- Research Article
- Open Access

# An Algorithm for Finding a Common Solution for a System of Mixed Equilibrium Problem, Quasivariational Inclusion Problem, and Fixed Point Problem of Nonexpansive Semigroup

- M Liu
^{1}Email author, - SS Chang
^{1}Email author and - P Zuo
^{1}

**2010**:895907

https://doi.org/10.1155/2010/895907

© M. Liu et al. 2010

**Received:**31 March 2010**Accepted:**8 June 2010**Published:**15 June 2010

## Abstract

We introduce a hybrid iterative scheme for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for nonexpansive semigroup, and the set of solutions of the quasi-variational inclusion problem with multivalued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extend some recent results announced by some authors.

## Keywords

- Variational Inequality
- Nonexpansive Mapping
- Multivalued Mapping
- Maximal Monotone
- Common Fixed Point

## 1. Introduction

Throughout this paper we assume that is a real Hilbert space, and is a nonempty closed convex subset of .

In the sequel, we denote the set of fixed points of by .

*strongly positive*, if there exists a constant such that

*quasi-variational inclusion problem*(see, Chang [1, 2]) is to find an such that

A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions (see, e.g., [3]).

The set of solutions of variational inclusion (1.2) is denoted by .

Special Case

This problem is called *Hartman-Stampacchia variational inequality problem* (see, e.g., [4]). The set of solutions of (1.4) is denoted by
.

*monotone*, if for all , , and , then it implies that . A multivalued mapping is called

*maximal monotone*, if it is monotone and if for any

(the graph of mapping ) implies that .

Proposition 1.1 (see [5]).

Let be an -inverse strongly monotone mapping, then

(a) is a -Lipschitz continuous and monotone mapping;

(b)if is any constant in , then the mapping is nonexpansive, where is the identity mapping on .

Let be an equilibrium bifunction (i.e., ), and let be a real-valued function.

*mixed equilibrium problem*, that is, to find such that

*equilibrium problem*, that is, to find such that

Denote the set of solution of EP by .

On the other hand, Li et al. [7] introduced two steps of iterative procedures for the approximation of common fixed point of a nonexpansive semigroup on a nonempty closed convex subset in a Hilbert space.

Very recently, Saeidi [8] introduced a more general iterative algorithm for finding a common element of the set of solutions for a system of equilibrium problems and of the set of common fixed points for a finite family of nonexpansive mappings and a nonexpansive semigroup.

Recall that a family of mappings
is called *a nonexpansive semigroup*, if it satisfies the following conditions:

(a) for all and ;

(b) .

(c)the mapping is continuous, for each .

Motivated and inspired by Ceng and Yao [6], Li et al. [7], Saeidi [8], and [9–13], the purpose of this paper is to introduce a hybrid iterative scheme for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of the quasi-variational inclusion problem with multivalued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extend the recent results in Zhang et al. [5], S. Takahashi and W. Takahashi [14], Chang et al. [15], Ceng and Yao [6], Li et al. [7] and, Saeidi [8].

## 2. Preliminaries

In the sequel, we use and to denote the weak convergence and strong convergence of the sequence in , respectively.

Definition 2.1.

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

Proposition 2.2 (see [5]).

Definition 2.3.

A single-valued mapping
is said to be *hemicontinuous*, if for any
, the mapping
converges weakly to
(as
).

It is well known that every continuous mapping must be hemicontinuous.

Lemma 2.4 (see [16]).

Let be a real Banach space, the dual space of a maximal monotone mapping, and a hemicontinuous bounded monotone mapping with , then the mapping is a maximal monotone mapping.

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

for all ;

is monotone, that is, for all ;

for each , is concave and upper semicontinuous.

for each , is convex.

A differentiable function on a convex set is called

where
is the *Fréchet* derivative of
at
;

_{1})–(H

_{4}). Let be any given positive number. For a given point , consider the following

*auxiliary problem for*(for short, ) to find such that

*Fréchet*derivative of a functional at . Let be the mapping such that for each , is the set of solutions of , that is,

Then the following conclusion holds.

Proposition 2.5 (see [6]).

Let
be a nonempty closed convex subset of
a lower semicontinuous and convex functional. Let
be an equilibrium bifunction satisfying conditions (H_{1})–(H_{4}). Assume that

is Lipschitz continuous with constant such that

is affine in the first variable,

for each fixed , is continuous from the weak topology to the weak topology;

is -strongly convex with constant , and its derivative is continuous from the weak topology to the strong topology;

Then the following hold:

is single-valued;

is nonexpansive if is Lipschitz continuous with constant such that ;

;

is closed and convex.

Lemma 2.6 (see [17]).

Lemma 2.7 (see [7]).

Let be a nonempty bounded closed convex subset of , and let be a nonexpansive semigroup on . If is a sequence in such that and , then .

## 3. The Main Results

In order to prove the main result, we first give the following lemma.

Lemma 3.1 (see [5]).

If , then is a closed convex subset in .

In the sequel, we assume that satisfy the following conditions:

is a real Hilbert space, is a nonempty closed convex subset;

is a strongly positive linear bounded operator with a coefficient is a contraction mapping with a contraction constant , , is an -inverse-strongly monotone mapping, and is a multivalued maximal monotone mapping;

is a nonexpansive semigroup;

is a finite family of bifunctions satisfying conditions (H_{1})–(H_{4}), and
is a finite family of lower semicontinuous and convex functional;

is a finite family of Lipschitz continuous mappings with constant such that

is affine in the first variable,

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

is a finite family of -strongly convex with constant , and its derivative is not only continuous from the weak topology to the strong topology but also Lipschitz continuous with constant .

*auxiliary problem for a system of mixed equilibrium problems*:

and , is the mapping defined by (2.8).

In the sequel we denote by for and .

Theorem 3.2.

Let be the same as above. Let be a finite family of positive numbers, , and . If and the following conditions are satisfied:

, , and , then

the sequence converges strongly to some point , provided that is firmly nonexpansive;

Proof.

We observe that from condition (ii), we can assume, without loss of generality, that .

is well defined.

We divide the proof of Theorem 3.2 into 8 steps.

Step 1.

So, . This implies that is a bounded sequence in . Therefore , and are all bounded.

Step 2.

Step 3.

Since and , by condition (ii), it yields .

Step 4.

Since , and are bounded, these imply that .

Step 5.

Since , , , and are bounded, these imply that .

In fact, since , so . This together with (3.25) shows that .

Step 6.

- (a)
We first prove that . In fact, since is bounded, there exists a subsequence of such that . From Lemma 2.7 and Step 2, we obtain .

- (b)
Now we prove that .

_{2}) we know that the function and the mapping both are convex and lower semicontinuous, hence they are weakly lower semicontinuous. These together with and , we have

- (c)
Now we prove that .

- (d)
Now we prove that is the unique solution of variational inequality (3.6).

We first prove that .

Now, replacing in (3.53) with and letting and , we have .

Next we prove that is the unique solution of the variational inequality (3.6).

It follows from [18, Theorem ] that the solution of the variational inequality (3.6) is unique, that is, is a unique solution of (3.6).

Step 7.

Step 8.

Combining (3.61) and (3.69), we obtain that .

This completes the proof of Theorem 3.2.

Corollary 3.3.

Let be the same as in Theorem 3.2. Let be a finite family of positive parameter, and . If and conditions (i) and (ii) in Theorem 3.2 are satisfied, then

the sequence converges strongly to some point , provided that is firmly nonexpansive;

is the unique solution of variational inequality (3.6).

Proof.

The conclusion of Corollary 3.3 can be obtained from Theorem 3.2 immediately.

## 4. Applications to Optimization Problem

*optimization problem*:

where is the set of fixed points of in and is a potential function for (i.e., , ), where is a contractive mapping with a contractive constant . We have the following theorem.

Theorem 4.1.

and the sequence converges strongly to some point which is the unique minimal point of optimization problem (4.1).

Proof.

Since is the unique solution of (4.3), we have .

This completes the proof of Theorem 4.1.

## Declarations

### Acknowledgment

The authors would like to express their thanks to the referees for their valuable suggestions and comments.

## Authors’ Affiliations

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