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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 349158, 14 pages
http://dx.doi.org/10.1155/2010/349158
Research Article

A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Received 26 January 2010; Accepted 21 April 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Watcharaporn Cholamjiak 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 monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established.

1. Introduction

Let be a nonempty convex subset of a Banach spaces . Let be a bifunction from to , where is the set of all real numbers. The equilibrium problem for is to find such that for all . The set of such solutions is denoted by . The set is called proximal if for each , there exists an element such that , where . Let , , and denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of , respectively. The Hausdorff metric on is defined by for . A single-valued map is called nonexpansive if for all . A multivalued map is said to be nonexpansive if for all . An element is called a fixed point of (resp., ) if (resp., ). The set of fixed points of is denoted by . The mapping is called quasi-nonexpansive [1] if and for all and all . It is clear that every nonexpansive multivalued map with is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive; see [2].

The mapping is called hemicompact if, for any sequence in such that as , there exists a subsequence of such that . We note that if is compact, then every multivalued mapping is hemicompact.

A mapping is said to satisfy Condition (I) if there is a nondecreasing function with , for such that for all .

In 1953, Mann [3] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping in a Hilbert space : where the initial point is taken in arbitrarily and is a sequence in .

However, we note that Mann's iteration process (1.3) has only weak convergence, in general; for instance, see [46].

In 2003, Nakajo and Takahashi [7] introduced the method which is the so-called CQ method to modify the process (1.3) so that strong convergence is guaranteed. They also proved a strong convergence theorem for a nonexpansive mapping in a Hilbert space.

Recently, Tada and Takahashi [8] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space .

In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for multivalued map with a fixed point converge to a fixed point of under certain conditions. They also claimed that the fixed point may be different from . More precisely, they proved the following result for nonexpansive multivalued map with compact domain.

In 2007, Panyanak [10] extended the above result of Sastry and Babu [9] to uniformly convex Banach spaces but the domain of remains compact.

Later, Song and Wang [11] noted that there was a gap in the proofs of Theorem [10] and Theorem [9]. They further solved/revised the gap and also gave the affirmative answer to Panyanak [10] question using the following Ishikawa iteration scheme. In the main results, domain of is still compact, which is a strong condition (see [11, Theorem ]) and satisfies condition (I) (see [11, Theorem ]).

In 2009, Shahzad and Zegeye [2] extended and improved the results of Panyanak [10], Sastry and Babu [9], and Song and Wang [11] to quasi-nonexpansive multivalued maps. They also relaxed compactness of the domain of and constructed an iteration scheme which removes the restriction of , namely, for any . The results provided an affirmative answer to Panyanak [10] question in a more general setting. In the main results, satisfies Condition (I) (see [2, Theorem ]) and is hemicompact and continuous (see [2, Theorem ]).

A mapping is called -inverse-strongly monotone [12] if there exists a positive real number such that

Remark 1.1. It is easy to see that if is -inverse-strongly monotone, then it is a -Lipschitzian mapping.
Let be a mapping. The classical variational inequality problem is to find a such that The set of solutions of variational inequality (3.9) is denoted by .

Question 1. How can we construct an iteration process for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inequality problem, and the set of common fixed points of nonexpansive multivalued maps ?

In the recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points of single-valued nonexpansive mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; for instance, see [8, 1320] and the references cited theorems.

In this paper, we introduce a monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a countable family of nonexpansive multivalued maps, the set of variational inequality, and the set of solutions of an equilibrium problem in a Hilbert space.

2. Preliminaries

The following lemmas give some characterizations and a useful property of the metric projection in a Hilbert space.

Let be a real Hilbert space with inner product and norm . Let be a closed and convex subset of . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . We know that is a nonexpansive mapping of onto .

Lemma 2.1 (see [21]). Let be a closed and convex subset of a real Hilbert space and let be the metric projection from onto . Given and , then if and only if the following holds:

Lemma 2.2 (see [7]). Let be a nonempty, closed and convex subset of a real Hilbert space and the metric projection from onto . Then the following inequality holds:

Lemma 2.3 (see [21]). Let be a real Hilbert space. Then the following equations hold: (i), for all ;(ii), for all and .

Lemma 2.4 (see [22]). Let be a nonempty, closed and convex subset of a real Hilbert space . Given and also given , the set is convex and closed.

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

Lemma 2.5 (see [13]). Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4) and let and . Then, there exists such that

Lemma 2.6 (see [18]). For , , defined a mapping as follows: Then the following holds:(1) is a single value;(2) is firmly nonexpansive, that is, for any , (3);(4) is closed and convex.

In the context of the variational inequality problem, A set-valued mapping is said to be monotone if for all , , and imply that . A monotone mapping is said to be maximal [23] if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , , imply that . Let be an inverse strongly monotone mapping and let be the normal cone to at , that is, and define Then is maximal monotone and if and only if (see, e.g., [24]).

In general, the fixed point set of a nonexpansive multivalued map is not necessary to be closed and convex (see [25, Example ]). In the next Lemma, we show that is closed and convex under the assumption that for all .

Lemma 2.7. Let be a closed and convex subset of a real Hilbert space . Let be a nonexpansive multivalued map with and for each . Then is a closed and convex subset of .

Proof. First, we will show that is closed. Let be a sequence in such that as . We have It follows that , so . Next, we show that is convex. Let where and . Let ; by Lemma 2.3, we have Hence . Therefore, .

3. Main Results

In the following theorem, we introduce a new monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a countable family of nonexpansive multivalued maps, the set of variational inequality, and the set of solutions of an equilibrium problem in a Hilbert space, and we prove strong convergence theorem without the condition (I).

Theorem 3.1. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4), let be an -inverse strongly monotone mapping, and let be nonexpansive multivalued maps for all with and , . Assume that with for all , for some , and for some . For an initial point with and , let , , , and be sequences generated by where . Then, , , and converge strongly to .

Proof. We split the proof into six steps.
Step 1. Show that is well defined for every .
Since for all , we get that is nonexpansive for all . Hence, is closed and convex. By Lemma 2.6(4), we know that is closed and convex. By Lemma 2.7, we also know that is closed and convex. Hence, is a nonempty, closed and convex set. By Lemma 2.4, we see that is closed and convex for all . This implies that is also closed and convex. Therefore, is well defined. Let and . From , we have for every . From this, we have So, we have , hence , . This shows that .
Step 2. Show that exists.
Since is a nonempty closed convex subset of , there exists a unique such that From , and , , we get On the other hand, as , we obtain It follows that the sequence is bounded and nondecreasing. Therefore, exists.
Step 3. Show that as .
For , by the definition of , we see that . By Lemma 2.2, we get From Step 2, we obtain that is Cauchy. Hence, there exists such that as .
Step 4. Show that .
From Step 3, we get as . Since , we have as for all , as . Hence, as . It follows from (3.9) and (3.10) that as for all . For each , we have From (3.11), we obtain . Hence .
Step 5. Show that .
By the nonexpansiveness of and the inverse strongly monotonicity of , we obtain This implies It follows from (3.10) that Since is firmly nonexpansive, we have This implies that It follows that From (3.10) and (3.15), we get It follows from (3.10) and (3.19) that Since , we have From the monotonicity of , we have hence From (3.20) and condition (A4), we have For with and , let . Since and is convex, then and hence . So, we have Dividing by , we obtain Letting and from (A3), we get Therefore, we obtain .
Step 6. Show that .
Since is the maximal monotone mapping defined by (2.10), For any given , hence . It follows that On the other hand, since , we have and so From (3.29), (3.31), and the -inverse monotonicity of , we have It follows that Again since is maximal monotone, hence . This shows that .
Step 7. Show that .
Since and , we obtain By taking the limit in (3.34), we obtain This shows that .
From Steps 3 to 5, we obtain that , , and converge strongly to . This completes the proof.

Theorem 3.2. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be nonexpansive multivalued maps for all with and , for all . Assume that with and for some . For an initial point with and , let , , and be sequences generated by where . Then, and converge strongly to .

Proof. Putting for all in Theorem 3.1, we obtain the desired result directly from Theorem 3.1.

Theorem 3.3. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be nonexpansive multivalued maps for all with and , for all . Assume that with . For an initial point with and , let and be sequences generated by where . Then, converge strongly to .

Proof. Putting in Theorem 3.2, we obtain the desired result directly from Theorem 3.2.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics and the Graduate School of Chiang Mai University.

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