#### Abstract

We study the existence and the algorithmic aspect of a System of Generalized Mixed Equilibrium Problems involving variational-like inequalities (SGMEPs) in the setting of Banach spaces. The approach adopted is based on the auxiliary principle technique and arguments from generalized convexity. A new existence theorem for the auxiliary problem is established; this leads us to generate an algorithm which converges strongly to a solution of (SGMEP) under weaker assumptions. When the study is reduced to the setting of reflexive Banach spaces, then it can be more relaxed by dropping the coercivity condition. The results obtained in this paper are new and improve some recent studies in this field.

#### 1. Introduction and the Problem Statement

Let be a nonempty closed convex subset of a Banach space , and let be a real-valued bifunction. By equilibrium problem, in short (EP), we mean the following problem: Equilibrium problems are suitable and common format for investigation of various applied problems arising in economics, mathematical physics, transportation, communication systems, engineering, and other fields. Moreover, equilibrium problems are closely related with other general problems in nonlinear analysis, such as fixed points, game theory, variational inequality, and optimization problems, see [1–4]. Various kinds of iterative algorithms to solve equilibrium problems and variational inequalities have been developed by many authors. There exists a vast literature on the approximation solvability of equilibrium problems and nonlinear variational inequalities using projection type methods, proximal-type methods, or resolvent operator type methods, see [5–11]. We observe that the projection method and its variant forms cannot be applied for constructing iterative algorithms of mixed variational-like inequalities or mixed equilibrium problems involving variational-like inequalities. This fact motivated many authors to develop the auxiliary principle technique to study the existence and algorithm of solutions for variational-like inequalities and its extensions to mixed equilibrium problems, see [12–15]. Kazmi and Khan [16] studied a system of generalized variational-like inequality problems in Hilbert spaces by using the auxiliary principle technique. Recently, Ding and Wang [17] and Ding [18] introduced new iterative algorithms for solving some class of system of generalized mixed variational-like inequalities and system of mixed equilibrium problems involving variational-like inequalities.

In this paper, we study the existence and the algorithmic aspect of a System of Generalized Mixed Equilibrium Problems involving variational-like inequalities (SGMEPs) in the setting of Banach spaces. A new existence theorem for the auxiliary problem is established, this leads us to generate an algorithm which converges strongly to a solution of (SGMEP) under weaker assumptions. When the study is reduced to the setting of reflexive Banach spaces, then it can be more relaxed by dropping the coercivity condition. The results obtained in this paper are new and improve some recent studies in this field.

Throughout this paper, let be an index set. For each , let be a Banach space, its dual, is the duality pairing between and , denotes the norm in and , and let be a nonempty closed convex subset of . We will denote by the family of all nonempty closed and bounded subsets of .

For each , let and be multivalued mappings, and be single-valued mappings, be two real-valued functions, and let . We consider the following System of Generalized Mixed Equilibrium Problem involving variational-like inequalities (SGMEP).

Find , , and such that

##### 1.1. Some Special Cases

(1)If for each , is a Hilbert space, , , and for each , and , then problem (1.2) reduces to the problem of finding such that Problem (1.3) has been introduced and studied by Kazmi and Khan in [16]. (2)If for each , is a reflexive Banach space, and , then problem (1.2) reduces to the problem of finding , and such that Problem (1.4) has been considered recently by Ding and Wang in [17]. (3)In finite dimensional spaces a particular problem of (1.2) has been considered by Mordukhovich et al. in [19] for finding a common solution of a variational inequality problem and an equilibrium problem by an approach based on an hybrid proximal point algorithm.

The organization of the paper is as follows. In Section 2, we give some definitions and preliminary results that we will need in the sequel. We introduce in Section 3 an auxiliary principle, for the problem studied and we show that under some suitable conditions this problem has a unique solution. Further, by using the auxiliary principle we consider an algorithm to approach the solution of the main problem studied in this paper and discuss its convergence. Finally, we end the paper by some commentaries on the approach used and we give some comparisons with some known results in this direction.

#### 2. Preliminaries

In this section, we present some basic concepts, properties, and notations that we will consider in the development of our work. Let be a real Banach space with norm , its dual space, and denote the duality pairing between and , and let be a nonempty closed convex subset of . For a finite subset of , we denote by the convex hull of . Let be the family of all nonempty closed and bounded subsets of , and let be the Hausdorff metric on defined for all by where .

*Definition 2.1. *A mapping is said to be(i)monotone if and only if for all ; (ii)-strongly monotone with if and only if
(iii)-Lipschitz continuous if there exists such that

*Definition 2.2. *Let be a bounded linear operator. is said to be -strongly positive if there exists such that

*Remark 2.3. * One can easily see that if a bounded linear operator is -strongly positive, then it is -strongly monotone and -Lipschitz continuous, where is the operator norm of .

*Definition 2.4. *Let and be two Banach spaces with respective norm and . A mapping is said to be -Lipschitz continuous if there exist constants such that

*Definition 2.5. *A mapping is said to be (i)affine in the second argument if
(ii)-Lipschitz continuous if there exists a constant such that

*Definition 2.6. *A real-valued function is said to be skew-symmetric if for all ,

The skew-symmetric functions have certain properties, see [20], which can be regarded as analogs of the conditions governing the gradient monotonicity and nonnegativity of the second derivative of convex functions.

*Definition 2.7. *Let be a real-valued bifunction. Then, (i) is said to be monotone if
(ii) is said to be -strongly monotone if there exists such that
(iii) is said to be upper-hemicontinuous if, for all , the mapping defined by is upper-semicontinuous.

*Remark 2.8. *Clearly, strong monotonicity of implies monotonicity of .

Now, let denote the solution set of the equilibrium problem (EP) associated to a bifunction , that is, . In many situations, we usually need some more information on the structure of the solution set when it is nonempty. As a preliminary result that we will need in the sequel, the following Lemma gives some sufficient conditions which insure that is convex and closed.

Lemma 2.9. *Let be a Banach space, and let be a closed convex subset of . Let be a real-valued bifunction such that for all . Assume that*(i)* is monotone and upper-hemicontinuous; *(ii)*for each fixed, the function is convex and lower semicontinuous. Then, the solution set is convex and closed whenever it is nonempty.*

*Proof. *Assume that . Let and for set . Let us show that . Since , it follows
From the monotonicity of , one deduces
Therefore, from the convexity of with respect to its second argument, it follows
Now for and , set . Since is an equilibrium bifunction and is convex with respect to its second argument, it follows that
Taking account of relation (2.13) with , it follows that for all , and therefore since
Since is upper-hemicontinuous, it follows that
Therefore, , and hence is convex.

Now, let us show that is closed. To this aim, let such that . Let us show that . One has for all , since is monotone, it follows that
From the lower semicontinuity of with respect to the second argument, it follows that
By a similar argument as above, one can easily show that for all . Therefore, , which completes the proof.

In the sequel, we will need the following result that we present in a more general setting and for which we refer to [21].

Lemma 2.10. *Let be a Hausdorff topological vector space and a closed convex subset of . Consider two real bifunctions such that*(i)*for each , if then ; *(ii)*for each fixed , the function is lower semicontinuous on every compact subset of ; *(iii)*for each finite subset of , one has
*(iv)* coercivity: there exists a nonempty compact convex subset of such that either (a) or (b) in the following holds(a) for all ;(b)there exists .*

*Then, there exists such that for all . Furthermore, the set of solution is compact.*

*Remark 2.11. * Condition (iii) in Lemma 2.10 is much more related to convexity assumptions of the bifunction , see Proposition 2.12 in the following.

If is compact, then condition (iv) in Lemma 2.10 can be dropped.

The following proposition gives some sufficient conditions which insure condition (iii) in Lemma 2.10.

Proposition 2.12. *Suppose that*(i) for each ;(ii)for each fixed, the set is convex. *
Then, condition (iii) of Lemma 2.10 is staisfied.*

*Proof. *Suppose by contradiction that condition (iii) of Lemma 2.10 is not satisfied. Then, there exist and , with , such that .

Therefore, by setting , it follows from (ii) that , which contradicts assumption (i).

As a consequence of Lemma 2.10, we obtain the following result on existence of mixed equilibrium problem that we will need in the sequel. We will include its proof for completeness.

Lemma 2.13. *Let be a Banach space and a closed convex subset of . Let be two real bifunctions such that*(i)* for all ; is monotone and upper-hemicontinuous; for each fixed, the function is convex and lower semicontinuous; *(ii)* for all ; for each fixed, the function is upper-semicontinuous; for each fixed, the function is convex and lower semicontinuous; *(iii)*coercivity: there exists a nonempty compact convex subset of and such that
**Then, there exists such that
**
Furthermore, the solution set of the mixed equilibrium problem (2.21) is compact and convex.*

*Proof. *The proof is a direct consequence of Lemma 2.10 by setting

*Remark 2.14. * Lemma 2.13 is in fact a slight extension of Theorem 1 in [2] and Theorem 4.5 in [21] where the equilibrium condition has been relaxed by assuming and for all .

If is a reflexive Banach space endowed with its weak topology , then the coercivity condition (iii) in Lemma 2.13 can be replaced by the following condition: (iii)′ there exists such that.

Ended, let , and set . One has a convex and -compact subset of . Since is a reflexive Banach space, is lower semicontinuous and is weakly compact, it follows that there exists such that for all . Let , and set

Since is convex, and , one deduces

It follows that

Thus,

Since is monotone, it follows from relation (2.26) that for all

Since when , then there exists such that for with one has

Take and set , then from relations (2.27) and (2.28), one deduces that for each one has

Hence, condition (iii) in Lemma 2.13 is satisfied.

We end this section by the following result related to the Hausdorff metric that we will need in the sequel and for which we refer to [22].

Lemma 2.15. *Let be a complete metric space and a set-valued mapping. Then, for any given and any given and , there exists such that
*

#### 3. Approximation by an Auxiliary Principle

In order to get approximate solutions for the system (1.2) of generalized mixed equilibrium problem involving generalized mixed variational-like inequality problems (SGMEP), we consider the following *auxiliary problem*: for and for given mappings , , , , and ,

In this section, we give some existence results of solutions for the auxiliary problem (AP). The results obtained will be needed in the sequel to generate a unified algorithm to approach solutions of the system (1.2) under some weaker assumptions in comparison with some known results in literature.

Theorem 3.1. *For each , let be a Banach space and a nonempty closed convex subset of , two real-valued bifunctions, a bounded linear operator, and single-valued mappings such that *(i)* for all ; is monotone and upper-hemicontinuous; for each fixed, the function is convex and lower semicontinuous; *(ii)* is affine in the first argument and continuous in the second argument such that
*(iii)* is skew symmetric and continuous; for each fixed, the function is convex; *(iv)* is -strongly positive; *(v)*coercivity: for each , and , there exists a nonempty compact convex subset of and such that **
Then, the auxiliary problem (AP) has a unique solution.*

*Proof. *The proof of this lemma is a direct application of Lemma 2.13 by considering, for each , the bifunctions defined by
where , , and are given. We need only to show that the solution is unique. To this aim, suppose that problem (AP) has two solutions and , then for and for all , we have
Take in relation (3.5) and in relation (3.6) and adding the two inequalities, one obtain
Since for each , is monotone, is skew symmetric, and is -strongly positive, it follows that
Therefore, for , which completes the proof.

Theorem 3.2. *For each , let be a reflexive Banach space and a nonempty closed convex subset of , two real-valued bifunctions, a bounded linear operator, and single-valued mappings such that *(i)* for all ; is monotone and upper-hemicontinuous; for each fixed, the function is convex and lower semicontinuous; *(ii)* is affine in the first argument and continuous in the second argument such that
*(iii)* is skew symmetric and continuous; for each fixed, the function is convex;*(iv)* is -strongly positive linear operator.**
Then, the auxiliary problem (AP) has a unique solution.*

*Proof. *For each and given , , and , define the following bifunctions by
One can easily see that conditions (i)–(iii) above imply conditions (i) and (ii) of Lemma 2.13. In order to get the conclusion, we need only to show that the coercivity condition (iii) of Lemma 2.13 is satisfied. To this aim, taking into account Remark 2.14 (2), we need only to show that for some one has when . Let . Then,
Therefore,
It follows that when which completes the proof.

*Remark 3.3. *Theorem 3.2 improves recent results given by Ding [18, Theorem 3.1] since the bifunction is not needed to be -Lipschitz continuous and weakly upper semicontinuous with respect to the first argument. We mention also that all the results obtained in [18] are under the assumption ; in our approach, this assumption is not needed.

Theorem 3.2 shows that the auxiliary problem (AP) has a unique solution; we can define the following general iterative method to approach the solution of system (1.2) of generalized mixed equilibrium problem involving variational-like inequalities (SGMEP).

*Algorithm 3.4. * For a given , , and . By Theorem 3.1, the auxiliary problem (AP) has a unique solution , that is, for each , we have
Since for each , , and , by Lemma 2.15, there exist and such that
where and are the Hausdorff metrics on and , respectively.

By using Theorem 3.1 again, the auxiliary problem (AP) has a unique solution such that
By induction, we can construct an iterative algorithm to compute the approximate solution for the system (1.2) as follows: for given , , and , there exist sequences , and such that for each

The following convergence analysis is presented for the algorithm above.

Theorem 3.5. *Under the hypotheses of Theorem 3.2, further assume that for each ,*(i)* is -strongly monotone and upper-hemicontinuous; is -mixed Lipschitz continuity; is --Lipschitz continuous, and is --Lipschitz continuous;*(ii)* is -Lipschitz continuous; *(iii)*the sequence of positive real numbers is increasing and . **
Furthermore, assume that the following condition holds:
**
Then, the sequences , and generated by Algorithm 3.4 converge strongly to , and , respectively, where ,, and is a solution of the System of Generalized Mixed Equilibrium Problem involving variational-like in equalities (1.2).*

*Proof. *By the definition of Algorithm 3.4, we have for ,
By considering in relation (3.18) and in relation (3.19), and taking into account , one obtains by adding the two inequalities
Since is monotone, is skew-symmetric, and is -strongly positive and -Lispchitz continuous, it follows from relation (3.20) that
Note that is -mixed Lipschitz continuous, is --Lipschitz continuous, and is --Lipschitz continuous, it follows by Algorithm 3.4
Since is -Lipschitz continuous, it follows from relation (3.21) considered for that
Hence,
with
Similarly, by the assumptions on , , , , , and , it follows from relation (3.21) considered for that
with
Adding the inequalities (3.24) and (3.26), we obtain
where .

On , let us consider the norm defined by for all , that is, is a Banach space. It follows from relation (3.28) that
Taking account of the assumptions, it is easy to see that as . From condition (C_{1}) (relation (3.17)), we know that . Hence, there exist and such that for all . Therefore, it follows from (3.29) that
This implies that is a Cauchy sequence in . Thus, converges strongly to some .

By Algorithm 3.4 and the Lipschitz continuity assumption on ,, , and , we have
It follows, for each , that is a Cauchy sequence in and is a Cauchy sequence in . Thus, there exists such that converges strongly to .

Noting that , it follows that
Hence, we must have . Similarly, one can show that , , and . By Algorithm 3.4, we have that, for all ,
Since is -mixed Lipschitz continuous and is continuous in the second argument, one has, for each ,
Hence,
From the monotonicity of and the linearity of , one has
Taking into account result (3.35) and since, for each , is a lower semicontinuous function, is continuous, and the sequence converges strongly to , it follows from relation (3.36) by passing to the limit when goes to infinity that
Now, for each and for and , set . Since is convex, then for ; it follows that
Taking into account the convexity of , the fact that is affine with respect to the first argument and , one has