#### Abstract

We study and establish the existence of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space. Furthermore, we also modify Halpern-Mann iterations for finding a common solution of a generalized mixed equilibrium problem and a fixed point problem. Under suitable conditions of the purposed iterative sequences, the strong convergence theorems are established by using sunny generalized nonexpansive retraction in Banach spaces. Our results extend and improve various results existing in the current literature.

#### 1. Introduction

In the past years, variational inequalities is among the most important and interesting mathematical problems, since they have wide applications in the optimization and control, economics and equilibrium, engineering science and physical sciences.

Equilibrium problem represents an important area of mathematical sciences such as optimization, operations research, game theory, financial mathematics, and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, and fixed point problems as special cases.

Throughout this paper, we denote the strong convergence and weak convergence by , , respectively.

Let be a closed and convex subset of a real Banach spacewith the dual space . Let be a closed and convex subset of . We recall the following definitions.(1)A mapping is said to be *monotone* if for each such that
(2)A mapping is said to be , if there exists a constant such that
(3)A mapping is said to be , if there exists a constant such that
(4)A mapping is said to be *skew monotone* if for each such that
(5)A mapping is said to be -*inverse strongly skew monotone* if there exists a constant such that

*Definition 1. *Letbe a Banach space. Then,(1) is said to be *strictly convex* if for all with ;(2) is said to be *uniformly convex* if for each , there exists such that , for all with ;(3) is said to be *smooth* if the limit
â€‰exists, for each ;(4) is said to be *uniformly smooth* if the limit (6) is attained uniformly, for all ;(5) is said to have *uniformly GÃ¢teaux differentiable norm* if for all , the the limit (6) converges uniformly, for .

*Definition 2. *Let be a Banach space. Then, a function is said to be the *modulus of smoothness* of if
(1) is said to be *smooth * if .(2) is said to be *uniformly smooth * if and only if .

*Definition 3. *Let be a Banach space. Then, the *modulus of convexity* of is the function defined by
(1) is said to be *uniformly convex* if and only if for all .(2)Let be a fixed real number . Then, is said to be -*uniformly convex* if there exists a constant such that for all .

Observe that every -uniformly convex is uniformly convex. One should note that no a Banach space is -uniformly convex, for . It is well known that a Hilbert space is -uniformly convex and uniformly smooth.

For any , the *generalized duality mapping * is defined by

In particular, is called the *normalized duality mapping*. If is a Hilbert space, then , where is the identity mapping. That is,

*Remark 4. *The basic properties below hold (see [1â€“3]). (1)If is uniformly smooth real Banach space, then is uniformly continuous on each bounded subset of .(2)If is uniformly smooth real Banach space, then is a normalized duality mapping on , and then , , and , where on and are the identity mappings on and , respectively.(3)Let be a smooth, strictly convex reflexive Banach space, and let be the duality mapping from into . Then, is also single-valued, one-to-one, and onto, and it is also the duality mapping from into .(4)If is a reflexive, strictly convex Banach space, then is hemicontinuous; that is, is norm-to-weak*-continuous. (5)If is a reflexive, smooth, and strictly convex Banach space, then is single-valued, one-to-one, and onto. (6)A Banach space is uniformly smooth if and only if is uniformly convex. (7)Each uniformly convex Banach space has the *Kadec-Klee property*; that is, for any sequence , if , and , then .(8)A Banach space is strictly convex if and only if is strictly monotone; that is,
(9)Both uniformly smooth Banach space and uniformly convex Banach space are reflexive. (10)If is uniformly convex, then is uniformly norm-to-norm continuous on each bounded subset of .(11)If is strictly convex Banach space, then is one-to-one; that is, implies .

Let be the *normalized duality mapping*; then is said to be *weakly sequentially continuous* if the strong convergence of a sequence to implies the weak* convergence of a sequence to in .

Let be a smooth and strictly convex reflexive Banach space, and let be a nonempty, closed, and convex subset of . We assume that the *Lyapunov functional * is defined by [3, 4]

Let is nonempty, closed, and convex subset of a Banach space . The *generalized projection* [3] is defined by for each ,

*Remark 5. *From the definition of . It is easy to see that (1), for all ;(2), for all ;(3), for all ;(4)if is a real Hilbert space , then , and (the metric projection of onto ).

Lemma 6 (see [3, 4]). *If is a nonempty, closed, and convex subset of a smooth and strictly convex reflexive real Banach space , then*(1)*for and , one has
*(2)*,, and;*(3)* if and only if , . *

Let be a nonempty, closed subset of a smooth, strictly convex, and reflexive Banach space such that is closed and convex. For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:(DA1), for all ; (DA2)is monotone, that is, , for all (DA3) for all ,(DA4) for all ,â€‰ is convex and lower semicontinuous.

The following result is in Blum and Oettli [5], and see proof in [6].

Let be the set of real numbers, let be a real Banach space with the norm , and let that is the dual pair between and by be the dual space of . Let be a nonempty, closed, and convex subset of , let be the duality mapping from into such that is closed and convex of , let us assume that a bifunction satisfies suitable conditions, let be a skew monotone operator from into , and let be a real-valued function.

The *generalized mixed equilibrium problem* is to find such that

The set of solutions of (16) is denoted by ; that is,

If , then the problem (16) reduces to the *mixed equilibrium problem* which is to find such that

The set of solution of problem (18) is denoted by ; that is,

If and , then the problem (16) reduces to the *equilibrium problem* which is to find such that

The set of solution of problem (20) is denoted by ; that is,

The above formulation (20) was considered in Takahashi and Zembayashi [7], and they proved a strong convergence theorem for finding a solution of the equilibrium problem (20) in Banach spaces.

If and , then the problem (16) reduces to *variational inequality*, which is to find such that

The set of solution of problem (22) is denoted by ; that is,

In the sequel, let be a mapping, we denote by the set of fixed points of ; that is,

In 1953, Mann [8] introduced an iterative algorithm which is defined by the initial point is taken in arbitrarily and where the sequence . Mannâ€™s iteration can yield only weak convergence.

In 1967, Halpern [9] introduced another iterative algorithm which is defined by the initial point is taken in arbitrarily and which satisfied the conditions and . Then, the sequence is converges strongly to a fixed point of .

In 2007, Takahashi and Zembayashi [7] introduced an iterative algorithm for finding a solution of an equilibrium problem with a bifunction defined on the dual space of a Banach space by using the shrinking projection method, and they established the strong convergence of the following results.

Theorem TZ. *Let be a uniformly convex Banach space whose norm is uniformly GÃ¢teaux differentiable, and let be a nonempty, closed and convex subset of such that is closed, and convex of . Assume that a mapping satisfied the conditions (DA1)â€“(DA4) such that . Let be a sequence generated by the following algorithm:
**
where is the duality mapping on , the sequence such that, for some , and is the sunny generalized nonexpansive retraction from onto . Then, the sequence converges strongly to some point , where is the sunny generalized nonexpansive retraction from to .*

In 2010, Plubtieng and Sriprad [10] proved the existence theorem of the variational inequality problem for skew monotone operator defined on the dual space of a smooth Banach space, and they established weak convergence theorem for finding a solution of the variational inequality problem using projection algorithm method with a new projection which was introduced by Ibaraki and Takahashi [11] and Iiduka and Takahashi [12] in Banach spaces.

Let be a smooth, strictly convex, and reflexive Banach space, let be the dual space of , and let be a nonempty, closed, and convex subset of such that is closed and convex of , where is the duality mapping on . Let be a skew monotone operator from into . Then, the *variational inequality problem* is to find such that

The set of solution of problem (28) is denoted by ; that is,

Theorem PS. *Let be a uniformly convex and 2-uniformly smooth Banach space whose duality mapping is weakly sequentially continuous. Let be a nonempty, closed and convex subset of such that is closed, and convex, and let be an -inverse-strongly-skew-monotone operator of into such that , and , for all and . Let be a sequence defined by and
**
for every , where is the sunny generalized nonexpansive retraction of into ,â€‰ and for some , with , where is a constant that satisfies , for all . Then, the sequence converges weakly to some element . Further .*

In 2011, Chen et al. [13] introduced a new iterative method for finding a solution of equilibrium with a bifunction defined on the dual space of a Banach space. They established the strong convergence theorem by using the sunny generalized nonexpansive retraction in Banach spaces.

Theorem CCW. *Let be a nonempty, closed and convex subset of a uniformly convex and uniformly smooth Banach space such that is closed and convex. Assume that a bifunction satisfies the conditions (DA1)â€“(DA4). Define a sequence in C by the following algorithm:
**
where is the duality mapping on , the sequences , , and , for some such that
**
Then, the sequence converges strongly to some point , where is the sunny generalized nonexpansive retraction from to .*

In 2012, Saewan et al. [14] introduced a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of fixed points for a --nonexpansive mapping in Banach spaces by using sunny generalized nonexpansive retraction in Banach spaces.

Theorem SCK. *Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty, closed, and convex subset of such that is closed and convex of . Let be a bifunction that satisfies the conditions (DA1)â€“(DA4), and let be a closed and --nonexpansive mapping. Assume that . For an initial point and define a sequence inby the following algorithm:
**
where is the duality mapping on , the sequence , , and for someand. If the following conditions are satisfied:
**
Then, the sequence converges strongly to some point , where is the sunny generalized nonexpansive retraction from onto .*

In this paper, Motivated and inspired by the previously mentioned above results, we study and investigate the existence of theorem for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space, and we construct an iterative procedure generated by the conditions for solving the common solution of a generalized mixed equilibrium problem and a fixed point problem by using the sunny generalized nonexpansive retraction. Under some suitable assumptions, the strong convergence theorem are established in Banach spaces. The results obtained in this paper extend and improve several recent results in this area.

#### 2. Preliminaries

*Definition 7. *Let be a nonempty, closed subset of a smooth Banach space. (1)A mapping is said to be *closed* if for each , and imply .(2)A mapping is said to be *nonexpansive* if
(3)A mapping is said to be --*nonexpansive *if , and
(4)A mapping is said to be *generalized nonexpansive *[15] if and

*Definition 8 (see [15]). *Let be a nonempty, closed subset of a smooth Banach space . A mapping is called(1)a retraction if ;(2)a sunny if , for all and .

We also know that if is a smooth, strictly convex, and reflexive Banach space and is nonempty, closed, and convex subset of , then there exists a sunny generalized nonexpansive retraction of onto if and only if is closed and convex. In this case is given by

*Definition 9 (see [15]). *A nonempty, closed subset of a smooth Banach space is said to be a *sunny generalized nonexpansive retraction *of if there exists a sunny generalized nonexpansive from onto .

Lemma 10 (see [11]). *Let be a nonempty, closed, and subset of a smooth and strictly convex Banach space , and let be a retraction from onto . Then, the following are equivalent: *(1)* is sunny generalized nonexpansive; *(2)*, for all and .*

Lemma 11 (see [11]). *Let be a nonempty, closed, and sunny generalized nonexpansive retraction of a smooth and strictly convex Banach space . Then, the sunny generalized nonexpansive retraction from onto is uniquely determined.*

Lemma 12 (see [11]). *Let be a nonempty, closed, and subset of a smooth and strictly convex Banach space such that there exists a sunny generalized nonexpansive retraction from onto . Let and . Then, the following hold: *(1)* if and only if , for all ; *(2)*. *

Lemma 13 (see [16]). *Let be a nonempty, closed, and subset of a smooth, strictly convex, and reflexive Banach space . Then, the following are equivalent: *(1)* is sunny generalized nonexpansive retraction of ; *(2)* is closed and convex. *

*Remark 14. *Let be a Hilbert space. By the Lemmas 11 and 13, a sunny generalized nonexpansive retraction from onto reduces to a metric projection operator from onto .

Lemma 15 (see [16]). *Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty, closed, and sunny generalized nonexpansive retraction of , and let be the sunny generalized nonexpansive retraction from onto . Let and . Then, the following are equivalent: *(1)*; *(2)*. *

Lemma 16 (see [4]). *Let E be a uniformly smooth and strictly convex real Banach space, and let and be two sequences of . If and either or is bounded, then .*

Lemma 17 (see [17]). *Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let be a nonempty, closed, and convex subset of . Let and be two sequences in and . If and , then .*

Lemma 18 (see [18]). *Let and be two sequences of nonnegative real numbers satisfying the inequality
**
If , then exists.*

Lemma 19 (see [19]). *Let be a uniformly convex Banach space. Then, for any , there exists a strictly increasing, continuous, and convex function such that and
**
where .*

Lemma 20 (see [4]). *Let be a smooth and uniformly convex Banach space. Then, for any , there exists a strictly increasing, continuous, and convex function such that and
**
where .*

Now, let us recall the following well-known concept and result.

*Definition 21 (see [20]). *Let be a subset of a topological vector space . A mapping is called a KKM mapping if for and , where denotes the convex hull of the set .

In [21], Fan gave the following famous infinite-dimensional generalization of Knaster, Kuratowski, and Mazurkiewiczâ€™s classical finite-dimensional result.

Lemma 22 (see [21]). *Let be a subset of a Hausdorff topological vector space , and let be a KKM mapping. If is closed, for all and is compact for at least one , then .*

Lemma 23 (see [14]). *Let be a nonempty, closed, and convex subset of a smooth and strictly convex Banach space, and let be a closed and --nonexpansive mapping. Then, is a closed and convex subset of .*

#### 3. Existence Theorem

In this section, we prove the existence theorem of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space.

Lemma 24. *Let be a nonempty, compact, and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , let be the duality mapping from into such that is closed and convex, let us assume that a bifunction satisfies the following conditions (DA1)â€“(DA4), let be a nonempty, closed, and convex subset of , let be an -inverse strongly skew monotone and let be a convex and lower semicontinuous. Let be given real number and be any point. Then, there exists such that
*

*Proof. *Let be any point in . For each , we define the mapping as follows:
It is easy to see that , and hence .(a) First, we will show that is a KKM mapping.

Suppose that is not a KKM mapping. Then, there exists a finite subset of and with such that for all .

It follows from the definition of a mapping that
By the assumptions of (DA1) and (DA4), we get
which is a contradiction. Thus, is a KKM mapping on .(b) Next, we will show that is closed, for all .Let be a sequence in such that as .

It then follows from that
By the assumption (DA3), the continuity of , and the lower semicontinuity of and , it follows from (46) that
Now, we get
Therefore, , and so is closed, for all .(c) We will show that is weakly compact.

Now, we know that is closed and subset of .

Since is compact. Therefore, is compact, and then is weakly compact.

By using (a), (b), and (c) and Lemma 22, we can conclude that .

Therefore, there exists such that

Theorem 25. *Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space such that is closed and convex, let us assume that a bifunction satisfies the following conditions (DA1)â€“(DA4), let be a nonempty, closed, and convex subset of , let be an -inverse strongly skew monotone and let be a convex and lower semicontinuous. Let be given real number and be any point. We define a mapping as follows:
*

*Then, the following conclusion hold:*(1)

*is single-valued;*(2)

*, ;*(3)

*;*(4)

*is closed and convex;*

*Proof. *We will complete this proof by the following four items.

(1) *We will show that ** is single-valued. *

Indeed, for any and , let . Then,
Adding the two inequalities, we have
Therefore,
From the condition (DA2), is an -inverse strongly skew monotone, and we have
Since , is monotone, is strictly convex, and we obtain
This implies that is single-valued.

(2) *We will show that *, for all .

Indeed, for any and , we have
Adding the two inequalities, we have
Therefore,
From the condition (DA2), is an -inverse strongly skew monotone, and we have
Since , we have
Therefore,
This implies that

(3) *We will show that *.

It is easy to see that
This implies that .

(4) *We will show that ** is closed and convex. *

For each , we define the mappingas follows:
It is easy to see that , so that .

Next, we will show that is a KKM mapping.

Suppose that there exists a finite subset of and with such that , for all . Then,
It follows from (DA1) and (DA4) that
which is the contradiction. Hence, is a KKM mapping on .

(4.1) Next, we will show that is closed, for each .

For any , let be any sequence in such that as .

Hence, as . Next, we will show that . Then, for each , we have

It follows from the assumption (DA3) that

This implies that , and hence is closed, for each .

Therefore, is closed.

(4.2) Next, we will show that is convex.

Let ; then, we have and , where .

For , let , and for any , we set .

It follows from (DA1) and (DA4) that