`Journal of Applied MathematicsVolume 2012 (2012), Article ID 282094, 17 pageshttp://dx.doi.org/10.1155/2012/282094`
Research Article

The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

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

Received 19 February 2012; Accepted 23 March 2012

Copyright © 2012 Rabian Wangkeeree and Panatda Boonman. 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 first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.

1. Introduction

Let be a real Hilbert space with inner product and norm , let be 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 quasinonexpansive if for all and . It is well known that the set of fixed points of a quasi-nonexpansive mapping is a closed and convex set [1]. A mapping is said to be firmly nonexpansive [2] if for all , and it is an important example of nonexpansive mappings in a Hilbert space.

Let be a real-valued function, and let be an equilibrium bifunction, that is, for each . The mixed equilibrium problem is to find such that Denote the set of solution of (1.2) by . In particular, if , this problem reduces to the equilibrium problem, which is to find such that The set of solution of (1.3) is denoted by .

The problem (1.2) 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 [3] and Moudafi [4]. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3).

Let be two monotone bifunctions and is constant. In 2009, Moudafi [5] introduced an alternating algorithm for approximating a solution of the system of equilibrium problems, finding such that For such mappings and and two given positive constants , Plubtieng and Sombut [6] considered the following system of mixed equilibrium problem, finding such that In particular, if and , then problem (SMEP) reduces to (SEP). Furthermore, Plubtieng and Sombut [6] introduced the following iterative procedure to approximate a common element of the fixed-point set of a quasi-nonexpansive mapping and the solution set of (SMEP) in a Hilbert space . Let , and be given by where for some and satisfying appropriate conditions. The weak convergence theorems are obtained in a real Hilbert space.

On the other hand, in 1953, Mann [7] 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 .

For two nonexpansive mappings of into itself, Moudafi [4] studied weak convergence theorems in the following iterative process: for all , where and are appropriate sequences in and . Recently, Iemoto and Takahashi [8] also considered this iterative procedure for is a nonexpansive mapping and is a nonspreading mapping. Very recently, Kim [9] studied the weak and strong convergence for the Moudafi’s iterative scheme (1.8) of two quasi-nonexpansive mappings.

In this paper, inspired and motivated by Plubtieng and Sombut [6], Moudafi [4], Iemoto and Takahashi [8], and Kim [9], we first introduce the iterative procedure to approximate a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of SMEP in a real Hilbert space. Next, we prove the weak convergence theorem for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed point of asymptotic nonspreading mapping and asymptotic mapping and the solution set of SMEP in a real Hilbert space.

2. Preliminaries

Throughout this paper, let be the set of positive integers, and let be the set of real numbers. Let be a real Hilbert space with inner product and norm , respectively, and let be a closed convex subset of . We denote the strong convergence and the weak convergence of to by and , respectively.

From [10], for each and , we have For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is well know that is a nonexpansive mapping of onto and satisfies Moreover, is characterized by the following properties: , Further, for all and if and only if , for all .

Lemma 2.1 (see [11]). Let be a nonempty closed convex subset of a real Hilbert space . Let be the metric projection of onto , and let be in . If for all and . Then, converges strongly to an element of .

Theorem 2.2 (Opial’s theorem, [10]). Let be a real Hilbert space, and suppose that , then for all with .

All Hilbert space and satisfy Opial’s condition, while with do not.

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

The following lemma appears implicitly in [3, 12].

Lemma 2.3 (see [3]). Let be a nonempty closed convex subset of , and let be a bifunction satisfying (A1)–(A4). Let and , then there exists such that

The following lemma was also given in [12].

Lemma 2.4 (see [12]). Let be a nonempty closed closed convex subset of and let be a bifunction satisfying (A1)–(A4), then, for any and , define a mapping as follows: for all. Then the following hold:(i) is single valued,(ii) is firmly nonexpansive, that is, (iii),(iv) is closed and convex.

We note that Lemma 2.4 is equivalent to the following lemma.

Lemma 2.5 (see [6]). Let be a nonempty closed convex subset of a real Hilbert space . Let be an equilibrium bifunction satisfying (A1)–(A4) and let be a lower semicontinuous and convex functional. For each and , define a mapping Then, the following results hold:(i)for each , ,(ii) is single valued,(iii) is firmly nonexpansive, that is, for any , (iv),(v) is closed and convex.

3. Main Results

In this section, we prove the weak convergence for approximating a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of the system of mixed equilibrium problems in a Hilbert space.

To begin with, let us state and proof the following characterizations of the solution set of GMEP.

Lemma 3.1. Let be a closed convex subset of a real Hilbert space . Let and be two mappings from satisfying (A1)–(A4), and let and be defined as in Lemma 2.5 associated to and , respectively. For given is a solution of problem (1.5) if and only if is a fixed point of the mapping defined by where .

Proof. For given , we observe the following equivalency: This completes the proof.

We note from Lemma 3.1 that the mapping is nonexpansive. Moreover, if is a closed bounded convex subset of , then the solution of problem (1.5) always exists. Throughout this paper, we denote the set of fixed points of by .

Theorem 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and and be defined as in Lemma 2.5 associated to and , respectively. Let , , be two quasi-nonexpansive mappings such that are demiclosed at zero, that is, if , , and , then , with . Let the sequences , and , be given by where , for some , and satisfy then and is a solution of problem (1.5), where .

Proof. Let , then and .
Putting , and , we have Next, we prove that Since and are quasi-nonexpansive, we obtain that which gives that Hence, is a nonincreasing sequence, and hence, exists. This implies that , and are bounded. From (3.8), we have Since , this implies that Furthermore, since , we have From (3.10), we conclude that From (3.7), we have where is a constant satisfying . Again from (3.10), we conclude that Using , we have Now, we prove that We observe that which gives that Since , we have Using (3.12) and (3.15), we conclude that which gives that since . Similarly, we have which implies that Thus, we have Hence, . Since , we have Next, we prove that Since and are firmly nonexpansive, it follows that which gives that This implies that By the convexity of , we have Thus, Since , we have Since exists, we have Similarly, we have which gives that This implies that By the convexity of , we have Thus, Since , we have Since exists, we have Hence, It follow from (3.10), (3.33), and (3.40) that from which it follows that that is, Thus, Similarly, we have . Since is bounded sequence, there exists a subsequence of such that as . Since and are demiclosed at 0, we conclude that . Let be a mapping which is defined as in Lemma 3.1. Thus, we have and hence, This together with implies that , if is another subsequence of such that as . Since and are demiclosed at 0, we conclude that . From and , we will show that . Assume that . Since exists for all , by Opial’s Theorem 2.2, we have This is a contradiction. Thus, we have . This implies that . Since , we have . Put . Finally, we show that . Now from (2.4) and , we have Since is nonnegative and nonincreasing for all , it follows by Lemma 2.1 that converges strongly to some . By (3.49), we have Therefore, .

Setting in Theorem 3.2, we have the following result.

Corollary 3.3 (see [6]). Let be a closed convex subset of a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let , and let and be defined as in Lemma 2.5 associated to and , respectively. Let be a quasi-nonexpansive mapping such that is demiclosed at zero and . Suppose that and are given by for all , where for some , and satisfy , then converges weakly to and is a solution of problem (1.5), where .

Setting , in Theorem 3.2, we have the following result.

Corollary 3.4 (see [9]). Let be a Hilbert space, let be a nonempty, closed, and convex subset of , and let be two quasi-nonexpansive mappings of into itself such that , are demiclosed at zero with . For any in C, let be defined by where and are chosen so that then .

4. Applications

In this section, we apply our results to approximate a common element of the set of common fixed points of an asymptotic nonspreading mapping and an asymptotic mapping and the solution set of SMEP in a real Hilbert space. We recall the following definitions. A mapping is called nonspreading [13] if Furthermore, Takahashi and Yao [14] also introduced two nonlinear mappings in Hilbert spaces. A mapping is called a mapping [14] if for all . A mapping is called a [14] mapping if for all . For these two nonlinear mappings, and mappings, Takahashi and Yao [14] studied the existence results of fixed points in Hilbert spaces. Very recently, Lin et al. [15] introduced the following definitions of new mappings.

Definition 4.1. Let be a nonempty closed convex subset of a Hilbert space . We say that is an asymptotic nonspreading mapping if there exist two functions and , such that(A1) for all ,(A2) for all .

Remark 4.2. The class of asymptotic nonspreading mappings contains the class of nonspreading mappings and the class of mappings in a Hilbert space. Indeed, in Definition 4.1, we know that(i)if for all , then is a nonspreading mapping,(ii)if and for all , then is a mapping.

Definition 4.3. Let be a nonempty closed convex subset of a Hilbert space . We say is an asymptotic mapping if there exists two functions and , such that(B1) for all ,(B2) for all .

Remark 4.4. The class of asymptotic mappings contains the class of mappings and the class of nonexpansive mappings in a Hilbert space. Indeed, in Definition 4.3, we know that(i)if and for each , then is a nonexpansive mapping,(ii)if for each , then is a mapping.

It is well known that the set of fixed points of a quasi-nonexpansive mapping is a closed and convex set [1]. Hence, if is an asymptotic nonspreading mapping (resp., asymptotic mapping) with , then is a quasi-nonexpansive mapping, and this implies that is a nonempty closed convex subset of .

Theorem 4.5 (see [15]). Let be a nonempty closed convex subset of a real Hilbert space , and let be an asymptotic nonspreading mapping, then is demiclosed at 0.

Theorem 4.6 (see [15]). Let be a nonempty closed convex subset of a real Hilbert space , and let be an asymptotic mapping, then is demiclosed at 0.

Applying the above results, we have the following theorem.

Theorem 4.7. Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and and be defined as in Lemma 2.5 associated to and , respectively. Let , , be any one of asymptotic nonspreading mapping and asymptotic mapping such that . Let , and be given by where for some and satisfying then , and is a solution of problem (1.5), where .

Setting and in the above theorem, we have the following result.

Corollary 4.8. Let be a nonempty closed convex subset of a real Hilbert space . Let be the bifunctions from satisfying (A1)–(A4). Let , , be any one of asymptotic nonspreading mapping and asymptotic mapping such that . For given and , let the sequences and be defined by where , are two sequences in satisfying then for some .

Setting and in Theorem 4.7, we have the following result.

Corollary 4.9 (see [15]). Let be a nonempty closed convex subset of a real Hilbert space , and let , , be any one of asymptotic nonspreading mapping and asymptotic mapping. Let . Let and be two sequences in . Let be defined by Assume that and , then for some .

Acknowledgment

This paper is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.

References

1. S. Itoh and W. Takahashi, “The common fixed point theory of singlevalued mappings and multivalued mappings,” Pacific Journal of Mathematics, vol. 79, no. 2, pp. 493–508, 1978.
2. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1990.
3. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
4. A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,” Inverse Problems, vol. 23, no. 4, pp. 1635–1640, 2007.
5. A. Moudafi, “From alternating minimization algorithms and systems of variational inequalities to equilibrium problems,” Communications on Applied Nonlinear Analysis, vol. 16, no. 3, pp. 31–35, 2009.
6. S. Plubtieng and K. Sombut, “Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space,” Journal of Inequalities and Applications, vol. 2010, Article ID 246237, 12 pages, 2010.
7. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
8. S. Iemoto and W. Takahashi, “Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2082–e2089, 2009.
9. G. E. Kim, “Weak and strong convergence theorems of quasi-nonexpansive mappings in a Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 152, no. 3, pp. 727–738, 2010.
10. W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009.
11. W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
12. S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.
13. F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008.
14. W. Takahashi and J. C. Yao, “Fixed point theorems and ergodic theorems for nonlinear mappings in hilbert spaces,” Taiwanese Journal of Mathematics, vol. 15, no. 2, pp. 457–472, 2011.
15. L. J. Lin, C. H. Chuang, and Z. T. Yu, “Fixed point theorems for some new nonlinear mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, no. 1, article 51, 2011.