- Research Article
- Open Access

# A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of Quasi-Nonexpansive Mappings

- Wiyada Kumam
^{1, 2}, - Chaichana Jaiboon
^{3}Email author, - Poom Kumam
^{2, 4}and - Akarate Singta
^{1}

**2010**:458247

https://doi.org/10.1155/2010/458247

© Wiyada Kumam et al. 2010

**Received:**21 March 2010**Accepted:**29 June 2010**Published:**14 July 2010

## Abstract

The purpose of this paper is to consider a 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 quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S.Takahashi and W. Takahashi (2008), and many others.

## Keywords

- Equilibrium Problem
- Nonexpansive Mapping
- Real Hilbert Space
- Variational Inequality Problem
- Nonempty Closed Convex Subset

## 1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively.

Recall that the following definitions.

(1)A mapping
is said to be *nonexpansive* if

We denote be the set of fixed points of .

*variational inclusion problem*is to find such that

where is the zero vector in . The set of solutions of problem (1.3) is denoted by .

Definition 1.1.

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

Remark 1.2.

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

A set-valued mapping
is called *monotone* if for all
, and
implying
. 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 1.3.

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

Remark 1.4.

(R2)The resolvent operator is 1-inverse strongly monotone; see [1], that is,

(R3)The solution of problem (1.3) is a fixed point of the operator for all ; see also [2], that is,

(R4)If , then the mapping is nonexpansive.

(R5) is closed and convex.

Let
be a nonlinear mapping, let
be a real-valued function and
a bifunction from
to
. We consider the following *generalized mixed equilibrium problem*.

It is easy to see that is solution of problem (1.9) implies that .

(i)In the case of
(:the zero mapping), then the generalized mixed equilibrium problem (1.9) is reduced to *the mixed equilibrium problem*. Finding
such that

The set of solution of (1.11) isdenoted by

*the generalized equilibrium problem*. Finding such that

The set of solution of (1.12) is denoted by

*the equilibrium problem*. Finding such that

The set of solution of (1.13) is denoted by

*the variational inequality problem*. Finding such that

The set of solution of (1.14) is denoted by

The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the generalized mixed equilibrium problems, generalized equilibrium problems and equilibrium problems; see, for instance, [9–22].

where . They proved that the sequence generated by (1.15) converges weakly to , where

They proved that under certain appropriate conditions imposed on , and , the sequence generated by (1.16) converges strongly to

where is the resolvent operator associated with and a positive number is a sequence in the interval .

They proved that under certain appropriate conditions imposed on and , the sequence generated by (1.18) converges strongly to

In 2010, Katchang and Kumam [27] introduced an iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings, and inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space.

In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme 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 quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.

## 2. Preliminaries

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1.

Lemma 2.2 (see [1]).

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 [28]).

Let be a closed convex subset of and 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 [29]).

holds for each with .

Lemma 2.5 (see [30]).

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

For solving the generalized equilibrium problems, let us give the following assumptions for and the set :

(A1) for all

(A2) is monotone, that is, for all

(A3)for each is weakly upper semicontinuous;

(A4)for each is convex and lower semicontinuous;

(B1)for each and there exists a bounded subset and such that for any

(B2)C is bounded set.

Lemma 2.6 (see [31]).

Assume that either or holds. Then, the following conclusions hold:

(1)for each

(2) is single-valued;

(3) is firmly nonexpansive, that is, for any

(4)

(5) is closed and convex.

Remark 2.7.

## 3. Main Results

In this section, we will introduce an iterative scheme by using 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 quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space.

Such a mapping
is called the *K-mapping* generated by
and
; see [32].

We have the following crucial Lemma 3.1 and Lemma 3.2 concerning -mapping which can be found in [14]. Now we only need the following similar version in Hilbert spaces.

Lemma 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasi-nonexpansive mappings and -Lipschitz mappings of into itself with and let be real numbers such that for every , and . Let be the K-mapping generated by and . Then, the followings hold:

(1) is quasi-nonexpansive and Lipschitz,

(2) .

Lemma 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasi-nonexpansive mappings and -Lipschitz mappings of into itself and sequences in such that Moreover, for every , let and be the K-mappings generated by and , and and , respectively. Then, for every , we have

Now we study the strong convergence theorem concerning the shrinking projection method.

Theorem 3.3.

where satisfy the following conditions:

(i) for some with ;

(ii) , for some with ;

(iii) for some with .

Then, and converge strongly to

Proof.

Next, we will divide the proof into six steps.

Step 1.

We first 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 2.

Next, we show by induction that for each .

It follows that This implies that for each .

Step 3.

Next, we show that and .

Step 4.

Next, we show that

Step 5.

Next, we show that

- (a)
First, we prove that .

- (b)

- (c)
Now, we prove that .

which is a contradiction. Thus, we get .

The conclusion is .

Step 6.

Finally, we show that and , where

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

Corollary 3.4.

Let be the -mapping generated by and . Let , , , and be sequences generated by (3.3) satisfying the following conditions in Theorem 3.3. Then, and converge strongly to

From Theorem 3.3, we can obtain the following results.

Theorem 3.5.

where satisfy the following conditions:

(i) for some with ;

(ii) , for some with ;

(iii) for some with .

Then, and converge strongly to

Proof.

We can obtain the desired conclusion from Theorem 3.3 immediately.

Next, we consider another class of important nonlinear mappings: strict pseudocontractions.

Definition 3.6.

If , then is nonexpansive.

This shows that is -inverse-strongly monotone mapping.

Now, we get the following result.

Theorem 3.7.

where satisfy the following conditions:

(i) for some with ;

(ii) , for some with ;

(iii) for some with .

Then, and converge strongly to

Proof.

By using Theorem 3.5, it is easy to obtain the desired conclusion.

## Declarations

### Acknowledgments

The authors would like to express their thank to the referees for helpful suggestions. The first author was supported by the National Research Council of Thailand and the Faculty of Science and Technology RMUTT Research Fund. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute. The third author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044.

## Authors’ Affiliations

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