- Research Article
- Open Access

# Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces

- Somyot Plubtieng
^{1}Email author and - Sukanya Chornphrom
^{1}

**2010**:296759

https://doi.org/10.1155/2010/296759

© Somyot Plubtieng and Sukanya Chornphrom. 2010

**Received: **30 June 2010

**Accepted: **13 December 2010

**Published: **21 December 2010

## Abstract

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with the work of S. Takahashi and W. Takahashi (2007) and Iemoto and Takahashi (2009).

## Keywords

- Hilbert Space
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme
- Common Element

## 1. Introduction

The set of solution of (1.1) is denoted by . Given a mapping , let for all . Then, if and only if for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); see, for example, [1–9] and the references therein.

*nonexpansive*if for all , and a mapping is said to be

*firmly*

*nonexpansive*if for all . Let be a smooth, strictly convex and reflexive Banach space, and let be the duality mapping of and a nonempty closed convex subset of . A mapping is said to be

*nonspreading*if

for all
. Let
be the set of fixed points of
, and
nonempty; a mapping
is said to be *quasi-nonexpansive* if
for all
and
.

Remark 1.1.

In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and that if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [10, 11].

for all , where and is a sequence of . Strong convergence of this type iterative sequence has been widely studied: Wittmann [15] discussed such a sequence in a Hilbert space.

Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, [1, 2, 6, 7, 17–20] and the references therein. In 1997, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Recently, S. Takahashi and W. Takahashi [8] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let be a nonexpansive mapping. In 2008, Plubtieng and Punpaeng [7] introduced a new iterative sequence for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space which is the optimality condition for the minimization problem. Very recently, S. Takahashi and W. Takahashi [9] introduced an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets.

In this paper, motivated by S. Takahashi and W. Takahashi [8] and Lemoto and Takahashi [16], we introduce an iterative sequence and prove a strong convergence theorem for finding solution of equilibrium problems and the set of fixed points of a nonspreading mapping in Hilbert spaces.

## 2. Preliminaries

holds for every with ; see [21, 22] for more details.

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 (see [23]).

Lemma 2.2 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then the following are equivalent.

(1)There exists such that is bounded;

Lemma 2.3 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then is closed and convex.

Lemma 2.4.

Let be a real Hilbert space. Then for all ,

Lemma 2.5 (see [24]).

Let , and let be sequences of real numbers such that

Lemma 2.6 (see [16]).

Let be a Hilbert space, a closed convex subset of , and a nonspreading mapping with . Then is demiclosed, that is, and imply .

Lemma 2.7 (see [16]).

Lemma 2.8 (see [25]).

where is a sequence in and is a sequence in such that

For solving the equilibrium problems for a bifunction , let us assume that satisfies the following conditions:

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

The following lemma appears implicitly in [26].

Lemma 2.9 (see [26]).

The following lemma was also given in [4].

Lemma 2.10 (see [4]).

for all . Then, the following hold:

(2) is firmly nonexpansive, that is, for any , ;

Lemma 2.11 (see [27]).

## 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a nonspreading mapping and the set of solutions of the equilibrium problems.

Theorem 3.1.

Then converges strongly to , where .

Proof.

for all . Put . We divide the proof into several steps.

Step 1.

By induction, we obtain that for all . So, is bound. Hence, , , and are also bounded.

Step 2.

Step 3.

Step 4.

Putting , we claim that the sequence converges strongly to . Indeed, we discuss two possible cases.

Case 1.

By (3.28) and , we immediately deduce by Lemma 2.8 that .

Case 2.

This completes the proof.

As direct consequences of Theorem 3.1, we obtain corollaries.

Corollary 3.2.

## Declarations

### Acknowledgments

The authors would like to thank the referees for the insightful comments and suggestions. Moreover, the authors gratefully acknowledge the Thailand Research Fund Master Research Grants (TRF-MAG, MRG-WII515S029) for funding this paper.

## Authors’ Affiliations

## References

- Chang S-S, Joseph Lee HW, Chan CK:
**A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization.***Nonlinear Analysis Theory: Methods & Applications*2009,**70**(9):3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleMATHGoogle Scholar - Colao V, Marino G, Xu H-K:
**An iterative method for finding common solutions of equilibrium and fixed point problems.***Journal of Mathematical Analysis and Applications*2008,**344**(1):340–352. 10.1016/j.jmaa.2008.02.041MathSciNetView ArticleMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*1994,**63:**123–145.Google Scholar - Peng J-W, Wang Y, Shyu DS, Yao J-C:
**Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems.***Journal of Inequalities and Applications*2008,**2008:**-15.Google Scholar - Plubtieng S, Punpaeng R:
**A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings.***Applied Mathematics and Computation*2008,**197**(2):548–558. 10.1016/j.amc.2007.07.075MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Punpaeng R:
**A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**336**(1):455–469. 10.1016/j.jmaa.2007.02.044MathSciNetView ArticleMATHGoogle 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.2006.08.036MathSciNetView ArticleMATHGoogle 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.2008.02.042MathSciNetView ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W:
**Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces.***Archiv der Mathematik*2008,**91**(2):166–177. 10.1007/s00013-008-2545-8MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Sombut K:
**Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space.***Journal of Inequalities and Applications*2010,**2010:**-12.Google Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Iemoto S, Takahashi W:
**Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(12):e2082-e2089. 10.1016/j.na.2009.03.064MathSciNetView ArticleMATHGoogle 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 - 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, Thammathiwat T:
**A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities.***Journal of Global Optimization*2010,**46**(3):447–464. 10.1007/s10898-009-9448-5MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Thammathiwat T: A viscosity approximation method for finding a common solution of fixed points and equilibrium problems in Hilbert spaces. Journal of Global Optimization. In pressGoogle Scholar
- Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Osilike MO, Igbokwe DI:
**Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations.***Computers & Mathematics with Applications*2000,**40**(4–5):559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Blum E, Oettli W:
**From optimization and variationnal inequalities to equilibrium problems.***Mathematics Students*2005,**6:**117–136.Google 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-zMathSciNetView ArticleMATHGoogle 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.