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Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 215261, 17 pages

http://dx.doi.org/10.1155/2012/215261

## Modified Hybrid Block Iterative Algorithm for Uniformly Quasi--Asymptotically Nonexpansive Mappings

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 6 June 2012; Accepted 11 July 2012

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Qiansheng Feng 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

Saewan and Kumam (2010) have proved the convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space *E* with Kadec-Klee property. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results.

#### 1. Introduction

Let be a nonempty closed convex subspace of a real Banach space . A mapping is said to be monotone if for each , the following inequality holds:

A mapping is called -inverse-strongly monotone if there exists such that

A monotone mapping is said to be maximal monotone if , for all , where is the normalized duality mapping. We denote by the set of zero points of .

*Remark 1.1. *It is well know that if is an -inverse-strongly monotone mapping, then it is -Lipschitzian, and hence uniformly continuous. Clearly, the class of monotone mappings includes the class of -inverse strongly monotone mappings.

Let be a nonempty closed convex subspace of a real Banach space with dual and is the pairing between and . Let be a bifunction and be a monotone mapping. The generalized equilibrium problem means that finding a such that

The set of solutions of (1.3) is denoted by , that is,

If , then the problem (1.3) is equivalent to that of finding a such that which is called the equilibrium problem. The solution of (1.5) is denoted by . If , then the problem (1.3) is equivalent to that of finding a such that which is called the variational inequality of Browder type. The solution of (1.6) is denoted by .

The problem (1.3) was shown in [1] to cover variational inequality problems, monotone inclusion problems, vector equilibrium problems, numerous problems in physics, minimization problems, saddle point problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, fixed point problem, the complementarity problem, and optimization problem, which can also be written in the form of an . In other words, the is a unifying model for several problems arising in physics, engineering, science, optimization, economics, and so on. In the past two decades, Some methods have been modified for solving the generalized equilibrium problem and the equilibrium problem in Hilbert space and Banach space, see [2β9].

The convex feasibility problem (CFP) is the problem for computing points that lay in the intersection of a finite family of closed convex subsets , of a Banach space . This problem appears in many fields of applied mathematics, such as the theory of optimization [1], Image Reconstruction from projections [10], and Game Theory [11] and plays an important role in these domains. There is a considerable investigation of (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [12]. Also the projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [13] deal with the convex feasibility problem by convex and sunny nonexpansive retractions in a uniformly convex Banach space.

We note that the block iterative method is a common method by many authors to solve (CFP) [14]. In 2008, Plubtieng and Ungchittrakool [15] established block iterative methods for a finite family of relatively nonexpansive mappings and got some strong convergence theorems in a Banach space by using the hybrid method.

In 2009, Takahashi and Zembayashi [16] introduced the following iterative scheme in the case that is uniformly smooth and uniformly convex Banach space: where is a relatively nonexpansive mapping and is a bifunction from into . They prove that the sequence converges strongly to under appropriate conditions.

In the same year, Qin et al. [7] introduced a hybrid projection algorithm to two quasi--nonexpansive mappings in Banach spaces as follows: where is the generalized projection from onto . They proved that the sequence converges strongly to . Then Petrot et al. [17] improved the notion from a relatively nonexpansive mapping or a quasi--nonexpansive mapping to two relatively quasi-nonexpansive mappings; they also proved some strong convergence theorems to find a common element of the set of fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces.

In 2010, Saewan and Kumam [18] introduced the following iterative method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property.

They proved that converges strongly to under the proper conditions. The same year, Chang et al. [19] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi--asymptotically nonexpansive mapping; they obtain the strong convergence theorems in a Banach space.

Motivated by Saewan and Kumam [18], in this paper we use some new conditions to prove strong convergence theorems for modified block hybrid projection algorithm for finding a common element of the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings which is more general than closed quasi--nonexpansive mappings in a uniformly smooth and strictly convex Banach space . In (1.9) we find iterative step is not essential, so we combine with of (1.9), and use an equally continuous mapping that is more weak than uniformly -Lipschitz mapping in a uniformly smooth and strictly convex Banach space , but the Banach space does not have Kadec-Klee property, under the circumstances we prove strong convergence theorems and get some results same as the results of Saewan and Kumam [18]. The results presented in this paper improve some well-known results in the literature.

#### 2. Preliminaries

The space is said to be smooth if the limit exists for all , and is said to be uniformly smooth if the limit (2.1) exists uniformly for all . Then a Banach space is said to be strictly convex if for all and . It is said to be uniformly convex if for each , there exists such that for all with .

Let be a Banach space and let be the topological dual of . For all and , we denote the value of at by . Then, the duality mapping is defined by for every . By the Hahn-Banach theorem, is nonempty.

The following basic properties can be found in Cioranescu [20].(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 the normalized duality mapping is single-valued, one-to-one, and onto.(iv)A Banach space is uniformly smooth if and only if is uniformly convex.

Let be a smooth, strictly convex, and reflexive Banach space and let be a nonempty closed convex subset of . Throughout this paper, we denote by the function defined by Following Alber [21], the generalized projection from onto is defined by

If is a Hilbert space, then and is the metric projection of onto . We know the following lemmas for generalized projections.

Lemma 2.1 (see Alber [21] and Kamimura and Takahashi [22]). * Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space . Then
*

Lemma 2.2 (see Alber [21], Kamimura and Takahashi [22]). *Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space and let and . Then
*

Lemma 2.3 (see Kamimura and Takahashi [22]). *Let be a smooth and uniformly convex Banach space and let and be sequences in such that either or is bounded. If , then .*

Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space and let be a mapping from into itself. We denoted by the set of fixed points of , that is . A point is said to be an asymptotic fixed point of if there exists in which converges weakly to and . We denote the set of all asymptotic fixed points of by .

A mapping from into itself is said to be relatively nonexpansive [23] if the following conditions are satisfied:(1) is nonempty,(2),(3).

A mapping from into itself is said to be relatively quasi-nonexpansive if the following conditions are satisfied:(1) is nonempty,(2),

The asymptotic behavior of a relatively nonexpansive mapping was studied in [24]. is said to be -nonexpansive, if for . is said to be quasi--asymptotically nonexpansive if the following conditions are satisfied:(1) is nonempty,(2) and ,where is a real sequence within and as .

A mapping is said to be closed if for any sequence with and , then . It is easy to know that each relatively nonexpansive mapping is closed. 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 (see more details [24, 25]).

By using the similar method as in Su et al. [26], the following Lemma is not hard to prove.

Lemma 2.4. *Let be a strictly convex and uniformly smooth real Banach space, let be a closed convex subset of , and let be a closed and quasi--asymptotically nonexpansive mapping from into itself with a sequence and as . Then is a closed and convex subset of .*

For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:(A1),(A2) is monotone, that is, ,(A3)for all , ,(A4)for all , is convex and lower semicontinuous.

Lemma 2.5 (see Blum and Oettli [1]). * Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)β(A4), and let and . Then, there exists such that
*

Lemma 2.6 (see Kumam [5]). *Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space . Let be a bifunction from to satisfying (A1)β(A4) and let be a monotone mapping from into . For , define a mapping as follows:
**
for all . Then, the following hold:*(1)* is single-valued,*(2)* is a firmly nonexpansive-type mapping [6], that is, for all ,
*(3)*,*(4)* is closed and convex. *

Lemma 2.7 (see Kumam [5]). * Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a bifunction from to satisfying (A1)β(A4) and let be a monotone mapping from into . For , , then the following holds:
*

Lemma 2.8 (see Chang et al. [19]). * Let E 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 number with , there exists a continuous, strictly increasing, and convex function with such that, for any positive integer with ,
*

*Definition 2.9. *A mapping from into itself is said to be equally continuous if it is follows that

A mapping from into itself is said to be uniformly -Lipschitz continuous if there exists a constant such that

It is easy to know that each -Lipschitz continuous mapping is equally continuous, but the converse may be not true.

*Definition 2.10. *Let be a sequence of mapping. is said to be a family of uniformly quasi--asymptotically nonexpansive mappings, if , and there exists a sequence with such that for each ,

#### 3. Main Results

Theorem 3.1. *Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying and let be a continuous monotone mapping of into . Let be an infinite family of closed equally continuous and uniformly quasi--asymptotically nonexpansive mappings with a sequence such that is a nonempty and bounded subset in . Let be a sequence generated by
**
where is the duality mapping on , are sequences in which satisfies , , and for some . If for all , then converges strongly to , where is the generalized projection from onto .*

*Proof. *We first show that is closed and convex. It is obvious that is closed. In addition, since

so is convex, therefore, is a closed convex subset of for all .

Next, we show that for all . It is clear that . Suppose for , by the property of , , Lemmas 2.6 and 2.8, and uniformly quasi--asymptotically nonexpansive of for each , then we have

This shows that implies that for all by induction. On the one hand, since and for all , we have

Therefore is nondecreasing. In the other hand, by Lemma 2.1, we have
for each for all . Therefore, is bounded; this together with (3.4) implies that the limit of exists.

Since is bounded, so is bounded by (1.7), together with , we have that

From Lemma 2.1, we have, for any positive integers , that

Because the limit of exists, then we have
uniformly for positive integers . Since is a bounded sequence, by using Lemma 2.4, we have
uniformly for positive integers . Hence is a Cauchy sequence, therefore, there exists a point such that converges strongly to .

In addition, from (3.7) we have , this together with the fact implies that

Taking limit on both side of (3.10) and from (3.6),we get that

By using Lemma 2.4, we have
which implies that converges strongly to .

From (3.3), we have , together with and Lemma 2.7, we have

for any . This implies that

Therefore, we have
which implies that converges strongly to . Thus we have proved that
as , where . From (3.1)
and hence

Taking limit on both side of above inequality, by and from (3.16), we have

Since is uniformly norm-to-norm continuous on bounded sets, we have
for each , together with (3.16), we get that
for each . Since is equally continuous, we have

Together with (3.16) and (3.20), we have . From (3.21), we have , that is, . In view of closeness of , we have , for all . This implies that .

Next we show . By , we have

From (A2), we get that
and hence

For with and , let , then , from (3.25) we have
Since is uniformly norm-to-norm continuous on bounded sets, is monotone and (3.16), we have

It follows from (A4) that

From the conditions (A1) and (A4), we have

Letting , we get

This implies that .

Finally, we show that . Let , from , we have

Since and ,

together with above two hands and , we obtain

that is . The proof is completed.

By using the similar method of proof as in Theorem 3.1, the following theorem is not hard to prove.

Theorem 3.2. *Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying . Let be an infinite family of closed equally continuous and uniformly quasi--asymptotically nonexpansive mappings with a sequence such that is a nonempty and bounded subset in . Let be a sequence generated by
**
where is the duality mapping on , are sequences in which satisfies , , and for some . If for all , then converges strongly to , where is the generalized projection from onto .*

*Proof. *In Theorem 3.1, put we can obtain the conclusion of Theorem 3.2.

Theorem 3.3. *Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying and let be a continuous monotone mapping of into . Let be an infinite family of closed equally continuous and quasi--asymptotically nonexpansive mappings with a sequence such that is a nonempty and bounded subset in . Let be a sequence generated by
**
where is the duality mapping on , are sequences in which satisfies for all , and for some . Then converges strongly to , where is the generalized projection from onto .*

*Proof. *In Theorem 3.1, put , for , we can obtain the conclusion of Theorem 3.3.

#### 4. Application for Optimization Problem

In this section, we study a kind of optimization problem by using the result of this paper. that is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions: where is a convex and lower semicontinuous functional defined on a closed convex subset of a Banach space . We denoted by the set of solutions of (4.1). Let be a bifunction from to defined by . We consider the following equilibrium problem, that is, to find such that

It is obvious that , where denote the set of solutions of equilibrium problem (4.2). In addition, it is easy to see that satisfies the conditions (A1)β(A4) in the Section 2. Therefore, from the Theorem 3.1, we can obtain the following theorem.

Theorem 4.1. *Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying and let be a continuous monotone mapping of into . Let be an infinite family of closed equally continuous and uniformly quasi--asymptotically nonexpansive mappings with a sequence such that is a nonempty and bounded subset in . Let be a sequence generated by
**
where is the duality mapping on , are sequences in which satisfies , , and for some . If for all , then converges strongly to , where is the generalized projection from onto .*

*Proof. *By the proof of Theorem 3.2, we can obtain Theorem 4.1.

It is easy to see that, this paper has some new methods and conditions than the conditions of Takahashi and Zembayashi [16]. In this paper, we prove the convergence theorems for uniformly quasi--asymptotically nonexpansive mappings and do not need the Kadec-Klee property of Banach space and use the condition of equally continuous that is more weak different from the condition of uniformly -Lipscitz.

#### Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

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