ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 595263, 18 pagesdoi:10.5402/2011/595263

Research Article

A New Hybrid Algorithm for a System of Generalized Mixed Equilibrium Problems and Fixed Point of Quasi--Asymptotically Nonexpansive Mappings

Jinhua Zhu and Shih-Sen Chang

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Received 6 March 2011; Accepted 11 April 2011

Academic Editors: S. Deng and K. A. Lurie

Copyright © 2011 Jinhua Zhu and Shih-Sen Chang. 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.

The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family 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 . In the sequel, we denote the strong convergence and weak convergence of a sequence by and , respectively.

Let be a real-valued function, a nonlinear mapping, and a finite family of equilibrium bifunctions, that is, for each . The “so-called” system of generalized mixed equilibrium problems (SGMEP) for functions is to find a common element such that

We denote the set of solutions of (1.1) by , where is the set of solutions to the generalized mixed equilibrium problem:

Special examples are as follows.(i)If and , the problem (1.1) is equivalent to finding such that

which is called the mixed equilibrium problem (MEP) [1]. The set of solutions to (1.3) is denoted by MEP.(ii)If and , the problem (1.1) is equivalent to finding such that

which is called the mixed variational inequality of Browder type (VI) [2]. The set of solutions to (1.4) is denoted by .

Recently, many authors studied the problems of finding a common element of the set of fixed point for a nonexpansive mapping and the set of solutions for an equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [3–5] and the references therein).

Motivated and inspired by the researches going on in this direction, the purpose of this paper is using a hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family 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 improve and extend the corresponding results in [6–12].

2. Preliminaries

First, we recall some definitions and conclusions.

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

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 [13]. (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 norm-weak continuous.(iii)If is a smooth, strictly convex, and reflexive Banach space, then is single-valued, 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 [14], the generalized projection is defined by

Lemma 2.2 (see [14, 15]). Let be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Then the following conclusions hold: (a), for alland;(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 [16]). (1) A mapping is said to be relatively nonexpansive if , and A mappingis said to beclosed if, for any sequence with and, then.

Definition 2.5 (see [9]). (1) A mapping is said to be quasi--nonexpansive if and A mapping is said to be quasi--asymptotically nonexpansive if and there exists a real sequence with such that 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 [6]). Let be a uniformly convex Banach space, a positive number, and a closed ball of . Then, for any given sequence and for any given sequence of positive numbers with , then there exists a continuous, strictly increasing, and convex function with such that for any positive integers with ,

Lemma 2.8 (see [6]). Let be a real uniformly smooth and strictly convex Banach space with 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 system of generalized mixed equilibrium problems (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:; is monotone, that is, ;;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 be a bifunction satisfying conditions . Let and . Then, the following hold. (i)(Blum and Oettli [17]) There exists such that (ii)(Takahashi and Zembayashi [3]) Define a mapping by Then, the following conclusions hold: (a) is single valued;(b) is firmly nonexpansive-type mapping, that is, for all , (c); (d) is closed and convex;(e).

Lemma 2.10 (see [7]). 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 the conditions . Let be any given number and be 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) is single valued;(b) is a firmly nonexpansive-type mapping, that is, (c); (d) is closed and convex;(e)

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

3. Main Results

In this section, we will use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (1.1) and the set of common fixed points for an infinite family of quasi--asymptotically nonexpansive mappings in Banach spaces.

Theorem 3.1. Let be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and a nonempty closed convex subset of . Let be a continuous and monotone mapping, a lower semicontinuous and convex function, and be a finite family of bifunction satisfying conditions . Let an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence and . Suppose that for each is uniformly -Lipschitz continuous and that is a nonempty and bounded subset in , where . Let , and , be the sequences generated by where and , is the mapping defined by (2.16), for some , , is the generalized projection of onto the set , and for each , , are sequences in satisfying the following conditions: (a) for all ;(b) for all ;(c) for some . Then converges strongly to , where is the generalized projection from onto G.

Proof. We divide the proof of Theorem 3.1 into five steps.(I)We first prove that and both are closed and convex subset of for all .In fact, it follows from Lemmas 2.8 and 2.10 that , , and both are closed and convex. Therefore is a closed and convex subset in . Furthermore, it is obvious that is closed and convex. Suppose that is closed and convex for some . Since the inequality is equivalent to therefore, we haveThis implies that is closed and convex. The desired conclusions are proved. These in turn show that and are well defined.(II)We prove that is a bounded sequence in .By the definition of , we have for all . It follows from Lemma 2.2(a) that This implies that is bounded. By virtue of (2.4), is bounded. Denote by In view of the structure of , we have , , and . This implies that and Therefore, is convergent. Without loss of generality, we can assume that (III)Next, we prove that for all .Indeed, it is obvious that . Suppose that for some . Since , by Lemma 2.10 and Remark 2.6, is quasi--asymptotically nonexpansive. Since is uniformly smooth, is uniformly convex. For any given and for any positive integer , from Lemma 2.7 we have Hence , and so for all . By the way, from the definition of , (2.4), and (3.6), it is easy to see that (IV)Now, we prove that converges strongly to some point .First, 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 definition of , we have Since we have This implies that , that is, . In view of the Kadec-Klee property of , we obtain that .Now we prove that . In fact, if there exists a subsequence such that , then we have Therefore we have . This implies that Now we prove that . In fact, by the construction of , we have that and . Therefore, by Lemma 2.2(a) we have In view of and noting the construction of we obtain that From (2.4) it yields . Since , we have Hence we have This implies that is bounded in . Since is reflexive, and so is reflexive, there exists a subsequence such that . In view of the reflexive of , we see that . Hence there exists such that . Since taking on both sides of above equality and in view of the weak lower semicontinuity of norm , it yields that That is, . This implies that , and so . It follows from (3.19) and the Kadec-Klee property of that (as ). Noting that is norm-weak continuous, it yields that . It follows from (3.18) and the Kadec-Klee property of that .By the similar way as given in the proof of (3.15), we can also prove that From (3.15) and (3.22) we have that Since is uniformly continuous on any bounded subset of , we have For any and any , it follows from (3.9), (3.15), and (3.22) that Since From (3.23) and (3.24), it follows that In view of conditions (b) and (c) , we see that It follows from the property of that Since and is uniformly continuous, it yields . Hence from (3.29) we have Since is norm-weak continuous, it follows that On the other hand, for each we have This together with (3.31) shows that Furthermore, by the assumption that for each , is uniformly -Lipschitz continuous, hence we have This together with (3.15) and (3.33) yields (as . Hence from (3.33) we have , that is, . In view of (3.33) and the closeness of , it yields that for all. This implies that .Next, we prove that . Since , from (2.18), (3.9), and (3.27), we have From (2.4) it yields . Since , we have Hence we have This implies that is bounded in . Since is reflexive, and so is reflexive, there exists a subsequence such that . In view of the reflexive of , we see that . Hence there exists such that . Since taking on both sides of 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 (3.37) and the Kadec-Klee property of that (as ). Note that is hemi-continuous, it yields that . It follows from (3.36) and the Kadec-Klee property of that .By the similar way as given in the proof of (3.15), we can also prove that From (3.22) and (3.40) we have that Since is uniformly continuous on any bounded subset of , we have Since by the similar way as above, we can also prove that From (3.44) and the assumption that , we have Now, we define functions by By Lemma 2.9, we know that the function satisfies conditions . Since by condition , we have By the assumption that is convex and lower semicontinuous, letting in (3.48), from (3.45) and (3.48), for each , we have , for all .For and , letting , there are and . By conditions and , we have Dividing both sides of the above equation by , we have ,for all . Letting , from condition , we have , for all , that is, , that is, , and .(V)Now, we prove .Let . From and , we have , for all . This implies that By the definition of and (3.50), we have . Therefore, . This completes the proof of Theorem 3.1.

Theorem 3.2. Let be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and a nonempty closed convex subset of . Let be a continuous and monotone mapping, a lower semicontinuous and convex function, and a finite family of bifunction satisfying conditions . Let be an infinite family of closed and quasi--nonexpansive mappings. Suppose that , where . Let , , , and , , be the sequences generated by where for some and for are sequences in satisfying the following conditions: (a), for all ;(b) for all ;(c) for some .Then converges strongly to .

Proof. Since is an infinite family of closed quasi--nonexpansive mappings, it is an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings with sequence . Hence . Therefore, the conditions appearing in Theorem 3.1—“ is bounded subset in ” and for each is uniformly -Lipschitz continuous— are of no use here. In fact, by the same methods as given in the proofs of (3.15), (3.22), (3.31), (3.37), and (3.42), we can prove that , , , and (as ), for each , that is, . Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 is obtained from Theorem 3.1 immediately.

Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results in [10–12, 18].(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 Kadec-Klee property).(b)For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or quasi--nonexpansive mapping to an infinite family of quasi--asymptotically nonexpansive mappings.(c)We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.

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