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Abstract and Applied Analysis
Volume 2010, Article ID 876819, 16 pages
http://dx.doi.org/10.1155/2010/876819
Research Article

A Strong Convergence Theorem for a Common Fixed Point of Two Sequences of Strictly Pseudocontractive Mappings in Hilbert Spaces and Applications

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 17 July 2010; Accepted 27 October 2010

Academic Editor: Chaitan Gupta

Copyright © 2010 Kasamsuk Ungchittrakool. 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 prove a strong convergence theorem for a common fixed point of two sequences of strictly pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main theorem to find a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. The results extend and improve the recent ones announced by Marino and Xu (2007) and others.

1. Introduction

Let be a real Hilbert space and a nonempty closed convex subset of . Let be a self-mapping of . Recall that is said to be a strict pseudocontraction if there exists a constant such that for all . (We also say that is a -strict pseudocontraction if satisfies (1.1)). We use to denote the set of fixed points of (i.e., ). Note that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings which are mappings on such that for all . That is, is nonexpansive if and only if is a 0-strict pseudocontraction.

In 1953, Mann [1] introduced the following iterative scheme: where the sequence is chosen in . Mann's iteration process (1.3) has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [2]. In an infinite-dimensional Hilbert space, the Mann's iteration (1.3) can conclude only weak convergence [3, 4]. In 1967, Browder and Petryshyn [5] established the first convergence result for a -strict pseudocontraction in a real Hilbert space. They proved weak and strong convergence theorems by using (1.3) with a constant control sequence for all . However, this scheme has only weak convergence even in a Hilbert space. Therefore, many authors try to modify the normal Mann's iteration process to have strong convergence; see, for example, [610] and the references therein.

Attempts to modify (1.3) so that strong convergence is guaranteed have been made. In 2003, Nakajo and Takahashi [9] proposed the following modification of (1.3) for a single nonexpansive mapping by using the hybrid projection method in a Hilbert space where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one, then defined by (1.4) converges strongly to .

In 2007, Marino and Xu [11] proved the following strong convergence theorem by using the hybrid projection method for a strict pseudocontraction. They defined a sequence as follows: They proved that if , then defined by (1.5) converges strongly to .

Motivated and inspired by the above-mentioned results, it is the purpose of this paper to improve and generalize the algorithm (1.5) to the new general process of two sequences of strictly pseudocontractive mappings in Hilbert spaces. Let be a closed convex subset of a Hilbert space and two sequences of strictly pseudocontractive mappings such that . Define in the following ways: where , are sequences in .

We prove that the algorithm (1.6) converges strongly to a common fixed point of two sequences of strictly pseudocontractive mappings and provided that , , , and satisfy some appropriate conditions, and then we apply the result for finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. Our results extend and improve the corresponding ones announced by Marino and Xu [11] and others.

Throughout the paper, we will use the following notation:(i) for strong convergence and for weak convergence, (ii) denotes the weak -limit set of .

2. Preliminaries

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Lemma 2.1. Let be a real Hilbert space. There holds the following identity: (i) for all .(ii) for all , for all .

Lemma 2.2. Let be a real Hilbert space. Given a closed convex subset and . Given also a real number . The set is convex (and closed).

Recall that given a closed convex subset of a real Hilbert space , the nearest point projection from onto assigns to each its nearest point denoted by which is a unique point in with the property

Lemma 2.3. Let be a closed convex subset of real Hilbert space . Given and . Then, if and only if there holds the relation

Lemma 2.4 (Martinez-Yanes and Xu [8]). Let be a closed convex subset of real Hilbert space . Let be a sequence in and . Let . If is such that and satisfies the condition Then, .

Given a closed convex subset of a real Hilbert space and a mapping . Recall that is said to be a quasistrict pseudocontraction if is nonempty and there exists a constant such that for all and .

Proposition 2.5 (Marino and Xu [11, Proposition 2.1]). Assume is a closed convex subset of a Hilbert space , and let be a self-mapping of . (i)If is a -strict pseudocontraction, then satisfies Lipschitz condition (ii)If is a -strict pseudocontraction, then the mapping is demiclosed (at 0). That is, if is a sequence in such that and , then .(iii)If is a -quasistrict pseudocontraction, then the fixed-point set of is closed and convex so that the projection is well defined.

Lemma 2.6 (Plubtieng and Ungchittrakool [12, Lemma 3.1]). Let be a nonempty subset of a Banach space and a sequence of mappings from into . Suppose that for any bounded subset of there exists continuous increasing function from into such that and where for all . Then, for each , converges strongly to some point of . Moreover, let be a mapping from into defined by Then, .

Lemma 2.7. Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be a sequence such that for each , is -strict pseudo contraction from into with and Then, is closed and convex so that the projection is well defined.

Proof. To see that is closed, assume that is a sequence in such that . Since is a -quasistrict pseudocontraction, we get, for each , Taking the limit as yields , where . Since , we have .

3. Main Result

In this section, we prove a strong convergence theorem by using the hybrid projection method (some authors call this the method) for finding a common element of the set of fixed points of two sequences of strictly pseudocontractive mappings in Hilbert spaces.

Theorem 3.1. Let be a closed convex subset of a Hilbert space . For each , let be -strict pseudocontractions for some , with , respectively, and assume that . Let be the sequence generated by Assume that and are chosen so that and for all . Suppose that for any bounded subset of there exists an increasing, continuous, and convex function from into such that , and Let such that and for all , respectively, and suppose that and . Then, converges strongly to a common fixed point .

Proof. It is not hard to check that and are closed and convex for all (via Lemma 2.2 and the properties of the inner product). Then, if is nonempty for all , the sequence is well defined. Now, we will show that for all . Let , we observe that By (3.3) and (3.4) we obtain Thus, we have for all . Next, we will show that for all . If , then . Assume that . Since is the projection of onto , by Lemma 2.3 we have Noting that by the induction assumption, it implies that , thus by induction for all . Hence, for all . So, is well defined.
Notice that the definition of actually implies . This together with the fact further implies In particular, is bounded and where .
The fact asserts that . This together with Lemma 2.1(i) and Lemma 2.3 implies It turns out that By the fact , we get Observe that With simple calculation by using (3.12) and (3.4), we have So, when we combine (3.11) and (3.13) and compute, we obtain Since for all , we have Notice that By (3.15), (3.16), and (3.17), we have Since , , and are bounded, for all and , , it follows from (3.10) and (3.18) that Since is bounded, there exists a bounded subset of such that . From Lemma 2.6, we are able to set for all , and then observe that Since is an increasing, continuous, and convex function from into such that , we discover that By Lemma 2.6 and the continuity of , we have . And then the properties of yield By the same argument, we have Now Proposition 2.5 guarantees that . This fact, the inequality (3.8), and Lemma 2.4 ensure the strong convergence of to .

If and for all , then and for all . So, Theorem 3.1 reduces to the following corollary.

Corollary 3.2. Let be a closed convex subset of a Hilbert space . Let be -strict pseudocontractions for some , , respectively, and assume that . Let be the sequence generated by where and be as in Theorem 3.1. Then, converges strongly to a common fixed point .

In particular, if , then and So, Corollary 3.2 reduces to the following corollary.

Corollary 3.3 (Marino and Xu [11, Theorem 4.1]). Let be a closed convex subset of a Hilbert space . Let be a -strict pseudocontraction for some , and assume that the fixed-point set . Let be the sequence generated by Assume that the control sequence is chosen so that for all . Then, converges strongly to a fixed-point .

If for all and is a sequences of nonexpansive mappings, then and for all . So, Theorem 3.1 reduces to the following corollary.

Corollary 3.4. Let be a closed convex subset of a Hilbert space . Let be a -strict pseudocontraction for some , for each and a nonexpansive mapping, and assume that . Let be the sequence generated by where and be as in Theorem 3.1. Suppose that for any bounded subset of there exists an increasing, continuous, and convex function from into such that , and Let be such that for all , and suppose . Then, converges strongly to a common fixed point .

4. Equilibrium Problem

In this section, we have an application of the main result for finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem.

Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that The set of solution of (4.1) is denoted by . Many problems in physics, optimization, and economics reduce to find some elements of .

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions: (A1) for all ; (A2) is monotone, that is, for all ; (A3) for each , (A4) for each , is convex and lower semicontinuous.

The following lemma appears implicitly in [13].

Lemma 4.1 (Blum and Oettli [13]). Let be a nonempty closed convex subset of , and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

The following lemma was also given in [14].

Lemma 4.2 (Combettes and Hirstoaga [14]). Assume that satisfies (A1)–(A4). For and , define a mapping as follows: for all . Then, the following hold:(1) is single-valued; (2) is firmly nonexpansive, that is, for any , ; (3); (4) is closed and convex.

The following corollary is an application of Corollary 3.4 in the case of finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem.

Corollary 4.3. Let be a closed convex subset of a Hilbert space . Let be a -strict pseudocontraction for some and a bifunction from into satisfying (A1)–(A4). Suppose that . Let be the sequence generated by where and be as in Theorem 3.1 and is chosen so that with and . Then, converges strongly to a common fixed point .

Proof. Obviously, , where are mappings as in Lemma 4.2. Next, we want to show that for any bounded subset of there exists an increasing, continuous, and convex function from into such that , and Let be a bounded subset of . For each , let . Then, by Lemma 4.2, we have Put in (4.7) and in (4.8), we have So, from (A2), we have and hence . Thus, Let . Thus, we have Put . Observe that for all , and then Let , we have . Then, by Lemma 2.6, we can define a mapping by Next, we will show that Since , and (4.14), it is easy to see that Let . By the definition of , we have By (A2), we have From (4.14) and the lower semicontinuity of , we have . Let and set , for . Then, we have So . Letting and using (A3), we get , and hence . Hence, we have (4.15). Then, applying Corollary 3.4, .

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand. The author would like to thank the referees for reading this paper carefully and providing valuable suggestions and comments of this paper.

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