# 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

## 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

## References

- Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(11):2400-2411. 10.1016/j.na.2005.08.018MATHMathSciNetView ArticleGoogle Scholar - Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings.
*Fixed Point Theory and Applications*2008, 2008:-11.Google Scholar - Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings.
*Fixed Point Theory and Applications*2008, 2008:-8.Google Scholar - Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics*.*Volume 178*. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15-50.Google Scholar - Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces.
*Journal of Applied Analysis*2001, 7(2):151-174. 10.1515/JAA.2001.151MATHMathSciNetView ArticleGoogle Scholar - Matsushita S-Y, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space.
*Journal of Approximation Theory*2005, 134(2):257-266. 10.1016/j.jat.2005.02.007MATHMathSciNetView ArticleGoogle Scholar - Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space.
*SIAM Journal on Optimization*2002, 13(3):938-945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar - Xu HK: Inequalities in Banach spaces with applications.
*Nonlinear Analysis: Theory, Methods & Applications*1991, 16(12):1127-1138. 10.1016/0362-546X(91)90200-KMATHMathSciNetView ArticleGoogle Scholar - Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*1994, 63(1–4):123-145.MATHMathSciNetGoogle Scholar - Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(1):45-57. 10.1016/j.na.2007.11.031MATHMathSciNetView ArticleGoogle Scholar - Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces. to appear in SIAM Journal on OptimizationGoogle Scholar
- Rockafellar RT: On the maximality of sums of nonlinear monotone operators.
*Transactions of the American Mathematical Society*1970, 149: 75-88. 10.1090/S0002-9947-1970-0282272-5MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.