Research Article | Open Access

# Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems

**Academic Editor:**Bin Liu

#### Abstract

We introduce a new general hybrid iterative algorithm for finding a common
element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of
generalized mixed equilibrium problem, and the set of solution of the variational inclusion for a *Î²*-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), Yao and Liou (2010), Tan and Chang
(2011), and other authors.

#### 1. Introduction

In the theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance, and applied sciences.

Equilibrium theory represents an important area of mathematical sciences such as optimization, operations research, game theory, complementarity problems, financial mathematics, and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, fixed point problems, and complementarity problems as special cases; for example, see the references herein. 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, be a mapping and be a real-valued function. The *generalized mixed equilibrium problem* for finding such that
The set of solutions of (1.1) is denoted by , that is
If and , the problem (1.1) is reduced into the *equilibrium problem* [1] for finding such that
The set of solutions of (1.3) is denoted by . This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem, please consult [2â€“4].

If and , the problem (1.1) is reduced into the *Hartmann-Stampacchia variational inequality* [5] for finding such that
The set of solutions of (1.4) is denoted by . The variational inequality has been extensively studied in the literature [6].

If and , the problem (1.1) is reduced into the *minimize problem* for finding such that
The set of solutions of (1.5) is denoted by .

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 [7].

Recall, 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 [8]. We denote weak convergence and strongly 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 mappings 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 *contraction* 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 .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. Let be a single-valued nonlinear mapping and be 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 (1.12) is denoted by . The variational inclusion has been extensively studied in the literature. See, for example, [9â€“12] 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 [13].

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 (1.12) is a fixed point of the operator for all , see [14], that is, .

In 2000, Moudafi [15] 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 chosen arbitrarily, where satisfies certain conditions, converges strongly to a fixed point of (say ) which is the unique solution of the following variational inequality:

In 2006, Marino and Xu [7] 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 prove that if and the sequence satisfies appropriate conditions, then the sequence generated by (1.17) 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 ).

In 2010, Yao and Liou [16] introduced the following composite iterative scheme in a real Hilbert space: for all , where . Furthermore, they proved and converge strongly to the same point , where is the projection of onto .

In 2011, Tan and Chang [11] introduced the following iterative process for be a sequence of nonexpansive mappings. Let be the sequence defined by where , and . Then, the sequence defined by (1.21) converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solution of the variational inequality and the generalized equilibrium problem.

In this paper, we modify the iterative methods (1.17), (1.20), and (1.21) by purposing the following new general viscosity iterative method: , for all , where , , and satisfy some appropriate conditions. Consequently, we show that under some control conditions the sequence strongly converge to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problem, and the set of solution of the variational inclusion in a real Hilbert space.

#### 2. Preliminaries

Let be a real Hilbert space and be 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 (1.4) if and only if satisfies the relation for all .*

Lemma 2.2. *For a given , , .**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 [17]). *Let be a maximal monotone mapping and let be a monotone and Lipschitz continuous mapping. Then the mapping is a maximal monotone mapping.*

Lemma 2.4 (see [18]). *Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality , hold for each with .*

Lemma 2.5 (see [19]). *Assume 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 [20]). *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 [21]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction mapping satisfies (A1)â€“(A4) and let is convex and lower semicontinuous such that . Assume that either (B1) or (B2) holds. For and , then 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[7]). *Assume 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 inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. *Let be a real Hilbert space, be a closed convex subset of . Let be a bifunction of into satisfying (A1)â€“(A4) and be -inverse-strongly monotone mappings, is convex and lower semicontinuous function, be a contraction with coefficient , be a maximal monotone mapping and be a strongly positive linear bounded operator of into itself with coefficient , assume that . Let be a nonexpansive mapping of into itself and assume that either (B1) or (B2) holds such that
**
Suppose is a sequences generated by the following algorithm arbitrarily:
**
where such that and with satisfy the following conditions:*(C1)*, and ,*(C2)* and .**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. *Because of condition (C1), we may assume without loss of generality, then for all . By Lemma 2.8, we have . Next, we will assume that .*Step 1. *We will show are bounded.

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

Put . It follows that
By Lemma 2.7, we have for all . Then, we have
Put for all . From (3.2), we deduce that
It follows from induction that
Therefore is bounded, so are , , , , and .*Step 2. *We claim that . From (3.2), we have
We estimate , so we have
Substituting (3.12) into (3.11) that
We note that
Next, we estimate , then we get
Substituting (3.15) into (3.14), we obtain that
And substituting (3.12), (3.16) into (3.11), we get
where is a constant satisfying
This together with (C1), (C2), and Lemma 2.5, implies that
From (3.15), we also have as .*Step 3. *We show the following: (i);
(ii).
For and , then we get
It follows that
By the convexity of the norm , we have
Substituting (3.8), (3.21) into (3.22), we obtain
So, we obtain
where . Since condition (C1), (C2) and then we obtain that as . We consider this inequality in (3.21) that
Substituting (3.20) into (3.25), we have
Substituting (3.8) and (3.26) into (3.22), we obtain
So, we also have
where . Since condition (C1), (C2), and then we obtain that as .*Step 4. *We show the following:(i);
(ii);
(iii).
Since is firmly nonexpansive, we observe that
Hence, we have
Since is 1-inverse-strongly monotone, we have
which implies that
Substituting (3.32) into (3.25), we have
Substituting (3.30) and (3.33) into (3.22), we obtain
Then, we derive
By condition (C1), (C2), , and . So, we have as . It follows that
We note that . From , and hence
Since
So, by (3.37) and , we obtain
Therefore, we observe that
By condition (C1), we have as . Next, we observe that
By (3.39) and (3.40), we have as .*Step 5. *We show that and . It is easy to see that is a contraction of into itself. Indeed, since we have