Abstract and Applied Analysis

Volume 2013 (2013), Article ID 178053, 9 pages

http://dx.doi.org/10.1155/2013/178053

Research Article

A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Abdul Latif¹ and Mohammad Eslamian²

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran

Received 31 October 2013; Accepted 15 December 2013

Academic Editor: Ljubomir B. Ćirić

Copyright © 2013 Abdul Latif and Mohammad Eslamian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

“Dedicated to Professor Miodrag Mateljevi’c on the occasion of his 65th birthday”

1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of , and let be a nearest point projection of into ; that is, for , is the unique point in with the property . It is well known that iff for all .

Let be a bifunction from into , such that for all . Consider the Fan inequality [1]: find a point such that

where is convex and subdifferentiable on for every . The set of solutions of this problem is denoted by . In fact, the Fan inequality can be formulated as an equilibrium problem. Such problems arise frequently in mathematics, physics, engineering, game theory, transportation, economics, and network. Due to importance of the solutions of such problems, many researchers are working in this area and studying the existence of the solutions of such problems; for example, see, [2–4]. Further, if for every , where is a mapping from into , then the Fan inequality problem (equilibrium problem) becomes the classical variational inequality problem which is formulated as finding a point such that

Variational inequalities were introduced and studied by Stampacchia [5]. It is well known that this area covers many branches of mathematics, such as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance; see [6–11].

Here we recall some useful notions.

A mapping is said to be -Lipschitz on if there exists a constant such that for each ,

In particular, if , then is called a contraction on ; if , then is called a nonexpansive mapping on . The set of fixed points of is denoted by .

A family of mappings on a closed convex subset of a Hilbert space is called a nonexpansive semigroup if it satisfies the following: (i) for all ; (ii) for all ; (iii) for all , is continuous; (iv) for all and .

We use to denote the common fixed point set of the semigroup ; that is, . It is well known that is closed and convex [12]. A nonexpansive semigroup on is said to be uniformly asymptotically regular (in short, u.a.r.) on if for all and any bounded subset of ,

For each , define . Then

Provided that is closed bounded convex subset of . It is known that the set is a u.a.r. nonexpansive semigroup; see [13]. The other examples of u.a.r. operator semigroup can be found in [14].

A bifunction is said to be (i) strongly monotone on with if , ; (ii) monotone on if , ; (iii) pseudomonotone on if , ; (iv) Lipschitz-type continuous on with constants and if , .

Note that if is -Lipschitz on , then for each , the function is a Lipschitz-type continuous with constants .

An operator on is called strongly positive if there is a constant such that

Recently, iterative methods for nonexpansive mappings have been applied to solve convex minimization problems. In [15], Xu defined an iterative sequence in which converges strongly to the unique solution of the minimization problem under some suitable conditions. A well-known typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping on a real Hilbert space :

where is a given point in and is strongly positive operator.

For solving the variational inequality problem, Marino and Xu [16] introduced the following general iterative process for nonexpansive mapping based on the viscosity approximation method (see [17]):

where is strongly positive bounded linear operator on , is contraction on , and . They proved that, under some appropriate conditions on the parameters, the sequence generated by (8) converges strongly to the unique solution of the variational inequality

which is the optimality condition for the minimization problem

where is a potential function for (i.e., , ).

Iterative process for approximating common fixed points of a nonexpansive semigroup has been investigated by various authors (see [13, 14, 18–21]). Recently, Li et al. [19] introduced the following iterative procedure for the approximation of common fixed points of a nonexpansive semigroup on a closed convex subset of a Hilbert space :

where is a strongly positive bounded linear operator on and is a contraction on . Imposing some appropriate conditions on the parameters, they proved that the iterative sequence generated by (11) converges strongly to the unique solution of the variational inequality , .

For obtaining a common element of and the set of fixed points of a nonexpansive mapping , S. Takahashi and W. Takahashi [9] first introduced an iterative scheme by the viscosity approximation method. They proved that under certain conditions the iterative sequences converge strongly to .

During last few years, iterative algorithms for finding a common element of the set of solutions of Fan inequality and the set of fixed points of nonexpansive mappings in a real Hilbert space have been studied by many authors (see, e.g., [2, 4, 22–28]). Recently, Anh [22] studied the existence of a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of Fan inequality for monotone and Lipschitz-type continuous bifunctions. He introduced the following new iterative process:

where , is nonempty, closed convex subset of a real Hilbert space , is monotone, continuous, and Lipschitz-type continuous bifunction, is self-contraction on with constant , and is self nonexpansive mapping on . He proved that, under some appropriate conditions over positive sequences , , , and , the sequences , , and converge strongly to which is a solution of the variational inequality , .

In this paper, we introduce a new iterative scheme based on the viscosity method and study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative scheme for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh [22], Cianciaruso et al. [18], and many others.

2. Preliminaries

In this section we collect some lemmas which are crucial for the proofs of our results.

Let be a sequence in and . In the sequel, denotes that weakly converges to and denotes that weakly converges to .

Lemma 1. Let be a real Hilbert space. Then the following inequality holds:

Lemma 2 (see [15]). Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence in such that

(i), (ii) or .

Then .

Lemma 3 (see [16]). Let be a strongly positive linear bounded self-adjoint operator on with coefficient and . Then .

Lemma 4 (see [29]). Let be a Hilbert space and , . Then for any given with and for any positive integer with ,

Lemma 5 (see [30]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :

In fact

Lemma 6 (see [2]). Let be a nonempty closed convex subset of a real Hilbert space and let be a pseudomonotone and Lipschitz-type continuous bifunction with constants . For each , let be convex and subdifferentiable on . Let and let , , and be sequences generated by

Then for each ,

3. Main Results

In this section, we prove the main strong convergence result which solves the problem of finding a common element of three sets , , and for finite family of monotone, continuous, and Lipschitz-type continuous bifunctions in a real Hilbert space .

Theorem 7. Let be a nonempty closed convex subset of a real Hilbert space and let be a finite family of monotone, continuous, and Lipschitz-type continuous bifunctions with constants and . Let and be two u.a.r. nonexpansive self-mapping semigroups on such that . Assume that is a -contraction self-mapping of and is a strongly positive bounded linear self-adjoint operator on with coefficient and . Let and let , , and be sequences generated by

where and , , , , , , and satisfy the following conditions:

(i), (ii), , and ,(iii), where ,(iv), , , , and for each .

Then, the sequence converges strongly to which solves the variational inequality:

Proof. Since , , and are closed and convex, is well-defined. We claim that is a contraction from into itself. Indeed, for each , we have

Therefore, by the Banach contraction principle, there exists a unique element such that . We show that is bounded. Since , we can assume, with no loss of generality, that , for all . By Lemma 6, for each , we have

This implies that

It follows from Lemma 3 that

This implies that is bounded and so are , , , and . Next, we show that . Indeed, by Lemma 6, for each , we have

Applying Lemma 4 and inequality (26) for we have that

We now compute

Therefore,

In order to prove that as , we consider the following two cases.

Case 1. Assume that is a monotone sequence. In other words, for large enough , is either nondecreasing or nonincreasing. Since is bounded, it is convergent. Since and and are bounded, from (29), we have

and by assumption we get

By similar argument we can obtain that

Further, for all and , we see that

Since is u.a.r. nonexpansive semigroup and , we have

Similarly, for all and , we obtain that

Next, we show that

To show this inequality, we choose a subsequence of such that

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . Consider

Thus, we have

By the Opial property of the Hilbert space we obtain for all . Similarly we have that for all . This implies that . Now we show that . For each , since is convex on for each , we see that

if and only if

where is the (outward) normal cone of at . This follows that

where and . By the definition of the normal cone we have

Since is subdifferentiable on , by the well-known Moreau-Rockafellar theorem [31] (also see [6]), for , we have

Combining this with (43), we have

In particular, we have

Since , it follows from (32) that . And thus we have

This implies that and hence . Since and , we have

From Lemma 1, it follows that

This implies that

where

It is easy to see that , , and . Hence, by Lemma 2 the sequence converges strongly to . From (32) we have that and converge strongly to .

Case 2. Assume that is not a monotone sequence. Then, we can define an integer sequence for all (for some large enough ) by

Clearly, is a nondecreasing sequence such that as , and for all ,

From (33) we obtain that

Following an argument similar to that in Case 1 we have

where , , and . Hence, by Lemma 2, we obtain and . Now Lemma 5 implies that

Therefore, converges strongly to . This completes the proof.

4. Application

In this section, we consider a particular Fan inequality corresponding to the function defined by the following: for every

where . Then, we obtain the classical variational inequality as follows.

The set of solutions of this problem is denoted by . In that particular case, the solution of the minimization problem

can be expressed as

Let be -Lipschitz continuous on . Then

Therefore,

hence, satisfies Lipschitz-type continuous condition with .

Using Theorem 7 we obtain the following convergence theorem.

Theorem 8. Let be a nonempty closed convex subset of a real Hilbert space and let    be functions such that, for each , is monotone and -Lipschitz continuous on . Let and be two u.a.r. nonexpansive self-mapping semigroups on such that . Assume that is a -contraction of into itself and is a strongly positive bounded linear self-adjoint operator on with coefficient and . Let and let , , and be sequences generated by

where and , , , , , and satisfy the following conditions:

(i), (ii), , and ,(iii),(iv), , , , and for each .

Then, the sequence converges strongly to which solves the variational inequality

In [32], Baillon proved a strong mean convergence theorem for nonexpansive mappings, and it was generalized in [33]. It follows from the above proof that Theorems 7 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorems for nonexpansive mappings in a Hilbert space.

Theorem 9. Let be a nonempty closed convex subset of a real Hilbert space and let    be functions such that for each , is monotone and -Lipschitz continuous on . Let and be two nonexpansive mappings on such that . Assume that is a -contraction of into itself and is a strongly positive bounded linear self-adjoint operator on with coefficient and . Let , , and be sequences generated by and by

where and , , , , , and satisfy the following conditions:

(i), , and ,(ii), where ,(iii), , , , and for each .

Then, the sequence converges strongly to which solves the variational inequality

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, for technical and financial support.

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