#### Abstract

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi--asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

#### 1. Introduction

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. We also assume that is a real Banach space, is the dual space of , is a nonempty closed convex subset of , and is the pairing between and .

Let be a real-valued function, a bifunction, and a nonlinear mapping. The βso-calledβ generalized mixed equilibrium problem is to find such that The set of solutions for (1.1) is denoted by , that is,

Special examples are as follows.(I)If , the problem (1.1) is equivalent to finding such that which is called the generalized equilibrium problem. The set of solutions for (1.3) is denoted by GEP.(II)If , the problem (1.1) is equivalent to finding such that which is called the mixed equilibrium problem (MEP) [1]. The set of solutions for (1.4) is denoted by MEP.(III)If , the problem (1.1) is equivalent to finding such that which is called the mixed variational inequality of Browder type () [2]. The set of solutions for (1.5) is denoted by .(IV)If and , the problem (1.1) is equivalent to finding such that which is called the equilibrium problem. The set of solutions for (1.6) is denoted by .(V)If and , the problem (1.1) is equivalent to finding such that which is called the variational inequality of Browder type. The set of solutions for (1.7) is denoted by .

The problem (1.1) is very general in the sense that numerous problems in physics, optimiztion and economics reduce to finding a solution for (1.1). Some methods have been proposed for solving the generalized equilibrium problem and the equilibrium problem in Hilbert space (see, e.g., [3β6]).

A mapping is called nonexpansive if We denote the fixed point set of by .

In 2008, S. Takahashi and W. Takahashi [6] proved some strong convergence theorems for finding an element or a common element of , or , respectively, in a Hilbert space.

Recently, Takahashi and Zembayashi [7, 8] proved some weak and strong convergence theorems for finding a common element of the set of solutions for equilibrium (1.6) and the set of fixed points of a relatively nonexpansive mapping in a Banach space.

In 2010, Chang et al. [9] proved a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem (1.3) and the set of common fixed points of a pair of relatively nonexpansive mappings in a Banach space.

Motivated and inspired by [4β9], we intend in this paper, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem (1.1) and the set of common fixed points of a pair of quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

#### 2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.

The mapping defined by is called the normalized duality mapping. By the Hahn-Banach theorem, for each .

In the sequel, we denote the strong convergence and weak convergence of a sequence by and , respectively.

A Banach space is said to be strictly convex if for all with . is said to be uniformly convex if, for each , there exists such that for all with . is said to be smooth if the limit exists for all . is said to be uniformly smooth if the above limit exists uniformly in .

*Remark 2.1. *The following basic properties can be found in Cioranescu [10].(i)If is a uniformly smooth Banach space, then is uniformly continuous on each bounded subset of .(ii)If is a reflexive and strictly convex Banach space, then is hemicontinuous.(iii)If is a smooth, strictly convex, and reflexive Banach space, then is singlevalued, one-to-one and onto.(iv)A Banach space is uniformly smooth if and only if is uniformly convex.(v)Each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence , if and , then .

Next we assume that is a smooth, strictly convex, and reflexive Banach space and is a nonempty closed convex subset of . In the sequel, we always use to denote the Lyapunov functional defined by

It is obvious from the definition of that

Following Alber [11], the generalized projection is defined by

Lemma 2.2 (see [11, 12]). *Let be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Then, the following conclusions hold:*(a)* for all and ;*(b)*if and , then *(c)*for , if and only .*

*Remark 2.3. *If is a real Hilbert space , then and is the metric projection of onto .

Let be a smooth, strictly, convex and reflexive Banach space, a nonempty closed convex subset of , a mapping, and the set of fixed points of . A point is said to be an asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all asymptotic fixed points of by .

*Definition 2.4 (see [13]). *(1) A mapping is said to be relatively nonexpansive if , , and

(2) A mapping is said to be *closed* if, for any sequence with and , .

*Definition 2.5 (see [14]). *(1) A mapping is said to be quasi--nonexpansive if and

(2) A mapping is said to be quasi--asymptotically nonexpansive if and there exists a real sequence with such that

(3) A pair of mappings is said to be uniformly quasi--asymptotically nonexpansive if and there exists a real sequence with such that for

(4) A mapping is said to be uniformly -Lipschitz continuous if there exists a constant such that

*Remark 2.6. *(1) From the definition, it is easy to know that each relatively nonexpansive mapping is closed.

(2) The class of quasi--asymptotically nonexpansive mappings contains properly the class of quasi--nonexpansive mappings as a subclass, and the class of quasi--nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

Lemma 2.7 (see [15]). *Let be a uniformly convex Banach space, a positive number, and a closed ball of . Then, for any given subset and for any positive numbers with , there exists a continuous, strictly increasing, and convex function with such that, for any with ,
*

Lemma 2.8 (see [15]). *Let be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property and a nonempty closed convex subset of . Let be a closed and quasi--asymptotically nonexpansive mapping with a sequence , . Then is a closed convex subset of .*

For solving the generalized mixed equilibrium problem (1.1), let us assume that the function is convex and lower semicontinuous, the nonlinear mapping is continuous and monotone, and the bifunction satisfies the following conditions:(A_{1})β, for all ,(A_{2})β is monotone, that is, , ,(A_{3}),(A_{4})the function is convex and lower semicontinuous.

Lemma 2.9. *Let be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Let a bifunction satisfying the conditions (A _{1})β(A_{4}). Let and . Then, the followings hold.*(i)

*(Blum and Oettli [3]) there exists such that*(ii)

*(Takahashi and Zembayashi [8]) Define a mapping by*

*Then, the following conclusions hold:*(a)

*β is single-valued,*(b)

*β is a firmly nonexpansive-type mapping, that is, ,*(c)

*β,*(d)

*β is closed and convex,*(e)

*β.*

Lemma 2.10 (see [16]). *Let be a smooth, strictly convex, and reflexive Banach space, and a nonempty closed convex subset of . Let be a continuous and monotone mapping, a lower semicontinuous and convex function, and a bifunction satisfying conditions (A _{1})β(A_{4}). Let be any given number and any given point. Then, the following hold.*(i)

*There exists such that*(ii)

*If we define a mapping by*

*Then, the mapping has the following properties:*(a)

*β is single valued,*(b)

*β is a firmly nonexpansive-type mapping, that is,*(c)

*β,*(d)

*β is closed and convex,*(e)

*β*

*Remark 2.11. *It follows from Lemma 2.9 that the mapping is a relatively nonexpansive mapping. Thus, it is quasi--nonexpansive.

#### 3. Main Results

In this section, we will prove a strong convergence theorem for finding a common element of the set of solutions for the generalized mixed equilibrium problem (1.1) and the set of common fixed points for a pair of quasi--asymptotically nonexpansive mappings in Banach spaces.

Theorem 3.1. *Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and a nonempty closed convex subset of . Let be a continuous and monotone mapping, a lower semicontinuous and convex, function, and a bifunction satisfying conditions (A _{1})β(A_{4}). Let be two closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence and . Suppose that and are uniformly -Lipschitz continuous and that is a nonempty and bounded subset in . Let be the sequence generated by
*

*where is the normalized duality mapping, and are sequences in and for some , . Suppose that the following conditions are satisfied:*(i)

*β,*(ii)

*β.*

*Then converges strongly to , where is the generalized projection of onto .*

*Proof. *Firstly, we define two functions and by
By Lemma 2.10, we know that the function satisfies conditions (A_{1})β(A_{4}) and has properties (a)β(e). Therefore, (3.1) is equivalent to
We divide the proof of Theorem 3.1 into five steps.

(I) First we prove that and are both closed and convex subsets of for all .

In fact, it is obvious that is closed and convex for all . Again we have that
Hence , , is closed and convex, and so is closed and convex for all .

(II) Next we prove that , .

Putting , , by Lemma 2.10 and Remark 2.11, is relatively nonexpansive. Again since and are quasi--asymptotically nonexpansive, for any given , we have that
From (3.5) we have that
This implies that , , and so , .

Now we prove that , .

In fact, from , we have that . Suppose that , for some . Now we prove that . In fact, since , we have that
Since , for any , we have that
This shows that , and so . The conclusion is proved.

(III) Now we prove that is bounded.

From the definition of , we have that , . Hence, from Lemma 2.2(1),
This implies that is bounded. By virtue of (2.4), is bounded. Denote
Since and , from the definition of , we have that
This implies that is nondecreasing, and so the limit exists. Without loss of generality, we can assume that
By the way, from the definition of , (2.4), and (3.10), it is easy to see that

(IV) Now, we prove that converges strongly to some point .

In fact, since is bounded in and is reflexive, there exists a subsequence such that . Again since is closed and convex for each , it is weakly closed, and so for each . Since , from the defintion of , we have that
Since
we have that
This implies that , that is, . In view of the Kadec-Klee property of , we obtain that .

Now we first prove that . In fact, if there exists a subsequence such that , then we have that
Therefore we have that . This implies that

Now we first prove that . In fact, by the construction of , we have that . Therefore, by Lemma 2.2(a) we have that
In view of and noting the construction of we obtain
From (3.13) and (3.19), we have that

From (2.4) it yields that and . Since , we have that
Hence, we have that

This implies that is bounded in . Since is reflexive, and so is reflexive, there exists a subsequence such that . In view of the reflexiveness of , we see that . Hence, there exists such that . Since
taking on both sides of the equality above and in view of the weak lower semicontinuity of norm , it yields that
that is, . This implies that , and so . It follows from (3.23) and the Kadec-Klee property of that (as ). Noting that is hemicontinuous, it yields that . It follows from (3.22) and the Kadec-Klee property of that .

By the same way as given in the proof of (3.18), we can also prove that

From (3.18) and (3.26), we have that
Since is uniformly continuous on any bounded subset of , we have that
For any , it follows from (3.5) that
Since
From (3.27) and (3.28), it follows that

In view of condition (i) and , we see that
It follows from the property of that
Since and is uniformly continuous, it yields that . Hence from (3.33) we have that
Since is hemicontinuous, it follows that
On the other hand, we have that
This together with (3.35) shows that

Furthermore, by the assumption that is uniformly -Lipschitz continuous, we have that

This together with (3.18) and (3.37), yields (as . Hence from (3.37) we have that , that is, . In view of (3.37) and the closeness of , it yields that . This implies that .

By the same way as given in the proof of (3.23) to (3.31), we can also prove that

Since , from (2.19), (3.6), (3.13), and (3.39), we have that
From (2.4) it yields that . Since , we have that

Hence we have that
By the same way as given in the proof of (3.26), we can also prove that
From (3.39) and (3.43) we have that
Since is uniformly continuous on any bounded subset of , we have that
For any , it follows from (3.6), (3.13), and (3.39) that
In view of condition (ii) and , we see that
It follows from the property of that
Since and is uniformly continuous, it yields, . Hence from (3.48) we have that
Since is hemicontinuous, it follows that
On the other hand, we have that
This together with (3.50) shows that

Furthermore, by the assumption that is uniformly -Lipschitz continuous, we have that

This together with (3.26) and (3.52), yields that (as . Hence from (3.52) we have that , that is, . In view of (3.52) and the closeness of , it yields that . This implies that .

Next we prove that . From (3.45) and the assumption that , , we have that
Since , we have that
Replacing by in (3.55), from condition , we have that
By the assumption that is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in (3.55), from (3.54) and condition , we have that , .

For and , letting , there are and . By conditions (A_{1}) and (A_{4}), we have that
Dividing both sides of the above equation by , we have that , . Letting , from condition (A_{3}), we have that , , that is, , . Therefore , and so .

(V) Finally, we prove that .

Let . From , and , we have that
Since the norm is weakly lower semicontinuous, this implies that
It follows from the definition of and (3.59) that we have . Therefore, . This completes the proof of Theorem 3.1.

*Remark 3.2. *Theorem 3.1 improves and extends the corresponding results in [7β9].(a)For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property(note that each uniformly convex Banach space must have the Kadec-Klee property).(b)For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or weak relatively nonexpansive mappings to a pair of quasi--asymptotically nonexpansive mappings.(c)For the equilibrium problem, we extend the generalized equilibrium problem to the generalized mixed equilibrium problem.

The following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.3. *Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and a nonempty closed convex subset of . Let be a continuous and monotone mapping and a bifunction satisfying conditions (A _{1})β(A_{4}). Let be two closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence and . Suppose that and are uniformly -Lipschitz continuous and that is a nonempty and bounded subset in . Let be the sequence generated by
*

*where is the normalized duality mapping, and are sequences in , and for some , . If and satisfy conditions (i)-(ii) in Theorem 3.1, then converges strongly to , where is the set for the solutions of generalized equilibrium problem (1.3).*

*Proof. *Putting in Theorem 3.1, the conclusion of Theorem 3.3 can be obtained from Theorem 3.1.

Theorem 3.4. *Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and a nonempty closed convex subset of . Let be a lower semicontinuous and convex function and a bifunction satisfying conditions (A _{1})β(A_{4}). Let be two closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence and . Suppose that and are uniformly -Lipschitz continuous and that is a nonempty and bounded subset in . Let be the sequence generated by
*

*where is the normalized duality mapping, and are sequences in , and for some . If and satisfy conditions (i)-(ii) in Theorem 3.1, then converges strongly to , where MEP is the set of solutions for mixed equilibrium problem (1.4).*

*Proof. *Putting in Theorem 3.1, the conclusion of Theorem 3.4 can be obtained from Theorem 3.1.

Theorem 3.5. *Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and a nonempty closed convex subset of . Let be a continuous and monotone mapping and a lower semicontinuous and convex function. Let be two closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence and . Suppose that and are uniformly -Lipschitz continuous and that is a nonempty and bounded subset in . Let be the sequence generated by
**
where is the normalized duality mapping, and are sequences in , and for some . If and satisfy conditions (i)-(ii) in Theorem 3.1, then converges strongly to , where is the set of solutions for the mixed variational inequality (1.5).*

*Proof. *Putting in Theorem 3.1, the conclusion of Theorem 3.5 can be obtained from Theorem 3.1.

Theorem 3.6. *Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and a nonempty closed convex subset of *