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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 236269, 9 pages
http://dx.doi.org/10.1155/2008/236269
Research Article

Strong Convergence Theorem for Two Commutative Asymptotically Nonexpansive Mappings in Hilbert Space

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 20 June 2008; Accepted 8 December 2008

Academic Editor: Enrico Obrecht

Copyright © 2008 Jianjun Liu et al. 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

is a bounded closed convex subset of a Hilbert space , and are two asymptotically nonexpansive mappings such that . We establish a strong convergence theorem for and in Hilbert space by hybrid method. The results generalize and unify many corresponding results.

1. Introduction

Let be a bounded closed convex subset of a Hilbert space . Recall that a mapping is said to be asymptotically nonexpansive mapping ifwhere We may assume that for all . Denote by the set of fixed points of . Throughout this paper and are two commutative asymptotically nonexpansive mappings with asymptotical coefficients and , respectively. Suppose that ([1,  Goebel and Kirk's theorem] makes it possible). It is well known that and are convex and closed [1, 2], so is . denotes the metric projection from onto a closed convex subset of and denotes the weak -limit set of . It is well known that a Hilbert space satisfies Opial's condition [3], that is, if a sequence converges weakly to an element and , then

Up to now, fixed points iteration processes for nonexpansive and asymptotically nonexpansive mappings have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities [46]. There are many strong convergence theorems for nonexpansive and asymptotically nonexpansive mappings in Hilbert space [7, 8].

Especially, Shimizu and Takahashi [7] studied the following iteration process of nonexpansive mappings for arbitrary :where And then they proved that converges strongly to . This result was extended to two commutative asymptotically nonexpansive mappings by Shioji and Takahashi [9].

Recently, some attempts to the modified Mann iteration method are made so that strong convergence is guaranteed. And for hybrid method proposed by Haugazeau [10], Kim and Xu [8] introduced the following iteration processes for asymptotically nonexpansive mapping :where as . Then proved that converges strongly to . This result was generalized to two asymptotically nonexpansive mappings by Plubtieng and Ungchittrakool [11].

On the basis of (1.3) and (1.4), we propose a new iteration processes for two commutative asymptotically nonexpansive mappings and :where , for every The purpose of this paper is to prove converges strongly to .

2. Auxiliary Lemmas

This section collects some lemmas which will be used to prove the main results in the next section.

Lemma 2.1 (see [7]). Letting , there holds the identity in a Hilbert space :for , and .

Lemma 2.2. Let be a bounded closed convex subset of a Hilbert space , and two commutative asymptotically nonexpansive mappings of into itself with asymptotical coefficients and , respectively. For any , put . Then

Proof. Put , and . It follows from Lemma 2.1 thatChoose , then there exists a constant such that for all nonnegative integer , and . Hence, for all nonnegative integer , and . SoSimilarly, we can prove

Remark 2.3. Lemma 2.2 extends [7, Lemma 1].

Lemma 2.4. Let and be two commutative asymptotically nonexpansive mappings defined on a bounded closed convex subset of a Hilbert space with asymptotical coefficients and , respectively. Let . If is a sequence in such that converges weakly to some and converges strongly to 0, then .

Proof. We claim that converges strongly to as . If not, there exist a positive number and a subsequence of such that for all . However, we haveBy Opial's condition, for any with , we haveLet and choose a positive number such thatThen, there exists a subsequence of such that and for all . By definition of , there exists a positive integer such thatfor all . Sinceand is bounded, there exists a positive integer such thatfor all and . By is bounded and Lemma 2.2, there exist and such thatfor all . By (2.7), (2.10), (2.12), and (2.13), we havefor all . However,for all . This contradicts (2.8). So converges strongly to and then . Similarly, we can get . Hence, is a common fixed point of and .

Lemma 2.5 (see [12]). Let be a bounded closed convex subset of a Hilbert space . The set is convex and closed for given and .

3. Main Results

In this section, we prove our main theorem.

Theorem 3.1. Let be a bounded closed convex subset of a Hilbert , and be two commutative asymptotically nonexpansive mappings with asymptotical coefficients and , respectively. Suppose that for all , where . If , then the sequence generated by (1.5) converges strongly to .

Proof. Note that is convex and closed for all by Lemma 2.5. On the other hand, is convex and closed. So is .
By definition of and , there exists such that for all . On the other hand, for arbitrary , there exists such that for all . HenceThus . Obviously, .
Next, we prove that . Indeed, first of allfor all . So . It suffices to show that for all We prove this by induction. For , we have Assume that Since is the projection of onto , we haveAs , (3.3) holds for all , in particular. This together with the definition of implies that . Hence, for all .
We will show that as By the definition of , we have that . It follows from that . This shows that the sequence is increasing. Since is bounded, we obtain that exists. Notice again that from and , we have . Hence
Now we claim that as By the definition of , we haveSince , as . So as . This implies that
Since is bounded closed convex, . It follows from (3.6) and Lemma 2.4 that . By the definition of , we have that for all It follows from the weak lower semi-continuity of the norm that for all . Since , we have for all . Thus . Then, converges to weakly. By the factwe have converges to strongly. This completes the proof.

The following corollary follows from Theorem 3.1.

Corollary 3.2. Let be a bounded closed convex subset of a Hilbert , and be two commutative nonexpansive mappings. Suppose that for all , where . If , then the sequence generated byconverges strongly to .

Acknowledgments

This work is supported by National Natural Science Foundation of China (10771173).

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