A Hybrid-Extragradient Scheme for System of Equilibrium Problems, Nonexpansive Mappings, and Monotone Mappings

Abstract

We introduce a new iterative scheme based on both hybrid method and extragradient method for finding a common element of the solutions set of a system of equilibrium problems, the fixed points set of a nonexpansive mapping, and the solutions set of a variational inequality problems for a monotone and -Lipschitz continuous mapping in a Hilbert space. Some convergence results for the iterative sequences generated by these processes are obtained. The results in this paper extend and improve some known results in the literature.

1. Introduction

In this paper, we always assume that is a real Hilbert space with inner product and induced norm and is a nonempty closed convex subset of , is a nonexpansive mapping; that is, for all , denotes the metric projection of onto , and denotes the fixed points set of .

Let be a countable family of bifunctions from to , where is the set of real numbers. Combettes and Hirstoaga [1] introduced the following system of equilibrium problems:

(1.1)

where is an arbitrary index set. If is a singleton, the problem (1.1) becomes the following equilibrium problem:

(1.2)

The set of solutions of (1.2) is denoted by . And it is easy to see that the set of solutions of (1.1) can be written as .

Given a mapping , let for all . Then, the problem (1.2) becomes the following variational inequality:

(1.3)

The set of solutions of (1.3) is denoted by .

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance, [1–4].

In 1953, Mann [5] introduced the following iteration algorithm: let be an arbitrary point, let be a real sequence in , and let the sequence be defined by

(1.4)

Mann iteration algorithm has been extensively investigated for nonexpansive mappings, for example, please see [6, 7]. Takahashi et al. [8] modified the Mann iteration method (1.4) and introduced the following hybrid projection algorithm:

(1.5)

where . They proved that the sequence generated by (1.5) converges strongly to .

In 1976, Korpelevič [9] introduced the following so-called extragradient algorithm:

(1.6)

for all , where , is monotone and -Lipschitz continuous mapping of into . She proved that, if is nonempty, the sequences and , generated by (1.6), converge to the same point .

Some methods have been proposed to solve the problem (1.2); see, for instance, [10, 11] and the references therein. S. Takahashi and W. Takahashi [10] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the set of the solution (1.2) and the set of fixed points of a nonexpansive mapping in a real Hilbert space: starting with an arbitrary initial , define sequences and recursively by

(1.7)

They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to , where .

Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to , and let be a relatively nonexpansive mapping from into itself such that . Takahashi and Zembayashi [11] introduced the following hybrid method in Banach space: let be a sequence generated by , and

(1.8)

for every , where is the duality napping on , for all , and for all . They proved that the sequence generated by (1.8) converges strongly to if satisfies and for some .

On the other hand, Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [4] introduced a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions of problem (1.1), the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping in a Hilbert space and obtained two strong convergence theorems. Colao et al. [3] introduced an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Peng et al. [12] introduced a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed points set of a family of infinitely nonexpansive mappings, and the solution set of a variational inequality for a relaxed coercive mapping in a Hilbert space and obtained a strong convergence theorem.

In this paper, motivated by the above results, we introduce a new hybrid extragradient method to find a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the variational inequality for monotone and -Lipschitz continuous mappings in a Hilbert space and obtain some strong convergence theorems. Our results unify, extend, and improve those corresponding results in [8, 11] and the references therein.

2. Preliminaries

Let symbols and denote strong and weak convergence, respectively. It is well known that

(2.1)

for all and .

For any , there exists a unique nearest point in denoted by such that for all . The mapping is called the metric projection of onto . We know that is a nonexpansive mapping from onto . It is also known that and

(2.2)

for all and .

It is easy to see that (2.2) is equivalent to

(2.3)

for all and .

A mapping of into is called monotone if for all . A mapping is called -Lipschitz continuous if there exists a positive real number such that for all .

Let be a monotone mapping of into . In the context of the variational inequality problem, the characterization of projection (2.2) implies the following:

(2.4)

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions which were imposed in [2]:

(A1) for all ;

(A2) is monotone; that is, for any ;

(A3)for each ,

(2.5)

(A4)for each is convex and lower semicontinuous.

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

Lemma 2.1 (See [2]).

Let be a nonempty closed convex subset of , and let be a bifunction from to satisfying (A1)–(A4). Let and . Then, there exists such that

(2.6)

Lemma 2.2 (See [1]).

Let be a nonempty closed convex subset of , and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:

(2.7)

for all . Then, the following statements hold:

(1) is single-valued;

(2) is firmly nonexpansive; that is, for any ,

(2.8)

(3);

(4) is closed and convex.

3. Main Results

In this section, we will introduce a new algorithm based on hybrid and extragradient method to find a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the variational inequality for monotone and -Lipschitz continuous mappings in a Hilbert space and show that the sequences generated by the processes converge strongly to a same point.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let , be a family of bifunctions from to satisfying (A1)–(A4), let be a monotone and -Lipschitz continuous mapping of into , and let be a nonexpansive mapping from into itself such that . Pick any , and set . Let , and be sequences generated by and

(3.1)

for each . If for some for some , and satisfies for each , then , , , and generated by (3.1) converge strongly to .

Proof.

It is obvious that is closed for each . Since

(3.2)

we also have that is convex for each . Thus, , , , and are welldefined. By taking for and , for each , where is the identity mapping on . Then, it is easy to see that . We divide the proof into several steps.

Step 1.

We show by induction that for . It is obvious that . Suppose that for some . Let . Then, by Lemma 2.2 and , we have

(3.3)

Putting for each , from (2.3) and the monotonicity of , we have

(3.4)

Moreover, from and (2.2), we have

(3.5)

Since is -Lipschitz continuous, it follows that

(3.6)

So, we have

(3.7)

From (3.7) and the definition of , we have

(3.8)

(3.9)

and hence . This implies that for all .

Step 2.

We show that and .

Let . From and , we have

(3.10)

Therefore, is bounded. From (3.3)–(3.9), we also obtain that , , and are bounded. Since and , we have

(3.11)

Therefore, exists.

From and , we have

(3.12)

So

(3.13)

which implies that

(3.14)

Since , we have , and hence

(3.15)

It follows from (3.14) that .

For , it follows from (3.9) that

(3.16)

which implies that .

Step 3.

We now show that

(3.17)

Indeed, let . It follows form the firmly nonexpansiveness of that we have, for each ,

(3.18)

Thus, we get

(3.19)

which implies that, for each ,

(3.20)

By (3.8), , and (3.20), we have, for each ,

(3.21)

which implies that

(3.22)

It follows from and that (3.17) holds.

Step 4.

We now show that .

It follows from (3.17) that . Since , we get

(3.23)

We observe that

(3.24)

which implies that

(3.25)

Since , we obtain

(3.26)

Since , we get

(3.27)

Step 5.

We show that , where .

As is bounded, there exists a subsequence which converges weakly to . From , we obtain that . It follows from , , and that , , and .

In order to show that , we first show that . Indeed, by definition of , we have that, for each ,

(3.28)

From (A2), we also have

(3.29)

And hence

(3.30)

From (A4), and imply that, for each ,

(3.31)

Since , and is closed and convex, is weakly closed, and hence . Thus, for with and , let . Since and , we have , and hence . So, from (A1) and (A4), we have, for each ,

(3.32)

and hence, for each , . From (A3), we have, for each , . Thus, .

We now show that . Assume that . Since and , from Opial's condition [13], we have

(3.33)

which is a contradiction. Thus, we obtain .

We next show that . Let

(3.34)

It is worth to note that in this case the mapping is maximal monotone and if and only if (see [14]). Let . Since and , we have . On the other hand, from and , we have , and hence . Therefore, we have

(3.35)

Since and is -Lipschitz continuous, we obtain that . From , , and , we obtain

(3.36)

Since is maximal monotone, we have , and hence , which implies that . Finally, we show that , where

(3.37)

Since and , we have . It follows from and the lower semicontinuousness of the norm that

(3.38)

Thus, we obtain and

(3.39)

From and the Kadec-Klee property of , we have , and hence . This implies that . It is easy to see that , , and . The proof is now complete.

By Theorem 3.1, we can easily obtain some new results as follows.

Corollary 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4), let be a monotone and -Lipschitz continuous mapping of into , and let be a nonexpansive mapping from into itself such that . Pick any , and set . Let , and be sequences generated by and

(3.40)

for each . If for some for some , and satisfies , then , , , and converge strongly to .

Proof.

Putting in Theorem 3.1, we obtain Corollary 3.2.

Corollary 3.3.

Let be a nonempty closed convex subset of a real Hilbert space . Let , be a family of bifunctions from to satisfying (A1)–(A4), and let be a nonexpansive mapping from into itself such that . Pick any , and set . Let , and be sequences generated by and

(3.41)

for each . If for some and satisfies for each , then , , and converge strongly to .

Proof.

Let in Theorem 3.1, then complete the proof.

Corollary 3.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a monotone and -Lipschitz continuous mapping of into , and let be a nonexpansive mapping from into itself such that . Pick any , and set . Let , and be sequences generated by and

(3.42)

for each . If for some for some , then , , and converge strongly to .

Proof.

Putting in Theorem 3.1, we obtain Corollary 3.4.

Remark 3.5.

Letting in Corollary 3.3, we obtain the Hilbert space version of Theorem  3.1 in [11]. Letting in Corollary 3.4, we recover Theorem  4.1 in [8]. Hence, Theorem 3.1 unifies, generalizes, and extends the corresponding results in [8, 11] and the references therein.