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

Fixed Point Theory and Applications20102011:392741

https://doi.org/10.1155/2011/392741

Received: 15 October 2010

Accepted: 18 November 2010

Published: 2 December 2010

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.

1. Introduction

Let be a real Hilbert space and be a mapping of into itself. is said to be nonexpansive if
(1.1)

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

Let be a mapping. If there exists a constant such that
(1.2)
then is called a contraction with the constant . Recall that an operator is called to be strongly positive with coefficient if
(1.3)

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

In 2003, Xu [2] introduced the following iterative scheme:
(1.4)
where is the identical mapping on and , and proved some strong convergence theorems for the iterative scheme to the solution of the quadratic minimization problem
(1.5)
under suitable hypotheses on and the additional hypothesis:
(1.6)
Recently, Marino and Xu [3] introduced a new iterative scheme from an arbitrary point by the viscosity approximation method as follows:
(1.7)
and prove that the scheme strongly converges to the unique solution of the variational inequality:
(1.8)
which is the optimality condition for the minimization problem:
(1.9)

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

Let be a finite family of nonexpansive mappings of into itself. In 2007, Yao [4] defined the mappings
(1.10)
and, by extending (1.10), proposed the iterative scheme:
(1.11)
Then he proved that the iterative scheme (1.10) strongly converges to the unique solution of the variational inequality:
(1.12)
where , which is the optimality condition for the minimization problem:
(1.13)

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

Let be a nonempty closed convex subset of and be a bifunction. The equilibrium problem for the function is to determine the equilibrium points, that is, the set
(1.14)
Let be a nonlinear mapping. Let denote the set of all solutions to the following equilibrium problem:
(1.15)

In the case of , is deduced to . In the case of , is also denoted by .

In 2007, S. Takahashi and W. Takahashi [6] introduced a viscosity approximation method for finding a common element of and from an arbitrary initial element
(1.16)

and proved that, under certain appropriate conditions over and , the sequences and both converge strongly to .

By combing the schemes (1.7) and (1.16), Plubtieg and Punpaeng [7] proposed the following algorithm:
(1.17)
and proved that the iterative schemes and converge strongly to the unique solution of the variational inequality:
(1.18)
which is the optimality condition for the minimization problem:
(1.19)

where is a potential function for .

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:
(1.20)
and proved under some certain hypotheses that both sequences and converge strongly to a point which is an equilibrium point for and is the unique solution of the variational inequality:
(1.21)

where .

The equilibrium problems have been considered by many authors; see, for example, [6, 819] 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 ( ) be a finite family of nonexpansive mappings of into itself, be   be an infinite family of bifunctions, and be   be an infinite family of -inverse-strongly monotone mappings. Let be a sequence such that with for each . Define the mapping by
(1.22)
Assume that . For an arbitrary initial point , we define the iterative scheme by
(1.23)
where , , and are three sequences in (0,1), and are both strongly positive linear bounded operators on , is defined by (1.10), and prove that, under some certain appropriate hypotheses on the control sequences, the sequence strongly converges to a point , which is the unique solution of the variational inequality:
(1.24)
If , and , then (1.23) is reduced to the iterative scheme:
(1.25)

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 be a closed convex subset of a Hilbert space . For any point , there exists a unique nearest point in , denoted by , such that
(2.1)
Then is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies the following:
(2.2)
Let be a mapping from into , then is called monotone if
(2.3)
for all . However, is called an α-inverse-strongly monotone mapping if there exists a positive real number α such that
(2.4)
for all . Let denote the identity mapping of , then for all and , one has [20]
(2.5)

Hence, if , then is a nonexpansive mapping of into .

If there exists such that
(2.6)

for all , then is called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by .

In this paper, we need the following lemmas.

Lemma 2.1 (see [21]).

Given and . Then if and only if there holds the inequality
(2.7)

Lemma 2.2 (see [22]).

Let be a sequence of nonnegative real numbers satisfying
(2.8)

where , , and satisfy the conditions:

(1)   , or, equivalently, ;

(2)   ;

(3)      , .

Then .

Let be a Hilbert space. For all , the following equality holds:
(2.9)

Therefore, the following lemma naturally holds.

Lemma 2.3.

Let be a real Hilbert space. The following identity holds:
(2.10)

Lemma 2.4 (see [3]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 2.5 (see [2]).

Assume that is a sequence of nonnegative numbers such that
(2.11)

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

(1)   ;

(2)   or .

Then .

Lemma 2.6 (see [23]).

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

(A1)   for all ;

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

(A3)  For each ,

(2.12)

(A4)  For each , is convex and lower semicontinuous.

For and , define a mapping by
(2.13)

Then is well defined and the following hold:

(1)   is single-valued;

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

(2.14)

(3)   ;

(4)   is closed and convex.

It is easy to see that if there exists some point such that , where is an α-inverse strongly monotone mapping, then . In fact, since , one has
(2.15)
that is,
(2.16)

Hence, .

Let be a nonempty convex subset of a Banach space. Let be a finite family of nonexpansive mappings of into itself and be real numbers such that for each . Define a mapping of into itself as follows:
(2.17)

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

Lemma 2.7 (see [26]).

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

Lemma 2.8 (see [5]).

Let be a nonempty convex subset of a Banach space. Let be a finite family of nonexpansive mappings of into itself and let be sequences in such that for each . Moreover, for each , let and be the -mappings generated by and and and , respectively. Then, for all , it follows that
(2.18)

3. Main Results

Now, we give our main results in this paper.

Theorem 3.1.

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

(E1)  

(E2)   ;

(E3)   ;

(E4)   with ;

(E5)   .

Then the sequence defined by (1.23) converges strongly to , which is the unique solution of the variational inequality: (1.24), that is,
(3.1)

Proof.

Since as by the condition (E1), we may assume, without loss of generality, that for all . Noting that and are both the linear bounded self-adjoint operators, one has
(3.2)
Observing that
(3.3)
we obtain that is positive for all . It follows that
(3.4)
For each , define a quadratic function in as follows:
(3.5)
Note that
(3.6)
(3.7)
Hence, for each satisfying the condition (E4), one has
(3.8)
Moreover, it follows from (3.7), and (E4) that
(3.9)

Next, we proceed the proof with following steps.

Step 1.

is bounded.

Let . Lemma 2.6 shows that every is firmly nonexpansive and hence nonexpansive. Since , is nonexpansive for each . Therefore, is nonexpansive for each . Noting that is strictly decreasing, , we have
(3.10)
and hence
(3.11)
Then, from (3.4) and (3.11), it follows that (noting that is linear and )
(3.12)
It follows from and that . Therefore, by the simple induction, we have
(3.13)

which shows that is bounded, so is .

Step 2.

as .

First, we prove
(3.14)
Let and set
(3.15)
It follows from the definition of that
(3.16)
for each . Thus, using the above recursive inequalities repeatedly, we have
(3.17)
Also, we have
(3.18)

where .

Next, we prove . Observe (noting that is linear) that
(3.19)
Hence, by (3.4) and (3.18), we get
(3.20)

where .

Set . It follows from and (due to ) that . Thus we have
(3.21)
Set
(3.22)
Then it follows from (3.21) that
(3.23)
It follows from the assumption condition (E1), (E3), (E5), and (3.14) that
(3.24)

By applying Lemma 2.2 to (3.23), we obtain as .

Step 3.

as .

For all , we have
(3.25)
and hence (noting (3.9))
(3.26)
It follows from the assumption conditions (E1), (E2), and Step 2 that
(3.27)

Step 4.

as .

Notice that, for any ,
(3.28)
Let and . By using (3.8), (3.9), (3.28), Lemmas 2.3, and 2.4, we have (noting that )
(3.29)
This shows that
(3.30)
and hence, for each ,
(3.31)
Since , and , we have
(3.32)
Now, for , we have, from Lemma 2.2,
(3.33)
and hence
(3.34)
Therefore,
(3.35)
By using (3.8), (3.9), (3.35), Lemmas 2.3 and 2.4, we have (noting that )
(3.36)
and hence
(3.37)
This shows that for, each ,
(3.38)
Since is strictly decreasing, , , and , we have, for each ,
(3.39)
Now, from we get
(3.40)
Since and for each , one has
(3.41)

Step 5.

.

To prove this, we pick a subsequence of such that
(3.42)

Without loss of generality, we may further assume that . Obviously, to prove Step 5, we only need to prove that .

Indeed, for each , since , and is nonexpansive, by demiclosed principle of nonexpansive mapping we have
(3.43)
Assume that for each . Let be the -mapping generated by and . Then, by Lemma 2.8, we have
(3.44)
Moreover, it follows from Lemma 2.7 that . Assume that . Then . Since for each , by Step 3, (3.44) and Opial's property of the Hilbert space , we have
(3.45)

which is a contradiction. Therefore, . Hence, .

Step 6.

The sequence strongly converges to some point .

By using Lemmas 2.3 and 2.4, we have
(3.46)
which implies that
(3.47)
where is an appropriate constant such that . Put
(3.48)
Then we have
(3.49)
It follows from the assumption condition (E1) and (3.42) that
(3.50)

Thus, applying Lemma 2.5 to (3.49), it follows that as . This completes the proof.

By Theorem 3.1, we have the following direct corollaries.

Corollary 3.2.

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

(E1)   and ;

(E2)   for each ;

(E3)   for all ;

(E4)   with ;

(E5)   , , .

Then the sequence defined by (1.25) converges strongly to , which is the unique solution of the variational inequality:
(3.51)

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 as (see [4, 5, 28]). The proof method of 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:
(4.1)

where and are both the convex and lower semicontinuous functions defined on a closed convex subset of of a Hilbert space .

We denote by the set of solutions of the problem (4.1) and assume that . Also, we denote the sets of solutions of the following two optimization problems by and , respectively,
(4.2)
and
(4.3)

Note that, if we find a solution , then one must have obviously.

Now, let and be two bifunctions from to defined by
(4.4)
respectively. It is easy to see that and , where denotes the set of solutions of the equilibrium problem:
(4.5)
respectively. In addition, it is easy to see that and satisfy the conditions (A1)–(A4). Let be a sequence in (0,1) and . Define a sequence by
(4.6)

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

Authors’ Affiliations

(1)
National Engineering Laboratory for Biomass Power Generation Equipment, North China Electric Power University
(2)
Department of Mathematics and Physics, North China Electric Power University

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Copyright

© S.Wang and B. Guo. 2011

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