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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 316276, 39 pages
http://dx.doi.org/10.1155/2012/316276
Research Article

A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 30 August 2011; Accepted 1 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Pongsakorn Sunthrayuth and Poom Kumam. 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 iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

1. Introduction

Throughout this paper, we denoted by and the set of all positive integers and all positive real numbers, respectively. We always assume that is a real Hilbert space with inner product and norm , respectively, and is a nonempty closed convex subset of . Let be a real-valued function, be an equilibrium bifunction, and be two nonlinear mappings. The generalized mixed equilibrium problem with perturbed mapping is to find such that The set of solutions of the problem (1.1) is denoted by . As special cases of the problem (1.1), we have the following. (1)If , then the problem (1.1) reduces to the generalized mixed equilibrium problem of finding such that which was introduced and studied by Peng and Yao [1]. The set of solutions of the problem (1.2) is denoted by . (2)If , then the problem (1.1) reduces to the mixed equilibrium problem of finding such that which was consider by Ceng and Yao [2]. The set of solutions of the problem (1.3) is denoted by . (3)If , then the problem (1.1) reduces to the generalized equilibrium problem of finding such that which was consider by S. Takahashi and W. Takahashi [3]. The set of solutions of the problem (1.4) is denoted by .(4)If , then the problem (1.1) reduces to the equilibrium problem of finding such that The set of solutions of the problem (1.5) is denoted by .(5)If , then the problem (1.1) reduces to the classical variational inequality problem of finding such that The set of solutions of the problem (1.6) is denoted by . It is known that is a solution of the problem (1.6) if and only if is a fixed point of the mapping , where is a constant and is the identity mapping.

The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [4, 5]). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.2). Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature (see, e.g., [634]).

Let be a strongly positive bounded linear operator on ; that is, there exists a constant such that

Recall that a mapping is said to be contractive if there exists a constant such that

A mapping is said to be (1)nonexpansive if (2)firmly nonexpansive if (3)-strictly pseudocontractive if there exists a constant such that We denote by the set of fixed points of , that is, .

Recall the following definitions of a nonlinear mapping ; the following is mentioned.

Definition 1.1. The nonlinear mapping is said to be (i)monotone if (ii)-strongly monotone if there exists a constant such that (iii)-inverse-strongly monotone if there exists a constant such that

Let be a set-valued mapping. The set defined by is said to be the domain of . The set defined by is said to be the range of . The set defined by is said to be the graph of .

Recall that is said to be monotone if

is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, is maximal monotone if for all . For a maximal monotone operator on and , we may define the single-valued resolvent . It is known that is firmly nonexpansive , where denotes the fixed point set of .

We discuss the following algorithms for solving the solutions of variational inequality problems and fixed point problems for a nonexpansive mapping (see, e.g., [29, 3543]).

In 2010, Chantarangsi et al. [44] introduced a new viscosity hybrid steepest descent method for solving the generalized mixed equilibrium problems (1.2), variational inequality problems, and fixed point problems of nonexpansive mappings in a real Hilbert space. More precisely, they proved the following theorem.

Theorem CCK [see [44]]
Let be a nonempty closed and convex subset of a real Hilbert space . Let , be two bifunctions satisfying condition (H1)–(H5), let , be -inverse-strongly monotone mapping and -inverse-strongly monotone mapping, respectively, and let be a nonexpansive mapping. Let be an -Lipschitz continuous and relaxed cocoercive mapping, a contraction mapping with coefficient , and a strongly positive linear bounded self-adjoint operator with coefficient and . Suppose that . Let , , , and be generated by where , , , , , and are three sequences in satisfying the following conditions: (C1) and , (C2), (C3) and , (C4) and , (C5), and . Then, converges strongly to .

Very recently, Yu and Liang [45] proved the following convergence theorem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and inverse strongly monotone mapping in a real Hilbert space.

Theorem YL [see [45]]
Let be a real Hilbert space and a nonempty close and convex subset of . Let and be two maximal monotone operators such that and , respectively. Let be a -strict pseudocontraction mapping, an -inverse-strongly monotone mapping, and an -inverse-strongly monotone mapping. Assume that . Let be a sequence generated by where is a fixed element, , is a sequence in is a sequence in , and , , , and are sequences in satisfying the following conditions: (C1) and , (C2), (C3) and , (C4) and , (C5) and . Then, the sequence converges strongly to .

On the other hand, the following optimization problem has been studied extensively by many authors: where are infinitely many closed convex subsets of such that , , is a real number, is a strongly positive linear bounded operator on , and is a potential function for (i.e., for all ). This kind of optimization problem has been studied extensively by many authors (see, e.g., [5, 4652]) for when and , where is a given point in .

The following questions naturally arise in connection with above the results.

Question 1. Could we weaken the control conditions of Theorems CCK and YL in and ?

Question 2. Can Theorem YL be extended to finding a common element of the set of solutions of a system generalized mixed equilibrium problems and the set of common fixed points of infinite family of nonexpansive mappings?

The purpose of this paper is to give the affirmative answers to these questions mentioned above. Motivated by the iterative process (1.16) and (1.17), we introduce a new iterative algorithm (3.2) below, for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Then, we prove the strong convergence theorem of these iterative process in a real Hilbert space. The results presented in the paper improve and extend the recent ones announced by many others.

2. Preliminaries

Definition 2.1 (see [53]). Let be a nonempty convex subset of a real Hilbert space . Let , , be mappings of into itself. For each , let , where and . For every , we define the mapping as follows: Such a mapping is nonexpansive from into itself, and it is called -mapping generated by and .

Lemma 2.2 (see [53]). Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself with and let , where , , , and for all . For all , let and be -mappings generated by and and and , respectively. Then, (i) is nonexpansive and ,  for all , (ii)for all and for all positive integer , the exists,(iii)the mapping defined by is a nonexpansive mapping such that , and it is called the -mapping generated by and , (iv) if is any bounded subset of C, then

Lemma 2.3 (see [54]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all integers and . Then, .

Lemma 2.4 (see [55]). Let be a real Hilbert space. Then, the following inequalities hold: (i),(ii).

Lemma 2.5 (see [56]). Let be a nonempty closed convex subset of a real Hilbert space , a mapping, and a maximal monotone mapping. Then,

Lemma 2.6 (see [57]). Let be a real Hilbert space and let be a maximal monotone operator on . For and , define the resolvent . Then, the following holds: for all and .

For solving the equilibrium problem for bifunction , let us assume that satisfies the following conditions: (H1) for all , (H2) is monotone; that is, for all ,(H3) for each , is concave and upper semicontinuous, (H4) for each , is convex, (H5) for each , is lower semicontinuous.

Definition 2.7. A differentiable function on a convex set is called (i)convex [2] ifwhere is the Fréchet differentiable of at ; (ii)strongly convex [2] if there exists a constant such that

It is easy to see that if is a differentiable strongly convex function with constant , then is strongly monotone with constant .

Let be an equilibrium bifunction satisfying the conditions (H1)–(H5). Let be any given positive number. For a given point , consider the auxiliary mixed equilibrium problem to finding such that where is the Fréchet differentiable of at . Let be the mapping such that for each , is the set of solutions of , that is, Then, the following conclusion holds.

Lemma 2.8 (see [58]). Let be a nonempty closed convex subset of a real Hilbert space , and let be a lower semicontinuous and convex functional. Let be a bifunction satisfying the conditions (H1)–(H5). Assume that (i) is strongly convex with constant and the function is weakly upper semicontinuous for each ;(ii) for each , there exist a bounded subset and such that for all , Then, the following holds: (a) is single-valued mapping; (b) is nonexpansive if is Lipschitz continuous with constant and where for ; (c); (d) is closed and convex.

In particular, whenever is a bifunction satisfying the conditions (H1)–(H5) and , for all , then is firmly nonexpansive; that is, for any , In this case, is rewritten as . If, in addition, , then is rewritten as (see [59, Lemma 2.1] for more details).

Remark 2.9. We remark that Lemma 2.8 is not a consequence of [2, Lemma 3.1] because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .

Lemma 2.10. Let , and be as in Lemma 2.8. Then, the following holds: for all and .

Proof. By similar argument as in the proof of Proposition 1 in [58], for all and , let and ; we have Let in (2.14) and in (2.15); we have Adding up the last two inequalities and from the monotonicity of , we obtain that It follows that We derive from (2.18) that Hence, we obtain that

The following lemma can be found in [60, 61] (see also [62, Lemma 2.2]).

Lemma 2.11. Let be a nonempty closed convex subset of a real Hilbert space and a proper lower semicontinuous differentiable convex function differentiable convex function. If is a solution to the minimization problem then In particular, if solves the optimization problem then where is a potential function for .

Lemma 2.12 (see [49]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i), (ii) or . Then, .

3. Main Results

Now, we give our main results in this paper.

Theorem 3.1. Let be a nonempty closed convex subset of a Hilbert space . Let be two lower semicontinuous and convex functionals, and let be two bifunctions satisfying conditions (H1)–(H5). Let be -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, and -inverse-strongly monotone mapping, respectively, and let be -inverse-strongly monotone mapping and -inverse-strongly monotone mapping, respectively. Let be an infinite family of nonexpansive mappings, and let , where , , , and for all . For all , let and be -mappings generated by and and and , respectively. Let be two maximal monotone operators such that and , respectively. Assume that Let be a contraction mapping with a coefficient , and let be a strongly positive linear bounded operator with a coefficient . Let and be two constants such that . Let be a sequence defined by and where , , , , , , and . Assume that the following conditions are satisfied: (C1) for all , is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each , (C2)for all and for each , there exist a bounded subset and such that, for all , (C3), (C4) and , (C5), (C6) and , (C7) and , (C8) and , C9 and . Then, the sequence defined by (3.2) converges strongly to , provided is firmly nonexpansive, where solves the following optimization problem:

Proof. By the conditions (C4) and (C5), we may assume, without loss of generality, that for all . Since is a linear bounded self-adjoint operator on , we have Observe that This shows that is positive. It follows that First, we show that is bounded. Take . Since and by Lemma 2.4  (i), we have In a similar way, we can get It follows from (3.8), and (3.9) that Setting . Since , we have In a similar way, we can get It follows from (3.10), (3.11) and (3.12) that Since , it follows from (3.13) that By induction, we have Hence, is bounded, so are , , and .
Next, we show that . Since and , we have In a similar way, we can get Substitution (3.16) into (3.17), we obtain where .
On the other hand, notice from Lemma 2.6 that where is an appropriate constant such that .
In a similar way, we can get from Lemma 2.6 that where is an appropriate constant such that . It follows from (3.18) and (3.19) that where .
Define the sequence by