- Research Article
- Open Access

# Strong Convergence Theorem by Monotone Hybrid Algorithm for Equilibrium Problems, Hemirelatively Nonexpansive Mappings, and Maximal Monotone Operators

- Yun Cheng
^{1}Email author and - Ming Tian
^{1}

**2008**:617248

https://doi.org/10.1155/2008/617248

© Y. Cheng and M. Tian. 2008

**Received:**17 June 2008**Accepted:**11 November 2008**Published:**18 November 2008

## Abstract

We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of hemirelatively nonexpansive mappings and the set of solutions of an equilibrium problem and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space. Using this theorem, we obtain three new results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.

## Keywords

- Banach Space
- Convex Function
- Equilibrium Problem
- Nonexpansive Mapping
- Strong Convergence

## 1. Introduction

The set of such solutions is denoted by .

In 2006, Martinez-Yanes and Xu [1] obtained strong convergence theorems for finding a fixed point of a nonexpansive mapping by a new hybrid method in a Hilbert space. In particular, Takahashi and Zembayashi [2] established a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a uniformly convex and uniformly smooth Banach space. Very recently, Su et al. [3] proved the following theorem by a monotone hybrid method.

Theorem 1.1 (see Su et al. [3]).

where is the duality mapping on . Then, converges strongly to , where is the generalized projection from onto .

In this paper, motivated by Su et al. [3], we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a hemirelatively nonexpansive mapping and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space by using the monotone hybrid method. Using this theorem, we obtain three new strong convergence results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.

## 2. Preliminaries

where denotes the generalized duality pairing. It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . In this case, is single valued and also one to one.

If is a Hilbert space, then and is the metric projection of onto .

If is a reflexive strict 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 . From the definition of , we have , that is, .

Let be a closed convex subset of and let be a mapping from into itself. We denote by the set of fixed points of . is called hemirelatively nonexpansive if for all and .

A point in is said to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A hemirelatively nonexpansive mapping from into itself is called relatively nonexpansive [1, 5, 6] if .

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

Lemma 2.1 (see Alber [4]).

Lemma 2.2 (see Alber [4]).

Lemma 2.3 (see Kamimura and Takahashi [7]).

Let be a smooth and uniformly convex Banach space and let and be sequences in such that either or is bounded. If . Then .

Lemma 2.4 (see Xu [8]).

Lemma 2.5 (see Kamimura and Takahashi [7]).

Lemma 2.6 (see Blum and Oettli [9]).

Lemma 2.7 (see Takahashi and Zembayashi [10]).

- (1)
- (2)
- (3)
- (4)

Lemma 2.8 (see Takahashi and Zembayashi [10]).

Lemma 2.9 (see Su et al. [3]).

Let be a strictly convex and smooth real Banach space, let be a closed convex subset of , and let be a hemirelatively nonexpansive mapping from into itself. Then, is closed and convex.

Recall that an operator in a Banach space is called closed, if , then .

## 3. Strong Convergence Theorem

Theorem 3.1.

for every , where is the duality mapping on are sequences in such that and for some . Then, converges strongly to , where is the generalized projection of onto .

Proof.

First, we can easily show that and are closed and convex for each .

As , by the induction assumptions, the last inequality holds, in particular, for all . This, together with the definition of , implies that . So, is well defined.

From (3.9), we can prove that is a Cauchy sequence. Therefore, there exists a point such that converges strongly to .

Since is a closed operator and , then is a fixed point of .

By the definition of , it follows that and , whence . Therefore, it follows from the uniqueness of that . This completes the proof.

Corollary 3.2.

for every , where is the duality mapping on and for some . Then, converges strongly to .

Proof.

Putting in Theorem 3.1, we obtain Corollary 3.2.

Corollary 3.3.

for every , where is the duality mapping on are sequences in such that . Then, converges strongly to .

Proof.

Putting for all and for all in Theorem 3.1, we obtain Corollary 3.3.

Corollary 3.4.

for every , where is the duality mapping on are sequences in such that and for some . Then, converges strongly to .

Proof.

Since every relatively nonexpansive mapping is a hemirelatively one, Corollary 3.4 is implied by Theorem 3.1.

Remark 3.5 (see Rockafellar [12]).

Let be a reflexive, strictly convex, and smooth Banach space and let be a monotone operator from to . Then, is maximal if and only if for all .

Let be a reflexive, strictly convex, and smooth Banach space and let be a maximal monotone operator from to . Using Remark 3.5 and strict convexity of , we obtain that for every and , there exists a unique such that If , then we can define a single-valued mapping by , and such a is called the resolvent of . We know that for all and is relatively nonexpansive mapping (see [2] for more details). Using Theorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space.

Theorem 3.6.

for every , where is the duality mapping on , is a sequences in such that and for some , Then, converges strongly to .

Proof.

Since is a closed relatively nonexpansive mapping and , from Corollary 3.4, we obtain Theorem 3.6.

## Declarations

### Acknowledgment

This work is supported by Tianjin Natural Science Foundation in China Grant no. 06YFJMJC12500.

## Authors’ Affiliations

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