Journal of Applied Mathematics
Volume 2012 (2012), Article ID 859492, 21 pages
http://dx.doi.org/10.1155/2012/859492
Research Article

## Existence and Strong Convergence Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava Zang, Zanjan 45137-66731, Iran

Received 5 April 2012; Accepted 10 May 2012

Copyright © 2012 Rabian Wangkeeree et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

#### 1. Introduction

Let be a real Banach space with the dual and be a nonempty closed convex subset of . We denote by and the sets of positive integers and real numbers, respectively. Also, we denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing. Recall that if is smooth, then is single valued and if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . We will still denote by the single-valued duality mapping.

A mapping is called nonexpansive if for all . Also a mapping is called asymptotically nonexpansive if there exists a sequence with as such that for all and for each . Denote by the set of fixed points of , that is, . The following example shows that the class of asymptotically nonexpansive mappings which was first introduced by Goebel and Kirk [1] is wider than the class of nonexpansive mappings.

Example 1.1 (see [2]). Let be the closed unit ball in the Hilbert space and a mapping defined by where is a sequence of real numbers such that and . Then That is, is Lipschitzian but not nonexpansive. Observe that Here as . Therefore, is asymptotically nonexpansive but not nonexpansive.

A mapping is said to be relaxed monotone if there exist a mapping and a function positively homogeneous of degree , that is, for all and such that where is a constant; see [3]. In the case of for all , , is said to be relaxed -monotone. In the case of for all and , where and , is said to be -monotone; see [46]. In fact, in this case, if , then is a -strongly monotone mapping. Moreover, every monotone mapping is relaxed monotone with for all and . The following is an example of monotone mapping which can be found in [3]. Let , , and where is a constant. Then, is relaxed monotone with A mapping is said to be -hemicontinuous if, for each fixed , the mapping defined by is continuous at . For a real Banach space with the dual and for a nonempty closed convex subset of , let be a bifunction, a real-valued function and be a relaxed monotone mapping. Recently, Kamraksa and Wangkeeree [7] introduced the following generalized mixed equilibrium problem (GMEP). The set of such is denoted by , that is,

Special Cases
(1) If is monotone that is is relaxed monotone with for all and , (1.8) is reduced to the following generalized equilibrium problem (GEP). The solution set of (1.10) is denoted by , that is,
(2) In the case of and , (1.8) is reduced to the following classical equilibrium problem The set of all solution of (1.12) is denoted by , that is,
(3) In the case of , (1.8) is reduced to the following variational-like inequality problem [3].

The generalized mixed equilibrium problem (GMEP) (1.8) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems. Using the KKM technique introduced by Kanster et al. [8] and monotonicity of the mapping , Kamraksa and Wangkeeree [7] obtained the existence of solutions of generalized mixed equilibrium problem (1.8) in a real reflexive Banach space.

Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, Blum and Oettli [9], Combettes and Hirstoaga [10], and Moudafi [11]. On the other hand, there are several methods for approximation fixed points of a nonexpansive mapping; see, for instance, [1217]. Recently, Tada and Takahashi [13, 16] and S. Takahashi and W. Takahashi [17] obtained weak and strong convergence theorems for finding a common elements in the solution set of an equilibrium problem and the set of fixed point of a nonexpansive mapping in a Hilbert space. In particular, Tada and Takahashi [16] established a strong convergence theorem for finding a common element of two sets by using the hybrid method introduced in Nakajo and Takahashi [18]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

On the other hand, in 1953, Mann [12] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping in a Hilbert space : where the initial point is taken in arbitrarily and is a sequence in . However, we note that Manns iteration process (1.15) has only weak convergence, in general; for instance, see [1921]. In 2003, Nakajo and Takahashi [18] proposed the following sequence for a nonexpansive mapping in a Hilbert space: where for all , and is the metric projection from onto . Then, they proved that converges strongly to . Recently, motivated by Nakajo and Takahashi [18] and Xu [22], Matsushita and Takahashi [14] introduced the iterative algorithm for finding fixed points of nonexpansive mappings in a uniformly convex and smooth Banach space: and where denotes the convex closure of the set , is a sequence in (0, 1) with . They proved that generated by (1.17) converges strongly to a fixed point of . Very recently, Dehghan [23] investigated iterative schemes for finding fixed point of an asymptotically nonexpansive mapping and proved strong convergence theorems in a uniformly convex and smooth Banach space. More precisely, he proposed the following algorithm: , and where is a sequence in with as and is an asymptotically nonexpansive mapping. It is proved in [23] that converges strongly to a fixed point of .

On the other hand, recently, Kamraksa and Wangkeeree [7] studied the hybrid projection algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a countable family of nonexpansive mappings in a uniformly convex and smooth Banach space.

Motivated by the above mentioned results and the on-going research, we first prove the existence results of solutions for GMEP under the new conditions imposed on the bifunction . Next, we introduce the following iterative algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a finite family of asymptotically nonexpansive mappings in a uniformly convex and smooth Banach space: , , and The above algorithm is called the hybrid iterative algorithm for a finite family of asymptotically nonexpansive mappings from into itself. Since, for each , it can be written as , where , is a positive integer and as . Hence the above table can be written in the following form: Strong convergence theorems are obtained in a uniformly convex and smooth Banach space. The results presented in this paper extend and improve the corresponding Kimura and Nakajo [24], Kamraksa and Wangkeeree [7], Dehghan [23], and many others.

#### 2. Preliminaries

Let be a real Banach space and let be the unit sphere of . A Banach space is said to be strictly convex if for any , It is also said to be uniformly convex if for each , there exists such that for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function called the modulus of convexity of as follows: Then is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . Let be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space . Then for any , there exists a unique point such that The mapping defined by is called the metric projection from onto . Let and . The following theorem is well known.

Theorem 2.1. Let be a nonempty convex subset of a smooth Banach space and let and . Then the following are equivalent: (a)is a best approximation to ,(b) is a solution of the variational inequality: where is a duality mapping and is the metric projection from onto .

It is well known that if is a metric projection from a real Hilbert space onto a nonempty, closed, and convex subset , then is nonexpansive. But, in a general Banach space, this fact is not true.

In the sequel one will need the following lemmas.

Lemma 2.2 (see [25]). Let be a uniformly convex Banach space, let be a sequence of real numbers such that for all , and let and be sequences in such that , and . Then .

Dehghan [23] obtained the following useful result.

Theorem 2.3 (see [23]). Let be a bounded, closed, and convex subset of a uniformly convex Banach space . Then there exists a strictly increasing, convex, and continuous function such that and for any asymptotically nonexpansive mapping of into with , any elements , any numbers with and each .

Lemma 2.4 (see [26, Lemma 1.6]). Let be a uniformly convex Banach space, be a nonempty closed convex subset of and be an asymptotically nonexpansive mapping. Then is demiclosed at , that is, if and , then .

The following lemma can be found in [7].

Lemma 2.5 (see [7, Lemma 3.2]). Let be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space , let be an -hemicontinuous and relaxed monotone mapping. Let be a bifunction from to satisfying (A1), (A3), and (A4) and let be a lower semicontinuous and convex function from to . Let and . Assume that (i) for all ; (ii)for any fixed , the mapping is convex and lower semicontinuous; (iii) is weakly lower semicontinuous, that is, for any net converges to in which implies that . Then there exists such that

Lemma 2.6 (see [7, Lemma 3.3]). Let be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space , let be an -hemicontinuous and relaxed monotone mapping. Let be a bifunction from to satisfying (A1)–(A4) and let be a lower semicontinuous and convex function from to . Let and define a mapping as follows: for all . Assume that (i), for all ; (ii)for any fixed , the mapping is convex and lower semicontinuous and the mapping is lower semicontinuous; (iii) is weakly lower semicontinuous; (iv)for any , . Then, the following holds: (1) is single valued; (2) for all ; (3); (4) is nonempty closed and convex.

#### 3. Existence of Solutions for GMEP

In this section, we prove the existence results of solutions for GMEP under the new conditions imposed on the bifunction .

Theorem 3.1. Let be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space , let be an -hemicontinuous and relaxed monotone mapping. Let be a bifunction from to satisfying the following conditions (A1)–(A4): (A1) for all ; (A2) for all ; (A3)for all , is weakly upper semicontinuous; (A4)for all , is convex. For any and , define a mapping as follows: where is a lower semicontinuous and convex function from to . Assume that (i), for all ; (ii)for any fixed , the mapping is convex and lower semicontinuous and the mapping is lower semicontinuous; (iii) is weakly lower semicontinuous. Then, the following holds: (1) is single valued; (2) for all ; (3); (4) is nonempty closed and convex.

Proof. For each . It follows from Lemma 2.5 that is nonempty.
(1) We prove that is single valued. Indeed, for and , let . Then Adding the two inequalities, from (i) we have Setting and using (A2), we have that is, Since is relaxed monotone and , one has In (3.5) exchanging the position of and , we get that is, Now, adding the inequalities (3.6) and (3.8), we have Hence, Since is monotone and is strictly convex, we obtain that and hence . Therefore is single valued.
(2) For , we have Setting and applying (A2), we get that is, In (3.13) exchanging the position of and , we get Adding the inequalities (3.13) and (3.14), we have It follows that Hence The conclusions (3), (4) follow from Lemma 2.6.

Example 3.2. Define and by It is easy to see that satisfies (A1), (A3), (A4), and (A2): , for  all .

Remark 3.3. Theorem 3.1 generalizes and improves [7, Lemma 3.3] in the following manners.(1)The condition has been weakened by (A2) that is for all .(2)The control condition imposed on the mapping in [7, Lemma 3.3] can be removed.
If is monotone that is is relaxed monotone with for all and , we have the following results.

Corollary 3.4. Let be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a monotone mapping and be a bifunction from to satisfying the following conditions (i)–(iv): (i) for all ; (ii) for all ; (iii)for all , is weakly upper semicontinuous; (iv)for all , is convex. For any and , define a mapping as follows: where is a lower semicontinuous and convex function from to . Then, the following holds: (1) is single valued; (2) for all ; (3);(4) is nonempty closed and convex.

#### 4. Strong Convergence Theorems

In this section, we prove the strong convergence theorem of the sequence defined by (1.20) for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings.

Theorem 4.1. Let be a uniformly convex and smooth Banach space and let be a nonempty, bounded, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4). Let be an -hemicontinuous and relaxed monotone mapping and a lower semicontinuous and convex function from to . Let, for each , be an asymptotically nonexpansive mapping with a sequence , respectively, such that as . Assume that is nonempty. Let be a sequence generated by (1.20), where and are real sequences in satisfying and . Then converges strongly, as , to , where is the metric projection of onto .

Proof. First, define the sequence by and so as and where if , and ; . Next, we rewrite the algorithm (1.20) as the following relation: where is the mapping defined by (3.19). We show that the sequence is well defined. It is easy to verify that is closed and convex and for all . Next, we prove that . Indeed, since , we also have . Assume that for . Utilizing Theorem 3.1 (2), we obtain which gives that hence . By the mathematical induction, we get that for each and hence is well defined. Now, we show that Put , since and , we have Since and , we have Hence the sequence is bounded and monotone increasing and hence there exists a constant such that Moreover, by the convexity of , we also have and hence This implies that By Lemma 2.2, we have Furthermore, we can easily see that Next, we show that Fix and put . Since , we have . Since , there exists and a nonnegative number with such that By the boundedness of and , we can put the following: This together with (4.14) implies that for all . Therefore, for each , we get Moreover, since each , , is asymptotically nonexpansive, we can obtain that It follows from Theorem 2.3 and the inequalities (4.17)–(4.19) that Since and , it follows from the above inequality that Hence (4.13) is proved. Next, we show that From the construction of , one can easily see that The boundedness of and implies that On the other hand, since for any positive integer , and , we have that is Thus, Applying the facts (4.11), (4.13), and (4.24) to the above inequality, we obtain Therefore, for any , we have which gives that as required. Since is bounded, there exists a subsequence of such that . It follows from Lemma 2.4 that for all . That is .
Next, we show that . By the construction of , we see from Theorem 2.1 that . Since , we get Furthermore, since , we have as . By (4.32), we also have . By the definition of , for each , we obtain By (A3), (4.32), (ii), the weakly lower semicontinuity of and -hemicontinuity of , we have Hence, This shows that and hence .
Finally, we show that as , where . By the weakly lower semicontinuity of the norm, it follows from (4.6) that This shows that and . Since is uniformly convex, we obtain that . It follows that . So we have as . This completes the proof.

#### 5. Corollaries

Setting , an asymptotically nonexpansive mapping, in Theorem 4.1 then we have the following result.

Theorem 5.1. Let be a uniformly convex and smooth Banach space and let be a nonempty, bounded, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4). Let be an -hemicontinuous and relaxed monotone mapping and a lower semicontinuous and convex function from to . Let be an asymptotically nonexpansive mapping with a sequence , such that as . Assume that is nonempty. Let be a sequence generated by where and are real sequences in satisfying and . Then converges strongly, as , to , where is the metric projection of onto .

It's well known that each nonexpansive mapping is an asymptotically nonexpansive mapping, then Theorem 4.1 works for nonexpansive mapping.

Theorem 5.2. Let be a uniformly convex and smooth Banach space and let be a nonempty, bounded, closed, and convex subset of . Let be a bifunction from to satisfying (A1)–(A4). Let be an -hemicontinuous and relaxed monotone mapping and a lower semicontinuous and convex function from to . Let be a nonexpansive mapping of into itself such that . Let be the sequence in generated by where and are real sequences in satisfying and . Then, the sequence converges strongly to .
If one takes and in Theorem 4.1, then one obtains the following result concerning an equilibrium problem in a Banach space setting.

Theorem 5.3. Let be a uniformly convex and smooth Banach space and let