- Research Article
- Open Access

# A New Hybrid Algorithm for a System of Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive Semigroup, and Variational Inclusion Problem

- Thanyarat Jitpeera
^{1}and - Poom Kumam
^{1}Email author

**2011**:217407

https://doi.org/10.1155/2011/217407

© Thanyarat Jitpeera and Poom Kumam. 2011

**Received:**14 December 2010**Accepted:**15 January 2011**Published:**23 January 2011

## Abstract

The purpose of this paper is to consider a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a -strict pseudocontraction, the set of solutions of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion problem. Strong convergence of the sequences generated by the proposed iterative scheme is obtained. The results presented in this paper extend and improve some well-known results in the literature.

## Keywords

- Equilibrium Problem
- Nonexpansive Mapping
- Real Hilbert Space
- Nonempty Closed Convex Subset
- Strong Convergence Theorem

## 1. Introduction

*system of mixed equilibrium problems*is to find such that

*mixed equilibrium problem*(see also the work of Flores-Bazán in [1]). For finding such that,

*system of equilibrium problems*. For finding such that,

*equilibrium problem*. For finding such that

The set of solution of (1.6) is denoted by EP .

The equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the (mixed) equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the equilibrium problem, see, for instance, [9–17].

We denote the set of *fixed points* of
by
, that is
.

Definition 1.1.

A family
of mappings of
into itself is called a *nonexpansive semigroup* on
if it satisfies the following conditions:

We denoted by the set of all common fixed points of , that is, . It is know that is closed and convex.

*variational inclusion problem*is to find such that

where
is the zero vecter in
. The set of solutions of problem (1.8) is denoted by
. A set-valued mapping
is called *monotone* if for all
and
imply
. A monotone mapping
is *maximal* if its graph
of
is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping
is maximal if and only if for
for all
imply
.

Definition 1.2.

Definition 1.3.

Remark 1.4.

It is obvious that any -inverse-strongly monotone mappings is monotone and -Lipschitz continuous. It is easy to see that for any constant is in , then the mapping is nonexpansive, where is the identity mapping on .

Definition 1.5.

Let be a differentiable functional on a convex set , which is called:

where is the Fréchet derivative of at ;

In particular, if
for all
, then
is said to be *strongly convex*.

Definition 1.6.

is called the *resolvent operator* associated with
, where
is any positive number and
is the identity mapping. The following characterizes the resolvent operator.

(R4) If , then the mapping is nonexpansive.

where . They proved that the sequence generated by (1.18) converges weakly to , where .

where is the resolvent operator associated with and a positive number and is a sequence in the interval . Peng et al. [25] introduced the iterative scheme by the viscosity approximation method for finding a common element of the set of solutions to the problem (1.8), the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in a Hilbert space.

They proved that under certain appropriate conditions imposed on , and , the sequence generated by (1.20) converges strongly to , where . Later, Kumam et al. [28] proved a strongly convergence theorem of the iterative sequence generated by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasinonexpansive mappings, and the set of solutions of variational inclusion problems.

Liu et al. [29] introduced a hybrid iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inverse-strongly monotone mappings. Recently, Jitpeera and Kumam [30] considered a shrinking projection method of finding the common element of the set of common fixed points for a finite family of a -strict pseudocontraction, the set of solutions of a systems of equilibrium problems and the set of solutions of variational inclusions. Then, they proved strong convergence theorems of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Very recently, Hao [18] introduced a general iterative method for finding a common element of solution set of quasi variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings.

In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a -strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.

## 2. Preliminaries

Let be a real Hilbert space and be a nonempty closed convex subset of . Recall that the (nearest point) projection from onto assigns to each the unique point in satisfying the property .

The following characterizes the projection . We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1.

Lemma 2.2 (see [20]).

Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.

Lemma 2.3 (see [31]).

Let be a closed convex subset of . Let be a bounded sequence in . Assume that

Then is weakly convergent to a point in .

Lemma 2.4 (see [32]).

Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality , hold for each with .

Lemma 2.5 (see [33]).

Each Hilbert space , satisfies the Kadec-Klee property, that is, for any sequence with and together imply .

For solving the system of mixed equilibrium problem, let us assume that function , satisfies the following conditions:

is monotone, that is, , for all ;

for each fixed , is convex and upper semicontinuous;

Lemma 2.6 (see [34]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex functional from to . Let be a bifunction from to satisfying (H1)–(H3). Assume that

(i) is Lipschitz continuous with constant such that;

(b) is affine in the first variable,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology,

(ii) is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology;

for all . Then the following hold

(2) is nonexpansive if is Lipschitz continuous with constant such that ;

Lemma 2.7 (see [35]).

Let be a -strict pseudocontraction, then

(1)the fixed point set of is closed convex so that the projection is well defined;

If , then is a nonexpansive mapping such that .

*a family of uniformly*

*-strict pseudocontractions*, if there exists a constant such that

Such a mapping is nonexpansive from to and it is called the -mapping generated by and . For each , let the mapping be defined by (2.8). Then we can have the following crucial conclusions concerning .

Lemma 2.8 (see [36]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, let be real numbers such that for every . Then, for every and , exists.

Lemma 2.9 (see [36]).

Let be a nonempty closed convex subset of a Hilbert space , be a countable family of nonexpansive mappings with , be a real sequence such that , for all . Then .

Lemma 2.10 (see [37]).

Lemma 2.11 (see [38]).

Lemma 2.12 (see [39]).

Let C be a nonempty bounded closed convex subset of H, be a sequence in C and be a nonexpansive semigroup on C. If the following conditions are satisfied:

## 3. Main Results

In this section, we will introduce an iterative scheme by using a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of -strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem in a real Hilbert space.

Theorem 3.1.

where , is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:

is -Lipschitz continuous with constant such that

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;

is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;

Then, and converge strongly to .

Proof.

Next, we will divide the proof into eight steps.

Step 1.

We first show by induction that for each .

Hence . This implies that for each .

Step 2.

Next, we show that is well defined and is closed and convex for any .

Thus is closed and convex. Then, is closed and convex for any . This implies that is well-defined.

Step 3.

Step 4.

Step 5.

Step 6.

Next, we show that and , where .

Step 7.

- (1)
- (2)

By the assumption and by condition (H1), we know that the function and the mapping both are convex and lower semicontinuous, hence they are weakly lower semicontinuous.

It follows from the maximal monotonicity of that , that is, . Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . In similar way, we can obtain , hence .

Step 8.

Finally, we show that and , where .

for all . From (3.59) and is bounded, so .

and hence in norm. Finally, noticing . We also conclude that in norm. This completes the proof.

Theorem 3.2.

where is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:

is -Lipschitz continuous with constant such that

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;

is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;

Then, and converge strongly to .

Proof.

Hence, we can obtain the desired conclusion from Theorem 3.1 immediately.

Next, we consider another class of important mappings.

Definition 3.3.

Then, is -inverse-strongly monotone mapping.

Now, we obtain the following result.

Theorem 3.4.

where , is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:

is -Lipschitz continuous with constant such that

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;

is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;

Then, and converge strongly to .

Proof.

By using Theorem 3.2, it is easy to obtain the desired conclusion.

## Declarations

### Acknowledgments

The authors would like to thank the Faculty of Science, King Monkut's University of Technology Thonburi for its financial support. Moreover, P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5380044.

## Authors’ Affiliations

## References

- Flores-Bazán F:
**Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case.***SIAM Journal on Optimization*2000,**11**(3):675–690.MATHMathSciNetView ArticleGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MATHMathSciNetGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MATHMathSciNetGoogle Scholar - Chadli O, Wong NC, Yao JC:
**Equilibrium problems with applications to eigenvalue problems.***Journal of Optimization Theory and Applications*2003,**117**(2):245–266. 10.1023/A:1023627606067MATHMathSciNetView ArticleGoogle Scholar - Chadli O, Schaible S, Yao JC:
**Regularized equilibrium problems with application to noncoercive hemivariational inequalities.***Journal of Optimization Theory and Applications*2004,**121**(3):571–596.MATHMathSciNetView ArticleGoogle Scholar - Konnov IV, Schaible S, Yao JC:
**Combined relaxation method for mixed equilibrium problems.***Journal of Optimization Theory and Applications*2005,**126**(2):309–322. 10.1007/s10957-005-4716-0MATHMathSciNetView ArticleGoogle Scholar - Moudafi A, Théra M:
**Proximal and dynamical approaches to equilibrium problems.**In*Lecture Notes in Econcmics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.Google Scholar - Zeng L-C, Wu S-Y, Yao J-C:
**Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems.***Taiwanese Journal of Mathematics*2006,**10**(6):1497–1514.MATHMathSciNetGoogle Scholar - Jitpeera T, Kumam P:
**An extra gradient 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: Theory & Applications*2010,**1**(1):71–91.MathSciNetGoogle Scholar - Cianciaruso F, Marino G, Muglia L, Yao Y:
**A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem.***Fixed Point Theory and Applications*2010,**2010:**-19.Google Scholar - Cholamjiak P, Suantai S:
**A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings.***Fixed Point Theory and Applications*2009,**2009:**-20.Google Scholar - Jaiboon C, Kumam P:
**Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities.***Journal of Inequalities and Applications*2010,**2010:**-43.Google Scholar - Jaiboon C, Chantarangsi W, Kumam P:
**A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings.***Nonlinear Analysis. Hybrid Systems*2010,**4**(1):199–215. 10.1016/j.nahs.2009.09.009MATHMathSciNetView ArticleGoogle Scholar - Kumam P, Jaiboon C:
**A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems.***Nonlinear Analysis. Hybrid Systems*2009,**3**(4):510–530. 10.1016/j.nahs.2009.04.001MATHMathSciNetView ArticleGoogle Scholar - Shehu Y:
**Fixed point solutions of variational inequality and generalized equilibrium problems with applications.***Annali dell'Universita di Ferrara*2010,**56**(2):345–368. 10.1007/s11565-010-0102-4MATHMathSciNetView ArticleGoogle Scholar - Yao Y, Liou Y-C, Yao J-C:
**A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems.***Fixed Point Theory and Applications*2008,**2008:**-15.Google Scholar - Yao Y, Liou Y-C, Wu Y-J:
**An extragradient method for mixed equilibrium problems and fixed point problems.***Fixed Point Theory and Applications*2009,**2009:**-15.Google Scholar - Hao Y:
**On variational inclusion and common fixed point problems in Hilbert spaces with applications.***Applied Mathematics and Computation*2010,**217**(7):3000–3010. 10.1016/j.amc.2010.08.033MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Yao JC:
**Iterative schemes for solving mixed variational-like inequalities.***Journal of Optimization Theory and Applications*2001,**108**(3):527–541. 10.1023/A:1017531323904MATHMathSciNetView ArticleGoogle Scholar - Brézis H:
**Opérateur maximaux monotones.**In*Mathematics Studies*.*Volume 5*. North-Holland, Amsterdam, The Netherlands; 1973.Google Scholar - Lemaire B:
**Which fixed point does the iteration method select?**In*Recent Advances in Optimization, Lecture Notes in Economics and Mathematical Systems*.*Volume 452*. Springer, Berlin, Germany; 1997:154–167. 10.1007/978-3-642-59073-3_11Google Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MATHMathSciNetView 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 - Zhang S-S, Lee JHW, Chan CK:
**Algorithms of common solutions to quasi variational inclusion and fixed point problems.***Applied Mathematics and Mechanics. English Edition*2008,**29**(5):571–581. 10.1007/s10483-008-0502-yMATHMathSciNetView ArticleGoogle 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 - Saeidi S:
**Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings.***Nonlinear Analysis. Theory, Methods & Applications*2009,**70**(12):4195–4208. 10.1016/j.na.2008.09.009MATHMathSciNetView ArticleGoogle Scholar - Katchang P, Kumam P:
**A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems.***Journal of Inequalities and Applications*2010,**2010:**-25.Google Scholar - Kumam W, Jaiboon C, Kumam P, Singta A:
**A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings.***Journal of Inequalities and Applications*2010,**2010:**-25.Google Scholar - Liu M, Chang SS, Zuo P:
**An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansive semigroup.***Journal of Inequalities and Applications*2010,**2010:**-23.Google Scholar - Jitpeera T, Kumam P: A new hybrid algorithm for a system of equilibrium problems and variational inclusion. Annali dell'Universita di Ferrara. In pressGoogle Scholar
- Acedo GL, Xu H-K:
**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 - 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/S0002-9904-1967-11761-0MATHMathSciNetView ArticleGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle 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 - Zhou Haiyun:
**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 - 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 - Chang SS:
*Variational Inequalities and Related Problems*. Chongqing Publishing House, China; 2007.Google Scholar - Shimizu T, Takahashi W:
**Strong convergence to common fixed points of families of nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1997,**211**(1):71–83. 10.1006/jmaa.1997.5398MATHMathSciNetView ArticleGoogle Scholar - Tan K-K, Xu HK:
**The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1992,**114**(2):399–404. 10.1090/S0002-9939-1992-1068133-2MATHMathSciNetView ArticleGoogle Scholar

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