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Fixed Point Theory and Applications

, 2011:392741 | Cite as

Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces

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  1. Equilibrium Problems and Fixed Point Theory

Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality. As an application, we use the result of this paper to solve a multiobjective optimization problem. Our result extends and improves the ones of Colao et al. (2008) and some others.

Keywords

Hilbert Space Variational Inequality Equilibrium Problem Nonexpansive Mapping Iterative Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be a real Hilbert space and Open image in new window be a mapping of Open image in new window into itself. Open image in new window is said to be nonexpansive if

If there exists a point Open image in new window such that Open image in new window , then the point Open image in new window is called a fixed point of Open image in new window . The set of fixed points of Open image in new window is denoted by Open image in new window . It is well known that Open image in new window is closed convex and also nonempty if Open image in new window has a bounded trajectory (see [1]).

Let Open image in new window be a mapping. If there exists a constant Open image in new window such that
then Open image in new window is called a contraction with the constant Open image in new window . Recall that an operator Open image in new window is called to be strongly positive with coefficient Open image in new window if

Let Open image in new window be a fixed point, Open image in new window be a strongly positive linear bounded operator on Open image in new window and Open image in new window be a finite family of nonexpansive mappings of Open image in new window into itself such that Open image in new window .

In 2003, Xu [2] introduced the following iterative scheme:
where Open image in new window is the identical mapping on Open image in new window and Open image in new window , and proved some strong convergence theorems for the iterative scheme to the solution of the quadratic minimization problem
under suitable hypotheses on Open image in new window and the additional hypothesis:
Recently, Marino and Xu [3] introduced a new iterative scheme from an arbitrary point Open image in new window by the viscosity approximation method as follows:
and prove that the scheme strongly converges to the unique solution Open image in new window of the variational inequality:
which is the optimality condition for the minimization problem:

where Open image in new window is a potential function for Open image in new window (i.e., Open image in new window for all Open image in new window ).

Let Open image in new window be a finite family of nonexpansive mappings of Open image in new window into itself. In 2007, Yao [4] defined the mappings
and, by extending (1.10), proposed the iterative scheme:
Then he proved that the iterative scheme (1.10) strongly converges to the unique solution Open image in new window of the variational inequality:
where Open image in new window , which is the optimality condition for the minimization problem:

where Open image in new window is a potential function for Open image in new window (However, Colao et al. pointed out in [5] that there is a gap in Yao's proof).

Let Open image in new window be a nonempty closed convex subset of Open image in new window and Open image in new window be a bifunction. The equilibrium problem for the function Open image in new window is to determine the equilibrium points, that is, the set
Let Open image in new window be a nonlinear mapping. Let Open image in new window denote the set of all solutions to the following equilibrium problem:

In the case of Open image in new window , Open image in new window is deduced to Open image in new window . In the case of Open image in new window , Open image in new window is also denoted by Open image in new window .

In 2007, S. Takahashi and W. Takahashi [6] introduced a viscosity approximation method for finding a common element of Open image in new window and Open image in new window from an arbitrary initial element Open image in new window

and proved that, under certain appropriate conditions over Open image in new window and Open image in new window , the sequences Open image in new window and Open image in new window both converge strongly to Open image in new window .

By combing the schemes (1.7) and (1.16), Plubtieg and Punpaeng [7] proposed the following algorithm:
and proved that the iterative schemes Open image in new window and Open image in new window converge strongly to the unique solution Open image in new window of the variational inequality:
which is the optimality condition for the minimization problem:

where Open image in new window is a potential function for Open image in new window .

Very recently, for finding a common element of the set of a finite family of nonexpansive mappings and the set of solutions of an equilibrium problem, by combining the schemes (1.11) and (1.17), Colao et al. [5] proposed the following explicit scheme:
and proved under some certain hypotheses that both sequences Open image in new window and Open image in new window converge strongly to a point Open image in new window which is an equilibrium point for Open image in new window and is the unique solution of the variational inequality:

where Open image in new window .

The equilibrium problems have been considered by many authors; see, for example, [6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and the reference therein. But, in these references, the authors only considered at most finite family of equilibrium problems and few of authors investigate the infinite family of equilibrium problems in a Hilbert space or Banach space. In this paper, we consider a new iterative scheme for obtaining a common element in the solution set of an infinite family of generalized equilibrium problems and in the common fixed-point set of a finite family of nonexpansive mappings in a Hilbert space. Let Open image in new window ( Open image in new window ) be a finite family of nonexpansive mappings of Open image in new window into itself, be   Open image in new window be an infinite family of bifunctions, and be   Open image in new window be an infinite family of Open image in new window -inverse-strongly monotone mappings. Let Open image in new window be a sequence such that Open image in new window with Open image in new window for each Open image in new window . Define the mapping Open image in new window by
Assume that Open image in new window . For an arbitrary initial point Open image in new window , we define the iterative scheme Open image in new window by
where Open image in new window , Open image in new window , Open image in new window and Open image in new window are three sequences in (0,1), Open image in new window and Open image in new window are both strongly positive linear bounded operators on Open image in new window , Open image in new window is defined by (1.10), and prove that, under some certain appropriate hypotheses on the control sequences, the sequence Open image in new window strongly converges to a point Open image in new window , which is the unique solution of the variational inequality:
If Open image in new window , Open image in new window and Open image in new window , then (1.23) is reduced to the iterative scheme:

The proof method of the main result of this paper is different with ones of others in the literatures and our result extends and improves the ones of Colao et al. [5] and some others.

2. Preliminaries

Let Open image in new window be a closed convex subset of a Hilbert space Open image in new window . For any point Open image in new window , there exists a unique nearest point in Open image in new window , denoted by Open image in new window , such that
Then Open image in new window is called the metric projection of Open image in new window onto Open image in new window . It is well known that Open image in new window is a nonexpansive mapping of Open image in new window onto Open image in new window and satisfies the following:
for all Open image in new window . However, Open image in new window is called an α-inverse-strongly monotone mapping if there exists a positive real number α such that

Hence, if Open image in new window , then Open image in new window is a nonexpansive mapping of Open image in new window into Open image in new window .

If there exists Open image in new window such that

for all Open image in new window , then Open image in new window is called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by Open image in new window .

In this paper, we need the following lemmas.

Lemma 2.1 (see [21]).

Given Open image in new window and Open image in new window . Then Open image in new window if and only if there holds the inequality

Lemma 2.2 (see [22]).

Let Open image in new window be a sequence of nonnegative real numbers satisfying

where Open image in new window , Open image in new window , and Open image in new window satisfy the conditions:

(1)   Open image in new window , Open image in new window or, equivalently, Open image in new window ;

(2)   Open image in new window ;

(3)   Open image in new window    Open image in new window , Open image in new window .

Then Open image in new window .

Let Open image in new window be a Hilbert space. For all Open image in new window , the following equality holds:

Therefore, the following lemma naturally holds.

Lemma 2.3.

Let Open image in new window be a real Hilbert space. The following identity holds:

Lemma 2.4 (see [3]).

Assume that Open image in new window is a strongly positive linear bounded operator on a Hilbert space Open image in new window with coefficient Open image in new window and Open image in new window . Then Open image in new window .

Lemma 2.5 (see [2]).

Assume that Open image in new window is a sequence of nonnegative numbers such that

where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(1)   Open image in new window ;

(2)   Open image in new window or Open image in new window .

Then Open image in new window .

Lemma 2.6 (see [23]).

Let C be a nonempty closed convex subset of a Hilbert space H and let Open image in new window be a bifunction which satisfies the following:

(A1)   Open image in new window for all Open image in new window ;

(A2)   Open image in new window is monotone, that is, Open image in new window for all Open image in new window ;

(A3)  For each Open image in new window ,

(A4)  For each Open image in new window , Open image in new window is convex and lower semicontinuous.

Then Open image in new window is well defined and the following hold:

(1)   Open image in new window is single-valued;

(2)   Open image in new window is firmly nonexpansive, that is, for any Open image in new window ,

(3)   Open image in new window ;

(4)   Open image in new window is closed and convex.

It is easy to see that if there exists some point Open image in new window such that Open image in new window , where Open image in new window is an α-inverse strongly monotone mapping, then Open image in new window . In fact, since Open image in new window , one has
that is,

Hence, Open image in new window .

Let Open image in new window be a nonempty convex subset of a Banach space. Let Open image in new window be a finite family of nonexpansive mappings of Open image in new window into itself and Open image in new window be real numbers such that Open image in new window for each Open image in new window . Define a mapping Open image in new window of Open image in new window into itself as follows:

Such a mapping Open image in new window is called the Open image in new window - Open image in new window generated by Open image in new window and Open image in new window (see [5, 24, 25]).

Lemma 2.7 (see [26]).

Let Open image in new window be a nonempty closed convex subset of a Banach space. Let Open image in new window be nonexpansive mappings of Open image in new window into itself such that Open image in new window and let Open image in new window be real numbers such that Open image in new window for each Open image in new window and Open image in new window . Let Open image in new window be the Open image in new window -mapping of Open image in new window generated by Open image in new window and Open image in new window . Then Open image in new window .

Lemma 2.8 (see [5]).

Let Open image in new window be a nonempty convex subset of a Banach space. Let Open image in new window be a finite family of nonexpansive mappings of Open image in new window into itself and let Open image in new window be sequences in Open image in new window such that Open image in new window for each Open image in new window . Moreover, for each Open image in new window , let Open image in new window and Open image in new window be the Open image in new window -mappings generated by Open image in new window and Open image in new window and Open image in new window and Open image in new window , respectively. Then, for all Open image in new window , it follows that

3. Main Results

Now, we give our main results in this paper.

Theorem 3.1.

Let Open image in new window be a Hilbert space and Open image in new window be a nonempty closed convex subset of Open image in new window . Let Open image in new window be a contraction with coefficient Open image in new window , Open image in new window , Open image in new window be strongly positive linear bounded self-adjoint operators with coefficients Open image in new window and Open image in new window , respectively, Open image in new window    Open image in new window be a finite family of nonexpansive mappings, Open image in new window be an infinite family of bifunctions satisfying Open image in new window be an infinite family of inverse-strongly monotone mappings with constants Open image in new window such that Open image in new window . Let Open image in new window and Open image in new window be two sequences in Open image in new window , Open image in new window be asequence in Open image in new window with Open image in new window ,   Open image in new window be a sequence in Open image in new window with Open image in new window and Open image in new window be a strictly decreasing sequence Open image in new window Set Open image in new window . Take a fixed number Open image in new window with Open image in new window . Assume that

(E1)   Open image in new window

(E2)   Open image in new window ;

(E3)   Open image in new window ;

(E4)   Open image in new window with Open image in new window ;

(E5)   Open image in new window .

Then the sequence Open image in new window defined by (1.23) converges strongly to Open image in new window , which is the unique solution of the variational inequality: (1.24), that is,

Proof.

Since Open image in new window as Open image in new window by the condition (E1), we may assume, without loss of generality, that Open image in new window for all Open image in new window . Noting that Open image in new window and Open image in new window are both the linear bounded self-adjoint operators, one has
Observing that
we obtain that Open image in new window is positive for all Open image in new window . It follows that
Note that
Hence, for each Open image in new window satisfying the condition (E4), one has
Moreover, it follows from (3.7), Open image in new window and (E4) that

Next, we proceed the proof with following steps.

Step 1.

Open image in new window is bounded.

Let Open image in new window . Lemma 2.6 shows that every Open image in new window is firmly nonexpansive and hence nonexpansive. Since Open image in new window , Open image in new window is nonexpansive for each Open image in new window . Therefore, Open image in new window is nonexpansive for each Open image in new window . Noting that Open image in new window is strictly decreasing, Open image in new window , we have
and hence
Then, from (3.4) and (3.11), it follows that (noting that Open image in new window is linear and Open image in new window )
It follows from Open image in new window and Open image in new window that Open image in new window . Therefore, by the simple induction, we have

which shows that Open image in new window is bounded, so is Open image in new window .

Step 2.

Open image in new window as Open image in new window .

First, we prove
It follows from the definition of Open image in new window that
for each Open image in new window . Thus, using the above recursive inequalities repeatedly, we have
Also, we have

where Open image in new window .

Next, we prove Open image in new window . Observe (noting that Open image in new window is linear) that
Hence, by (3.4) and (3.18), we get

where Open image in new window .

Then it follows from (3.21) that
It follows from the assumption condition (E1), (E3), (E5), and (3.14) that

By applying Lemma 2.2 to (3.23), we obtain Open image in new window as Open image in new window .

Step 3.

Open image in new window as Open image in new window .

and hence (noting (3.9))
It follows from the assumption conditions (E1), (E2), and Step 2 that

Step 4.

Open image in new window as Open image in new window .

Let Open image in new window and Open image in new window . By using (3.8), (3.9), (3.28), Lemmas 2.3, and 2.4, we have (noting that Open image in new window )
This shows that
Now, for Open image in new window , we have, from Lemma 2.2,
and hence
Therefore,
By using (3.8), (3.9), (3.35), Lemmas 2.3 and 2.4, we have (noting that Open image in new window )
and hence
This shows that for, each Open image in new window ,

Step 5.

Open image in new window .

To prove this, we pick a subsequence Open image in new window of Open image in new window such that

Without loss of generality, we may further assume that Open image in new window . Obviously, to prove Step 5, we only need to prove that Open image in new window .

Indeed, for each Open image in new window , since Open image in new window , Open image in new window and Open image in new window is nonexpansive, by demiclosed principle of nonexpansive mapping we have
Moreover, it follows from Lemma 2.7 that Open image in new window . Assume that Open image in new window . Then Open image in new window . Since Open image in new window for each Open image in new window , by Step 3, (3.44) and Opial's property of the Hilbert space Open image in new window , we have

which is a contradiction. Therefore, Open image in new window . Hence, Open image in new window .

Step 6.

The sequence Open image in new window strongly converges to some point Open image in new window .

By using Lemmas 2.3 and 2.4, we have
which implies that
where Open image in new window is an appropriate constant such that Open image in new window . Put
Then we have
It follows from the assumption condition (E1) and (3.42) that

Thus, applying Lemma 2.5 to (3.49), it follows that Open image in new window as Open image in new window . This completes the proof.

By Theorem 3.1, we have the following direct corollaries.

Corollary 3.2.

Let Open image in new window be a Hilbert space and Open image in new window be a nonempty closed convex subset of Open image in new window . Let Open image in new window be a contraction with coefficient Open image in new window , Open image in new window be strongly positive linear bounded self-adjoint operator with coefficient Open image in new window , Open image in new window    Open image in new window be a finite family of nonexpansive mappings, Open image in new window be a bifunction satisfying (A1)–(A4), and Open image in new window be an α-inverse strongly monotone mapping such that Open image in new window . Let Open image in new window and Open image in new window be two sequences in Open image in new window , Open image in new window be a sequence in Open image in new window with Open image in new window , Open image in new window be a number in Open image in new window , and Open image in new window be a sequence Open image in new window . Take a fixed number Open image in new window with Open image in new window . Assume that

(E1)   Open image in new window and Open image in new window ;

(E2)   Open image in new window for each Open image in new window ;

(E3)   Open image in new window for all Open image in new window ;

(E4)   Open image in new window with Open image in new window ;

(E5)   Open image in new window , Open image in new window , Open image in new window .

Then the sequence Open image in new window defined by (1.25) converges strongly to Open image in new window , which is the unique solution of the variational inequality:

Remark 3.3.

In the proof process of Theorem 3.1, we do not use Suzuki's Lemma (see [27]), which was used by many others for obtaining Open image in new window as Open image in new window (see [4, 5, 28]). The proof method of Open image in new window is simple and different with ones of others.

4. Applications for Multiobjective Optimization Problem

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. That is, we will give an iterative algorithm of solution for the following multiobjective optimization problem with the nonempty set of solutions:

where Open image in new window and Open image in new window are both the convex and lower semicontinuous functions defined on a closed convex subset of Open image in new window of a Hilbert space Open image in new window .

We denote by Open image in new window the set of solutions of the problem (4.1) and assume that Open image in new window . Also, we denote the sets of solutions of the following two optimization problems by Open image in new window and Open image in new window , respectively,

Note that, if we find a solution Open image in new window , then one must have Open image in new window obviously.

respectively. It is easy to see that Open image in new window and Open image in new window , where Open image in new window denotes the set of solutions of the equilibrium problem:
respectively. In addition, it is easy to see that Open image in new window and Open image in new window satisfy the conditions (A1)–(A4). Let Open image in new window be a sequence in (0,1) and Open image in new window . Define a sequence Open image in new window by

By Theorem 3.1 with Open image in new window , Open image in new window , Open image in new window and Open image in new window for all Open image in new window , the sequence Open image in new window converges strongly to a solution Open image in new window , which is a solution of the multiobjective optimization problem (4.1).

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Copyright information

© S.Wang and B. Guo. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.National Engineering Laboratory for Biomass Power Generation EquipmentNorth China Electric Power UniversityBaodingChina
  2. 2.Department of Mathematics and PhysicsNorth China Electric Power UniversityBaodingChina

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