Journal of Applied Mathematics

Volume 2012 (2012), Article ID 395760, 24 pages

doi:10.1155/2012/395760

Research Article

The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems

Kriengsak Wattanawitoon¹ and Poom Kumam²

¹Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

²Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 3 March 2012; Accepted 17 June 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Kriengsak Wattanawitoon 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.

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.

1. Introduction

The equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, networks, elasticity, and optimization, and it has been extended and generalized in many directions.

In the theory of equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. This theory combines theoretical and algorithmic advances with novel domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis, and numerical analysis.

Let be a Banach space with norm , be a nonempty closed convex subset of , and let denote the dual of . Let be a bifunction, be a real-valued function, where is denoted by the set of real numbers, and be a nonlinear mapping. The goal of the system of generalized mixed equilibrium problem is to find such that

If , , and , the problem (1.1) is reduced to the generalized mixed equilibrium problem, denoted by , to find such that

The set of solutions to (1.2) is denoted by , that is,

If , the problem (1.2) is reduced to the mixed equilibrium problem for , denoted by , to find such that

If , the problem (1.2) is reduced to the mixed variational inequality of Browder type, denoted by , is to find such that

If and , the problem (1.2) is reduced to the equilibrium problem for , denoted by , to find such that

The above formulation (1.6) was shown in [1] to cover monotone inclusion problems, saddle-point problems, variational inequality problems, minimization 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, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions to ; see, for example [1–4] and references therein. Some solution methods have been proposed to solve the ; see, for example, [2, 4–15] and references therein. In 2005, Combettes and Hirstoaga [5] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty, and they also proved a strong convergence theorem.

A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then the Banach space is said to be smooth, provided

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . 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 Banach space, be a closed convex subset of , a mapping is said to be nonexpansive if

for all . We denote by the set of fixed points of . If is a bounded closed convex set and is a nonexpansive mapping of into itself, then is nonempty (see [16]). A point in is said to be an asymptotic fixed point of [17] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by  . A point is said to be a strong asymptotic fixed point of , if there exists a sequence such that and . The set of strong asymptotic fixed points of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [18–20] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [21, 22]. is said to be -nonexpansive, if for . is said to be quas-ϕ-nonexpansive if and for and . A mapping is said to be asymptotically relatively nonexpansive, if , and there exists a real sequence with such that , , and . is said to be a countable family of weak relatively nonexpansive mappings [23] if the following conditions are satisfied:

(i); (ii); (iii).

A mapping is said to be uniformly L-Lipschitz continuous, if there exists a constant such that

A mapping is said to be closed if for any sequence with and , then . Let be a sequence of mappings. is said to be a countable family of uniformly asymptotically relatively nonexpansive mappings, if , and there exists a sequence with such that for each

In 2009, Petrot et al. [24], introduced a hybrid projection method for approximating a common element of the set of solutions of fixed points of hemirelatively nonexpansive (or quasi--nonexpansive) mappings in a uniformly convex and uniformly smooth Banach space:

They proved that the sequence converges strongly to , where and is the generalized projection from onto . Kumam and Wattanawitoon [25], introduced a hybrid iterative scheme for finding a common element of the set of common fixed points of two quasi--nonexpansive mappings and the set of solutions of an equilibrium problem in Banach spaces, by the following manner:

They proved that the sequence converges strongly to , where under the assumptions (C1) , (C2) , and (C3) .

Recently, Chang et al. [26], introduced the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi--asymptotically nonexpansive mappings,

where . Then, they proved that under appropriate control conditions the sequence converges strongly to .

Very recently, Tan and Chang [27], introduced a new hybrid iterative scheme for finding a common element between set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi--asymptotically nonexpansive mappings (which is more general than quasi--nonexpansive mappings), and null spaces of finite family of -inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space.

In this paper, motivated and inspired by Petrot et al. [24], Kumam and Wattanawitoon [25], Chang et al. [26], and Tan and Chang [27], we introduce the new hybrid block algorithm for two countable families of closed and uniformly Lipschitz continuous and uniformly asymptotically relatively nonexpansive mappings in a Banach space. Let be a sequence defined by , and

Under appropriate conditions, we will prove that the sequence generated by algorithms (1.15) converges strongly to the point . Our results extend many known recent results in the literature.

2. Preliminaries

Let be a real Banach space with norm , and let be the normalized duality mapping from into given by

for all , where denotes the dual space of and the generalized duality pairing between and . It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

We know the following (see [28, 29]):

(i)if is smooth, then is single valued;(ii)if is strictly convex, then is one-to-one and holds for all with ;(iii)if is reflexive, then is surjective;(iv)if is uniformly convex, then it is reflexive;(v)if is a reflexive and strictly convex, then is norm-weak-continuous;(vi) is uniformly smooth if and only if is uniformly convex;(vii)if is uniformly convex, then is uniformly norm-to-norm continuous on each bounded subset of ;(viii)each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence , if and , then .

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 [30], the generalized projection is a map that assigns to an arbitrary point the minimum point of the function , that is, , where is the solution to the minimization problem

Existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping . It is obvious from the definition of function that (see [30])

If is a Hilbert space, then .

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 (2.4), we have . This implies that . From the definition of , one has . Therefore, we have ; see [28, 29] for more details.

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

Lemma 2.1 (see Kamimura and Takahashi [31]). Let be a uniformly convex and smooth real Banach space, and let be two sequences of . If and either or is bounded, then .

Lemma 2.2 (see Alber [30]). Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if

Lemma 2.3 (see Alber [30]). Let be a reflexive, strictly convex, and smooth Banach space, let be a nonempty closed convex subset of , and let . Then

Lemma 2.4 (see Chang et al. [26]). 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 integers with ,

Lemma 2.5 (see Chang et al. [26]). Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and be a nonempty closed convex subset of . Let be a closed and asymptotically relatively nonexpansive mapping with a sequence . Then is closed and convex subset of .

For solving the generalized mixed equilibrium problem (or a system of generalized mixed equilibrium problem), let us assume that the bifunction and is convex and lower semicontinuous 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 2.6 (see Chang et al. [26]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a continuous and monotone mapping, is convex and lower semicontinuous and be a bifunction from to satisfying (A1)–(A4). For and , then there exists such that

Define a mapping as follows:

for all . Then, the following hold:

(i) is singlevalued; (ii) is firmly nonexpansive, that is, for all , ;(iii); (iv) is a solution of variational equation (2.9) if and only if is a fixed point of ; (v); (vi) is closed and convex; (vii), , .

3. Main Results

Theorem 3.1. Let be a uniformly smooth and uniformly convex Banach space, let be a nonempty, closed, and convex subset of . Let be a continuous and monotone mapping, be a lower semi-continuous and convex function, be a bifunction from to satisfying (A1)–(A4), is the mapping defined by (2.10) where , and let , be countable families of closed and uniformly , -Lipschitz continuous and asymptotically relatively nonexpansive mapping with sequence ;   such that . Let be a sequence generated by and ,

where , , and . The coefficient sequences and satisfy the following:

(i); (ii); (iii), ; (iv), ,

is the set of solutions to the following generalized mixed equilibrium problem:

Then the sequence converges strongly to .

Proof. We first show that , is closed and convex. Clearly is closed and convex. Suppose that is closed and convex for some . For each , we see that is equivalent to

By the set of , we have

Hence, is also closed and convex.

By taking for any and for all . We note that .

Next, we show that . For , we have . For any given . By (3.1) and Lemma 2.4, we have

where .

By (3.1) and (3.5), we note that

where , . By assumptions on and , we have

where .

So, we have . This implies that and also is well defined.

From Lemma 2.2 and , we have

From Lemma 2.3, one has

for all and . Then, the sequence is also bounded. Thus is bounded. Since and , we have

Therefore, is nondecreasing. Hence, the limit of exists. By the construction of , one has that and for any positive integer . It follows that

Letting in (3.12), we get . It follows from Lemma 2.1, that as . That is, is a Cauchy sequence.

Since is bounded and is reflexive, there exists a subsequence such that . Since is closed and convex and , this implies that is weakly closed and for each . since , we have

Since

We have

This implies that . That is, . Since , by the Kadec-klee property of , we obtain that

If there exists some subsequence such that , then we have

Therefore, we have . This implies that

Since

for all , we also have

Since and by the definition of , for , we have

Noticing that , we obtain

It then yields that . Since , we have

Hence,

From Lemma 2.1 and (3.22), we have

By the triangle inequality, we get

Since is uniformly norm-to-norm continuous on bounded sets, we note that

Now, we prove that . From the construction of , we obtain that

From (3.7) and (3.20), we have

By Lemma 2.1, we also have

Since is uniformly norm-to-norm continuous on bounded sets, we note that

From (2.4) and (3.29), we have . Since , it yields that

Since is uniformly norm-to-norm continuous on bounded sets, it follows that

This implies that is bounded in . Since is reflexive, there exists a subsequence such that . Since is reflexive, we see that . Hence, there exists such that . We note that

Taking the limit interior of both side and in view of weak lower semicontinuity of norm , we have

that is, . This implies that and so . It follows from , as and the Kadec-Klee property of that as . Note that is hemicontinuous, it yields that . It follows from , as and the Kadec-Klee property of that .

By similar, we can prove that

By (3.20) and (3.30), we obtain

Since is uniformly norm-to-norm continuous on bounded sets, we note that

So, from (3.27) and (3.31), by the triangle inequality, we get

Since is uniformly norm-to-norm continuous on bounded sets, we note that

Since

From (3.37) and (3.38), we obtain

On the other hand, we observe that, for .

From (3.22) and (3.27), we have

For any , it follows from (3.5) that

From condition, , property of , (3.7), and (3.42), we have that

Since and is uniformly norm-to-norm continuous. It yields . Hence from (3.46), we have

Since , this implies that as . Since is hemicontinuous, it follows that

On the other hand, for each , we have

from this, together with (3.48) and the Kadec-Klee property of , we obtain

On the other hand, by the assumption that is uniformly -Lipschitz continuous, we have

By (3.18) and (3.50), we obtain

and , that is, , . By the closeness of , we have , . This implies that .

By the similar way, we can prove that for each

Since and is uniformly norm-to-norm continuous. it yields . Hence from (3.53), we have

Since , this implies that as . Since is hemicontinuous, it follows that

On the other hand, for each , we have

From this, together with (3.54) and the Kadec-Klee property of , we obtain

On the other hand, by the assumption that is uniformly -Lipschitz continuous, we have

By (3.36) and (3.57), we obtain

and , that is, . By the closeness of , we have , . This implies that . Hence .

Next, we prove that . For any , for each , we have

It then yields that . Since , , we have

Hence,

This together with show that for each ,

where . On the other hand, we have

and is a solution of the following variational equation

By condition , we note that

By , (3.63), and for each , we have

For and , define . Noticing that , we obtain , which yields that

In view of the convexity of it yields

It follows from and that

Let , from , we obtain the following:

This implies that is a solution of the system of generalized mixed equilibrium problem (3.2), that is, . Hence, .

Finally, we show that . Indeed from and , we have the following:

This implies that

From the definition of and (3.73), we see that . This completes the proof.

Since every asymptotically relatively nonexpansive mappings is quasi--nonexpansive mappings, hence we obtain the following corollary.

Corollary 3.2. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty, closed, and convex subset of . Let be a continuous and monotone mapping, be a lower semi-continuous and convex function, be a bifunction from to satisfying (A1)–(A4), is the mapping defined by (2.10) where , and let , be countable families of closed and quasi--nonexpansive mapping such that . Let be a sequence generated by and , such that

where is the generalized projection from onto , is the duality mapping on . The coefficient sequences and , satisfying:

(i); (ii); (iii), ; (iv), .

is the set of solutions to the following generalized mixed equilibrium problem:

Then the sequence converges strongly to .

If , and for all in Theorem 3.1, we obtain the following corollary.

Corollary 3.3. Let be a uniformly smooth and uniformly convex Banach space, let be a nonempty, closed, and convex subset of . Let be a continuous and monotone mapping, be a lower semicontinuous and convex function, be a bifunction from to satisfying (A1)–(A4), be the mapping define by (2.10) where , and let , be countable families of closed and uniformly , -Lipschitz continuous, and asymptotically relatively nonexpansive mappings with sequence ;   such that . Let be a sequence generated by and , such that

where , , and . The coefficient sequences and , satisfying:

(i); (ii); (iii), ; (iv), .

Then the sequence converges strongly to .

If in Theorem 3.1, then we obtain the following corollary.

Corollary 3.4. Let be a uniformly smooth and uniformly convex Banach space, let be a nonempty, closed, and convex subset of . Let be a continuous and monotone mapping, be a lower semicontinuous and convex function, be a bifunction from to satisfying (A1)–(A4), is the mapping define by (2.10) where , and let are two closed and uniformly , -Lipschitz continuous and asymptotically relatively nonexpansive mappings with sequence ;   such that . Let be a sequence generated by and , we have

where , , and . The coefficient sequences and , satisfying

(D1); (D2).

Then the sequence converges strongly to .

Remark 3.5. Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results of Petrot et al. [24], Kumam and Wattanawitoon [25], and Chang et al. [26] in the following senses:

(i)for the mappings, we extend the mappings from nonexpansive mappings, hemirelatively nonexpansive mappings to two infinite family of closed asymptotically relatively nonexpansive mappings; (ii)from a solution of the classical equilibrium problem to a system of generalized mixed equilibrium problems and the generalized mixed equilibrium problem with an infinite family of closed relatively nonexpansive mappings.

Remark 3.6. Corollary 3.4 improves and extends the corresponding results of Theorem 3.1 in Kumam and Wattanawitoon [25] and Corollary 3.3 in Saewan et al. [11] in the following senses:

(i)the mapping in [11] and [25] (ii)the conditions (D1) and (D2) of the parameters and are weaker and not complicated than the conditions (C1)–(C3) in [[25], Theorem  3.1] and [[11], Theorem  3.1] which are easy to compute.

Acknowledgments

This first author was supported by The Hands-On Research and Development, Rajamangala University of Technology Lanna (Grant no. UR2L003). Moreover, this work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC Grant no. 55000613).

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