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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 147915, 16 pages
http://dx.doi.org/10.1155/2012/147915
Research Article

Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi--Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

1Department of Mathematics, Yibin University, Yibin 644007, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China

Received 30 December 2011; Accepted 2 February 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 J. F. Tan and S. S. 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 main purpose of this paper is by using a hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a infinite family of total quasi--asymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results announced by some authors.

1. Introduction

Throughout this paper, we always assume that is a real Banach space with the dual , is a nonempty closed convex subset of , and is the normalized duality mapping defined by In the sequel, we use to denote the set of fixed points of a mapping and use and to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by and the strong convergence and weak convergence of a sequence , respectively.

Let be a bifunction, a real valued function, and a nonlinear mapping. The so-called generalized mixed equilibrium problem is to find such that The set of solutions to (1.2) is denoted by , that is, Special examples are follows.(i)If , the problem (1.2) is equivalent to finding such that which is called the mixed equilibrium problem (MEP) [1].(ii)If , the problem (1.2) is equivalent to finding such that which is called the mixed variational inequality of Browder type (VI) [2].

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 is attained uniformly in .

Remark 1.1. The following basic properties of a Banach space can be found in Cioranescu [3].(i)If is uniformly smooth, 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 .

Let be a smooth Banach space. We always use to denote the Lyapunov functional defined by It is obvious from the definition of the function that Following Alber [4], the generalized projection is defined by

Lemma 1.2 (see [4]). 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 if .
Let be a smooth, strictly convex, and reflexive Banach space, a nonempty closed convex subset of , and a mapping. A point is said to be an asymptotic fixed point of T if there exists a sequence such that and . We denoted the set of all asymptotic fixed points of by .

Definition 1.3. A mapping is said to be relatively nonexpansive [5] if and
A mapping is said to be closed if, for any sequence with and , .
Let be a nonempty closed convex subset of a Banach space . Let be the family of nonempty subsets of .

Definition 1.4. Let be a multivalued mapping and a point in . The definitions of are as follows:
Let be a multivalued mapping. 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 .
A multivalued mapping is said to be relatively nonexpansive [5] if and
A multivalued mapping is said to be closed if, for any sequence with and with , then .

Definition 1.5. A multivalued mapping is said to be quasi-ϕ-nonexpansive if and
A multivalued mapping is said to be quasi-ϕ-asymptotically nonexpansive if and there exists a real sequence with such that
A multivalued mapping is said to be total quasi-ϕ-asymptotically nonexpansive if and there exist nonnegative real sequences with (as ) and a strictly increasing continuous function with such that for all

Definition 1.6. Let be a sequence of mappings. is said to be a family of uniformly total quasi-ϕ-asymptotically nonexpansive multivalued mappings if and there exist nonnegative real sequences with (as ) and a strictly increasing continuous function with such that for all
A total quasi--asymptotically nonexpansive multivalued mapping is said to be uniformly L-Lipschitz continuous if there exists a constant such that

In 2005, Matsushita and Takahashi [5] proved weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space . In 2008, Plubtieng and Ungchittrakool [6] proved the strong convergence theorems to approximate a fixed point of two relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space . In 2010, Chang et al. [7] obtained the strong convergence theorem for an infinite family of quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. In 2011, Chang et al. [8] proved some approximation theorems of common fixed points for countable families of total quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. In 2011, Homaeipour and Razani [9] proved weak and strong convergence theorems for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space . On the other hand, In 2009, Zhang [10] proved the strong convergence theorem for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a finite family of quasi--asymptotically nonexpansive mappings in a uniformly smooth and uniformly convex Banach space. Recently, Tang [11], Cho et al. [1221], and Noor et al. [2226] extended the finite family of quasi--asymptotically nonexpansive mappings to infinite family of quasi--asymptotically nonexpansive mappings.

Motivated and inspired by the researches going on in this direction, the purpose of this paper is by using the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi--asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the strong convergence theorems, the hybrid algorithms are presented and used to approximate the fixed point. The results presented in the paper improve and extend some recent results announced by some authors.

2. Preliminaries

Lemma 2.1 (see [8]). Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and a nonempty closed convex set of . Let and be two sequences in such that and , where is the function defined by (1.7), and then .

Lemma 2.2. Let and be as in Lemma 2.1. Let be a closed and total quasi--asymptotically nonexpansive multivalued mapping with nonnegative real sequences and a strictly increasing continuous function such that (as ), and . If , then the fixed point set is a closed and convex subset of .

Proof. Letting be a sequence in with (as ), we prove that . In fact, by the assumption that is a total quasi--asymptotically nonexpansive multivalued mapping and , we have Furthermore, we have By Lemma 1.2(c), . Hence, . This implies that , that is, is closed.
Next, we prove that is convex. For any , putting , we prove that . Indeed, let be a sequence generated by In view of the definition of , for all , we have since Substituting (2.5) into (2.4) and simplifying it, we have By Lemma 2.1, we have (as ). This implies that (as ). Since is closed, we have , that is, .
This completes the proof of Lemma 2.2.

Lemma 2.3 (see [7]). 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 , there exists a continuous, strictly increasing, and convex function with such that for any positive integers with ,
For solving the generalized mixed equilibrium problem, 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:(A1), (A2) is monotone, that is, ,(A3), (A4) the function is convex and lower semicontinuous.

Lemma 2.4. Let be a smooth, strictly convex and reflexive Banach space and a nonempty closed convex subset of . Let be a bifunction satisfying conditions (A1)–(A4). Let and . Then, the following hold.(i) [27] There exists such that (ii) [28] Define a mapping by Then, the following conclusions hold:(a) is single-valued,(b) is a firmly nonexpansive-type mapping, that is, , (c),(d) is closed and convex,(e).

Lemma 2.5 (see [10]). 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 (A1)–(A4). Let be any given number and any given point. Then, the following hold.
(i) There exists such that
(ii) If one defines 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 a closed convex set of ,(e).

Remark 2.6. It follows from Lemma 2.4 that the mapping defined by (2.12) is a relatively nonexpansive mapping. Thus, it is quasi--nonexpansive.

3. Main Results

In this section, we will use the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi--asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property.

Theorem 3.1. Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and a nonempty closed and convex subset of . Let be a bifunction satisfying conditions (A1)–(A4), a continuous and monotone mapping, and a lower semicontinuous and convex function. Let be an infinite family of closed and uniformly total quasi--asymptotically nonexpansive multivalued mappings with nonnegative real sequences and a strictly increasing continuous function such that , (as ) and and for each , is uniformly -Lipschitz continuous. Let , , and let be a sequence generated by where , is the generalized projection of onto , and and are sequences in satisfying the following conditions:(a)for each , ,(b) for any ,(c) for some .
If is a nonempty and bounded subset of , then the sequence converges strongly to .

Proof. First, we define two functions and by
By Lemma 2.5, we know that the function satisfies conditions (A1)–(A4) and has properties (a)–(e). Therefore, (3.1) is equivalent to
Now we divide the proof of Theorem 3.1 into six steps.
(i)   and are closed and convex for each .
In fact, it follows from Lemma 2.2 that , is a closed and convex subset of . Therefore, is a closed and convex subset .
Again by the assumption, is closed and convex. Suppose that is closed and convex for some . Since the condition is equivalent to the set is closed and convex. Therefore, is closed and convex for each .
(ii)   is bounded and is a convergent sequence.
Indeed, it follows from (3.1) and Lemma 1.2(a) that for all This implies that is bounded. By virtue of (1.3), we know that is bounded.
In view of the structure of , we have and . This implies that and Therefore, is a convergent sequence.
(iii)   for all .
Indeed, it is obvious that . Suppose that for some . Since , by Lemma 2.5 and Remark 2.6, is quasi--nonexpansive. Hence, for any given and we have Furthermore, it follows from Lemma 2.3 that for any , and we have Substituting (3.9) into (3.8) and simplifying it, we have for all that is, and so for all .
By the way, in view of the assumption on we have
(iv) converges strongly to some point .
In fact, since is bounded and is reflexive, there exists a subsequence such that (some point in ). Since is closed and convex and , this implies that is weakly closed and for each . In view of , we have Since the norm is weakly lower semicontinuous, we have and so This implies that , and so . Since , by virtue of Kadec-Klee property of , we obtain that Since is convergent, this together with , shows that . If there exists some sequence such that , then from Lemma 1.2(a) we have that This implies that and
(v) Now we prove that .
First, we prove that . In fact, since , it follows from (3.1) and (3.17) that By the virtue of Lemma 2.1, we have From (3.10), for any and , we have that is, By conditions (b) and (c) it is shown that . In view of property of , we have Since , this implies that . From Remark 1.1(ii) it yields Again since this together with (3.23) and the Kadec-Klee property of shows that Let be a sequence generated by By the assumption that each is uniformly -Lipschitz continuous, for any and we have This together with (3.17) and (3.27) shows that and . In view of the closeness of , it yields that , that is, . By the arbitrariness of , we have
Next, we prove that . Since , it follows from (3.1) and (3.17) that Since , by virtue of Lemma 2.1 we have This together with (3.19) shows that and . By the assumption that , we have Since , by condition (A1), we have By the assumption that is convex and lower semicontinuous, letting in (3.32), from (3.30) and (3.31), we have .
For and , letting , there are and . By conditions (A1) and (A4), we have Dividing both sides of the above equation by , we have . Letting , from condition (A3), we have , that is, , and .
(vi) We prove that .
Let . Since and , we have This implies that In view of the definition of , from (3.35) we have . Therefore, .
This completes the proof of Theorem 3.1.

4. Conclusions

Recently the extended general variational inequalities have been introduced and studied in Noor [24, 25]. We would like to point out that the results and the methods presented in this paper will be used to study this kind of extended general variational inequalities and its multivalued version.

Acknowledgment

The authors would like to express their thanks to the referees for their helpful suggestions and comments.

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