Abstract and Applied Analysis

Volume 2012 (2012), Article ID 161897, 22 pages

doi:10.1155/2012/161897

Research Article

Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

Rabian Wangkeeree and Uraiwan Boonkong

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

Received 3 September 2012; Accepted 1 November 2012

Academic Editor: Xiaolong Qin

Copyright © 2012 Rabian Wangkeeree and Uraiwan Boonkong. 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 new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the systems of generalized equilibrium problems and the common fixed point set 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 . It is well known that the set of fixed points of a quasi-nonexpansive mapping is closed and convex; see Itoh and Takahashi [1]. A mapping is called nonspreading [2] if

for all . We remark that nonlinear every nonspreading mappings are quasi-nonexpansive mappings if the set of fixed points is nonempty.

Recall that a mapping is said to be -inverse strongly monotone if there exists a positive real number such that

If is a -inverse strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous.

Let be a bifuction and be -inverse strongly monotone mapping. The generalized equilibrium problem (for short, ) for and is to find such that

The problem (1.3) was studied by Moudafi [3]. The set of solutions for problem (1.3) is denoted by , that is,

If in (1.3), then reduces to the classical equilibrium problem and is denoted by , that is,

If in (1.3), then reduces to the classical variational inequality and is denoted by , that is,

The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, Min–Max problems, the Nash equilibrium problems in noncooperative games, and others; see, for example, Blum and Oettli [4] and Moudafi [3].

In 2005, Combettes and Hirstoaga [5] 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 [6] 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 [7] 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. Mainge and Moudafi [8] introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree [9] 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 [10] 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 1967, Wittmann [11] (see also [12]) proved the strong convergence theorem of Halpern’s type [13] defined by, for any ,

where satisfies , and . In [14], Kurokawa and Takahashi also studied the following Halpern’s type for nonspreading mappings in a Hilbert space; see also Hojo and Takahashi [15]. Let be a nonspreading mapping of into itself. Let and define two sequences and in as follows: and

for all , where , and . If is nonempty, they proved that and converge strongly to , where is the metric projection of onto . Recently, Yao and Shahzad [16] gave the following iteration process for nonexpansive mappings with perturbation: and

where and are sequences in , and the sequence is a small perturbation for the -step iteration satisfying as . In fact, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate.

On the other hand, very recently, Chuang et al. [17] considered the following iteration process for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points for a quasi-nonexpansive mapping with perturbation

where is a nonempty closed convex subset of , is a function, and are real sequences in , and is a convergent sequence and for some . They obtained a strong convergence theorem for such iterations.

In this paper, inspired and motivated by Yao and Shahzad [16], S. Takahashi and W. Takahashi [18] and Chuang et al. [17], we introduce a new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the system of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in a Hilbert space.

2. Preliminaries

Let be a real Hilbert space with inner product and norm . We denote the strongly convergence and the weak convergence of to by and , respectively. In a Hilbert space, it is known that

for all and ; see [19]. Furthermore, we have that for any

Let be a nonempty closed convex subset of and . We know that there exists a unique nearest point such that . We denote such a correspondence by . The mapping is called the metric projection of onto . It is known that is nonexpansive and

for all and ; see [19, 20] for more details.

Let be a nonempty, closed and convex subset of and let be a bifunction. For solving the generalized equilibrium problem, let us assume that the bifunction satisfies the following conditions:

(A1) for all ; (A2) is monotone, that is, for any ;(A3) for each

(A4) for each is convex and lower semicontinuous.

We know the following lemma which appears implicitly in Blum and Oettli [4].

Lemma 2.1 (see [4]). Let be a nonempty closed convex subset of and let be a bifunction of into satisfying . Let and . Then, there exists a unique such that

The following lemma was also given in Combettes and Hirstoaga [5].

Lemma 2.2 (see [5]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction which satisfies conditions . For and , define a mapping as follows:

for all . Then the following hold:

(i) is single-valued;(ii) is firmly nonexpansive, that is, for any ,

(iii) is a closed convex subset of ;(iv).

Remark 2.3. For any and , by Lemma 2.2 (i), there exists such that

Replacing with in (2.8), we have

where is an inverse strongly monotone mapping.

Lemma 2.4 (see [21]). Let be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence of which satisfies for all . Define the sequence of integers as follows:

where such that . Then, the following hold:

(i) and ;(ii) and .

Lemma 2.5 (see [22]). Let be a sequence of nonnegative real numbers, a sequence of real numbers in with a sequence of nonnegative real numbers with a sequence of real numbers with . Suppose that

Then .

3. Main Results

Let be a nonempty closed convex subset of a Hilbert space . For each , let be a bifunction satisfying and a -inverse strongly monotone mapping. For each , let be two mappings. Let be a sequence generated in the following manner:

where are sequences in (0, 1) and is a sequence and for some and for all . Under certain appropriate assumptions imposed on the sequences , the strong convergence theorem of defined by (3.1) is studied in the following theorem.

Theorem 3.1. Let be a nonempty closed convex subset of a Hilbert space . For each , let be a bifunction satisfying and a -inverse strongly monotone mapping. For each , let be two quasi-nonexpansive mappings such that are demiclosed at zero with . Let the sequences ,  , and be defined by (3.1), where , and satisfy the following conditions:

(C1) and ;(C2);(C3); (C4) for some .

Then converges strongly to , where .

Proof. We first have that for all , is a nonexpansive mapping. Indeed, for all , we obtain

Thus is nonexpansive for each . Now, let be arbitrary. By (C4), is a bounded sequence, there exists such that

For each and , we have from that

which gives also that

Since is quasi-nonexpansive, we have

So, we have from (3.5) and (3.6) and the quasi-nonexpansiveness of that

By Induction, we have that

Thus we obtain that is bounded, so also , and are bounded. Since is closed and convex, we can take . It follows that

From (3.9), we have

Hence we have from (3.5), (3.9), and (3.10) that

We also have that

Furthemore, we have from that

On the other hand, since , we have

We also have that

Moreover, for any , we have from that

It follows that

This implies that

and hence

Furthermore, we have from Lemma 2.2 that for any , we have

This implies that

Then we have from (3.22) that

Hence we have from (3.23) that

It follows that

Next, we will consider the following two cases.

Case A. Put for all . Suppose that for all . In this case exists and then . By (C1), (C3), and (3.12), we have

Similarly by (C1), (C2), and (3.13), we also have

So, we have from (3.14), (3.26), and (3.27) that

Since exists, we have from (3.11) and (3.26)

We also have from (C1), (3.16), (3.26), and (3.27) that

Since exists we have from (C1) and (3.20) that

This together with (3.25) and the existence of implies that

which gives that

So, from (3.30), . Furthermore, we have from (3.33) that

that is

Now, since is a bounded sequence, there exists a subsequence of such that

Without loss of generality, we may assume that . Since is demiclosed at zero and by (3.26), we conclude that . Similarly, since is demiclosed at zero and by (3.28), we have . Therefore, we get that

Next, we show that . For each , since , we have

From , we also have

Replacing by , we have

Put for all and . Since , then and

Since as , we obtain that as . Furthermore, by the monotonicity of , we obtain that

Taking in (3.41), we have from (A4) that

Now, from (A1), (A4), and (3.43), we also have

which yields that

Taking , we have, for each

This shows , for all . Then, . Hence we have . So, we have from (3.36) that

By (C1), (C4), (3.15), (3.30), (3.47), and Lemma 2.5, we obtain that . Hence we have from (3.29) that converges to , where .

Case B. Assume that there exists a subsequence of such that for all . In this case, it follows from Lemma 2.4 that there exists a subsequence of such that , where is defined by

So, from (3.12), that

Since , we have

By (C1) and (C3), we have

By (3.15), we have

Now, in view of , we see that

Furthermore, we also have from (3.13) that

Applying (C1) and (C2) to the last inequality, we get that

By (C1), (3.16), (3.51), and (3.55), we have

By (3.33), we have

It follows from (3.56) and (3.57) that

Since is a bounded sequence, there exists a subsequence such that

Following the same argument as the proof of Case A for , we have that

Using (C4), (3.53), (3.56), and (3.60), we have that

By (3.58) and (3.61), we have that

By Lemma 2.4 (ii), we get ; that is . We observe that

Applying (C1), (C4), and , we have immediately

that is, converges strongly to , where . This completes the proof.

Setting for all in Theorem 3.1, we obtain the following result.

Corollary 3.2. Let be a nonempty closed convex subset of a Hilbert space . For each , let be a bifunction satisfying. For each , let be two quasi-nonexpansive mappings such that are demiclosed at zero with . Let the sequences ,   , and be defined by

where satisfy the following conditions.

(C1) and ;(C2);(C3); (C4) for some .

Then converges strongly to , where .

In the next results, using Theorem 3.1, we have new strong convergence theorems for two nonexpansive mappings in a Hilbert space.

Corollary 3.3. Let be a nonempty closed convex subset of a Hilbert space . For each , let be a bifunction satisfying (A1)–(A4) and a -inverse strongly monotone mapping. For each , let be two nonexpansive mappings such that . Let the sequences ,  , and be defined by (3.1), where satisfy the following conditions.

(C1) and ;(C2);(C3); (C4) for some .

Then converges strongly to , where .

4. Applications

In this section, we present some convergence theorems deduced from the results in the previous section. Recall that a mapping is said to be nonspreading [2] if

for all . Further, a mapping is said to be hybrid [23] if

for all . These mappings are deduced from a firmly nonexpansive mapping in a Hilbert space.

A mapping is said to be firmly nonexpansive if

for all ; see, for instance, Browder [24] and Goebel and Kirk [25]. We also know that a firmly nonexpansive mapping can be deduced from an equilibrium problem in a Hilbert space.

Recently, Kocourek et al. [26] introduced a more broad class of nonlinear mappings call generalized hybrid if there are such that

for all . Very recently, they defined a more broad class of mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping is called super hybrid if there are such that

for all . We call such a mapping an -super hybrid mapping. We notice that an -super hybrid mapping is -generalized hybrid. So, the class of super hybrid mappings contains the class of generalized hybrid mappings. A super hybrid mapping is not quasi-nonexpansive generally. For more details, see [27]. Before proving, we need the following lemmas.

Lemma 4.1 (see [27]). Let be a nonempty subset of a Hilbert space and let and be real numbers with . Let and be mappings of into such that . Then, is -super hybrid if and only if is -generalized hybrid. In this case, .

Lemma 4.2 (see [27]). Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a generalized hybrid mapping. Then is demiclosed on .

Setting in Theorem 3.1, where is a super hybrid mapping and is a real number, we obtain the following result.

Theorem 4.3. Let be a nonempty closed convex subset of a Hilbert space . For each , let be a bifunction satisfying (A1)–(A4) and a -inverse strongly monotone mapping. For each , let be -super hybrid mappings such that . Let the sequences ,  , and be defined by

where are sequences in and is a sequence and for some and for all . Suppose the following conditions are satisfied.

(C1) and ;(C2);(C3); (C4) for some .

Then converges strongly to , where .

Proof. For each , setting

we have from Lemma 4.1 that each is a generalized hybrid mapping and . Since , we have that each is quasi-nonexpansive. Following the proof of Theorem 3.1 and applying Lemma 4.2, we have the desired result. This completes the proof.

Setting in Theorem 4.3, we obtains the following result.

Corollary 4.4. Let be a nonempty closed convex subset of a Hilbert space . For each , let be a bifunction satisfying . For each , let be -super hybrid mappings such that . Let the sequences ,, and be defined by

where are sequences in (0, 1) and is a sequence and for some . Suppose the following conditions are satisfied.

(C1) and ;(C2);(C3); (C4) for some .

Then converges strongly to , where .

In Corollary 4.4, put for all and for all . Then we have that for all , which gives that . Thus we obtain the following results from Corollary 4.4.

Corollary 4.5. Let be a nonempty closed convex subset of a Hilbert space . For each , let be -super hybrid mappings such that . Let the sequences ,   , and be defined by

where are sequences in and is a sequence. Suppose the following conditions are satisfied.

(C1) and ;(C2);(C3) ; (C4) for some .

Then converges strongly to , where .

In Corollary 4.5, put , the identity mapping, and , an -super hybrid mapping. Thus we obtain the following results.

Corollary 4.6. Let be a nonempty closed convex subset of a Hilbert space . Let be an -super hybrid mapping such that . Let the sequences ,, and be defined by

where and are sequences in and is a sequence. Suppose the following conditions are satisfied.

(C1) and ;(C2); (C3) for some .

Then converges strongly to , where .

Acknowledgments

The first author is supported by the National Research Council of Thailand. The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

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