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International Journal of Mathematics and Mathematical Sciences

Volume 2011 (2011), Article ID 619813, 25 pages

http://dx.doi.org/10.1155/2011/619813

## A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 1 December 2010; Revised 28 January 2011; Accepted 17 February 2011

Academic Editor: Vittorio Colao

Copyright © 2011 Thanyarat Jitpeera 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 general iterative method for finding a common element
of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized
mixed equilibrium problems, and the set of solutions of the variational inclusion for a *β*-inverse-strongly
monotone mapping in a real Hilbert space. We prove that the sequence converges strongly
to a common element of the above three sets under some mild conditions. Our results improve and
extend the corresponding results of Marino and Xu (2006), Su et al. (2008), Klin-eam and Suantai
(2009), Tan and Chang (2011), and some other authors.

#### 1. Introduction

Let be a closed convex subset of a real Hilbert space with the inner product and the norm . Let be a bifunction of into , where is the set of real numbers, a mapping, and a real-valued function. The *generalized mixed equilibrium problem* is for finding such that
The set of solutions of (1.1) is denoted by GMEP, that is,
If , the problem (1.1) is reduced into the *mixed variational inequality of Browder type* [1] for finding such that
The set of solutions of (1.3) is denoted by MVI.

If and , the problem (1.1) is reduced into the *equilibrium problem * [2] for finding such that
The set of solutions of (1.4) is denoted by EP. This problem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; see [3–5].

If and , the problem (1.1) is reduced into the *Hartmann-Stampacchia variational inequality* [6] for finding such that
The set of solutions of (1.5) is denoted by VI. The variational inequality has been extensively studied in the literature [7].

If and , the problem (1.1) is reduced into the *minimize problem* for finding such that
The set of solutions of (1.6) is denoted by .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is a linear bounded operator, is the fixed point set of a nonexpansive mapping , and is a given point in [8].

Recall that a mapping is said to be *nonexpansive* if
for all . If is bounded closed convex and is a nonexpansive mapping of into itself, then is nonempty [9]. We denote weak convergence and strong convergence by notations and , respectively. A mapping of into is called *monotone* if
for all . A mapping of into is called -*inverse-strongly monotone* if there exists a positive real number such that
for all . It is obvious that any -inverse-strongly monotone mapping is monotone and Lipschitz continuous mapping. A linear bounded operator is *strongly positive* if there exists a constant with the property
for all . A self mapping is a *contractions* on if there exists a constant such that
for all . We use to denote the collection of all contraction on C. Note that each has a unique fixed point in .

Let be a single-valued nonlinear mapping and a set-valued mapping. The *variational inclusion problem* is to find such that
where is the zero vector in . The set of solutions of problem (1.13) is denoted by . The variational inclusion has been extensively studied in the literature, see, for example, [10–13] and the reference therein.

A set-valued mapping is called *monotone* if for all , , and imply . A monotone mapping is *maximal* if its graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for all imply .

Let be an inverse-strongly monotone mapping of into , and let be normal cone to at , that is, , and define Then, is a maximal monotone and if and only if [14].

Let be a set-valued maximal monotone mapping; then the single-valued mapping defined by
is called the *resolvent operator* associated with , where is any positive number and is the identity mapping. It is worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone and that a solution of problem (1.13) is a fixed point of the operator for all [15].

In 2000, Moudafi [16] introduced the viscosity approximation method for nonexpansive mapping and proved that, if is a real Hilbert space, the sequence defined by the iterative method below, with the initial guess , is chosen arbitrarily, where satisfies certain conditions and converges strongly to a fixed point of (say ) which is the unique solution of the following variational inequality:

In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansive mapping. They defined the sequence generated by the algorithm : where and is a strongly positive linear bounded operator. They proved that, if and the sequence satisfies appropriate conditions, then the sequence generated by (1.18) converges strongly to a fixed point of (say ) which is the unique solution of the following variational inequality: which is the optimality condition for the minimization problem where is a potential function for (i.e., for ).

For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities, let be the projection of onto . In 2005, Iiduka and Takahashi [17] introduced following iterative process for : where , , and for some with . They proved that under certain appropriate conditions imposed on and , the sequence generated by (1.21) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say ) which solve some variational inequality

In 2008, Su et al. [18] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: for all , where and satisfy some appropriate conditions. Furthermore, they proved that and converge strongly to the same point , where .

In 2011, Tan and Chang [12] introduced following iterative process for which is a sequence of nonexpansive mappings. Let be the sequence defined by where , , and . The sequence defined by (1.24) converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solutions of the variational inequality, and the generalized equilibrium problem.

In this paper, we modify the iterative methods (1.18), (1.23), and (1.24) by proposing the following new general viscosity iterative method: , for all , where , , and satisfy some appropriate conditions. The purpose of this paper is to show that under some control conditions the sequence strongly converges to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space.

#### 2. Preliminaries

Let be a real Hilbert space and a nonempty closed convex subset of . Recall that the (nearest point) projection from onto assigns to each the unique point in satisfying the property The following characterizes the projection . We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1. *The function is a solution of the variational inequality (1.5) if and only if satisfies the relation for all .*

Lemma 2.2. *For a given , , , for all .**It is well known that is a firmly nonexpansive mapping of onto and satisfies
**
Moreover, is characterized by the following properties: and, for all , ,
*

Lemma 2.3 (see [19]). *Let be a maximal monotone mapping, and let be a monotone and Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.*

Lemma 2.4 (see [20]). *Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality holds for each with .*

Lemma 2.5 (see [21]). *Assume that is a sequence of nonnegative real numbers such that
**
where and is a sequence in such that*(i)*,*(ii)* or .**Then .*

Lemma 2.6 (see [22]). *Let be a closed convex subset of a real Hilbert space , and let be a nonexpansive mapping. Then is demiclosed at zero, that is,
**
implies .*

For solving the generalized mixed equilibrium problem, let us assume that the bifunction , the nonlinear mapping is continuous monotone, and satisfies the following conditions: (A1) for all ; (A2) is monotone, that is, for any ; (A3)for each fixed , is weakly upper semicontinuous; (A4)for each fixed , is convex and lower semicontinuous; (B1)for each and , there exist a bounded subset and such that, for any , (B2) is a bounded set.

Lemma 2.7 (see [23]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction mapping satisfying (A1)–(A4), and let be convex and lower semicontinuous such that . Assume that either (B1) or (B2) holds. For and , there exists such that
**
Define a mapping as follows:
**
for all . Then, the following hold: *(i)* is single valued; *(ii)* is firmly nonexpansive, that is, for any ;*(iii)*;
*(iv)* is closed and convex.*

Lemma 2.8 (see [8]). *Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and ; then .*

#### 3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of , , and of an inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. *Let be a real Hilbert space, a closed convex subset of , , be , -inverse-strongly monotone mappings, respectively. Let be a convex and lower semicontinuous function, a contraction with coefficient , a maximal monotone mapping, and a strongly positive linear bounded operator of into itself with coefficient . Assume that . Let be a nonexpansive mapping of into itself such that
**
Suppose that is a sequences generated by the following algorithm for arbitrarily:
**
for all , where*(C1)*, , , and ,*(C2)* with and ,*(C3)*.**Then converges strongly to , where which solves the following variational inequality:
**
which is the optimality condition for the minimization problem
**
where is a potential function for (i.e., for ).*

*Proof. *Due to condition (C1), we may assume without loss of generality, then, that for all . By Lemma 2.8, we have that . Next, we will assume that .

Next, we will divide the proof into six steps.*Step 1. *We will show that are bounded.

Since , are , -inverse-strongly monotone mappings, we have that
In a similar way, we can obtain
It is clear that if , , then , are all nonexpansive.

Put , . It follows that
By Lemma 2.7, we have that for all . Then, we have that
Hence, we have that
From (3.2), we deduce that
It follows by induction that
Therefore is bounded, so are , , , , and .*Step 2. *We claim that . From (3.2), we have that

Since are nonexpansive, we also have that
On the other hand, from and , it follows that
Substituting into (3.14) and into (3.15), we get
From (A2), we obtain
and then
So
It follows that
Without loss of generality, let us assume that there exists a real number such that , for all . Then, we have that
and hence
where . Substituting (3.22) into (3.13), we have that
Substituting (3.23) into (3.12), we get
where . By conditions (C1)-(C2) and Lemma 2.5, we have that as . From (3.23), we also have that as .*Step 3. *We show the following: (i);
(ii). For and , by (3.5) and (3.8), we get
It follows that
So, we obtain
where . By conditions (C1) and (C3) and , we obtain that as .

Substituting (3.8) into (3.25), we get
From (3.26), we have that
So, we also have that
where . By conditions (C1)–(C3), and , we obtain that as .*Step 4. *We show the following: (i);
(ii);
(iii). Since is firmly nonexpansive and by (2.2), we observe that
Hence, we have that
Since is 1-inverse-strongly monotone and by (2.2), we compute
which implies that
Substituting (3.32) into (3.34), we have that
Substituting (3.35) into (3.26), we get
Then, we derive
By condition (C1), , , and . So, we have that , as . It follows that
From (3.2), we have that
By condition (C1) and , we obtain that as . Next, we observe that
Since is bounded and by condition (C1), we have that as , and
Since and , it implies that as . Hence, we have that
By (3.38) and , we obtain as . Moreover, we also have that
By (3.38) and , we obtain as .*Step 5. *We show that and . It is easy to see that is a contraction of into itself. Indeed, since , we have that
Hence is complete, and there exists a unique fixed point such that . By Lemma 2.2, we obtain that for all .

Next, we show that , where is the unique solution of the variational inequality , for all . We can choose a subsequence of such that
As is bounded, there exists a subsequence of which converges weakly to . We may assume without loss of generality that .

We claim that . Since , , and and by Lemma 2.6, we have that .

Next, we show that . Since , we know that
It follows by (A2) that
Hence,
For and , let . From (3.48), we have that
From , we have that . Further, from (A4) and the weakly lower semicontinuity of , and , we have that
From (A1), (A4), and (3.50), we have that
and hence
Letting , we have, for each , that
This implies that .

Lastly, we show that . In fact, since is a -inverse-strongly monotone, is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that is a maximal monotone. Let , since . Again since , we have that , that is, . By virtue of the maximal monotonicity of , we have that
and hence