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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 235474, 30 pages
http://dx.doi.org/10.1155/2012/235474
Research Article

Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive Mappings for Generalized Equilibrium Problems

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 16 July 2012; Accepted 27 August 2012

Academic Editor: Juan Torregrosa

Copyright © 2012 Rabian Wangkeeree and Nimit Nimana. 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 viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

1. Introduction

Let be a real Hilbert space with inner product and norm and a nonempty closed convex subset of and let be a mapping of into . Then, is said to be nonexpansive if for all . A mapping is said to be quasi-nonexpansive if for all and . Recall that a mapping is said to be -inverse strongly monotone if there exists a positive real number such that If is an -inverse strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous.

Let be a bifunction and be -inverse strongly monotone mapping. The generalized equilibrium problem (for short, ) for and is to find such that The problem (1.2) was studied by Moudafi [1]. The set of solutions for the problem (1.2) is denoted by , that is, If in (1.2), then reduces to the classical equilibrium problem and is denoted by , that is, If in (1.2), then reduces to the classical variational inequality and is denoted by , that is, The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, min-max problems, and the Nash equilibrium problems in noncooperative games, see, for example, Blum and Oettli [2] and Moudafi [3].

In 2005, Combettes and Hirstoaga [4] introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi [5] introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao [6] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings. Maingé and Moudafi [7] introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree [8] introduced a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. Wangkeeree and Kamraksa [9] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Their results extend and improve many results in the literature.

In 1953, Mann [10] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping in a Hilbert space as follows: where the initial point is taken in arbitrarily and is a sequence in . Wittmann [11] obtained the strong convergence results of the sequence defined by (1.6) to under the following assumptions: (C1); (C2); (C3), where is the metric projection of onto . In 2000, Moudafi [12] introduced the viscosity approximation method for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let be a contraction on . Starting with an arbitrary initial , define a sequence recursively by where is a sequence in . It is proved [12, 13] that under conditions (C1), (C2), and (C3) imposed on , the sequence generated by (1.7) strongly converges to the unique fixed point of which is a unique solution of the variational inequality Suzuki [14] considered the Meir-Keeler contractions, which is extended notion of contractions and studied equivalency of convergence of these approximation methods.

Using the viscosity approximation method, in 2007, S. Takahashi and W. Takahashi [5] introduced an iterative scheme for finding a common element of the solution set of the classical equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial , define sequences and recursively by They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to , where .

On the other hand, in 2008, Takahashi et al. [15] has adapted Nakajo and Takahashi’s [16] idea to modify the process (1.6) so that strong convergence has been guaranteed. They proposed the following modification for a family of nonexpansive mappings in a Hilbert space: , , and where for all . They proved that if satisfies the appropriate conditions, then generated by (1.10) converges strongly to a common fixed point of .

Very recently, Kimura and Nakajo [17] considered viscosity approximations by using the shrinking projection method established by Takahashi et al. [15] and the modified shrinking projection method proposed by Qin et al. [18], for finding a common fixed point of countably many nonlinear mappings, and they obtained some strong convergence theorems.

Motivated by these results, we introduce the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

2. Preliminaries

Throughout this paper, we denote by the set of positive integers and by the set of real numbers. Let be a real Hilbert space with inner product and norm . We denote the strong convergence and the weak convergence of to by and , respectively. From [19], we know the following basic properties. For and we have We also know that for , we have

For every point , there exists a unique nearest point of , denoted by , such that for all . is called the metric projection from onto . It is well known that , for all and . We also know that is firmly nonexpansive mapping from onto , that is, and so is nonexpansive mapping.

For solving the generalized equilibrium problem, 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. In order to prove our main results, we also need the following lemmas.

Lemma 2.1 (see [2]). Let be a nonempty closed convex subset of and let be a bifunction from into satisfying (A1), (A2), (A3), and (A4). Then, for any and , there exists a unique such that

Lemma 2.2 (see [4]). Let be a nonempty closed convex subset of and let be a bifunction from into satisfying (A1), (A2), (A3), and (A4). Then, for any and , define a mapping as follows: Then the following hold:(i) is single-valued;(ii) is firmly nonexpansive, that is, (iii); (iv) is closed and convex.

Remark 2.3 (see [20]). Using (ii) in Lemma 2.2 and (2.2), we have So, for and , we have

Remark 2.4. For any and , by Lemma 2.1, there exists such that Replacing with in (2.9), we have where is an inverse-strongly monotone mapping.

For a sequence of nonempty closed convex subsets of a Hilbert space , define and as follows.

if and only if there exists such that and that for all .

if and only if there exists a subsequence of and a subsequence such that and that for all .

If satisfies it is said that converges to in the sense of Mosco [21] and we write . It is easy to show that if is nonincreasing with respect to inclusion, then converges to in the sense of Mosco. For more details, see [21]. Tsukada [22] proved the following theorem for the metric projection.

Theorem 2.5 (see Tsukada [22]). Let be a Hilbert space. Let be a sequence of nonempty closed convex subsets of . If exists and is nonempty, then for each , converges strongly to , where and are the metric projections of onto and , respectively.

On the other hand, a mapping of a complete metric space into itself is said to be a contraction with coefficient if for all . It is well known that has a unique fixed point [23]. Meir-Keeler [24] defined the following mapping called Meir-Keeler contraction. Let be a complete metric space. A mapping is called a Meir-Keeler contraction if for all , there exists such that implies for all . It is well known that Meir-Keeler contraction is a generalization of contraction and the following result is proved in [24].

Theorem 2.6 (see Meir-Keeler [24]). A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.

We have the following results for Meir-Keeler contractions defined on a Banach space by Suzuki [14].

Theorem 2.7 (see Suzuki [14]). Let be a Meir-Keeler contraction on a convex subset of a Banach space . Then, for every , there exists such that implies for all .

Lemma 2.8 (see Suzuki [14]). Let be a convex subset of a Banach space . Let be a nonexpansive mapping on , and let be a Meir-Keeler contraction on . Then the following hold. (i) is a Meir-Keeler contraction on .(ii)For each , a mapping is a Meir-Keeler contraction on .

3. Main Results

In this section, using the shrinking projection method by Takahashi et al. [15], we prove a strong convergence theorem for a quasi-nonexpansive mapping with a generalized equilibrium problem in a Hilbert space. Before proving it, we need the following lemmas.

Lemma 3.1. Let be a nonempty closed convex subset of a Hilbert space and and let be -inverse strongly monotone. If , then is a nonexpansive mapping.

Proof. For , we can calculate Therefore is nonexpansive. This completes the proof.

Lemma 3.2. Let be a nonempty closed convex subset of , and let be a quasi-nonexpansive mapping of into . Then, is closed and convex.

Proof. We first show that is closed. Let be any sequence in with . We claim that . Since is closed, we have . We observe that Since , we obtain that and hence . This show that .
Next, we show that is convex. Let and . We claim that . Putting , we have Hence is convex. This completes the proof.

Theorem 3.3. Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction satisfying (A1), (A2), (A3), and (A4) and let be a -inverse strongly monotone mapping from into . Let be a quasi-nonexpansive mapping which is demiclosed on , that is, if and , then . Assume that and is a Meir-Keeler contraction of into itself. Let the sequence be defined by where is the metric projection of onto and and are real sequences satisfying for some . Then, converges strongly to , which satisfies .

Proof. Since is a closed convex subset of , we have that is well defined and nonexpansive. Furthermore, we know that is Meir-Keeler contraction and we know from Lemma 2.8 (i) that of onto is a Meir-Keeler contraction on . By Theorem 2.6, there exists a unique fixed point such that . Next, we observe that for each and take . From and Lemma 2.2, we have that for any , Next, we divide the proof into several steps.
Step 1. is closed convex and is well defined for every .
It is obvious from the assumption that is closed convex and . For any , suppose that is closed and convex, and . Note that for all , It is easy to see that is closed. Next, we prove that is convex. For any and , we claim that . Since , we have and so , that is, . Similarly, , we get .
Thus, Combining the above inequalities, we obtain Therefore . This shows that and hence is convex. Therefore is closed and convex for all .
Next, we show that , for all . For any , suppose that . Since is quasi-nonexpansive and from (3.6), we have So, we have . By principle of mathematical induction, we can conclude that is closed and convex, and , for all . Hence, we have for all . Therefore is well defined.
Step 2. for some and for all .
Since is closed convex, we also have that is well defined and so is a Meir-Keeler contraction on . By Theorem 2.6, there exists a unique fixed point of . Since is a nonincreasing sequence of nonempty closed convex subsets of with respect to inclusion, it follows that Setting and applying Theorem 2.5, we can conclude that Next, we will prove that . Assume to contrary that , there exists and a subsequence of such that which gives that We choose a positive number such that For such , by the definition of Meir-Keeler contraction, there exists with such that for all . Again for such , by Theorem 2.7, there exists such that Since , there exists such that By the idea of Suzuki [14] and Kimura and Nakajo [17], we consider the following two cases.
Case . Assume that there exists such that Thus, we get By induction on , we can obtain that for all . In particular, for all , we have and This implies that which is a contradiction. Therefore, we conclude that as .
Case . Assume that By (3.19), we have Thus, we have for every . In particular, we have for every . Let us consider which gives a contradiction. Hence, we obtain that and therefore is bounded. Moreover, , and are also bounded. Since , we have Since , we get We have from that
Step 3. There exists a subsequence of such that as .
We have from (3.13) and (3.31) that From , we have that and so . We also have From , there exists a subsequence of and with such that . Since , we have Using Lemma 2.2 (ii) and (3.6), we have So, we have Let us consider In particular, we have Since with and , we obtain that Using (3.40), we have where .
So, we have We have from , (3.37), and (3.43) that
Step 4. Finally, we prove that .
Since , we have . So, from (3.38) we have Since , from (3.37), (3.46), and (3.47) we have Since , we have . So, from (3.48) and the demiclosed property of , we have We next show that . Since , for any we have From (A2), we have and so Replacing by , we have Note that is -Lipschitz continuous, and from (3.46), we have For and , let . Since is convex, we have . So, from (3.53) we have From , we have Thus, From Step  3 and (3.54), we obtain From (A1), (A4), and (3.58), we have and hence Letting and from (A3), we have that for each , This implies that . So, we have . We obtain from (3.34) that and hence, converges strongly to . This completes the proof.

By Theorem 3.3, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.

Setting in Theorem 3.3, we obtain the following result.

Corollary 3.4. Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction satisfying (A1), (A2), (A3), and (A4) and let be a -inverse strongly monotone mapping from into . Let be a quasi-nonexpansive mapping which is demiclosed on . Assume that and let and be a sequence generated by and where is the metric projection of onto and and are sequences such that for some . Then converges strongly to .

Setting in Theorem 3.3, we obtain the following result.

Corollary 3.5. Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction satisfying (A1), (A2), (A3), and (A4). Let be a quasi-nonexpansive mapping which is demiclosed on . Assume that and is a Meir-Keeler contraction of into itself. Let and be a sequence generated by and where is the metric projection of onto and and are sequences such that for some . Then converges strongly to .

Setting and for all in Theorem 3.3, we obtain the following result.

Corollary 3.6. Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction satisfying (A1), (A2), (A3), and (A4). Let be an quasi-nonexpansive mapping which is demiclosed on and assume that . Let and be a sequence generated by and where is the metric projection of onto and and are sequences such that for some . Then converges strongly to .

Next, using the CQ hybrid method introduced by Nakajo and Takahashi [16], we prove a strong convergence theorem of a quasi-nonexpansive mapping for solving the generalized equilibrium problem.

Theorem 3.7. Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction satisfying (A1), (A2), (A3), and (A4) and let be a -inverse strongly monotone mapping from into . Let be an quasi-nonexpansive mapping which is demiclosed on . Assume that and is a Meir-Keeler contraction of into itself. Let and be a sequence generated by and where is the metric projection of onto and and satisfy for some . Then, converges strongly to , which satisfies .

Proof. As in the proof of Theorem 3.3, we have that the mapping of onto is a Meir-Keeler contraction on . By Theorem 2.6, there exists a unique fixed point such that . Next, it is clear that is closed and convex. Next, we will show that is closed and convex for all . For any , let be a sequence in such that . For each , we observe that Taking , we get and then . Therefore is closed.
Next, we will show that is convex. For any , and , put . We claim that . Since , we have . Similarly, since , we have . Thus, It follows that , and therefore we have that is convex. We obtain from both and which are closed convex sets for every that is closed and convex for every .
Next, we will show that is nonempty. Let . We will show that for any . We notice that for each and . From which is an inverse strongly monotone mapping Lemma 2.2 (ii), Lemma 3.1, we obtain Since is quasi-nonexpansive with the fixed point and from (3.72), we have So, we have . Therefore , for all .
Next, we will show that It is obvious that . Hence For any , suppose that Since , we have In particular, for any , we obtain that This shows that . Hence . Therefore, we conclude that By principle of mathematical induction, we can conclude that Hence is well defined. Since is a Meir-Keeler contraction on , there exists a unique element such that . For each , let