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Abstract and Applied Analysis
Volume 2010, Article ID 357120, 22 pages
http://dx.doi.org/10.1155/2010/357120
Research Article

Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi- -Asymptotically Nonexpansive Mappings

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 14 May 2010; Revised 16 June 2010; Accepted 21 June 2010

Academic Editor: W. A. Kirk

Copyright © 2010 Siwaporn Saewan and Poom Kumam. 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 introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.

1. Introduction and Preliminaries

The convex feasibility problem (CFP) is the problem of computing points laying in the intersection of a finite family of closed convex subsets , of a Banach space This problem appears in various fields of applied mathematics. The theory of optimization [1], Image Reconstruction from projections [2], and Game Theory [3] are some examples. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [4]. The advantage of a Hilbert space is that the projection onto a closed convex subset of is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [5] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection (see, Alber [6] or Kamimura and Takahashi [7]) from onto is relatively nonexpansive, whereas the metric projection from onto is not generally nonexpansive.

We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [8, 9], etc.). In 2008, Plubtieng and Ungchittrakool [10] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming.

Let be a nonempty closed convex subset of a real Banach space with and being the dual space of . Let be a bifunction of into and a monotone mapping. The generalized equilibrium problem, denoted by , is to find such that The set of solutions for the problem (1.1) is denoted by , that is If , the problem (1.1) reducing into the equilibrium problem for , denoted by , is to find such that If , the problem (1.1) reducing into the classical variational inequality, denoted by , is to find such that The above formulation (1.3) was shown in [11] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point 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 forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of ; see, for example [11] and references therein. Some solution methods have been proposed to solve the ; see, for example, [1229] and references therein.

Consider the functional defined by where is the duality mapping from into . It is well known that if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function that If is a Hilbert space, then , for all On the other hand, the generalized projection (Alber [6]) is a map that assigns to an arbitrary point the minimum point of the functional that is, where is the solution to the minimization problem and existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, for example, [6, 7, 3032]).

Remark 1.1. If is a reflexive, strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (1.5), we have . This implies that From the definition of one has . Therefore, we have see [31, 32] for more details.
Let be a closed convex subset of ; a mapping is said to be nonexpansive if A point is a fixed point of provided . Denote by the set of fixed points of ; that is, . Recall that a point in is said to be an asymptotic fixed point of [33] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by .
A mapping from into itself is said to be relatively nonexpansive [3436] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [3739]. is said to be -nonexpansive, if for . is said to be relatively quasi-nonexpansive if and for all and . is said to be quasi- -asymptotically nonexpansive if and there exists a real sequence with such that for all and . A mapping is said to be closed if for any sequence with and , . 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 [3741]).
A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then a Banach space is said to be smooth if the limit exists for each It is also said to be uniformly smooth if the limit is attained uniformly for . Let be a Banach space. The modulus of convexity of is the function defined by A Banach space is uniformly convex if and only if for all . Let be a fixed real number with . A Banach space is said to be -uniformly convex if there exists a constant such that for all ; see [42] for more details. Observe that every -uniform convex is uniformly convex. One should note that no Banach space is -uniform convex for . It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each , the generalized duality mapping is defined by for all . In particular, is called the normalized duality mapping. If is a Hilbert space, then , where is the identity mapping. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .
The following basic properties can be found in Cioranescu [31].(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. (v)Each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence if and

In 2005, Matsushita and Takahashi [40] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space : They proved that converges strongly to , where is the generalized projection from onto . In 2007, Plubtieng and Ungchittrakool [43] generalized the processes (1.8) to the new general processes of two relatively nonexpansive mappings in a Banach space. Let be a closed convex subset of a Banach space and relatively nonexpansive mappings such that . Define in the following way: where , , , and are sequences in with for all .

In 2009, Qin et al. [26] introduced a hybrid projection algorithm to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two quasi- -nonexpansive mappings in the framework of Banach spaces: where is the generalized projection from onto . They proved that the sequence converges strongly to In the same year, Wattanawitoon and Kumam [44] and Petrot et al. [45] using the idea of Takahashi and Zembayashi [46, 47] and Plubtieng and Ungchittrakool [43] extend the notion from relatively nonexpansive mappings or quasi- -nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2010, Chang et al. [48] 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. Recently, many authors considered the iterative methods for finding a common element of the set of solutions to the problem (1.3) and of the set of fixed points of nonexpansive mappings; see, for instance, [1227] and the references therein.

Motivated by Chang et al. [48], Qin et al. [26, 49], Wattanawitoon and Kumam [44], Petrot et al. [45], Zegeye [50], and other recent works, in this paper we introduce a new 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 with Kadec-Klee property. The results presented in this paper improve and generalize some well-known results in the literature.

2. Basic Results

We also need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [7]). Let be a uniformly convex and smooth Banach space and let and be two sequences of . If and either or is bounded, then

Lemma 2.2 (Alber [6]). Let be a nonempty closed convex subset of a smooth Banach space and . Then if and only if

Lemma 2.3 (Alber [6]). Let be a reflexive, strictly convex and smooth Banach space, let be a nonempty closed convex subset of , and let Then

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

Lemma 2.4 (Blum and Oettli [11]). Let be a closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying , and let and . Then there exists such that

Lemma 2.5 (Zegeye [50]). Let be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space and let be a bifunction from to satisfying and let be a monotone mapping from into . For and , define a mapping as follows: for all . Then the following hold: (1) is single-valued; (2) is a firmly nonexpansive-type mapping, for all , (3) (4) is closed and convex.

Lemma 2.6 (Zegeye [50]). (Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying and let be a monotone mapping from into . For , and , we have that

Lemma 2.7 (Chang et al. [48]). 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 number with , there exists a continuous, strictly increasing, and convex function with such that, for any positive integer with ,

Lemma 2.8 (Chang et al. [48]). Let be a real uniformly smooth and strictly convex Banach space, 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

Definition 2.9 (Chang et al. [48]). (1) 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
(2) A mapping is said to be uniformly -Lipschitz continuous if there exists a constant such that

3. Main Results

In this section, we prove the new 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 with Kadec-Klee property.

Theorem 3.1. Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space with Kadec-Klee property. Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be an infinite family of closed uniformly -Lipschitz continuous and uniformly quasi- -asymptotically nonexpansive mappings with a sequence , such that is a nonempty and bounded subset in For an initial point with and , we define the sequence as follows: where is the duality mapping on , , are sequences in and for some . If for all and for all , then converges strongly to , where .

Proof. We first show that is closed and convex for each Clearly is closed and convex. Suppose that is closed and convex for each . Since for any , we know So, is closed and convex. Therefore, and are well defined. Next, we show that for all Indeed, put for all . It is clear that . Suppose for , by the convexity of , property of , Lemma 2.7, and uniformly quasi- -asymptotically nonexpansive of for each , we observed that and  Substituting (3.4) into (3.3), we get This show that implies that and hence, for all . Since is nonempty, is a nonempty closed convex subset of , and hence exist for all . This implies that the sequence is well defined. From definition of that and we have By Lemma 2.3, we also have From (3.6) and (3.7), then are nondecreasing and bounded. So, we obtain that exists. In particular, by (1.6), the sequence is bounded. This implies that is also bounded. Denote Moreover, by the definition of and (3.8), it follows that
Next, we show that is a Cauchy sequence in . Since , for , by Lemma 2.3, we have Since exists and we take , then, we get From Lemma 2.1, we have . Thus is a Cauchy sequence and by the completeness of , and there exists a point such that as .
Now, we claim that , as . By definition of , one has Since exists, we have By Lemma 2.1, we obtain that Since is uniformly norm-to-norm continuous on bounded subsets of , we get From and the definition of , we have By (3.9) and (3.12), we also have Applying Lemma 2.1, we obtain Since we get Since is uniformly norm-to-norm continuous on bounded subsets of , we have
Next, we will show that
(i) First, we show that It follows from (3.3) and (3.4) we observe that By Lemma 2.6 and , we obtain By Lemma 2.1, (3.9), (3.18), and (3.19), we have Again since is uniformly norm-to-norm continuous, we also have From (A2), we note that and hence For with and let Then and hence It follows that Since as from (3.18) and (3.21), we can get and as . Furthermore, it follows from the continuity of that as . From and (3.22), we have as . Since is monotone, we know that Thus, it follows from (A4) that From the conditions (A1) and (A4), we have and hence Letting , we get This implies that .
We show that . From definition of and since , we have Form Lemma 2.1 and (3.9), we obtain that Since is uniformly norm-to-norm continuous, we also have From (3.1), we compute and hence From (3.14), (3.33), and we get Since is uniformly norm-to-norm continuous on bounded sets, we obtain that By the triangle inequality, Hence from (3.13) and (3.37), we have Since is uniformly continuous on any bounded subset of we obtain Since and is uniformly continuous, it yields . Thus from (3.40), we get Since is -continuous, we have On the other hand, for each we observe that In view of (3.41), we obtain for each . Since has the Kadec-Klee property, we get By the assumption that for each is uniformly -Lipschitz continuous, we have From (3.13) and (3.39), it yields that . From , we get that is In view of closeness of we have for all This implies that
Finally, we show that . Let . From and , we have This implies that By definition of , we have . Therefore, . This completes the proof.

f for each then Theorem 3.1 is reduced to the following corollary.

Corollary 3.2. Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space with Kadec-Klee property. Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be a closed uniformly -Lipschitz continuous and quasi- -asymptotically nonexpansive mappings with a sequence , , such that is a nonempty and bounded subset in For an initial point with and , we define the sequence as follows: where is the duality mapping on , , are sequences in and for some . If , then converges strongly to where

For a special case that , we can obtain the following results on a pair of quasi- -asymptotically nonexpansive mappings immediately from Theorem 3.1.

Corollary 3.3. Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space with Kadec-Klee property. Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be two closed quasi- -asymptotically nonexpansive mappings and uniformly -Lipschitz continuous, respectively with a sequence , such that is a nonempty and bounded subset in For an initial point with and , we define the sequence as follows: where is the duality mapping on , , are sequences in and for some . If for all and and , then converges strongly to , where .

Remark 3.4. Corollary 3.3 improves and extends [44, Theorem ] in the following senses: (i)for the mappings, we extend the mappings from two closed relatively quasi-nonexpansive mappings to an infinite family of closed and uniformly quasi- -asymptotically mappings, (ii)from a solution of the classical equilibrium problem to the generalized equilibrium problem, (iii)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.

Corollary 3.5. Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space with Kadec-Klee property. Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be an infinite family of closed quasi- -nonexpansive mappings such that For an initial point with and , we define the sequence as follows: where is the duality mapping on , are sequences in and for some . If for all and for all , then converges strongly to , where .

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 the conditions appearing in Theorem 3.1 is a bounded subset in and for each is uniformly -Lipschitz continuous are of no use here. By virtue of the closeness of mapping for each , it yields that for each , that is, . Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Corollary 3.5 is obtained from Theorem 3.1 immediately.

Corollary 3.6. Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space with Kadec-Klee property. Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings with a sequence , and uniformly -Lipschitz continuous such that is a nonempty and bounded subset in For an initial point with and , we define the sequence as follows: where is the duality mapping on , , is a sequence in , and for some If for all and for all , then converges strongly to , where .

Proof. Setting in Theorem 3.1, then, we get that Thus, the method of proof of Theorem 3.1 we obtain Corollary 3.6 immediately.

Remark 3.7. Theorem 3.1, Corollary 3.3, and Corollary 3.5, improve and extend the corresponding results in Qin et al. [49] and Zegeye [50] in the following senses: (i)from a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family of quasi- -asymptotically mappings, (ii)for the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or quasi- -nonexpansive mappings and a finite family of closed relatively quasi-nonexpansive mappings to an infinite family of quasi- -asymptotically nonexpansive mappings, (iii)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.

4. Applications

If , a Hilbert space, then is uniformly smooth and strictly convex Banach space with Kadec-Klee property and closed relatively quasi-nonexpansive mappings reducing to closed quasi-nonexpansive mappings. Moreover, , identity operator on and , projection mapping from into Thus, the following corollaries hold.

Theorem 4.1. Let be a nonempty closed and convex subset of a Hilbert space . Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings with a sequence , and uniformly -Lipschitz continuous such that is a nonempty and bounded subset in For an initial point with and , we define the sequence as follows: where , are sequences in , and for some . If for all and for all , then converges strongly to , where .

Corollary 4.2. Let be a nonempty closed and convex subset of a Hilbert space . Let be a bifunction from to satisfying . Let be a continuous monotone mapping of into . Let be an infinite family of closed and quasi- -nonexpansive mappings with a sequence , and uniformly -Lipschitz continuous such that is a nonempty and bounded subset in For an initial point with and , we define the sequence as follows: where are sequences in and for some . If for all and for all , then converges strongly to , where .

Remark 4.3. Theorem 4.1 improves and extends the Corollary  3.7 in Zegeye [50] in the aspect for the mappings; we extend the mappings from a finite family of closed relatively quasi-nonexpansive mappings to more general a infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings.

Acknowledgments

The authors would like to thank the referees for their careful readings and valuable suggestions to improve the writing of this paper. The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Commission on Higher Education and the second author is supported by the Thailand Research Fund and the Commission on Higher Education under the project no. MRG5380044.

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