srivastava, H. M., Et, M. 2017. Lacunary statistical convergence and strongly lacunary summable functions of order α. Filomat, 31(6)(2017), 1573–1582. [10] Pobyvanets, I. P. 1980. Asymptotic equivalence of some linear transformation defined by a non-negative matrix and reduced to generalized equivalence in the sense of Cesaro and Abel. Math. Fiz., 28(123)(1980), 83–87. [11] Fridy, J. A. 1978. Minimal rates of summability. Canad. J. Math., 30(4)(1978), 808–816. [12] Marouf, M. 1993. Asymptotic equivalence and summability. Internat. J. Math. Math. Sci., 16(1993), 755–762. [13] Patterson, R. F. 2003. On asymptotically statistical equivalent sequences. Demonstratio Math., 36(1)(2003), 149–153. [14] Patterson, R. F., Savas, E. 2006. On asymptotically lacunary statistical equivalent sequences. Thai J. Math., 4(2006), 267–272. [15] Basarir, M., Altundag, S. 2008. On ∆-lacunary statistical asymptotically equivalent sequences. Filomat, 22(1)(2008), 161–172. [16] Basarir, M., Altundag, S. 2011. On asymptotically equivalent difference sequences with respect to a modulus function. Ricerche di matematica, 60(2)(2011), 299–311. [17] Kosar, C., Kucukaslan, M., Et, M. 2017. On asymptotically deferred statistical equivalence of sequences. Filomat, 31(16)(2017), 5139–5150. [18] Savas, E. 2019. On asymptotically lacunary statistical equivalent functions via ideals. J. Math. Computer Sci., 19(2019), 35–40. [19] Konca, S., Kucukaslan, M. 2018 On asymptotically f -statistical equivalent sequences. J. Indonesian Math. Soc., 24(2)(2018), 54–61. [20] Hilger, S. 1990. Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math., 18(1-2)(1990), 18–56. [21] Bohner, M., Peterson, A. 2001. Dynamic equations on time scales: An introduction with applications. Boston, Birkhäuser. [22] Guseinov, G. S. 2003. Integration on time scales. J. Math. Anal. Appl., 285(1)(2003), 107–127.

89 /

[23] Cabada, A., Vivero, D. R. 2006. Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral: Application to the calculus of ∆-antiderivatives. Math. Comput. Model., 43(1-2)(2006), 194–207. [24] Seyyidoglu, M. S., Tan, N. O. 2012. A note on statistical convergence on time scale. J. Inequal. Appl., 2012(1)(2012), 219. [25] Turan, C., Duman, O. 2013. Statistical convergence on time scales and its characterizations. Springer Proc. Math. Stat., 41(2013), 57–71. [26] Turan, C., Duman, O. 2013. Convergence methods on time scales. AIP Conf. Proc., 1558(2013), 1120–1123. [27] Turan, C., Duman, O. 2017. Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat, 31(14)(2017), 4455–4467. [28] Altin, Y., Koyunbakan, H., Yilmaz, E. 2014. Uniform statistical convergence on time scales. J. Appl. Math., 2014(2014). [29] Yilmaz, E., Altin, Y., Koyunbakan, H. 2016. λ-statistical convergence on time scales. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 23(1)(2016), 69–78. [30] Sozbir, B., Altundag, S. 2019. Weighted statistical convergence on time scale. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 26(2)(2019), 137–143. [31] Sozbir, B., Altundag, S. 2020. αβ-statistical convergence on time scales. Facta Univ. Ser. Math. Inform., 35(1)(2020), 141–150. [32] Sozbir, B., Altundag, S., Basarir, M. 2020. On the (Delta,f)-lacunary statistical convergence of the functions. Maltepe Journal of Mathematics, 2(1)(2020), 1–8.

[33] Turan, N., Basarir, M. 2019. On the ∆g-statistical convergence of the function deﬁned time scale. AIP Conference Proceedings, 2183, 040017(2019), https://doi.org/10.1063/1.5136137.