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Abstract and Applied Analysis
Volume 2010, Article ID 706587, 15 pages
http://dx.doi.org/10.1155/2010/706587
Research Article

A New Approximation Method for Common Fixed Points of a Finite Family of Generalized Asymptotically Quasinonexpansive Mappings in Banach Spaces

Department of Mathematics, Faculty of Science, Chiangmai University, Chiangmai 50200, Thailand

Received 24 June 2010; Accepted 6 September 2010

Academic Editor: Nikolaos Papageorgiou

Copyright © 2010 Pornsak Yatakoat 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 a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasinonexpansive mappings. Several strong and weak convergence theorems of the proposed iteration in Banach spaces are established. The main results obtianed in this paper generalize and refine many known results in the current literature.

1. Introduction

Let be a convex subset of a Banach space , and let be a family of self-mappings of . Suppose that for all and

For , let be the sequence generated by the following algorithm: where for all . The iterative process (1.1) for a finite family of mappings introduced by Khan et al. [1], and the iterative process is the generalized form of the modified Mann (one-step) iterative process by Schu [2], the modified Ishikawa (two-step) iterative process by Tan and Xu [3], and the three-step iterative process by Xu and Noor [4].

Common fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities. Many researchers [1, 519] are interested in studying approximation method for finding common fixed points of nonlinear mapping. Also, approximation methods for finding fixed points for nonexpansive mappings can be seen in [1216, 20, 21].

In 2003, Sun [17] studied an implicit iterative scheme initiated by Xu and Ori [22] for a finite family of asymptotically quasinonexpansive mappings. Shahzand and Udomene [18], in 2006, proved some convergence theorems for the modified Ishikawa iterative process of two asymptotically quasinonexpence mappings to a common fixed point. Nammanee et al. [23] introduced a three-step iteration scheme for asymptotically nonexpansive mappings and proved weak and strong convergence theorems of that iteration scheme under some control conditions. In 2007, Fukhar-ud-din and Khan [24] studied a new three-step iteration scheme for approximating a common fixed point of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Shahzad and Zegeye [19] introduced a new concept of generalized asymptotially nonexpansive mappings and proved some strong convergence theorems for fixed points of finite family of this class. Recently, Khan et al. [1] introduced the iterative sequence (1.1) for a finite family of asymptotically quasinonexpansive mappings in Banach spaces.

Motivated by Khan et al. [1], we introduce a new iterative scheme for finding a common fixed point of a finite family of generalized asymptotically quasinonexpansive mappings as follows:

For , let be the sequence generated by where for all .

The aim of this paper is to obtain strong and weak convergence results for the iterative process (1.2) of a finite family of generalized asymptotically quasinonexpansive mappings in Banach spaces.

2. Preliminaries

In this section, we give some definitions and lemmas used in the main results.

Let be a nonempty subset of a real Banach space , and let be a self-mapping of . The fixed point set of is denoted by .

Then let is called(i)nonexpansive if , for all ; (ii)quasinonexpansive if and , for all and ; (iii)asymptotically nonexpansive if there exists a sequence in with and , for all and ; (iv)asymptotically quasinonexpansive if and there exists a sequence in with and , for all , and ; (v)generalized quasinonexpansive if and there exists a sequence in with as such that for all , and ;(vi)generalized asymptotically quasinonexpansive [19] if and there exist two sequences and in with and as such that for all , and ;(vii)uniformly -Lipschitzian if there exists constant such that , for all and ; (viii) uniform Lipschitz if there are constants and such that , for all and . (ix)semicompact if for a sequence in with , there exists a subsequence of such that

From the definition of these mappings, it can be seen that (i)a quasinonexpansive mapping is generalized quasinonexpansive;(ii)an asymptotically quasinonexpansive mapping is generalized asymptotically quasinonexpansive;(iii)a generalized quasinonexpansive mapping is generalized asymptotically quasinonexpansive;(iv)a uniformly -Lipschitzian mapping is uniform Lipschitz.

The map is said to be demiclosed at 0 if for each sequence in converging weakly to and converging strongly to 0, we get .

A Banach space is said to have Opial's property if for each sequence converging weakly to and , we have the condition

Condition 2 ( ). Let be a subset of a normed space . A family of self-mappings of is said to have Condition if there exists a nondecreasing function with and for all such that for some and for all where .

The condition defined above by the authors is the generalization of the condition (A) [25] when and condition [26] for .

The following lemmas are needed for proving our main results.

Lemma 2.1 (cf. [17, Lemma ]). Let the sequences , and of real numbers satisfy: and , . (i) exists;(ii)if then

Lemma 2.2 (see [2, Lemma ]). Let be a uniformly convex Banach space. Assume that , Let the sequences and in be such that , and , where . Then .

3. Convergence in Banach Spaces

The aim of this section is to establish the strong convergence of the iterative scheme (1.2) to converge to a common fixed point of a finite family of asymptotically quasinonexpansive mappings in a Banach space under some appropriate conditions.

Lemma 3.1. Let be a nonempty closed convex subset of a real Banach space , and be a finite family of generalized asymtotically quasinonexpansive self-mappings of , that is, for all with the sequence and , . Suppose that and the iterative sequence is defined by (1.2). Let and Then for , we get the following:(i) (ii) (iii) ; (iv) ; (v) where and (vi) If and , then exists.

Proof. Let . (i)For we have (ii)Similarly to part (i), we have (iii)By part (i) and , we have (iv)By part (ii) and part (iii), we get (v) Put and Then (v) is directly obtained by (iv).(vi)By , we have for all , where and From and , it follows that and . By Lemma 2.1, we get exists.

Theorem 3.2. Let be a nonempty closed convex subset of a real Banach space , and be a finite family of generalized asymtotically quasinonexpansive self-mappings of , that is, for all and . Suppose that is closed, and the iterative sequence is defined by (1.2). Assume that and , where and . Then converges strongly to a common fixed point of the family of mappings if and only if where .

Proof. The necessity is obvious and then we prove only the sufficiency. Let Since for , we obtain , for Thus by Lemma 3.1(iv) and (v), for positive integers and , we have where .
By Lemma 3.1(v), we have where and .
It follows that From the given condition and Lemma 2.1, we get
Next, we show that is a Cauchy sequence in . By (3.8) and , we get that for any , there exists a positive integer such that, for all ,
From there exists such that
For any positive integer , by (3.5), (3.9), and (3.10), we have Thus, is a Cauchy sequence in . Since is complete, . Actually, because and is a closed subset of . Next we show that . Since is closed, by the continuity of with and as , we get and then . Therefore, the proof is complete.

Since any asymptotically quasinonexpansive mapping is generalized asymptotically quasinonexpansive, the next corollary is obtained immediately from Theorem 3.2.

Corollary 3.3 (see [5, Theorem ]). Let be a nonempty closed convex subset of a real Banach space , and be a finite family of asymptotically quasinonexpansive self-mappings of , that is, for all and , . and the iterative sequence be defined by (1.2). Then converges strongly to a common fixed point of the family of mappings if and only if where .

4. Convergence in Uniformly Convex Banach Spaces

In this section, strong and weak convergence results for the iterative process (1.2) on uniformly convex Banach spaces are proved without using the condition appearing in Section 3.

Theorem 4.1. Let be a nonempty closed convex subset of an uniformly convex real Banach space . Let be a finite family of uniformly Lipschitzian and generalized asymptotically quasinonexpansive self-mappings of , that is, and for all and , . Suppose that satisfies condition and . Let and the iterative sequence be defined by (1.2) with , where . Assume that where and Then converges strongly to a common fixed point of the family of mappings.

Proof. Let By Lemma 3.1(vi), we get that exists. Then there is a real number such that
By Lemma 3.1(iii), we have
By taking on both sides of the above inequality, we get Since and (4.3), we obtain Since , we have Using (4.1), (4.4), and Lemma 2.2, we conclude that We assume that It follows from (4.5) and (4.7) that By Lemma 3.1(iv), (1.2), and (4.8), we get Using (4.3), (4.4) and Lemma 2.2, we conclude that Therefore, by mathematical induction, we obtain
From (1.2), we have By (4.11), we obtain that From for , it follows by (4.13) that From (4.11), when , we get . For , we have
From (4.11) and (4.15), we conclude that where . From (1.2), we have From (4.11) and (4.15), For we have Using (4.17) and (4.19), we obtain Therefore, by using condition , there exists a nondecreasing function with and for all such that for some that is By Theorem 3.2, we conclude that converges strongly to a point .

Lemma 4.2. Let be a nonempty closed convex subset of an uniformly convex real Banach space , and be a family of uniform Lipschitz and generalized asymtotically quasinonexpansive self-mappings of , that is, and for all and , . Suppose that Let and the iterative sequence be defined by (1.2) with , where . Assume that , where and Then, (i) , for all ; (ii) , for all .

Proof. (i) Let . By Lemma 3.1(vi), we obtain that exists and we then suppose that By (4.24) and Lemma 3.1(iii), we have
By (1.2), we have for . It follows that From (4.25) and (4.27), we obtain and then for
Since , for , we have From (4.25), (4.29), (4.30) and Lemma 2.2, we obtain Now we want to show that (4.31) is also true for .
Since , and , it follows by (4.28), We also have Hence, by (4.25), (4.32), and Lemma 2.2, we obtain Then, (4.31) and (4.34) give us From it implies by (4.15) and (4.35) that for
(ii) From part (i), for , we have For we get By part (i) and (4.15), For , we obtain From (4.19) and (4.40), we then have

Theorem 4.3. Under the hypotheses of Lemma 4.2, assume that is semicompact for some positive integers and . Then converges strongly to a common fixed point of the family .

Proof. Suppose that is semicompact for some positive integers and . We have Then, by Lemma 4.2(ii), we get as . Since is bounded and is semicompact, there exists a subsequence of such that as .
By continuity of and Lemma 4.2(ii), we obtain Therefore, and then Theorem 3.2 implies that converges strongly to a common fixed point of the family .

We note that in practical Theorem 4.3 is very useful in the case that one of , is semicompact.

Theorem 4.4. Let be a nonempty closed convex subset of an uniformly convex real Banach space satisfying the Opial property, and be a family of uniform Lipschitz and generalized asymtotically quasinonexpansive self-mappings of , that is, and for all and . Suppose that . Let and the iterative sequence be defined by (1.2) with , where . Assume that , where If is demiclosed at 0, then converges weakly to a common fixed point of the family of mappings.

Proof. Let . By Lemma 3.1(v), we get exists. Then we follow the proof of Theorem 3.2 by Khan et al. [1] until we can conclude that converges weakly to a common fixed point .

Acknowledgments

This paper was supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Ph. D. Program Thai Doctoral degree from the Office of the Higher Education Commission, the Graduate School of Chiang Mai University, and the Thailand Research Fund.

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