- Research Article
- Open Access

# A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings

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

**2009**:350979

https://doi.org/10.1155/2009/350979

© P. Cholamjiak and S. Suantai. 2009

**Received:**12 June 2009**Accepted:**28 September 2009**Published:**13 October 2009

## Abstract

We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of -mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.

## Keywords

- Equilibrium Problem
- Monotone Mapping
- Nonexpansive Mapping
- Lipschitz Mapping
- Maximal Monotone

## 1. Introduction

Let be a real Hilbert space with inner product and inducted norm , and let be a nonempty closed and convex subset of . Then, a mapping is said to be

(1)*nonexpansive* if
, for all
;

(2)*quasi-nonexpansive* if
, for all
and
;

(3)
*-Lipschitzian* if there exists a constant
such that
, for all
. We denoted by
the set of fixed points of
.

where the initial point is taken in arbitrarily and is a sequence in .

However, we note that Mann's iteration process (1.1) has only weak convergence, in general; for instance, see [2, 3].

Many authors attempt to modify the process (1.1) so that strong convergence is guaranteed that has recently been made. Nakajo and Takahashi [4] proposed the following modification which is the so-called CQ method and proved the following strong convergence theorem for a nonexpansive mapping in a Hilbert space .

Theorem 1.1 (see [4]).

where . Then, converges strongly to .

The set of solutions of (1.3) is denoted by ; see also [5–7].

The set of solutions (1.5) is denoted by . Problem (1.5) includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem; see [9–12] and the reference cited therein.

Recently, Tada and Takahashi [13] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain the following theorem.

Theorem 1.2 (see [13]).

where and . Then, converges strongly to .

*-inverse strongly monotone*if there exists a constant such that

*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 . We define the resolvent operator associated with and as follows:

It is known that the resolvent operator is single-valued, nonexpansive, and 1-inverse strongly monotone; see [14], and that a solution of problem (1.7) is a fixed point of the operator for all ; see also [15]. If , it is easy to see that is a nonexpansive mapping; consequently, is closed and convex.

The equilibrium problems, generalized equilibrium problems, variational inequality problems, and variational inclusions have been intensively studied by many authors; for instance, see [8, 16–43].

Motivated by Tada and Takahashi [13] and Peng et al. [7], we introduce a new approximation scheme for finding a common element of the set of fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings, the set of solutions of a generalized equilibrium problem, and the set of solutions of a variational inclusion with set-valued maximal monotone and inverse strongly monotone mappings in the framework of Hilbert spaces.

## 2. Preliminaries and Lemmas

*metric projection*of on to . It is also known that for and , is equivalent to for all . Furthermore

for all and .

Lemma 2.1 (see [45]).

For solving the generalized equilibrium problem, 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;

(A5) for each , is lower semicontinuous;

(B2) is a bounded set.

Lemma 2.2 (see [7]).

Assume that either (B1) or (B2) holds. Then, the following conclusions hold:

(1)for each , ;

(2) is single-valued;

(4) ;

(5) is closed and convex.

Lemma 2.3 (see [14]).

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

Lemma 2.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a quasi-nonexpansive and -Lipschitz mapping of into itself. Then, is closed and convex.

Proof.

Since is -Lipschitz, it is easy to show that is closed.

which implies ; consequently, is convex. This completes the proof.

Lemma 2.5 (see [46]).

for all and , then .

where is a finite mapping of into itself and for all with .

Such a mapping
is called the
*-mapping* generated by
and
; see also [48–50]. Throughout this paper, we denote
.

Next, we prove some useful lemmas concerning the -mapping.

Lemma 2.6.

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

(i) is quasi-nonexpansive and Lipschitz;

(ii) .

This shows that is a quasi-nonexpansive mapping.

- (ii)

Applying Lemma 2.5 to (2.19), we get that and hence .

Applying Lemma 2.5 to (2.22), we get that and hence .

which yields that since . Hence .

Lemma 2.7.

Proof.

Since as , we obtain the result.

## 3. Strong Convergence Theorems

In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings.

Theorem 3.1.

where for some , for some and for some .

Then, , , , and converge strongly to .

Proof.

Since for all , we get that is nonexpansive for all . Hence, is closed and convex. By Lemma 2.2(5), we know that is closed and convex. By Lemma 2.4, we also know that is closed and convex. Hence, is a nonempty closed convex set; consequently, is well defined for every .

Next, we divide the proof into seven steps.

Step 1.

Show that for all .

It follows that , and hence for all .

Step 2.

Show that exists.

Hence is bounded; so are , , and .

From (3.3) and (3.4), we get that exists.

Step 3.

Show that is a Cauchy sequence.

as . Hence is a Cauchy sequence. By the completeness of and the closeness of , we can assume that .

Step 4.

Show that .

By Lemma 2.7, we also get that . From Lemma 2.6(i), we know that is Lipschitz. Since as , it is easy to verify that . Moreover, by Lemma 2.6(ii), we can conclude that .

Step 5.

Show that .

for all . Observe that if , then holds. Hence .

Step 6.

Show that .

By the maximal monotonicity of , we have ; consequently, .

Step 7.

Show that .

This shows that .

From Steps 1–7, we can conclude that , , , and converge strongly to . This completes the proof.

## 4. Applications

Theorem 4.1.

where for some , for some and for some .

Then, , , , and converge strongly to .

Proof.

In Theorem 3.1, take , where is the indicator function of . It is well known that the subdifferential is a maximal monotone operator. Then, problem (1.7) is equivalent to problem (4.1) and the resolvent operator for all . This completes the proof.

Next, we give a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings. In order to do this, let us assume that

Theorem 4.2.

where for some , for some , and for some .

Then, , , , and converge strongly to .

Proof.

In Theorem 3.1, take , for all . Then problem (1.3) reduces to the equilibrium problem (1.5).

Remark 4.3.

Theorem 3.1 improves and extends the main results in [4, 13] and the corresponding results.

## Declarations

### Acknowledgments

The authors would like to thank the referee for the valuable suggestions on the manuscript. The authors were supported by the Commission on Higher Education, the Thailand Research Fund, and the Graduate School of Chiang Mai University.

## Authors’ Affiliations

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