# A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces

- Weerayuth Nilsrakoo
^{1}Email author

**2011**:572156

https://doi.org/10.1155/2011/572156

© Weerayuth Nilsrakoo. 2011

**Received: **5 June 2010

**Accepted: **20 January 2011

**Published: **6 February 2011

## Abstract

We introduce a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.

## 1. Introduction

*equilibrium problem*is to find such that

The set of solutions of (1.1) is denoted by . The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases.

Using this functional, Matsushita and Takahashi [2, 3] studied and investigated the following mappings in Banach spaces. A mapping
is *relatively nonexpansive* if the following properties are satisfied:

where and denote the set of fixed points of and the set of asymptotic fixed points of , respectively. It is known that satisfies condition (R3) if and only if is demiclosed at zero, where is the identity mapping; that is, whenever a sequence in converges weakly to and converges strongly to 0, it follows that . In a Hilbert space , the duality mapping is an identity mapping and for all . Hence, if is nonexpansive (i.e., for all ), then it is relatively nonexpansive.

for every
,
is relatively nonexpansive,
is an appropriate sequence in
, and
is an appropriate positive real sequence. They proved that if
is weakly sequentially continuous, then
converges *weakly* to some element in
.

Motivated by S. Takahashi and W. Takahashi [17] and Takahashi and Zembayashi [19], we prove a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly smooth and uniformly convex Banach space.

## 2. Preliminaries

*strictly convex*if the following implication holds for :

It is known that if
is a uniformly convex Banach space, then
is reflexive and strictly convex. We say that
is *uniformly smooth* if the dual space
of
is uniformly convex. A Banach space
is *smooth* if the limit
exists for all norm one elements
and
in
. It is not hard to show that if
is reflexive, then
is smooth if and only if
is strictly convex.

for all and . The following lemma is an analogue of Xu's inequality [22, Theorem 2] with respect to .

Lemma 2.1.

It is also easy to see that if and are bounded sequences of a smooth Banach space , then implies that .

Lemma 2.2 (see [23, Proposition 2]).

Let be a uniformly convex and smooth Banach space, and let and be two sequences of such that or is bounded. If , then .

Remark 2.3.

Following Alber [1], we denote such an element
by
. The mapping
is called the *generalized projection* from
onto
. It is easy to see that in a Hilbert space, the mapping
coincides with the metric projection
. Concerning the generalized projection, the following are well known.

Lemma 2.4 (see [23, Propositions 4 and 5]).

Let be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space , , and . Then,

Remark 2.5.

The generalized projection mapping above is relatively nonexpansive and .

for all and . Obviously, for all and . We know the following lemma (see [1] and [24, Lemma 3.2]).

Lemma 2.6.

Lemma 2.7 (see [25, Lemma 2.1]).

for all , where the sequences in and in satisfy conditions: , , and . Then, .

Lemma 2.8 (see [26, Lemma 3.1]).

For solving the equilibrium problem, we usually assume that a bifunction satisfies the following conditions:

(A2) is monotone, that is, , for all ,

(A4)for all , is convex and lower semicontinuous.

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.9 (see [19, Lemma 2.8 ]).

for all . Then, the following hold:

(ii) is a firmly nonexpansive-type mapping [27], that is, for all

Lemma 2.10 (see [4, Lemma 2.3]).

Let be a nonempty closed convex subset of a Banach space , a bifunction from satisfying conditions (A1)–(A4) and . Then, if and only if for all .

Remark 2.11 (see [27]).

for all and . In particular, satisfies condition (R2).

Lemma 2.12 (see [3, Proposition 2.4]).

Let be a nonempty closed convex subset of a strictly convex and smooth Banach space and a relatively nonexpansive mapping. Then, is closed and convex.

## 3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly convex and uniformly smooth Banach space.

Theorem 3.1.

for all , where satisfying and , , and . Then, and converge strongly to .

Proof.

where for all . Notice that satisfying and .

The rest of the proof will be divided into two parts.

Case 1.

It follows from Lemma 2.7 and (3.8) that . Then, and so .

Case 2.

But for all , we conclude that , and .

From two cases, we can conclude that and converge strongly to and the proof is finished.

Applying Theorem 3.1 and [28, Theorem 3.2], we have the following result.

Theorem 3.2.

Then, and converge strongly to .

Setting and in Theorem 3.1, we have the following result.

Corollary 3.3.

for all , where satisfying and , . Then, and converge strongly to .

Letting in Corollary 3.3, we have the following result.

Corollary 3.4.

for all , where satisfying and , . Then converges strongly to .

Let be the identity mapping in Theorem 3.1, we also have the following result.

Corollary 3.5.

for all , where satisfying and , , and . Then, and converge strongly to .

## 4. Deduced Theorems in Hilbert Spaces

In Hilbert spaces, every nonexpansive mappings are relatively nonexpansive, and is the identity operator. We obtain the following result.

Theorem 4.1.

for all , where is the resolvent of , satisfying and , , and . Then, converges strongly to .

Remark 4.2.

In Theorem 4.1, we have the same conclusion if the mapping is only quasinonexpansive (i.e., and for all and ) such that is demiclosed at zero.

Letting in Theorem 4.1, we have the following result.

Corollary 4.3.

for all , where satisfying , , and . Then, converges strongly to .

Let be the identity mapping in Theorem 4.1, we have the following result.

Corollary 4.4.

for all , where is the resolvent of , satisfying , , and . Then converges strongly to .

Proof.

, and . Applying Theorem 4.1, converges strongly to .

Remark 4.5.

Corollary 4.4 improves and extends [29, Corollary 5.3]. More precisely, the conditions and are removed.

Applying Corollary 4.4 and [30, Theorem 8], we have the following result.

Corollary 4.6.

for all , where is the resolvent of , satisfying and and . Then, converges strongly to .

Remark 4.7.

Corollary 4.6 improves and extends [16, Corollary 3.4]. More precisely, the conditions and are removed.

## Declarations

### Acknowledgment

The author would like to thank the referees for their comments and helpful suggestions.

## Authors’ Affiliations

## References

- 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*. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar - Matsushita S-Y, Takahashi W:
**Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces.***Fixed Point Theory and Applications*2004, (1):37–47.Google 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 - Boonchari D, Saejung S:
**Approximation of common fixed points of a countable family of relatively nonexpansive mappings.***Fixed Point Theory and Applications*2010,**2010:**-26.Google Scholar - Ceng L-C, Yao J-C:
**A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.***Journal of Computational and Applied Mathematics*2008,**214**(1):186–201. 10.1016/j.cam.2007.02.022MATHMathSciNetView ArticleGoogle Scholar - Ceng LC, Petruşel A, Yao JC:
**Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings.***Journal of Optimization Theory and Applications*2009,**143**(1):37–58. 10.1007/s10957-009-9549-9MATHMathSciNetView ArticleGoogle Scholar - Kangtunyakarn A, Suantai S:
**A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(10):4448–4460. 10.1016/j.na.2009.03.003MATHMathSciNetView ArticleGoogle Scholar - Nilsrakoo W, Saejung S:
**Weak and strong convergence theorems for countable Lipschitzian mappings and its applications.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(8):2695–2708. 10.1016/j.na.2007.08.044MATHMathSciNetView ArticleGoogle Scholar - Nilsrakoo W, Saejung S:
**Weak convergence theorems for a countable family of Lipschitzian mappings.***Journal of Computational and Applied Mathematics*2009,**230**(2):451–462. 10.1016/j.cam.2008.12.013MATHMathSciNetView ArticleGoogle Scholar - Nilsrakoo W, Saejung S:
**Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications.***Journal of Mathematical Analysis and Applications*2009,**356**(1):154–167. 10.1016/j.jmaa.2009.03.002MATHMathSciNetView ArticleGoogle Scholar - Nilsrakoo W, Saejung S:
**Equilibrium problems and Moudafi's viscosity approximation methods in Hilbert spaces.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis*2010,**17**(2):195–213.MATHMathSciNetGoogle Scholar - Plubtieng S, Sriprad W:
**Hybrid methods for equilibrium problems and fixed points problems of a countable family of relatively nonexpansive mappings in Banach spaces.***Fixed Point Theory and Applications*2010,**2010:**-17.Google Scholar - Plubtieng S, Sriprad W:
**A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-20.Google Scholar - Qin X, Cho YJ, Kang SM:
**Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.***Journal of Computational and Applied Mathematics*2009,**225**(1):20–30. 10.1016/j.cam.2008.06.011MATHMathSciNetView ArticleGoogle Scholar - Tada A, Takahashi W:
**Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.***Journal of Optimization Theory and Applications*2007,**133**(3):359–370. 10.1007/s10957-007-9187-zMATHMathSciNetView ArticleGoogle Scholar - Takahashi S, Takahashi W:
**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**331**(1):506–515. 10.1016/j.jmaa.2006.08.036MATHMathSciNetView ArticleGoogle Scholar - Takahashi S, Takahashi W:
**Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(3):1025–1033. 10.1016/j.na.2008.02.042MATHMathSciNetView 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 - Takahashi W, Zembayashi Kei:
**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 - Wattanawitoon K, Kumam P:
**Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings.***Nonlinear Analysis. Hybrid Systems*2009,**3**(1):11–20. 10.1016/j.nahs.2008.10.002MATHMathSciNetView ArticleGoogle Scholar - Yao Y, Liou Y-C, Yao J-C:
**Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.***Fixed Point Theory and Applications*2007,**2007:**-12.Google 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 - 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 ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstract and Applied Analysis 2004, (3):239–249.Google Scholar
- Xu H-K:
**Another control condition in an iterative method for nonexpansive mappings.***Bulletin of the Australian Mathematical Society*2002,**65**(1):109–113. 10.1017/S0004972700020116MATHMathSciNetView ArticleGoogle Scholar - Maingé P-E:
**Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization.***Set-Valued Analysis*2008,**16**(7–8):899–912. 10.1007/s11228-008-0102-zMATHMathSciNetView ArticleGoogle Scholar - Kohsaka F, Takahashi W:
**Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.***SIAM Journal on Optimization*2008,**19**(2):824–835. 10.1137/070688717MATHMathSciNetView ArticleGoogle Scholar - Nilsrakoo W, Saejung S:
**On the fixed-point set of a family of relatively nonexpansive and generalized nonexpansive mappings.***Fixed Point Theory and Applications*2010,**2010:**-14.Google Scholar - Song Y, Zheng Y:
**Strong convergence of iteration algorithms for a countable family of nonexpansive mappings.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(7–8):3072–3082. 10.1016/j.na.2009.01.219MATHMathSciNetView ArticleGoogle Scholar - Suzuki T:
**Moudafi's viscosity approximations with Meir-Keeler contractions.***Journal of Mathematical Analysis and Applications*2007,**325**(1):342–352. 10.1016/j.jmaa.2006.01.080MATHMathSciNetView 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.