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Journal of Function Spaces and Applications Some Classes of Function Spaces, Their Properties, and Applications Guest Editors: Józef Banaś, Janusz Matkowski, Nelson Merentes, Manuel Pinto, and Jose Luis Sanchez Some Classes of Function Spaces, Their Properties, and Applications Journal of Function Spaces and Applications Some Classes of Function Spaces, Their Properties, and Applications Guest Editors: Jozef´ Bana´s, Janusz Matkowski, Nelson Merentes, Manuel Pinto, and Jose Luis Sanchez Copyright © 2013 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in “Journal of Function Spaces andlications.” App All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board John R. Akeroyd, USA Norimichi Hirano, Japan L.E. Persson, Sweden Gerassimos Barbatis, Greece Henryk Hudzik, Poland Adrian Petrusel, Romania Ismat Beg, Pakistan Pankaj Jain, India Konrad Podczeck, Austria Bjorn Birnir, USA Krzysztof Jarosz, USA JoseRodr´ ´ıguez, Spain Messaoud Bounkhel, Saudi Arabia Anna Kaminfhska, USA Natasha Samko, Portugal Huy Qui Bui, New Zealand H. Turgay Kaptanoglu, Turkey Carlo Sbordone, Italy Victor I. Burenkov, Italy Valentin Keyantuo, Puerto Rico Simone Secchi, Italy Jiecheng Chen, China Vakhtang M. Kokilashvili, Georgia Naseer Shahzad, Saudi Arabia Jaeyoung Chung, Korea Alois Kufner, Czech Republic Mitsuru Sugimoto, Japan Sompong Dhompongsa, Thailand David R. Larson, USA Rodolfo H. Torres, USA Luisa Di Piazza, Italy Yuri Latushkin, USA Wilfredo Urbina, USA Lars Diening, Germany Young Joo Lee, Republic of Korea Nikolai L. Vasilevski, Mexico Dragan Djordjevic, Serbia Hugo Leiva, Venezuela P. Veeramani, India Miroslav Engliˇs, Czech Republic Guozhen Lu, USA Igor E. Verbitsky, USA Jose A. Ezquerro, Spain Dag Lukkassen, Norway Dragan Vukotic, Spain Dashan Fan, USA Qiaozhen Ma, China Bruce A. Watson, South Africa Xiang Fang, USA Mark A. McKibben, USA Anthony Weston, USA Hans G. Feichtinger, Austria Mihail Megan, Romania Quanhua Xu, France Alberto Fiorenza, Italy Alfonso Montes-Rodriguez, Spain Gen-Qi Xu, China Ajda Foˇsner, Slovenia Dumitru Motreanu, France Dachun Yang, China Eva A. Gallardo Gutierrez,´ Spain Sivaram K. Narayan, USA Kari Ylinen, Finland Aurelian Gheondea, Turkey Renxing Ni, China Chengbo Zhai, China AntonioS.Granero,Spain Kasso A. Okoudjou, USA Ruhan Zhao, USA Yongsheng S. Han, USA Gestur Olafsson,´ USA Kehe Zhu, USA Seppo Hassi, Finland Josip E. Pecariˇ c,´ Croatia William P. Ziemer, USA Stanislav Hencl, Czech Republic Jose´ Afh´ Pelaez,´ Spain Contents Some Classes of Function Spaces, Their Properties, and Applications,Jozef´ Bana´s, Janusz Matkowski, Nelson Merentes, Manuel Pinto, and Jose Luis Sanchez Volume 2013, Article ID 360980, 3 pages Sensitivity Analysis for Nonlinear Set-Valued Variational Equations in Banach Framework, A. Farajzadeh and Salahuddin Volume 2013, Article ID 258543, 6 pages Nonlinear Kato Class and Unique Continuation of Eigenfunctions for -Laplacian Operator, ReneErl´ ´ın Castillo and Julio C. Ramos Fernandez´ Volume 2013, Article ID 512050, 7 pages Generalized Lorentz Spaces and Applications, Hatem Mejjaoli Volume 2013, Article ID 302941, 14 pages Density in Spaces of Interpolation by Hankel Translates of a Basis Function, Cristian Arteaga and Isabel Marrero Volume 2013, Article ID 813502, 9 pages 2 On a New Space (,,,)of Double Sequences,CenapDuyarandOguz˘ Ogur˘ Volume 2013, Article ID 509613, 8 pages Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity, Mariusz Gil and Stanisław We¸drychowicz Volume 2013, Article ID 692879, 12 pages Frequency-Uniform Decomposition, Function Spaces ,, and Applications to Nonlinear Evolution Equations, Shaolei Ru and Jiecheng Chen Volume 2013, Article ID 176596, 12 pages The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions,LuisAngel´ Gutierrez´ Mendez,´ Juan Alberto Escamilla Reyna, Francisco Javier Mendoza Torres, and Mar´ıa Guadalupe Morales Mac´ıas Volume 2013, Article ID 715789, 5 pages Generalized Virtually Stable Maps and Their Associated Sequences,P.Chaoha,S.Iampiboonvatana, and J. Intrakul Volume 2013, Article ID 237858, 8 pages Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions,ManfredMoller¨ and Bertin Zinsou Volume 2013, Article ID 280970, 8 pages The Uniqueness of Strong Solutions for the Camassa-Holm Equation,MengWuandChongLai Volume 2013, Article ID 409760, 7 pages On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Several Functions, Banyat Sroysang Volume 2013, Article ID 921828, 6 pages Positive Solutions for Some Competitive Fractional Systems in Bounded Domains,ImedBachar, Habib Maagli,ˆ and Noureddine Zeddini Volume 2013, Article ID 140130, 6 pages Commutators of Higher Order Riesz Transform Associated with Schrodinger¨ Operators,YuLiu, Lijuan Wang, and Jianfeng Dong Volume 2013, Article ID 842375, 15 pages On the Space of Functions with Growths Tempered by a Modulus of Continuity and Its Applications, Jozef´ Bana´sandRafałNalepa Volume 2013, Article ID 820437, 13 pages Boundary Value Problems for a Class of Sequential Integrodifferential Equations of Fractional Order, Bashir Ahmad and Juan J. Nieto Volume 2013, Article ID 149659, 8 pages Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms, Ana Portilla, Yamilet Quintana, JoseM.Rodr´ ´ıguez, and Eva Tour´ıs Volume 2013, Article ID 628031, 11 pages Functions of Bounded -Variation in the Sense of Riesz-Korenblum, Mariela Castillo, Sergio Rivas, Mar´ıa Sanoja, and Ivan´ Zea Volume 2013, Article ID 718507, 12 pages Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups,GuorongHu Volume 2013, Article ID 475103, 16 pages A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces, Canqin Tang Volume 2013, Article ID 865835, 12 pages The Space of Continuous Periodic Functions Is a Set of First Category in (), Zhe-Ming Zheng, Hui-Sheng Ding, and Gaston M. N’Guer´ ekata´ Volume 2013, Article ID 275702, 3 pages Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 360980, 3 pages http://dx.doi.org/10.1155/2013/360980 Editorial Some Classes of Function Spaces, Their Properties, and Applications Józef BanaV,1 Janusz Matkowski,2 Nelson Merentes,3 Manuel Pinto,4 and Jose Luis Sanchez3 1 Department of Mathematics, Rzeszow´ University of Technology, al. Powstanc´ ow´ Warszawy 8, 35-959 Rzeszow,´ Poland 2 Division of Functional Equations, Zielona GoraUniversity,ul.Prof.Z.Szafrana4a,65-516ZielonaG´ ora,´ Poland 3 Department of Mathematics, Central University of Venezuela, Paseo Los Ilustres, Urb. Valle Abajo, Apartado Postal 20513, Caracas 1020-A, Venezuela 4 Department of Mathematics, University of Chile, Casilla Postal 653, Santiago, Chile Correspondence should be addressed to Jozef´ Bana´s; [email protected] Received 10 October 2013; Accepted 10 October 2013 Copyright © 2013 Jozef´ Bana´s et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Function spaces create the basis of almost all investigations integral, and so on). We describe below the results contained in several branches of mathematics such as functional anal- in the papers published in this special issue. ysis, nonlinear analysis, operator theory, and the theories At the beginning we present eight papers which are of differential and integral equations, among others. Those mainly devoted to describe some function spaces and their spaces are frequently an object of the intensive study, in which various properties. properties of the mentioned function spaces are considered In the paper of C. Tang a weighted Besov-type space and described. First of all, such an approach is presented andweightedTriebel-Lizorkin-typespaceareintroduced. in functional analysis. But some topics of nonlinear analysis Moreover, the author obtained some characterization of those and operator theory are also closely related to the study of spaces expressed in terms of the so-called -transforms. function spaces and their properties. The paper of G. Hu is dedicated to some topics of homoge- The second direction of investigations connected with neous Triebel-Lizorkin spaces with full range of parameters. the theory of function spaces depends on the application ThesespacesareintroducedonstratifiedLiegroupsinterms of that theory in the study of equations of various type, of Littlewood-Paley-type decomposition. The main result of such as ordinary and partial differential equations, integral the paper asserts that the scale of the considered Triebel- equations, functional differential, functional integral and Lizorkin spaces is independent of the choice of Littlewood- functional equations, operator equations, and so forth. The Paley-type decomposition and the sub-Laplacian used for the study of the mentioned equations requires to place considera- construction of the decomposition. tions in some function space. Consequently, all investigations The next paper written by H. Mejjaoli discusses the associated with a considered equation are closely linked with Lorentz spaces associated with the Dunkl operators on the a function
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  • Upper Estimates for Gâteaux Differentiability of Bump Functions in Orlicz-Lorentz Spaces

    Upper Estimates for Gâteaux Differentiability of Bump Functions in Orlicz-Lorentz Spaces

    Volume 8 (2007), Issue 4, Article 113, 8 pp. UPPER ESTIMATES FOR GÂTEAUX DIFFERENTIABILITY OF BUMP FUNCTIONS IN ORLICZ-LORENTZ SPACES B. ZLATANOV DEPARTMENT OF MATHEMATICS AND INFORMATICS PLOVDIV UNIVERSITY, 24 “TZAR ASSEN” STR. PLOVDIV, 4000 BULGARIA [email protected] Received 29 January, 2007; accepted 17 November, 2007 Communicated by C.P. Niculescu ABSTRACT. Upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz– Lorentz spaces d(w, M, Γ), Γ uncountable, are obtained. The best possible order of Gâteaux differentiability in the class of all equivalent norms in d(w, M, Γ) is found. Key words and phrases: Orlicz function, Orlicz–Lorentz space, Smooth bump functions. 2000 Mathematics Subject Classification. 46B25, 40E30. 1. INTRODUCTION The existence of higher order Fréchet smooth norms and bump functions and its impact on the geometrical properties of a Banach space have been subject to many investigations begin- ning with the classical result for Lp–spaces in [1] and [6]. An extensive study and bibliography may be found in [2]. As any negative result on the existence of Gâteaux smooth bump func- tions immediately applies to the problem of existence of Fréchet smooth bump functions and norms, the question arises of estimating the best possible order of Gâteaux smoothness of bump functions in a given Banach space. A variational technique (the Ekeland variational principle) was applied in [2] to show that in `1(Γ), Γ uncountable, there is no continuous Gâteaux dif- ferentiable bump function. Following the same idea and using Stegall’s variational principle, an extension of this result to Banach spaces with uncountable unconditional basis was given in [4] and to Banach spaces with uncountable symmetric basis in [9].