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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 825130, 8 pages
http://dx.doi.org/10.1155/2013/825130
Research Article

Strong Convergence Results for Equilibrium Problems and Fixed Point Problems for Multivalued Mappings

1Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 21 July 2013; Accepted 10 October 2013

Academic Editor: Mohamed Amine Khamsi

Copyright © 2013 J. Vahidi 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

Using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition () in the setting of Hilbert space. Our results improve and extend some recent results in the literature.

1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset . A subset is called proximal if, for each , there exists an element such that

A single-valued mapping is said to be nonexpansive, if

Let be a nearest point projection of into ; that is, for , is a unique nearest point in with the property

We denote by , , and the collection of all nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of respectively. The Hausdorff metric on is defined by for all .

Let be a multivalued mapping. An element is said to be a fixed point of , if and the set of fixed points of is denoted by .

A multivalued mapping is called (i)nonexpansive if (ii)quasi-nonexpansive if and for all and all .

Recently, García-Falset et al. [1] introduced a new condition on single-valued mappings, called condition , which is weaker than nonexpansiveness.

Definition 1. A mapping is said to satisfy condition provided that
We say that satisfies condition whenever satisfies for some .

Recently, Abkar and Eslamian [2, 3] generalized this condition for multivalued mappings as follows.

Definition 2. A multivalued mapping is said to satisfy condition provided that for some .

It is obvious that every nonexpansive multivalued mapping satisfies the condition , and every mapping which satisfies the condition with nonempty fixed point set is quasi-nonexpansive.

Example 3. Let us define a mapping on by It is easy to see that satisfies the condition but is not nonexpansive. Indeed, for , . Let and . Then . If and , then, we have and ; hence Thus, satisfies the condition . However, is not nonexpansive; indeed for and , .
Let be a bifunction. The equilibrium problem associated with the bifunction and the set is: Such a point is called the solution of the equilibrium problem. The set of solutions is denoted by .
A broad class of problems in optimization theory, such as variational inequality, convex minimization, and fixed point problems, can be formulated as an equilibrium problem; see [4, 5]. In the literature, many techniques and algorithms have been proposed to analyze the existence and approximation of a solution to equilibrium problem; see [6]. Many researchers have studied various iteration processes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of a class of nonlinear mappings. For example, see [722].
Fixed points and fixed point iteration process for nonexpansive mappings have been studied extensively by many authors to solve nonlinear operator equations, as well as variational inequalities; see, for example, [2328]. In the recent years, fixed point theory for multivalued mappings has been studied by many authors; see [2940] and the references therein.
In this paper, using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition in the setting of Hilbert space. Our results improve and extend some recent results in the literature.

2. Preliminaries

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

Lemma 4 (see [4]). Let be a nonempty closed convex subset of and let be a bifunction of into satisfying . Let and . Then, there exists such that

Lemma 5 (see [6]). Assume that satisfies . For and , define a mapping as follows: Then, the following hold: (i) is single valued; (ii) is firmly nonexpansive; that is, for any , (iii); (iv) is closed and convex.

Lemma 6 (see [41]). Let be a real Hilbert space. Then, for all and with one has

Lemma 7. For every and in a Hilbert space , the following inequality holds:

Lemma 8 (see [42]). Let be a sequence of nonnegative real numbers, a sequence in with , a sequence of nonnegative real numbers with , and a sequence of real numbers with . Suppose that the following inequality holds: Then, .

Lemma 9 (see [43]). Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of such that for all . For every sufficiently large number , define an integer sequence as Then, as and for all ,

Lemma 10 (see [20]). Let be a closed convex subset of a real Hilbert space . Let be a quasi-nonexpansive multivalued mapping. If and for all . Then is closed and convex.

Lemma 11 (see [20]). Let be a closed convex subset of a real Hilbert space . Let be a multivalued mapping such that is quasi-nonexpansive and , where . Then, is closed and convex.

Lemma 12 (see [16, 20]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a multivalued mapping satisfying the condition . If converges weakly to and , then .

3. A Strong Convergence Theorem

Theorem 13. Let be a nonempty closed convex subset of a real Hilbert space and a bifunction of into satisfying . Let () be a finite family of multivalued mappings, each satisfying condition . Assume further that and for each . Let be a -contraction of into itself. Let and be sequences generated the following algorithm: where , for , and , , , , and satisfy the following conditions: (i), (ii), , , (iii), and . Then, the sequences and converge strongly to , where .

Proof. Let . It is easy to see that is a contraction. By Banach contraction principle, there exists a such that . From Lemma 5 for all , we have We show that is bounded. Since, for each , satisfies the condition and we have
By continuing this process, we obtain
This implies that By induction, we get for all . This implies that is bounded and we also obtain that , and are bounded. Next, we show that for each . By Lemma 6, we have Applying Lemma 6 once more, we have By continuing this process we have which implies that Therefore, we have that In order to prove that as , we consider the following two cases.
Case 1. Suppose that there exists such that is nonincreasing, for all . Boundedness of implies that is convergent. From (31) and conditions (i), (ii) we have that By a similar argument, for , we obtain that Hence, Therefore, we have For , we have Using the previous inequality for , we have Next, we show that where . To show this inequality, we choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that converges weakly to . Since , we have converges weakly to . We show that . Let us show . Since , we have From (A2), we have Replacing with , we have From (A4), we have For and , let . Since , and is convex, we have and hence . So, from (A1) and (A4) we have which gives . Letting , we have, for each , Also, since and , by Lemma 12 we have . Hence, . Since and , it follows that By using Lemma 7 and inequality (31) we have
This implies that From Lemma 8, we conclude that the sequence converges strongly to .
Case 2. Assume that there exists a subsequence of such that for all . In this case, from Lemma 9, there exists a nondecreasing sequence of for all (for some large enough) such that as and the following inequalities hold for all From (31) we obtain , and . Following an argument similar to that in Case 1, we have Thus, by Lemma 9 we have Therefore, converges strongly to . This completes the proof.

Now, we remove the condition that for all , and state the following theorem.

Theorem 14. Let be a nonempty closed convex subset of a real Hilbert space and a bifunction of into satisfying . Let, for each , be multivalued mappings such that satisfies the condition . Assume that . Let be a -contraction of into itself. Let and be sequences generated the following algorithm: where , for , and , , , and, satisfy the following conditions: (i),, (ii), , , (iii), and . Then, the sequences and converge strongly to , where .

Proof. Let ; then . Now by substituting instead of , and using a similar argument as in the proof of Theorem 13, the desired result follows.

As a corollary for single-valued mappings, we obtain the following result.

Corollary 15. Let be a nonempty closed convex subset of a real Hilbert space and a bifunction of into satisfying . Let, for each , be a finite family of mappings satisfying condition . Assume that . Let be a -contraction of into itself. Let and be sequences generated the following algorithm: where , ,, , and satisfy the following conditions: (i), ,(ii), , (iii), and . Then, the sequences and converge strongly to , where .

Remark 16. Our results generalize the corresponding results of S. Takahashi and W. Takahashi [9] from a single valued nonexpansive mapping to a finite family of multivalued mappings satisfying the condition . Our results also improve the recent results of Eslamian [16].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors therefore, acknowledge with thanks DSR for technical and financial support. The authors are also thankful to the referees for their valuable suggestion/comments.

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