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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 Applications volume 2011, Article number: 392741 (2011)
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
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
then is called a contraction with the constant . Recall that an operator is called to be strongly positive with coefficient if
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:
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
under suitable hypotheses on and the additional hypothesis:
Recently, Marino and Xu [3] introduced a new iterative scheme from an arbitrary point by the viscosity approximation method as follows:
and prove that the scheme strongly converges 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., for all ).
Let be a finite family of nonexpansive mappings of 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 of the variational inequality:
where , which is the optimality condition for the minimization problem:
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
Let be a nonlinear mapping. Let denote the set of all solutions to the following equilibrium problem:
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
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:
and proved that the iterative schemes and converge strongly to the unique solution of the variational inequality:
which is the optimality condition for the minimization problem:
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:
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:
where .
The equilibrium problems have been considered by many authors; see, for example, [6, 8–19] 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 () 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
Assume that . For an arbitrary initial point , we define the iterative scheme by
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:
If , and , 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 be a closed convex subset of a Hilbert space . For any point , there exists a unique nearest point in , denoted by , such that
Then is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies the following:
Let be a mapping from into , then is called monotone if
for all . However, is called an α-inverse-strongly monotone mapping if there exists a positive real number α such that
for all . Let denote the identity mapping of , then for all and , one has [20]
Hence, if , then is a nonexpansive mapping of into .
If there exists such that
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
Lemma 2.2 (see [22]).
Let be a sequence of nonnegative real numbers satisfying
where , , and satisfy the conditions:
(1)  , or, equivalently, ;
(2)  ;
(3)    , .
Then .
Let be a Hilbert space. For all , the following equality holds:
Therefore, the following lemma naturally holds.
Lemma 2.3.
Let be a real Hilbert space. The following identity holds:
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
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 ,
(A4)  For each , is convex and lower semicontinuous.
For and , define a mapping by
Then is well defined and the following hold:
(1)   is single-valued;
(2)   is firmly nonexpansive, that is, for any ,
(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
that is,
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:
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
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,
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
Observing that
we obtain that is positive for all . It follows that
For each , define a quadratic function in as follows:
Note that
Hence, for each satisfying the condition (E4), one has
Moreover, it follows from (3.7), and (E4) that
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
and hence
Then, from (3.4) and (3.11), it follows that (noting that is linear and )
It follows from and that . Therefore, by the simple induction, we have
which shows that is bounded, so is .
Step 2.
as .
First, we prove
Let and set
It follows from the definition of that
for each . Thus, using the above recursive inequalities repeatedly, we have
Also, we have
where .
Next, we prove . Observe (noting that is linear) that
Hence, by (3.4) and (3.18), we get
where .
Set . It follows from and (due to ) that . Thus we have
Set
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 as .
Step 3.
as .
For all , we have
and hence (noting (3.9))
It follows from the assumption conditions (E1), (E2), and Step 2 that
Step 4.
as .
Notice that, for any ,
Let and . By using (3.8), (3.9), (3.28), Lemmas 2.3, and 2.4, we have (noting that )
This shows that
and hence, for each ,
Since , and , we have
Now, for , 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 )
and hence
This shows that for, each ,
Since is strictly decreasing, , , and , we have, for each ,
Now, from we get
Since and for each , one has
Step 5.
.
To prove this, we pick a subsequence of such that
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
Assume that for each . Let be the -mapping generated by and . Then, by Lemma 2.8, we have
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
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
which implies that
where is an appropriate constant such that . 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 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:
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:
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,
and
Note that, if we find a solution , then one must have obviously.
Now, let and be two bifunctions from to defined by
respectively. It is easy to see that and , where denotes the set of solutions of the equilibrium problem:
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
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).
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Wang, S., Guo, B. 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 Appl 2011, 392741 (2011). https://doi.org/10.1155/2011/392741
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DOI: https://doi.org/10.1155/2011/392741