- Research Article
- Open Access

# Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping

- Atid Kangtunyakarn
^{1}Email author

**2011**:274820

https://doi.org/10.1155/2011/274820

© Atid Kangtunyakarn. 2011

**Received:**8 November 2010**Accepted:**14 December 2010**Published:**20 December 2010

## Abstract

The purpose of this paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method.

## Keywords

- Hilbert Space
- Variational Inequality
- Equilibrium Problem
- Monotone Mapping
- Nonexpansive Mapping

## 1. Introduction

In the case of , is denoted by . In the case of , is also denoted by . Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of (1.3); see, for instance, [1–3].

*inverse strongly monotone mapping*, see [4], if there exists a positive real number such that

for all . The following definition is well known.

Definition 1.1.

for all .

*mapping*. is

*strong pseudocontraction*if there exists a positive constant such that is pseudocontraction. In a real Hilbert space , (1.6) is equivalent to

The class of -strict pseudocontractions falls into the one between classes of nonexpansive mappings, and the pseudocontraction mappings, and the class of strong pseudocontraction mappings is independent of the class of -strict pseudocontraction.

converges weakly to a fixed point of provided the control sequence satisfies the conditions that for all and . In 1974, S. Ishikawa proved the following strong convergence theorem of pseudocontractive mapping.

Theorem 1.2 (see [9]).

where , are two real sequences in satisfying

(i) ,

(ii) ,

(iii) .

Then converges strongly to a fixed point of .

In order to prove a strong convergence theorem of Mann algorithm (1.12) associated with strictly pseudocontractive mapping, in 2006, Marino and Xu [7] proved the following theorem for strict pseudocontractive mapping in Hilbert space by using method.

Theorem 1.3 (see [7]).

Under suitable conditions of and , they proved that the sequence defined by (1.15) converges strongly to .

Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15]. The motivation of (1.14), (1.15), and the research in this direction, we prove the strong convergence theorem for finding solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.

## 2. Preliminaries

The following characterizes the projection .

Lemma 2.1 (see [5]).

The following lemma is well known.

Lemma 2.2.

Let be Hilbert space, and let be a nonempty closed convex subset of . Let be -strictly pseudocontractive, then the fixed point set of is closed and convex so that the projection is well defined.

Lemma 2.3 ((demiclosedness principle) (see [16]).

If is a -strict pseudocontraction on closed convex subset of a real Hilbert space , then is demiclosed at any point .

To solve the equilibrium problem for a bifunction , assume that satisfies the following conditions:

() for all ,

() is monotone, that is, ,

()for all ,

()for all is convex and lower semicontinuous.

The following lemma appears implicitly in [1].

Lemma 2.4 (see [1]).

for all .

Lemma 2.5 (see [11]).

for all . Then, the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is,

(3) ;

(4) is closed and convex.

Lemma 2.6 (see [17]).

then , as .

Lemma 2.7 (see [7]).

for all .

## 3. Main Result

Theorem 3.1.

*α*-inverse strongly monotone mapping, and let be a inverse strongly monotone mapping. Let be a strict pseudocontraction mapping with . Let be a sequence generated by and

where is sequence in , , and satisfy the following conditions:

(i) ,

(ii) .

Then converges strongly to .

Proof.

By Lemma 2.5, we have . By the same argument as above, we conclude that .

Then, we have that is convex. By Lemmas 2.5 and 2.2, we conclude that is closed and convex. This implies that is well defined. Next, we show that for every .

From (3.50), (3.52), and (3.53), we have . Hence . Therefore, by (3.12) and Lemma 2.6, we have that converges strongly to . The proof is completed.

## 4. Applications

By using our main result, we have the following results in Hilbert spaces.

Theorem 4.1.

where is sequence in , , and satisfy the following conditions:

(i) ,

(ii) .

Then converges strongly to .

Proof.

Putting in Theorem 3.1, we have the desired conclusions.

Theorem 4.2.

*α*-inverse strongly monotone mapping, and let be a sequence generated by and

where is sequence in , , and , for all . Then converges strongly to .

Proof.

Putting , , and , for all , in Theorem 3.1, we have the desired conclusions.

## Authors’ Affiliations

## References

- Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***Mathematics Student*1994,**63**(1–4):123–145.MATHMathSciNetGoogle Scholar - Moudafi A, Théra M:
**Proximal and dynamical approaches to equilibrium problems.**In*Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View 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 - Iiduka H, Takahashi W:
**Weak convergence theorems by Cesáro means for nonexpansive mappings and inverse-strongly-monotone mappings.***Journal of Nonlinear and Convex Analysis*2006,**7**(1):105–113.MATHMathSciNetGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis. Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MATHMathSciNetView ArticleGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MATHMathSciNetView ArticleGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MATHMathSciNetView ArticleGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MATHMathSciNetView ArticleGoogle Scholar - Tanga Y-C, Penga J-G, Liu L-W:
**Strong convergence theorem for pseudo-contractive mappings in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2011,**74:**380–385. 10.1016/j.na.2010.08.048View ArticleMathSciNetGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MATHMathSciNetGoogle 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 - Kangtunyakarn A, Suantai S:
**Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings.***Nonlinear Analysis. Hybrid Systems*2009,**3**(3):296–309. 10.1016/j.nahs.2009.01.012MATHMathSciNetView 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 - 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.2008.02.041MATHMathSciNetView ArticleGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MATHMathSciNetView ArticleGoogle Scholar - 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

## 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.