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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 391839, 9 pages
http://dx.doi.org/10.1155/2009/391839
Research Article

A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

Received 16 December 2008; Accepted 9 April 2009

Academic Editor: Jie Xiao

Copyright © 2009 Suwicha Imnang 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

Let be a real uniformly convex Banach space and a closed convex nonempty subset of . Let be a finite family of nonexpansive self-mappings of . For a given , let and , , be sequences defined , where for all , and . In this paper, weak and strong convergence theorems of the sequence to a common fixed point of a finite family of nonexpansive mappings are established under some certain control conditions.

1. Introduction

Let be a real Banach space, a nonempty closed convex subset of , and a mapping. Recall that is nonexpansive if for all Let be nonexpansive mappings. Let denote the fixed points set of that is, and let

For a given and a fixed ( denote the set of all positive integers), compute the iterative sequences by where for all , and . If for all , and , then (1.1) reduces to the iterative scheme where , for all and .

If for all , , and for all for all , then (1.1) reduces to the iterative scheme defined by Liu et al. [1] where , for all and . They showed that defined by (1.3) converges strongly to a common fixed point of , in Banach spaces, provided that satisfy condition A. The result improves the corresponding results of Kirk [2], Maiti and Saha [3] and Sentor and Dotson [4].

If and for all , then (1.1) reduces to a generalization of Mann and Ishikawa iteration given by Das and Debata [5] and Takahashi and Tamura [6]. This scheme dealts with two mappings: where are appropriate sequences in .

The purpose of this paper is to establish strong convergence theorems in a uniformly convex Banach space of the iterative sequence defined by (1.1) to a common fixed point of under some appropriate control conditions in the case that one of is completely continuous or semicompact or satisfies condition (B). Moreover, weak convergence theorem of the iterative scheme (1.1) to a common fixed point of is also established in a uniformly convex Banach spaces having the Opial's condition.

2. Preliminaries

In this section, we recall the well-known results and give a useful lemma that will be used in the next section.

Recall that a Banach space is said to satisfy Opial's condition [7] if weakly as and imply that A finite family of mappings with is said to satisfy condition (B) [8] if there is a nondecreasing function with and for all such that for all , where .

Lemma 2.1 (see [9, Theorem 2]). Let , be two fixed numbers. Then a Banach space is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function , such that for all in where

Lemma 2.2 (see [10, Lemma 1.6]). Let be a uniformly convex Banach space, a nonempty closed convex subset of , and nonexpansive mapping. Then is demiclosed at , that is, if weakly and strongly, then .

Lemma 2.3 (see [11, Lemma 2.7]). Let be a Banach space which satisfies Opial's condition and let be a sequence in . Let be such that and exist. If and are subsequences of which converge weakly to and , respectively, then .Lemma 2.4. Let be a uniformly convex Banach space and . Then for each , there exists a continuous, strictly increasing, and convex function such that for all and all with .Proof. Clearly (2.3) holds for , by Lemma 2.1. Next, suppose that (2.3) is true when . Let and with Then . By Lemma 2.1, we obtain that By the inductive hypothesis, there exists a continuous, strictly increasing and convex function such that for all and all with . It follows that Hence, we have the lemma.

3. Main Results

In this section, we prove weak and strong convergence theorems of the iterative scheme (1.1) for a finite family of nonexpansive mappings in a uniformly convex Banach space. In order to prove our main results, the following lemmas are needed.

The next lemma is crucial for proving the main theorems.

Lemma 3.1. Let be a Banach space and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of . Let for all , and . For a given , let the sequence be defined by (1.1). If , then for all and exists for all .

Proof. Let . For each we note that It follows from (3.1) that By (3.1) and (3.2), we have By continuing the above argument, we obtain that In particular, we get for all , which implies that exists.

Lemma 3.2. Let be a uniformly convex Banach space and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of with and for all , and such that are in for all and . For a given , let be defined by (1.1). If , then

(i)for all, (ii)for all, (iii)for all. Proof. (i) Let , by Lemma 3.1, Choose a number such that , it follows by (3.4) that for all .

By Lemma 2.4, there exists a continuous strictly increasing convex function such that for all with . By (3.4) and (3.5), we have for , Therefore for all . Since , it implies by Lemma 3.1 that . Since is strictly increasing and continuous at with , it follows that for all .

(ii) For , we have It follows from (i) that

(iii) For , it follows from (i) that Theorem 3.3. Let be a uniformly convex Banach space and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of with . Let the sequence be as in Lemma 3.2. For a given , let sequences and be defined by (1.1). If one of is completely continuous then and converge strongly to a common fixed point of for all .Proof. Suppose that is completely continuous where Then there exists a subsequence of such that converges.

Let for some . By Lemma 3.2 (ii), . It follows that . Again by Lemma 3.2(ii), we have for all . It implies that . By continuity of , we get , . So . By Lemma 3.1, exists, it follows that . By Lemma 3.2(iii), we have for each . It follows that for all .Theorem 3.4. Let be a uniformly convex Banach space and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of with . Let the sequence be as in Lemma 3.2. For a given , let sequences and be defined by (1.1). If the family satisfies condition (B) then and converge strongly to a common fixed point of for all .Proof. Let . Then by Lemma 3.1, exists and for all . This implies that therefore, we get exists. By Lemma 3.2(ii), we have for each . It follows, by the condition (B) that . Since is nondecreasing and , therefore, we get . Next we show that is a Cauchy sequence. Since , given any , there exists a natural number such that for all . In particular, . Then there exists such that . For all and , it follows by Lemma 3.1 that This shows that is a Cauchy sequence in , hence it must converge to a point of . Let . Since and is closed, we obtain . By Lemma 3.2(iii), for each . It follows that for all .

In Theorem 3.4, if for all , and , we obtain the following result. Corollary 3.5. Let be a uniformly convex Banach space and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of with and for all and such that are in for all . For a given , let the sequence be defined by (1.2). If the family satisfies condition (B) and , then the sequence converges strongly to a common fixed point of .Remark 3.6. In Corollary 3.5, if for all and for all , the iterative scheme (1.2) reduces to the iterative scheme (1.3) defined by Liu et al. [1] and we obtain strong convergence of the sequence defined by Liu et al. when satisfies condition (B) which is different from the condition (A) defined by Liu et al. and we note that the result of Senter and Dotson [4] is a special case of Theorem 3.4 when .

In the next result, we prove weak convergence for the iterative scheme (1.1) for a finite family of nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.Theorem 3.7. Let be a uniformly convex Banach space which satisfies Opial's condition and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of with . For a given , let be the sequence defined by (1.1). If the sequence is as in Lemma 3.2, then the sequence converges weakly to a common fixed point of .Proof. By Lemma 3.2(ii), for all Since is uniformly convex and is bounded, without loss of generality we may assume that weakly as for some . By Lemma 2.2, we have . Suppose that there are subsequences and of that converge weakly to and , respectively. From Lemma 2.2, we have . By Lemma 3.1, and exist. It follows from Lemma 2.3 that . Therefore converges weakly to a common fixed point of

For for all , and in Theorem 3.7, we obtain the following result.

Corollary 3.8. Let be a uniformly convex Banach space which satisfies Opial's condition and a nonempty closed and convex subset of . Let be a finite family of nonexpansive self-mappings of with and for all and such that are in for all . For a given , let be the sequence defined by (1.2). If , then the sequence converges weakly to a common fixed point of .Remark 3.9. In Corollary 3.8, if for all and for all , then we obtain weak convergence of the sequence defined by Liu et al. [1].

Acknowledgments

The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin University, and the Graduate School of Chiang Mai University, Thailand for their financial support.

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