- 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:

(1) , for all ;

(2) , for all ;

(3) , for all and ;

(4)for all , is continuous.

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.

*inverse-strongly monotone*if there exists a constant with the property

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.

(R5) is closed and convex.

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.

For a given , , , for all .

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

(1)the weak -limit set ,

(2)for each , exists.

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;

for each fixed , is convex.

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;

(a) , for all ,

(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

(1) is single-valued;

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

(3) ;

(4) is closed and convex.

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:

(1) ;

(2) , then .

## 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

(a) , for all ,

(b) is affine,

(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 ;

, for some ;

, for some ;

, for some ;

, for each .

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.

Next, we show that and .

Step 5.

Step 6.

Next, we show that and , where .

Step 7.

Next, we show that .

- (1)
First, we prove that . Indeed, from Lemma 2.12 and (3.47), we get , that is, , for all .

- (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.

- (4)
Lastly, we prove that .

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

(a) , for all ,

(b) is affine,

(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 ;

, for some ;

, for some ;

, for some ;

, for each .

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

(a) , for all ,

(b) is affine,

(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 ;

, for some ;

, for some ;

, for some ;

, for each

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

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