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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 435790, 6 pages
http://dx.doi.org/10.1155/2012/435790
Research Article

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

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 13 December 2011; Accepted 6 January 2012

Academic Editor: SimeonΒ Reich

Copyright Β© 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 {𝑇𝑖} 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 𝑇𝑖. Some strong convergence theorems of the proposed method are also obtained for the following cases: all 𝑇𝑖 are continuous and one of 𝑇𝑖 is hemicompact, and the domain 𝐾 is compact.

1. Introduction

Let 𝑋 be a real Banach space. A subset 𝐾 of 𝑋 is called proximinal if for each π‘₯βˆˆπ‘‹, there exists an element π‘˜βˆˆπΎ such that𝑑(π‘₯,π‘˜)=𝑑(π‘₯,𝐾),(1.1) where 𝑑(π‘₯,𝐾)=inf{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘¦βˆˆπΎ} is the distance from the point π‘₯ to the set 𝐾. It is clear that every closed convex subset of a uniformly convex Banach space is proximinal.

Let 𝑋 be a uniformly convex real Banach space, 𝐾 be a nonempty closed convex subset of 𝑋, 𝐢𝐡(𝐾) be a family of nonempty closed bounded subsets of 𝐾, and 𝑃(𝐾) be a nonempty proximinal bounded subsets of 𝐾. The Hausdorff metric on 𝐢𝐡(𝑋) is defined by 𝐻(𝐴,𝐡)=maxsupπ‘₯βˆˆπ΄π‘‘(π‘₯,𝐡),supπ‘¦βˆˆπ΅ξƒ°π‘‘(𝑦,𝐴),(1.2) for all 𝐴,𝐡∈𝐢𝐡(𝑋).

An element π‘βˆˆπΎ is called a fixed point of a single valued mapping 𝑇 if 𝑝=𝑇𝑝 and of a multivalued mapping 𝑇 if π‘βˆˆπ‘‡π‘. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇).

A single valued mapping π‘‡βˆΆπΎβ†’πΎ is said to be quasi-nonexpansive if ‖𝑇π‘₯βˆ’π‘β€–β‰€β€–π‘₯βˆ’π‘β€– for all π‘₯∈𝐾 and π‘βˆˆπΉ(𝑇).

A multivalued mapping π‘‡βˆΆπΎβ†’πΆπ΅(𝐾) is said to be:(i)quasi-nonexpansive if 𝐹(𝑇)β‰ βˆ… and 𝐻(𝑇π‘₯,𝑇𝑝)≀‖π‘₯βˆ’π‘β€– for all π‘₯∈𝐾 and π‘βˆˆπΉ(𝑇),(ii)nonexpansive if 𝐻(𝑇π‘₯,𝑇𝑦)≀‖π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπΎ.

It is well known that every nonexpansive multivalued mapping 𝑇 with 𝐹(𝑇)β‰ βˆ… is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive. It is clear that if 𝑇 is a quasi-nonexpansive multivalued mapping, then 𝐹(𝑇) is closed.

A map π‘‡βˆΆπΎβ†’πΆπ΅(𝐾) is called hemicompact if, for any sequence {π‘₯𝑛} in 𝐾 such that 𝑑(π‘₯𝑛,𝑇π‘₯𝑛)β†’0 as π‘›β†’βˆž, there exists a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} such that π‘₯π‘›π‘˜β†’π‘βˆˆπΎ. We note that if 𝐾 is compact, then every multivalued mapping 𝑇 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 (πœ€,πœ†)-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 𝑝 and the initial choice of π‘₯0. 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 𝑇 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 𝑇 with 𝑇𝑝={𝑝} for any π‘βˆˆπΉ(𝑇) which relaxed compactness of the domain of 𝑇.

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 {π‘‡π‘–βˆΆπ‘–βˆˆβ„•} of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space.

Let {𝑇𝑖} be a countable family of multivalued quasi-nonexpansive mapping from a bounded and closed convex subset 𝐾 of a Banach space into 𝑃(𝐾) with β‹‚πΉβˆΆ=βˆžπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and let π‘βˆˆπΉ. For π‘₯1∈𝐾, we defineπ‘₯𝑛+1=𝛼𝑛,0π‘₯𝑛+βˆžξ“π‘–=1𝛼𝑛,𝑖π‘₯𝑛,𝑖,(1.3) where the sequences {𝛼𝑛,𝑖}βŠ‚[0,1) satisfying βˆ‘βˆžπ‘–=0𝛼𝑛,𝑖=1 and π‘₯𝑛,π‘–βˆˆπ‘‡π‘–π‘₯𝑛 such that 𝑑(𝑝,π‘₯𝑛,𝑖)=𝑑(𝑝,𝑇𝑖π‘₯𝑛) for π‘–βˆˆβ„•. The main purpose of this paper is to prove strong convergence of the iterative scheme (1.3) to a common fixed point of 𝑇𝑖.

2. Preliminaries

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

Lemma 2.1 (see [24]). Suppose that 𝑋 is a uniformly convex Banach space and 0<π‘β‰€π‘‘π‘›β‰€π‘ž<1 for all positive integers 𝑛. Also suppose that {π‘₯𝑛} and {𝑦𝑛} are two sequences of 𝑋 such that limsupπ‘›β†’βˆžβ€–π‘₯π‘›β€–β‰€π‘Ÿ, limsupπ‘›β†’βˆžβ€–π‘¦π‘›β€–β‰€π‘Ÿ, and limπ‘›β†’βˆžβ€–π‘‘π‘›π‘₯𝑛+(1βˆ’π‘‘π‘›)𝑦𝑛‖=π‘Ÿ hold for some π‘Ÿβ‰₯0. Then limsupπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–=0.

Lemma 2.2 (see [25]). Let 𝐸 be a uniformly convex Banach space. For arbitrary π‘Ÿ>0, let π΅π‘Ÿ(0)∢={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–β‰€π‘Ÿ}. Then, for any given sequence {π‘₯𝑛}βˆžπ‘›=1βŠ‚π΅π‘Ÿ(0) and for any given sequence {πœ†π‘›}βˆžπ‘›=1 of positive numbers such that βˆ‘βˆžπ‘–=1πœ†π‘–=1, there exists a continuous strictly increasing convex function []π‘”βˆΆ0,2π‘ŸβŸΆβ„,𝑔(0)=0,(2.1) such that for any positive integers 𝑖, 𝑗 with 𝑖<𝑗, the following inequality holds: β€–β€–β€–β€–βˆžξ“π‘›=1πœ†π‘›π‘₯𝑛‖‖‖‖2β‰€βˆžξ“π‘›=1πœ†π‘›β€–β€–π‘₯𝑛‖‖2βˆ’πœ†π‘–πœ†π‘—π‘”ξ€·β€–β€–π‘₯π‘–βˆ’π‘₯𝑗‖‖.(2.2)

3. Main Results

We first prove that the sequence {π‘₯𝑛} generated by (1.3) is an approximating fixed point sequence of each 𝑇𝑖(π‘–βˆˆβ„•).

Lemma 3.1. Let 𝐾 be a nonempty bounded and closed convex subset of a uniformly convex Banach space 𝑋. For π‘–βˆˆβ„•, let {𝑇𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐾 into 𝑃(𝐾) with β‹‚πΉβˆΆ=βˆžπ‘–=1𝐹(𝑇𝑖)β‰ βˆ…and π‘βˆˆπΉ. Let {π‘₯𝑛} be a sequence defined by (1.3). Then(i)β€–π‘₯𝑛+1βˆ’π‘β€–β‰€β€–π‘₯π‘›βˆ’π‘β€–,(ii)limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘β€– exists.

Proof. By (1.3), we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›Όπ‘›,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖π‘₯𝑛,π‘–β€–β€–βˆ’π‘=𝛼𝑛,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘βˆžξ“π‘–=1𝛼𝑛,𝑖𝑑𝑇𝑖π‘₯𝑛,𝑝≀𝛼𝑛,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘βˆžξ“π‘–=1𝛼𝑛,𝑖𝐻𝑇𝑖π‘₯𝑛,𝑇𝑖𝑝≀𝛼𝑛,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖π‘₯𝑛‖‖=β€–β€–π‘₯βˆ’π‘π‘›β€–β€–.βˆ’π‘(3.1) So (i) is obtained. (ii) follows from (i).

Theorem 3.2. Let 𝐾 be a nonempty bounded and closed convex subset of a uniformly convex Banach space 𝑋. For π‘–βˆˆβ„•, let {𝑇𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐾 into 𝑃(K) with β‹‚πΉβˆΆ=βˆžπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and π‘βˆˆπΉ. Let {π‘₯𝑛} be a sequence defined by (1.3) with liminfπ‘›β†’βˆžπ›Όπ‘›,0𝛼𝑛,𝑗>0 for all π‘—βˆˆβ„•. Then limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝑇𝑖π‘₯𝑛)=0 for all π‘–βˆˆβ„•.

Proof. For π‘—βˆˆβ„•, by Lemma 2.2, we get β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2=‖‖‖‖𝛼𝑛,0ξ€·π‘₯𝑛+βˆ’π‘βˆžξ“π‘–=1𝛼𝑛,𝑖π‘₯𝑛,π‘–ξ€Έβ€–β€–β€–β€–βˆ’π‘2≀𝛼𝑛,0β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖π‘₯𝑛,π‘–β€–β€–βˆ’π‘2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖‖=𝛼𝑛,0β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+βˆžξ“π‘–=1𝛼𝑛,𝑖𝑑𝑇𝑖π‘₯𝑛,𝑝2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖‖≀𝛼𝑛,0β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+βˆžξ“π‘–=1𝛼𝑛,𝑖𝐻𝑇𝑖π‘₯𝑛,𝑇𝑖𝑝2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖‖≀𝛼𝑛,0β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2+βˆžξ“π‘–=1𝛼𝑛,𝑖‖‖π‘₯π‘›β€–β€–βˆ’π‘2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖‖=β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2βˆ’π›Όπ‘›,0𝛼𝑛,𝑗𝑔‖‖π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖‖.(3.2) Thus, 0<𝛼𝑛,0𝛼𝑛,𝑗𝑔(β€–π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖)≀‖π‘₯π‘›βˆ’π‘β€–2βˆ’β€–π‘₯𝑛+1βˆ’π‘β€–2. It follows that limπ‘›β†’βˆžπ‘”(β€–π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖)=0. By property of 𝑔, we have limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘₯𝑛,𝑗‖=0. Thus limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝑇𝑗π‘₯𝑛)=0 for π‘–βˆˆβ„•.

Theorem 3.3. Let 𝑋 be a uniformly convex real Banach space and 𝐾 be a bounded and closed convex subset of 𝑋. For π‘–βˆˆβ„•, let {𝑇𝑖} be a sequence of multivalued quasi-nonexpansive and continuous mappings from 𝐾 into 𝑃(𝐾) with β‹‚πΉβˆΆ=βˆžπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and π‘βˆˆπΉ. Let {π‘₯𝑛} be a sequence defined by (1.3) with liminfπ‘›β†’βˆžπ›Όπ‘›,0𝛼𝑛,𝑗>0 for all π‘—βˆˆβ„•. Assume that one of 𝑇𝑖 is hemicompact. Then {π‘₯𝑛} converges strongly to a common fixed point of {𝑇𝑖}.

Proof. Suppose that 𝑇𝑖 is hemicompact for some 𝑖0βˆˆβ„•. By Theorem 3.2, we have limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝑇𝑖π‘₯𝑛)=0 for all π‘–βˆˆβ„•. Then there exists a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} such that limπ‘˜β†’βˆžπ‘₯π‘›π‘˜=π‘žβˆˆπΎ. From continuity of 𝑇𝑖, we get 𝑑(π‘₯π‘›π‘˜,𝑇𝑖π‘₯π‘›π‘˜)→𝑑(π‘ž,π‘‡π‘–π‘ž). This implies that 𝑑(π‘ž,π‘‡π‘–π‘ž)=0 and π‘žβˆˆπΉ. Since limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘žβ€– exists, it follows that {π‘₯𝑛} converges strongly to π‘ž.

Theorem 3.4. Let 𝑋 be a uniformly convex real Banach space and 𝐾 be a compact convex subset of 𝑋. For π‘–βˆˆβ„•, let {𝑇𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐾 into 𝑃(𝐾) with β‹‚πΉβˆΆ=βˆžπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and π‘βˆˆπΉ. Let {π‘₯𝑛} be a sequence defined by (1.3) with liminfπ‘›β†’βˆžπ›Όπ‘›,0𝛼𝑛,𝑗>0 for all π‘—βˆˆβ„•. Then {π‘₯𝑛} converges strongly to a common fixed point of {𝑇𝑖}.

Proof. From the compactness of 𝐾, there exists a subsequence {π‘₯π‘›π‘˜}βˆžπ‘›=π‘˜ of {π‘₯𝑛}βˆžπ‘›=1 such that limπ‘˜β†’βˆžβ€–π‘₯π‘›π‘˜βˆ’π‘žβ€–=0 for some π‘žβˆˆπΎ. Thus, it follows by Theorem 3.2 that, π‘‘ξ€·π‘ž,π‘‡π‘–π‘žξ€Έξ€·β‰€π‘‘π‘ž,π‘₯π‘›π‘˜ξ€Έξ€·π‘₯+π‘‘π‘›π‘˜,𝑇𝑖π‘₯π‘›π‘˜ξ€Έξ€·π‘‡+H𝑖π‘₯π‘›π‘˜,π‘‡π‘–π‘žξ€Έβ€–β€–π‘₯≀2π‘›π‘˜β€–β€–ξ€·π‘₯βˆ’π‘ž+π‘‘π‘›π‘˜,𝑇𝑖π‘₯π‘›π‘˜ξ€ΈβŸΆ0asπ‘˜βŸΆβˆž.(3.3) Hence π‘žβˆˆπΉ. By Lemma 3.1(ii), limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘žβ€– exists. Hence limπ‘›β†’βˆžπ‘₯𝑛=π‘ž. 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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