Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings

Abstract

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.

1. Introduction

Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization, which has been extended and generalized in many directions using novel and innovative techniques; see [1–8]. Inspired and motivated by the research and activities going in this fascinating area, we introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems.

Let be a nonempty closed convex subset of a real Hilbert space and a multivalued mapping. Let be a real-valued function and an equilibrium-like function, that is,

(1.1)

We consider the problem of finding and such that

(1.2)

which is called the generalized mixed equilibrium problem (for short, GMEP). If is a single-valued mapping, then problem (1.2) is equivalent to finding such that

(1.3)

We denote for the set of solutions of GMEP (1.2). This class is a quite general and unifying one and includes several classes of equilibrium problems and variational inequalities as special cases. In recent years, several numerical techniques including projection, resolvent, and auxiliary principle have been developed and analyzed for solving variational inequalities. It is well known that projection- and resolvent-type methods cannot be extended for equilibrium problems. To overcome this drawback, one usually uses the auxiliary principle technique. Glowinski et al. [9] have used this technique to study the existence of a solution of mixed variational inequalities. The viscosity approximation method is one of the important methods for approximation fixed points of nonexpansive type mappings. It was first discussed by Moudafi [10]. Recently, Hirstoaga [11] and S. Takahashi and W. Takahashi [12] applied viscosity approximation technique for finding a common element of set of solutions of an equilibrium problem (EP) and set of fixed points of a nonexpansive mapping. Very recently, Yao et al. [13] introduced and studied an iteration process for finding a common element of the set of solutions of the EP and the set of common fixed points of infinitely many nonexpansive mappings in . Let be a sequence of nonexpansive mappings of into itself and let be a sequence of nonnegative numbers in . For any , define a mapping of into itself as follows:

(1.4)

Such a mapping is called the -mapping generated by and , see [14].

The purpose of this paper is to develop an iterative algorithm for finding a common element of set of solutions of GMEP (1.2) and set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. The result presented in this paper improves and extends the main result of S. Takahashi and W. Takahashi [12].

2. Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by , such that

(2.1)

is called metric projection of onto . It is well known that is nonexpansive. Furthermore, for and ,

(2.2)

We denote by the set of fixed points of a self-mapping on , that is, . It is well known that if is nonempty, bounded, closed, and convex and is nonexpansive, then is nonempty; see [15]. Let be a sequence of nonexpansive mappings of into itself, where is a nonempty closed convex subset of a real Hilbert space . Given a sequence in , we define a sequence of self-mappings on by (1.4). Then we have the following lemmas which are important to prove our results.

Lemma 2.1 (see [14]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, for every and the limit exists.

Using Lemma 2.1, one can define mapping of into itself as follows:

(2.3)

for every . Such a mapping is called the -mapping generated by and Throughout this paper, we will assume that for every . Since is nonexpansive, is also nonexpansive.

Lemma 2.2 (see [14]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, .

Let be a convex subset of a real Hilbert space and a Fréchet differential function. Then is said to be -convex strongly convex if there exists a constant such that

(2.4)

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

Let be a nonempty subset of a real Hilbert space . A bifunction is said to be skew-symmetric if

(2.5)

If the skew-symmetric bifunction is linear in both arguments, then

(2.6)

We denote for weak convergence and for strong convergence. A function is called weakly sequentially continuous at , if as for each sequence in converging weakly to . The function is called weakly sequentially continuous on if it is weakly sequentially continuous at each point of .

Let denote the set of nonempty closed bounded subsets of . For , define the Hausdorff metric as follows:

(2.7)

Lemma 2.3 (see [16]).

Let and . Then for , there must exist a point such that .

Let be a nonempty closed convex subset of a real Hilbert space and a multivalued mapping. For , let . Let be a real-valued function satisfying the following:

is skew symmetric;

for each fixed , is convex and upper semicontinuous;

is weakly continuous on .

Let be a differentiable functional with Fréchet derivative at satisfying the following:

is sequentially continuous from the weak topology to the strong topology;

is Lipschitz continuous with Lipschitz constant .

Let be a function satisfying the following:

for all ;

is affine in the first coordinate variable;

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

Let us consider the equilibrium-like function which satisfies the following conditions with respect to the multivalued mapping :

for each fixed , is an upper semicontinuous function from to , that is, and imply ;

for each fixed , is a concave function;

for each fixed , is a convex function.

Let be a positive parameter. For a given element and , consider the following auxiliary problem for GMEP(1.2): find such that

(2.8)

It is easy to see that if , then is a solution of GMEP(1.2).

Lemma 2.4 (see [6]).

Let be a nonempty closed convex bounded subset of a real Hilbert space and a real-valued function satisfying the conditions . Let be a multivalued mapping and the equilibrium-like function satisfying the conditions . Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies the conditions and . For each , let . For , define a mapping by

(2.9)

Then one has the following:

• (a) the auxiliary problem (2.8) has a unique solution;

(b) is single valued;

• (c) if and for all and all , , it follows that is nonexpansive;

(d);

(e) is closed and convex.

We also need the following lemmas for our main results.

Lemma 2.5 (see [17]).

Let , and be three sequences of nonnegative numbers such that

(2.10)

If , , and , then exists.

Lemma 2.6.

Let and be sequences of nonnegative numbers such that

(2.11)

If and , then .

Proof.

It is easy to see that inequality (2.11) is equivalent to

(2.12)

where , and . It follows that

(2.13)

Note that Lemma 2.5 implies that exists. Suppose for some . It is obvious that and so inequality (2.12) implies that , which is a contradiction. Thus, . This completes the proof.

Lemma 2.7 (see [6]).

Let be a sequence in a normed space such that

(2.14)

where , and and are sequences satisfy the following conditions:

• (i) for all and ;

• (ii) for all and .

Then is a Cauchy sequence.

Lemma 2.8 (see [18]).

Let be a sequence of nonnegative real numbers such that

(2.15)

where , and are sequences of real numbers satisfying the following conditions:

• (i) , and ;

• (ii) ;

• (iii) for all and .

Then, .

3. Iterative Algorithm and Convergence Theorem

Let be a nonempty closed convex subset of a real Hilbert space , a multivalued mapping, a contraction mapping with constant , and an -mapping generated by and , where sequence is nonexpansive. Let be a sequence in and a sequence in . We can develop Algorithm 3.1 for finding a common element of a set of fixed points of -mapping and a set of solutions of GMEP(1.2).

Algorithm.

For given and , there exist sequences , in and in such that for all ,

(3.1)

We now prove the strong convergence of iterative sequence , , and generated by Algorithm 3.1.

Theorem 3.2.

Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued -Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying conditions and :

for all and , where , and .

Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies conditions and with . Let be an -mapping generated by and and , where sequence is nonexpansive. Let , and be sequences generated by Algorithm 3.1, where is a sequence in and in satisfying the following conditions:

, and ;

and ;

where .

Then the sequences and converge strongly to , and converges strongly to , where .

Proof.

It is easy to see from () that

(3.2)

for all and , where , , and . All the conclusions (a)–(e) of Lemma 2.4 hold.

Let . Then is a contraction of into itself. In fact,

(3.3)

Hence there exists a unique element such that . Noting that and , we get that .

Now, we prove that and as . Observe that

(3.4)

Noting that and , it follows from (3.1) that

(3.5)

(3.6)

Putting in (3.5) and in (3.6), respectively, we have

(3.7)

Adding up those inequalities, we obtain from (2.5), (), and () that

(3.8)

It follows that

(3.9)

since and are Lipschitz continuous wiht Lipschitz constants and , respectively. Noting that , without loss of generality, we assume that there exists a real number such that for all Thus,

(3.10)

which implies that

(3.11)

and hence

(3.12)

where . Set . Combining (3.4) and (3.12) yields

(3.13)

From conditions and ,

(3.14)

Set and

(3.15)

Then Lemmas 2.6 and 2.7 imply that and is a Cauchy sequence in . Hence from (3.12), we get

(3.16)

We know from that . It follows that

(3.17)

Thus,

Next, we prove that there exists , such that , , and as , where .

Let . Then

(3.18)

and so

(3.19)

By the convexity of , we have

(3.20)

It follows that

(3.21)

This implies that

(3.22)

Since is a Cauchy sequence in , there exists an element such that . Now implies that . From (3.1), we have

(3.23)

and for ,

(3.24)

(3.25)

Thus,

(3.26)

By (3.24) and (3.26), we have

(3.27)

It follows that is a Cauchy sequence in and so there exists an element in such that :

(3.28)

that is, . We conclude that as .

It follows that

(3.29)

and so , that is, . Since and , we know that . From (3.1) and (), we have

(3.30)

that is, . Thus, .

Since , we have for all . From , we have

(3.31)

and so

(3.32)

It follows from (3.19) that

(3.33)

Set

(3.34)

Then, , , and . It follows from Lemma 2.8 that and so . This completes the proof.

Remark 3.3.

Theorem 3.2 improves and extends the main results of S. Takahashi and W. Takahashi [12].

We now give some applications of Theorem 3.2. If the set-valued mapping in Theorem 3.2 is single-valued, then we have the following corollary.

Corollary 3.4.

Let be a nonempty closed convex bounded subset of a real Hilbert space , a Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying the conditions and :

for all and .

Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies the conditions and with . Let be an -mapping generated by and and , where sequence is nonexpansive. Let , , and be sequences generated by

(3.35)

where is a sequence in and in satisfying conditions . Then the sequences and converge strongly to , where .

Corollary 3.5.

Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued -Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying the conditions and . Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies the conditions and with . Let , , and be sequences generated by

(3.36)

where is a sequence in and in satisfying conditions and . Then the sequences and converge strongly to , and converges strongly to , where .

Proof.

Let in Theorem 3.2 for , where is an identity mapping. Then for Thus, the condition is satisfied. Now Corollary 3.5 follows from Theorem 3.2. This completes the proof.