Journal of Applied Mathematics

Volume 2012, Article ID 395760, 24 pages

http://dx.doi.org/10.1155/2012/395760

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

^{1}Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand^{2}Department 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.

#### 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 *