Abstract and Applied AnalysisVolume 2012 (2012), Article ID 435790, 6 pagesdoi:10.1155/2012/435790

Research Article

Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

Aunyarat Bunyawat1,2 and Suthep Suantai1,2

¹Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 13 December 2011; Accepted 6 January 2012

Academic Editor: Simeon Reich

Copyright © 2012 Aunyarat Bunyawat and Suthep Suantai. 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.

Abstract.

We introduce an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed point sequence of each . Some strong convergence theorems of the proposed method are also obtained for the following cases: all are continuous and one of is hemicompact, and the domain is compact.

1. Introduction

Let be a real Banach space. A subset of is called proximinal if for each , there exists an element such that where is the distance from the point to the set . It is clear that every closed convex subset of a uniformly convex Banach space is proximinal.

Let be a uniformly convex real Banach space, be a nonempty closed convex subset of , be a family of nonempty closed bounded subsets of , and be a nonempty proximinal bounded subsets of . The Hausdorff metric on is defined by for all .

An element is called a fixed point of a single valued mapping if and of a multivalued mapping if . The set of fixed points of is denoted by .

A single valued mapping is said to be quasi-nonexpansive if for all and .

A multivalued mapping is said to be:(i)quasi-nonexpansive if and for all and ,(ii)nonexpansive if for all .

It is well known that every nonexpansive multivalued mapping with is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive. It is clear that if is a quasi-nonexpansive multivalued mapping, then is closed.

A map is called hemicompact if, for any sequence in such that as , there exists a subsequence of such that . We note that if is compact, then every multivalued mapping is hemicompact.

In 1969, Nadler [1] proved a fixed point theorem for multivalued contraction mappings and convergence of a sequence. They extended theorems on the stability of fixed points of single-valued mappings and also given a counterexample to a theorem about -uniformly locally expansive (single-valued) mappings. Later in 1997, Hu et al. [2] obtained common fixed point of two nonexpansive multivalued mappings satisfying certain contractive condition.

In 2005, Sastry and Babu [3] proved the convergence theorem of multivalued maps by defining Ishikawa and Mann iterates and gave example which shows that the limit of the sequence of Ishikawa iterates depends on the choice of the fixed point and the initial choice of . In 2007, there is paper which generalized results of Sastry and Babu [3] to uniformly convex Banach spaces by Panyanak [4] and proved a convergence theorem of Mann iterates for a mapping defined on a noncompact domain.

Later in 2008, Song and Wang [5] shown that strong convergence for Mann and Ishikawa iterates of multivalued nonexpansive mapping under some appropriate conditions. In 2009, Shahzad and Zegeye [6] proved strong convergence theorems of quasi-nonexpansive multivalued mapping for the Ishikawa iteration. They also constructed an iteration scheme which removes the restriction of with for any which relaxed compactness of the domain of .

Recently, Abbas et al. [7] established weak- and strong-convergence theorems of two multivalued nonexpansive mappings in a real uniformly convex Banach space by one-step iterative process to approximate common fixed points under some basic boundary conditions.

A fixed points and common fixed points theorem of multivalued maps in uniformly convex Banach space or in complete metric spaces or in convex metric spaces have been intensively studied by many authors; for instance, see [8–23].

In this paper, we generalize and modify the iteration of Abbas et al. [7] from two mapping to the infinite family mappings of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space.

Let be a countable family of multivalued quasi-nonexpansive mapping from a bounded and closed convex subset of a Banach space into with and let . For , we define where the sequences satisfying and such that for . The main purpose of this paper is to prove strong convergence of the iterative scheme (1.3) to a common fixed point of .

2. Preliminaries

Before to say the main theorem, we need the following lemmas.

Lemma 2.1 (see [24]). Suppose that is a uniformly convex Banach space and for all positive integers . Also suppose that and are two sequences of such that , , and hold for some . Then .

Lemma 2.2 (see [25]). Let be a uniformly convex Banach space. For arbitrary , let . Then, for any given sequence and for any given sequence of positive numbers such that , there exists a continuous strictly increasing convex function such that for any positive integers , with , the following inequality holds:

3. Main Results

We first prove that the sequence generated by (1.3) is an approximating fixed point sequence of each .

Lemma 3.1. Let be a nonempty bounded and closed convex subset of a uniformly convex Banach space . For , let be a sequence of multivalued quasi-nonexpansive mappings from into with and . Let be a sequence defined by (1.3). Then (i),(ii) exists.

Proof. By (1.3), we have So (i) is obtained. (ii) follows from (i).

Theorem 3.2. Let be a nonempty bounded and closed convex subset of a uniformly convex Banach space . For , let be a sequence of multivalued quasi-nonexpansive mappings from into with and . Let be a sequence defined by (1.3) with for all . Then for all .

Proof. For , by Lemma 2.2, we get Thus, . It follows that . By property of , we have . Thus for .

Theorem 3.3. Let be a uniformly convex real Banach space and be a bounded and closed convex subset of . For , let be a sequence of multivalued quasi-nonexpansive and continuous mappings from into with and . Let be a sequence defined by (1.3) with for all . Assume that one of is hemicompact. Then converges strongly to a common fixed point of .

Proof. Suppose that is hemicompact for some . By Theorem 3.2, we have for all . Then there exists a subsequence of such that . From continuity of , we get . This implies that and . Since exists, it follows that converges strongly to .

Theorem 3.4. Let be a uniformly convex real Banach space and be a compact convex subset of . For , let be a sequence of multivalued quasi-nonexpansive mappings from into with and . Let be a sequence defined by (1.3) with for all . Then converges strongly to a common fixed point of .

Proof. From the compactness of , there exists a subsequence of such that for some . Thus, it follows by Theorem 3.2 that, Hence . By Lemma 3.1(ii), exists. Hence . The proof is complete.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author is supported by the Graduate School, Chiang Mai University, Thailand.

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