Abstract and Applied Analysis

Volume 2010 (2010), Article ID 123027, 31 pages

http://dx.doi.org/10.1155/2010/123027

## A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

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

Received 14 August 2010; Revised 30 September 2010; Accepted 10 October 2010

Academic Editor: Douglas Robert Anderson

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 new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

#### 1. Introduction

Let be a Banach space with norm , a nonempty closed convex subset of , and let denote the dual of . Let be a bifunction, be a real-valued function, and a mapping. The *generalized mixed equilibrium problem*, is to find such that
The set of solutions to (1.1) is denoted by , that is,
If , the problem (1.1) reduces into the *mixed equilibrium problem for *, denoted by , which is to find such that
If , the problem (1.1) reduces into the *mixed variational inequality* of Browder type, denoted by , which is to find such that
If and the problem (1.1) reduces into the *equilibrium problem for *, denoted by , which is to find such that
If , the problem (1.3) reduces into the *minimize problem*, denoted by , is to find such that
The above formulation (1.4) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, 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 an 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, [1, 2] and references therein. Some solution methods have been proposed to solve the ; see, for example, [1, 3–11] and references therein.

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 exists 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 [12, 13] for more details. Observe that every -uniformly convex is uniformly convex. One should note that no Banach space is -uniformly 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.

A set valued mapping with graph , domain , and rang . is said to be *monotone* if whenever . A monotone operator is said to be *maximal monotone* if its graph is not properly contained in the graph of any other monotone operator on the same space. We know that if is maximal monotone, then the solution set is closed and convex. It is knows that is a maximal monotone if and only if for all when is a reflexive, strictly convex and smooth Banach space (see Rockafellar [14]).

Let be a smooth, strictly convex and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying . Then we define the *resolvent * by for all . In other words, for all . is a single-valued mapping from to . Also, we know that for all , where is the set of all fixed points of . We can define, for , the *Yosida approximation* of by for all We know that for all and

It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . We consider the problem of finding:
where is an operator from into . Such is called a *zero point* of . Such a problem contains numerous problems in economics, optimization and physics. When is a maximal monotone operator, a well-know method for solving (1.9) in a Hilbert space is the *proximal point algorithm*: and,
where and , then Rockafellar [15] proved that the sequence converges weakly to an element of Let be a real Banach space and let be a nonempty closed convex subset of and be an operator. *The classical variational inequality problem* for an operator is to find such that
The set of solution of (1.11) is denote by . Recall that let be a mapping. Then is called (i)*monotone* if
(ii)*-inverse-strongly monotone* if there exists a constant such that

The class of inverse-strongly monotone mappings has been studied by many researchers to approximating a common fixed point; see [6, 7, 16, 17] for more details.

Let be a closed convex subset of , a mapping is said to be *nonexpansive* if , for all A point is a *fixed point *of provided . Denote by the set of fixed points of ; that is, .

Consider the functional defined by
Recall that a point in is said to be an *asymptotic fixed point* of [18] 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* [19–21] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [22–24]. is said to be *-nonexpansive*, if for . is said to be *relatively quasi-nonexpansive* (or *quasi*-*-nonexpansive*) if and for and . We note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [22–26] which requires the strong restriction: .

Let be 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. In this connection, Alber [27] recently introduced a *generalized projection * from in to as follows:
It is obvious from the definition of function that
If is a Hilbert space, then and becomes the metric projection of onto . Let be the generalized projection from a smooth, strictly convex and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed relatively quasi-nonexpansive mapping from onto with . On the author hand, the generalized projection 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
The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping (see, e.g., [27–31]).

*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 (1.14), we have . This implies that From the definition of one has . Therefore, we have see [29, 31] for more details.

In 2004, Matsushita and Takahashi [32] introduced the following iteration: a sequence defined by where the initial guess element is arbitrary, is a real sequence in , is a relatively nonexpansive mapping and denotes the generalized projection from onto a closed convex subset of . They proved that the sequence converges weakly to a fixed point of .

In 2005, Matsushita and Takahashi [25] proposed the following hybrid iteration method (it is also called the 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 2008, Iiduka and Takahashi [33] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator in a -uniformly convex and uniformly smooth Banach space : and for every , where is the generalized metric projection from onto , is the duality mapping from into and is a sequence of positive real numbers. They proved that the sequence generated by (1.20) converges weakly to some element of .

Recently, Takahashi and Zembayashi [34, 35], studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2008, Cholamjiak [36], proved the following iteration where is the duality mapping on . Assume that and are sequences in such that , and . Then converges strongly to , where In 2009, Wei et al. [37] proved the following iteration for two relatively nonexpansive mappings in a Banach space : if and are sequences in such that and for some then generated by (1.22) converges strongly to a point Where the mapping of onto is the generalized projection operator. Inoue et al. [38] proved strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method. After that, Klin-eam et al. [2], extend Inoue et al. [38] to obtain the strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method.

On the other hand, Nakajo et al. [39] introduced the following condition. Let be a nonempty closed convex subset of a Hilbert space , let be a family of mappings of into itself with and denotes the set of all weak subsequential limits of a bounded sequence in . is said to satisfy the NST-condition if for every bounded sequence in ,

Recall that a mapping is closed if for each in , if and then Let be a family of mappings of in to itself with , is said to satisfy the -condition if for each bounded sequence in , It follows directly from the definitions above that if satisfies NST-condition, then satisfies -condition. If and is closed, then satisfies -condition (see [40] for more details).

In this paper, we introduce a new hybrid projection method for finding a common solution of the set of common fixed points of two countable families of relatively quasi nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in a real uniformly smooth and -uniformly convex Banach space.

#### 2. Preliminaries

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

Lemma 2.1 (Xu [41]). *If be a 2-uniformly convex Banach space and , then for all , one has
**
where is the normalized duality mapping of .*

The best constant in lemma is called the *-uniformly convex constant* of .

Lemma 2.2 (Chidume [42, Corollary , pages 36-37]). *If be a p-uniformly convex Banach space and let be a given real number with , then for all , and **
where is the generalized duality mapping of and is the p-uniformly convexity constant of .*

Lemma 2.3 (Kamimura and Takahashi [30]). *Let be a uniformly convex and smooth Banach space and let and be two sequences of . If and either or is bounded, then *

Lemma 2.4 (Alber [27]). *Let be a nonempty closed convex subset of a smooth Banach space and . Then if and only if
*

Lemma 2.5 (Alber [27]). *Let be a reflexive, strictly convex and smooth Banach space, let be a nonempty closed convex subset of and let Then
*

Lemma 2.6 (Qin et al. [9]). *Let be a real uniformly smooth and strictly convex Banach space, and be a nonempty closed convex subset of . Let be a relatively quasi-nonexpansive mapping. Then is a closed convex subset of *

Let be a reflexive, strictly convex, smooth Banach space and the duality mapping from into . Then is also single valued, one-to-one, surjective, and it is the duality mapping from into . We make use of the following mapping studied in Alber [27] for all and , that is, .

Lemma 2.7 (Kohsaka and Takahashi [43, Lemma ]). *Let be a reflexive, strictly convex smooth Banach space and let be as in (2.5). Then
**
for all and *

Lemma 2.8 (Kohsaka and Takahashi [44]). *Let be a smooth, strictly convex and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying Let , let and be the resolvent and the Yosida approximation of , respectively. Then the following hold: *(i)*, for all *(ii)*, for all *(iii)

Let be a nonempty closed convex subset of a Banach space and let be an inverse-strongly monotone mapping of into which is said to be *hemicontinuous* if for all , the mapping of into , defined by , is continuous with respect to the weak topology of . We define by *the normal cone* for at a point , that is,

Lemma 2.9 (Rockafellar [14]). *Let be a nonempty, closed convex subset of a Banach space and a monotone, hemicontinuous operator of into . Let be an operator defined as follows:
**
Then is maximal monotone and *

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.

For example, let be a continuous and monotone operator of into and define Then, satisfies (A1)–(A4).

The following result is in Takahashi and Zembayashi ([34, 35]).

Lemma 2.10 (see [34, 35]). *Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), let and let . Then, there exists such that
*

Motivated by Combettes and Hirstoaga [4] in a Hilbert space and Takahashi and Zembayashi [34] in a Banach space, Zhang [45] obtain the following lemma.

Lemma 2.11 (Zhang [45, Lemma ]). *Let be a nonempty 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 followings hold: *(1)* is single-valued; *(2)* is firmly nonexpansive, that is, for all *(3)(4)* is closed and convex; *(5)* for all and *

#### 3. Main Results

In this section, by using the -condition, we prove the new convergence theorems for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of two countable families of relatively quasi-nonexpansive mappings, zeros of maximal monotone operators and the solution set of variational inequalities for an -inverse strongly monotone mapping in a *2*-uniformly convex and uniformly smooth Banach space.

Theorem 3.1. *Let be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a maximal monotone operator satisfying and let for all where is the duality mapping on . Let be a bifunction from to satisfying (A1)–( A4), and let be a proper lower semicontinuous and convex function. Let be an -inverse-strongly monotone mapping of into satisfy , for all and and let be a continuous and monotone mapping. Let be two families of relatively quasi-nonexpansive mappings with satisfy the -condition such that
**
For an initial point with and , we define the sequence as follows:
**
where are sequences in and for some and for some with , where is the 2-uniformly convexity constant of . If and then converges strongly to , where .*

*Proof. *We split the proof into seven steps.*Step 1. *We first show that is closed and convex for each

By Lemma 2.6, we know that is closed and convex. We also know that if and are closed and convex. From Lemma 2.11 (4), we have is closed and convex. Hence is a nonempty, closed and convex subset of . Consequently, is well defined.

Next, we prove that is closed and convex for each . It is obvious that is closed and convex. Suppose that is closed and convex for each . Since for any , we know is equivalent to
So, is closed and convex.*Step 2. *We show that for all

Next, we show by induction that for all . Indeed, let and for all . On the other hand, from Lemma 2.11 one has is relatively quasi-nonexpansive mapping and . Suppose that for some . Let Since is relatively quasi-nonexpansive mapping, we have
by nonexpansiveness of (see [31, Theorem , page 130]) and is relatively quasi-nonexpansive mappings, we also have
So, it follows that
It follows from Lemmas 2.5 and 2.7, that
Thus, since and is -inverse-strongly monotone, we have
By Lemma 2.1 and the fact that for all and , we obtain
Substituting (3.8) and (3.9) into (3.7), we have
Substituting (3.10) into (3.6), we get

This shows that which implies that and hence, for all . This implies that the sequence is well defined.*Step 3. *We prove that is bounded.

Since and we have
By Lemma 2.5, we get
From (3.12) and (3.13), then are nondecreasing and bounded. So, we obtain that exists. In particular, by (1.16), the sequence is bounded. This implies is also bounded. So, we have , , and are bounded.*Step 4. *We show that is a Cauchy sequence in . Since , for , by Lemma 2.5, we have

Taking , we have From Lemma 2.3, we get . Hence is a Cauchy sequence and by the completeness of and the closedness of , we can assume that there exists such that as .*Step 5. *We show that , as . We taking in Step 4, we also have
From Lemma 2.3, that
Since is uniformly norm-to-norm continuous on bounded subsets of , we have
Since and the definition of , we have
By (3.15), we obtain
Again applying Lemma 2.3, we get
From
It follows that
Since is uniformly norm-to-norm continuous on bounded subsets of , we also have
*Step 6. *We will show that , where
We show that . From definition of , for any , we have
Since , we get It follows from (3.15), that
again from Lemma 2.3, that
it follows that since is uniformly norm-to-norm continuous, we also have
Since
from (3.16) and (3.27), we also have
Since is uniformly norm-to-norm continuous, we obtain
From (3.4), (3.5) and (3.10), we get By Lemma 2.11 (5) and , we observe that
Since , and are bounded, it follows from (3.22), (3.23), and Lemma 2.3, we also have
Since is uniformly norm-to-norm continuous, we have
By using the triangle inequality, we obtain
By (3.20) and (3.33), we get
Since is uniformly norm-to-norm continuous, we obtain
Since
From (3.27) and (3.36), we have
Since is uniformly norm-to-norm continuous, we also have
From (3.2), we get
and hence
it follows that
Since (3.31) and (3.40), one has Since is uniformly norm-to-norm continuous, we get
Since and , then we get , hence it follows from -condition, that .

Since , we compute
and hence
From (3.17), (3.28) and we obtain that
Since is uniformly norm-to-norm continuous on bounded sets, we have
Using the triangle inequality, we have
From (3.16) and (3.48), we have
On the other hand, for we note that
Since and are bounded, it follows from and , that
Furthermore, from (3.4), (3.5), (3.6) and (3.10), that
and hence
From (3.52), and obtain that
From Lemmas 2.5, 2.7, and (3.9), we compute

Applying Lemma 2.3 and (3.55) it follows that
Since is uniformly norm-to-norm continuous, we also have
Again by the triangle inequality, we get
From (3.50) and (3.57), we have
From (3.5), we have it follows from Lemma 2.8 and (3.10), we note that