- Research Article
- Open Access

# Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions

- Prasit Cholamjiak
^{1}and - Suthep Suantai
^{1, 2}Email author

**2011**:941090

https://doi.org/10.1155/2011/941090

© P. Cholamjiak and S. Suantai. 2011

**Received:**18 October 2010**Accepted:**27 December 2010**Published:**4 January 2011

## Abstract

We introduce a new iterative algorithm for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions.

## Keywords

- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space
- Common Fixed Point
- Convex Banach Space

## 1. Introduction

The solutions set of (1.2) is denoted by . If , then the solutions set of (1.2) is denoted by , and if , then the solutions set of (1.2) is denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and the Nash equilibrium problem in noncooperative games; see also [1, 2]. Some methods have been constructed to solve the system of equilibrium problems (see, e.g., [3–7]). Recall that a mapping is said to be

It is easy to see that if
is *α*-inverse-strongly monotone, then
is monotone and
-Lipschitz.

For solving the equilibrium problem, let us assume that satisfies the following conditions:

(A1) for all ,

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

(A3)for each , ,

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

*-strict pseudocontraction*if there exists a constant such that

It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. It is also known that every -strict pseudocontraction is -Lipschitz; see [8].

where . If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence defined by (1.7) converges weakly to a fixed point of (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [10]).

In 1967, Browder and Petryshyn [11] introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann iterative algorithm (1.7) with a constant sequence for all . Recently, Marino and Xu [8] and Zhou [12] extended the results of Browder and Petryshyn [11] to Mann's iteration process (1.7). Since 1967, the construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors (see, e.g., [13–22]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping, a bifunction, and let be an inverse-strongly monotone mapping.

where and . He proved that the sequence generated by (1.8) converges weakly to an element in under suitable conditions.

Due to the weak convergence, recently, S. Takahashi and W. Takahashi [24] introduced another modification iterative method of (1.8) for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space. To be more precise, they proved the following theorem.

Theorem 1.1 (see [24]).

*α*-inverse-strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Let and , and let and be sequences generated by

where , and satisfy

(i) and ,

(ii) ,

(iii) ,

(iv) .

Then, converges strongly to .

Recently, Yao et al. [25] introduced a new modified Mann iterative algorithm which is different from those in the literature for a nonexpansive mapping in a real Hilbert space. To be more precise, they proved the following theorem.

Theorem 1.2 (see [25]).

Suppose that the following conditions are satisfied:

(i) and ,

(ii) ,

then, the sequence generated by (1.10) strongly converges to a fixed point of .

We know the following crucial lemmas concerning the equilibrium problem in Hilbert spaces.

Lemma 1.3 (see [1]).

Lemma 1.4 (see [26]).

Then, the following statements hold:

(1) is single-valued,

(3) ,

(4) is closed and convex.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let for each . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let be a countable family of -strict pseudocontractions. For each , denote the mapping by , where is the mapping defined as in Lemma 1.4.

where , and .

In this paper, we first prove a path convergence result for a nonexpansive mapping and a system of generalized equilibrium problems. Then, we prove a strong convergence theorem of the iteration process (1.14) for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the main results obtained by Yao et al. [25] in several aspects.

## 2. Preliminaries

for all and .

In the sequel, we need the following lemmas.

Let be a real uniformly convex Banach space, and let be a nonempty, closed, and convex subset of , and let be a nonexpansive mapping such that , then is demiclosed at zero.

Lemma 2.2 (see [29]).

where satisfies conditions: . If , then as .

Lemma 2.3 (see [30]).

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

(a) ; (b) or .Then, .

Lemma 2.4 (see [31]).

*α*-inverse-strongly monotone, and let be a constant. Then, we have

for all . In particular, if , then is nonexpansive.

*-condition*[32] if for each bounded subset of ,

Lemma 2.5 (see [32]).

The following results can be found in [33, 34].

Let be a closed, and convex subset of a Hilbert space . Suppose that is a family of -strictly pseudocontractive mappings from into with and is a real sequence in such that . Then, the following conclusions hold:

(1) is a -strictly pseudocontractive mapping,

(2) .

Lemma 2.7 (see [34]).

where is a family of nonnegative numbers satisfying

(i) for all ,

(ii) for all ,

(iii) .

Then,

(1)Each is a -strictly pseudocontractive mapping.

(2) satisfies -condition.

then and .

In the sequel, we will write satisfies the -condition if satisfies the -condition and is defined by Lemma 2.5 with .

## 3. Path Convergence Results

Theorem 3.1.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping. Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let . For each , let the mapping be defined by (3.1). Assume that . For each , let the net be generated by (3.3). Then, as , the net converges strongly to an element in .

Proof.

as since is bounded.

as .

Since is bounded, without loss of generality, we may assume that converges weakly to . Applying Lemma 2.1 to (3.21), we can conclude that .

Since , we have as . By using the same argument as in the proof of Theorem 3.1 of [25], we can show that as . This completes the proof.

## 4. Strong Convergence Results

Theorem 4.1.

Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings and let be a countable family of -strict pseudocontractions for some such that . Assume that , , and for each satisfy the following conditions:

(i) and ,

(ii) .

Suppose that satisfies the -condition. Then, generated by (1.14) converges strongly to an element in .

Proof.

Hence, is nonexpansive for each and so is .

Hence, by induction, is bounded and so are and .

for each .

By (i) and (4.24), it follows from Lemma 2.3 that . This completes the proof.

As a direct consequence of Lemmas 2.6 and 2.7 and Theorem 4.1, we obtain the following result.

Theorem 4.2.

where and are real sequences in which satisfy (i)-(ii) of Theorem 4.1 and is a real sequence which satisfies (i)–(iii) of Lemma 2.7. Then, converges strongly to an element in .

Remark 4.3.

Theorems 4.1 and 4.2 extend the main results in [25] from a nonexpansive mapping to an infinite family of strict pseudocontractions and a system of generalized equilibrium problems.

Remark 4.4.

If we take and for each , then Theorems 3.1, 4.1, and 4.2 can be applied to a system of equilibrium problems and to a system of variational inequality problems, respectively.

Remark 4.5.

Let be an infinite family of nonexpansive mappings of into itself, and let be real numbers such that for all . Moreover, let and be the -mappings [35] generated by and and and . Then, we know from [7, 35] that satisfies the -condition. Therefore, in Theorem 4.1, the mapping can be also replaced by .

## Declarations

### Acknowledgments

The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund. The first author is supported by the Royal Golden Jubilee Grant no. PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.

## Authors’ Affiliations

## References

- Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The 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, Lecture Notes in Economics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar - Colao V, Acedo GL, Marino G:
**An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(7–8):2708–2715. 10.1016/j.na.2009.01.115MATHMathSciNetView ArticleGoogle Scholar - Cholamjiak P, Suantai S:
**Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces.***Abstract and Applied Analysis*2010,**2010:**-17.Google Scholar - Jaiboon C:
**The hybrid steepest descent method for addressing fixed point problems and system of equilibrium problems.***Thai Journal of Mathematics*2010,**8:**275–292.MATHMathSciNetGoogle Scholar - Jitpeera T, Kumam P:
**An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings.***Journal of Nonlinear Analysis and Optimization*2010,**1:**71–91.MathSciNetGoogle Scholar - Peng J-W, Yao J-C:
**A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(12):6001–6010. 10.1016/j.na.2009.05.028MATHMathSciNetView 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 - 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)90024-6MATHMathSciNetView ArticleGoogle 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 - Zhou H:
**Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**343**(1):546–556. 10.1016/j.jmaa.2008.01.045MATHMathSciNetView ArticleGoogle Scholar - Acedo GL, Xu HK:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MATHMathSciNetView ArticleGoogle Scholar - Ceng LC, Shyu DS, Yao JC:
**Relaxed composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mappings.***Fixed Point Theory and Applications*2009,**2009:**-16.Google Scholar - Ceng L-C, Petruşel A, Yao J-C:
**Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings.***Applied Mathematics and Computation*2009,**209**(2):162–176. 10.1016/j.amc.2008.10.062MATHMathSciNetView ArticleGoogle Scholar - Ceng L-C, Al-Homidan S, Ansari QH, Yao J-C:
**An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings.***Journal of Computational and Applied Mathematics*2009,**223**(2):967–974. 10.1016/j.cam.2008.03.032MATHMathSciNetView ArticleGoogle Scholar - Cholamjiak P, Suantai S:
**Weak convergence theorems for a countable family of strict pseudocontractions in banach spaces.***Fixed Point Theory and Applications*2010,**2010:**-16.Google Scholar - Peng J-W, Yao J-C:
**Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping.***Journal of Global Optimization*2010,**46**(3):331–345. 10.1007/s10898-009-9428-9MATHMathSciNetView ArticleGoogle Scholar - Zhang Y, Guo Y:
**Weak convergence theorems of three iterative methods for strictly pseudocontractive mappings of Browder-Petryshyn type.***Fixed Point Theory and Applications*2008,**2008:**-13.Google Scholar - Zhang H, Su Y:
**Convergence theorems for strict pseudo-contractions in**-uniformly smooth Banach spaces.*Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4572–4580. 10.1016/j.na.2009.03.033MATHMathSciNetView ArticleGoogle Scholar - Zhou H:
**Convergence theorems for**-strict pseudo-contractions in 2-uniformly smooth Banach spaces.*Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(9):3160–3173. 10.1016/j.na.2007.09.009MATHMathSciNetView ArticleGoogle Scholar - Zhou HY:
**Convergence theorems for**-strict pseudo-contractions in -uniformly smooth Banach spaces.*Acta Mathematica Sinica*2010,**26**(4):743–758. 10.1007/s10114-010-7341-2MATHMathSciNetView ArticleGoogle Scholar - Moudafi A:
**Weak convergence theorems for nonexpansive mappings and equilibrium problems.***Journal of Nonlinear and Convex Analysis*2008,**9**(1):37–43.MATHMathSciNetGoogle 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 - Yao Y, Liou YC, Marino G:
**Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-7.Google Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MATHMathSciNetGoogle Scholar - Browder FE:
**Nonexpansive nonlinear operators in a Banach space.***Proceedings of the National Academy of Sciences of the United States of America*1965,**54:**1041–1044. 10.1073/pnas.54.4.1041MATHMathSciNetView ArticleGoogle Scholar - Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar - Nadezhkina N, Takahashi W:
**Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2006,**128**(1):191–201. 10.1007/s10957-005-7564-zMATHMathSciNetView ArticleGoogle Scholar - Aoyama K, Kimura Y, Takahashi W, Toyoda M:
**Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(8):2350–2360. 10.1016/j.na.2006.08.032MATHMathSciNetView ArticleGoogle Scholar - Boonchari D, Saejung S:
**Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings.***Journal of Computational and Applied Mathematics*2009,**233**(4):1108–1116. 10.1016/j.cam.2009.09.007MATHMathSciNetView ArticleGoogle Scholar - Boonchari D, Saejung S:
**Construction of common fixed points of a countable family of**-demicontractive mappings in arbitrary Banach spaces.*Applied Mathematics and Computation*2010,**216**(1):173–178. 10.1016/j.amc.2010.01.027MATHMathSciNetView ArticleGoogle Scholar - Shimoji K, Takahashi W:
**Strong convergence to common fixed points of infinite nonexpansive mappings and applications.***Taiwanese Journal of Mathematics*2001,**5**(2):387–404.MATHMathSciNetGoogle 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.