#### Abstract

We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.

#### 1. Introduction

Let be a real Hilbert space with inner product and norm , be a nonempty closed convex subset of and be the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called strongly positive on if there exists a constant such that A mapping is called -Lipschitz continuous if there exists a constant such that In particular, if then is called a nonexpansive mapping; if then is called a contraction.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP) [1] which is to find a point such that The solution set of VIP (3) is denoted by .

Let be a real-valued function, be a nonlinear mapping and be a bifunction. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (4) by . The system of equilibrium problems or generalized equilibrium problems is a tool to study Nash eequilibrium problems, see for example [3–8]. In fact, the GMEP (4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see for example, [9–15]. Here we also consider a system of two generalized equilibrium problem that could be usefull to study the Two players game problem, see [16].

Throughout this paper, it is assumed as in [2] that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (H5), where(H1) for all ;(H2) is monotone, that is, for any ;(H3) is upper-hemicontinuous, that is, for each , (H4) is convex and lower semicontinuous for each ;(H5)for each and , there exists a bounded subset and such that for any ,

Given a positive number . Let is the solution set of the auxiliary mixed equilibrium problem, that is, for each , In particular, whenever , is rewritten as .

Let be two bifunctions, and be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find such that where and are two constants. It is introduced and studied in [17]. Whenever , the SGEP reduces to a system of variational inequalities, which is considered and studied in [18]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.

In 2010, Ceng and Yao [17] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [17]). *Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse-strongly monotone for . Let for . Then, is a solution of (8) if and only if is a fixed point of the mapping defined by where . Here, we denote the fixed point set of by .*

Let be an infinite family of nonexpansive mappings on and be a sequence of nonnegative numbers in . For any , define a mapping on as follows: Such a mapping is called the -mapping generated by and .

In 2011, for the case where , Yao et al. [14] proposed the following hybrid iterative algorithm where be a contraction, is differentiable and strongly convex, and are given, for finding a common element of the set and the fixed point set of an infinite family of nonexpansive mappings on . They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (10) to a point under some appropriate conditions. This point also solves the following optimization problem: where is the potential function of .

Let be a contraction and be a strongly positive bounded linear operator on . Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)–(H4), and that are inverse-strongly monotone. Let the mapping be defined as in Proposition CY. Very recently, Ceng et al. [11] introduced the following hybrid extragradient-like iterative algorithm for finding a common solution of GMEP (4), SGEP (8) and the fixed point problem of an infinite family of nonexpansive mappings on , where , and are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (11) to a point under some suitable conditions. This point also solves the following optimization problem: where is the potential function of .

On the other hand, let be a single-valued mapping of into and be a set-valued mapping with . Consider the following variational inclusion: find a point such that We denote by the solution set of the variational inclusion (12). In particular, if , then . If , then problem (12) becomes the inclusion problem introduced by Rockafellar [19]. It is known that problem (12) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, and so forth. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows: where is a positive number.

In this paper, we will introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems. Such solution also solves an optimization problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement and generalization of the previously known results in this area.

#### 2. Preliminaries

Throughout this paper, we assume that is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence , that is,

*Definition 1. *A mapping is called

(i) monotone if

(ii) -strongly monotone if there exists a constant such that

(iii) -inverse-strongly monotone if there exists a constant such that

It is easy to see that the projection is -ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

*Definition 2. *A differentiable function is called:

(i) convex, if
where is the Frechet derivative of at ;

(ii) strongly convex, 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 .

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property

Some important properties of projections are gathered in the following proposition.

Proposition 3. * For given and :*(i);(ii), ;(iii), . (

*This implies that*

*is nonexpansive and monotone.)*

By using the technique of [20], we can readily obtain the following elementary result.

Proposition 4 (see [11, Lemma 1 and Proposition 1]). * Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)–(H4). Assume that *(i)

*is strongly convex with constant*

*and the function*

*is weakly upper semicontinuous for each*;(ii)

*for each*

*and*,

*there exists a bounded subset*

*and*

*such that for any*,

*Then the following hold:(a) for each*;*(b)**is single-valued*;*(c)**is nonexpansive if**is Lipschitz continuous with constant**and*

*where**for*;*(d) for all**and*

*(e)*;*(f)**is closed and convex.*

*In particular, whenever**is a bifunction satisfying the conditions (H1)–(H4) and*,*then that is, for any*,

*(**is firmly nonexpansive) and*

*In this case,**is rewritten as*.*If, in addition,*,*then**is rewritten as*.*Remark 5. *Suppose is strongly convex with constant and is Lipschitz continuous with constant . Then is -strongly monotone and -Lipschitz continuous with positive constants . Utilizing Proposition 4 (d) we can show that for all and ,

We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 6. *Let be a real inner product space. Then there holds the following inequality
*

Lemma 7. *Let be a real Hilbert space. Then the following hold:*(a)* for all ;*(b)* for all and with ;*(c)*If is a sequence in such that , it follows that
**We have the following crucial lemmas concerning the -mappings defined by (9).*

Lemma 8 (see [21, Lemma 3.2]). *Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and the limit exists, where is defined by (9).*

*Remark 9 (see [22, Remark 3.1]). *It can be known from Lemma 8 that if is a nonempty bounded subset of , then for there exists such that for all

*Remark 10 (see [22, Remark 3.2]). *Utilizing Lemma 8, we define a mapping as follows:
Such a is called the -mapping generated by and . Since is nonexpansive, is also nonexpansive. Indeed, observe that for each
If is a bounded sequence in , then we put . Hence, it is clear from Remark 5 that for an arbitrary there exists such that for all
This implies that

Lemma 11 (see [21, Lemma 3.3]). *Let be a sequence of nonexpansive self-mappings on such that , and let be a sequence in for some . Then, .*

Lemma 12 (see [23, Demiclosedness principle]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .*

Lemma 13. *Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3 (i)) implies
*

Lemma 14 (see [24]). *Let and be bounded sequences in a real Banach space and be a sequence in with . Suppose
**
Then, .*

Lemma 15 (see [25]). *Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a real sequence such that*(i)*;*(ii)* or .** Then .*

Recall that a set-valued mapping is called monotone if for all and imply A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for for every implies . Next we provide an example to illustrate the concept of maximal monotone mapping.

Let be a monotone, -Lipschitz-continuous mapping and let be the normal cone to at , that is, Define Then, is maximal monotone and if and only if ; see [19].

Assume that is a maximal monotone mapping. Let . In terms of Huang [26] (see also [27]), there holds the following property for the resolvent operator .

Lemma 16. * is single-valued and firmly nonexpansive, that is,
**
Consequently, is nonexpansive and monotone.*

Lemma 17 (see [28]). *Let be a maximal monotone mapping with . Then for any given is a solution of problem (12) if and only if satisfies
*

Lemma 18 (see [27]). *Let be a maximal monotone mapping with and let be a strongly monotone, continuous and single-valued mapping. Then for each , the equation has a unique solution for .*

Lemma 19 (see [28]). *Let be a maximal monotone mapping with and be a monotone, continuous and single-valued mapping. Then for each . In this case, is maximal monotone.*

Lemma 20 (see [29]). *Let be a nonempty closed convex subset of a real Hilbert space , and be a proper lower semicontinuous differentiable convex function. If is a solution the minimization problem
**
then,
**
In particular, if solves (OP), then
*

#### 3. Main Results

In this section, we introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.

Theorem 21. *Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:*(i)* is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;*(ii)*for each , there exist a bounded subset and such that for any ,
*(iii)*, and ;*(iv)*, and satisfies
*(v)*.**
Given arbitrarily, then the sequence generated iteratively by
**
converges strongly to which solves the following optimization problem provided is firmly nonexpansive:
**
where is the potential function of .*

*Proof. *Since and , we may assume, without loss of generality, that . Since is a -strongly positive bounded linear operator on , we know that
Observe that
that is, is positive. It follows that
Put
for all , and , where is the identity mapping on . Then we have that .

We divide the rest of the proof into several steps.*Step 1*. We show that is bounded. Indeed, take arbitrarily. Since , is -inverse strongly monotone and , we have, for any ,
Since and is -inverse strongly monotone, where , by Lemma 16 we deduce that for each
Combining (52) and (53), we have
Since , is -inverse-strongly monotone for , and for , we deduce that, for any ,
(This shows that is nonexpansive.) Thus, from (54), we get
Set . Then from (47) we have
By induction, we get
Therefore, is bounded and so are the sequences and .*Step 2.* We show that as .

Indeed, define
Then from the definition of , we obtain
It follows that
From (9), since and are all nonexpansive, we have
where is a constant such that
On the other hand, we estimate . Taking into account that and , we may assume, without loss of generality, that and . Utilizing Remark 5 and Lemma 16, we have
where for some .

Note that
Since is nonexpansive, from (62), (64) and (66) it follows that
Utilizing (61), and (68), we have