Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Dehaish, Kozlowski, Kozlowski Wm

Description

Let L ρ be a uniformly convex modular function space with a strong Opial property. Let T : C → C be an asymptotic pointwise nonexpansive mapping, where C is a ρ -a.e. compact convex subset of L ρ . In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T . In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point. MSC: 47H09, 47H10.

Sujets

• Fixed point
• Metric map
• Birnbaum?Orlicz space
• Opial property
• Uniformly convex space

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 8
• Langue: English

Extrait

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Dehaish and KozlowskiFixed Point Theory and Applications2012,2012:118 http://www.fixedpointtheoryandapplications.com/content/2012/1/118

R E S E A R C H

Open Access

Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Buthinah A Bin Dehaish1*and WM Kozlowski2 * Correspondence: [e-mail] Abstract 1 Department of Mathematics, King Abdulaziz University, P.O. Box 53909, LetL ρ be a uniformly convex modular function space with a strong Opial property. Jeddah 21593, Saudi Arabia LetT:C→Cbe an asymptotic pointwise nonexpansive mapping, whereCis a ρ -a.e. Full list of author information is available at the end of the article compact convex subset ofL ρ . In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point ofT. In addition, we prove that ifCis compact in the strong sense, then both processes converge strongly to a fixed point. MSC:Primary 47H09; Secondary 47H10 Keywords:fixed point; nonexpansive mapping; fixed point iteration process; Mann process; Ishikawa process; modular function space; Orlicz space; Opial property; uniform convexity

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23 1 Introduction 24 In , Kirk and Xu [] studied the existence of fixed points of asymptotic pointwise 25 nonexpansive mappingsT:C→C,i.e., 26 27  T n (x) –T n (y)  ≤α n (x)x–y, 28 29 wherelim sup n→∞ α n (x)≤, for allx,y∈CTheir main result (Theorem .) states that. 30 every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded 31 and convex subsetCof a uniformly convex Banach spaceXhas a fixed point. As pointed 32 out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization 33 of nonexpansive mappings. The conditions onα n can be for instance expressed in terms 34 of the derivatives of iterations ofTfor differentiableT. In  these results were gener- 35 alized by Hussain and Khamsi to metric spaces, [ ]. 36 In , Khamsi and Kozlowski [ ] extended their result proving the existence of 37 fixed points of asymptotic pointwiseρ-nonexpansive mappings acting in modular func- 38 s of the existential nature and does tion spaces. The proof of this important theorem i 39 not describe any algorithm for constructing a fixed point of an asymptotic pointwiseρ- 40 nonexpansive mapping. This paper aims at filling this gap. 41 Let us recall that modular function spaces are natural generalization of both func- 42 tion and sequence variants of many important, from applications perspective, spaces like 43 Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces 44 ©2012 Dehaish and Kozlowski; licensee Springer. This is an Open Access article distributed under the terms of the Creative Com- 45 mons Attribution License ( http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and repro- 46 duction in any medium, provided the original work is properly cited. 47

«FPTA  bmc-onecol-v v.// Prn://; fpta_.tex (audrone) class: /» : p.layout: Onecolumn v.. file: « reference style: mathphys»

Dehaish and KozlowskiFixed Point Theory and Applications2012, 2012:118 http://www.fixedpointtheoryandapplications.com/content/2012/1/118

and many others, see the book by Kozlowski [ ] for an extensive list of examples and special cases. There exists an extensive literature on the topic of the fixed point theory in modular function spaces, see,e.g., [–, , , , –, ] and the papers referenced there. It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful nu- merical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. There exists an extensive literature on the subject of iterative fixed point construction processes for asymptotically nonexpansive mappings in Hilbert, Banach and metric spaces, see,e.g., [, , , , , , , –, –] and the works referred there. Kozlowski proved conver- gence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [ ] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [ ]. Recently, weak and strong con- vergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [ ]. We would like to emphasize that all convergence theorems proved in this paper define constructive algorithms that can be actually implemented. When dealing with specific ap- plications of these theorems, one should take into consideration how additional properties of the mappings, sets and modulars involved can influence the actual implementation of the algorithms defined in this paper. The paper is organized as follows: (a) Section  provides necessary preliminary material on modular function spaces. (b) Section  introduces the asymptotic pointwise nonexpansive mappings and related notions. (c) Section  deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems. (d) Section  utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process. (e) Section  establishes the almost everywhere convergence theorem for generalized Ishikawa process. (f ) Section  provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact setC.

2 Preliminaries Letbe a nonempty set andbe a nontrivialσ-algebra of subsets of. LetPbe aδ- ring of subsets ofsuch thatE∩A∈Pfor anyE∈PandA∈. Let us assume that there exists an increasing sequence of setsK n ∈Psuch that=  K n . ByEwe denote the linear space of all simple functions with supports fromP. ByM ∞ we will denote the space of all extended measurable functions,i.e., all functionsf:→[–∞,∞] such that there exists a sequence{g n } ⊂E,|g n | ≤ |f|andg n (ω)→f(ω) for allω∈. By  A we denote the characteristic function of the setA. Definition .Letρ:M ∞ →[,∞] be a nontrivial, convex and even function. We say thatρis a regular convex function pseudomodular if: (i)ρ() = ;

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