A New Iterative Method for Solving Equilibrium Problems and Fixed Point Problems for Infinite Family of Nonexpansive Mappings
© Shenghua Wang et al. 2010
Received: 7 January 2010
Accepted: 11 July 2010
Published: 28 July 2010
We introduce a new iterative scheme for finding a common element of the solutions sets of a finite family of equilibrium problems and fixed points sets of an infinite family of nonexpansive mappings in a Hilbert space. As an application, we solve a multiobjective optimization problem using the result of this paper.
The set of solutions of the above inequality is denoted by . Many problems arising from physics, optimization, and economics can reduce to finding a solution of an equilibrium problem.
In 2007, S. Takahashi and W. Takahashi  first introduced an iterative scheme by the viscosity approximation method for finding a common element of the solutions set of equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem which is based on Combettes and Hirstoaga's result  and Wittmann's result . More precisely, they obtained the following theorem.
Theorem 1.1 (see ).
Let be a nonempty closed and convex subset of . Let be a bifunction which satisfies the following conditions:
(A1) for all ;
(A2) is monotone, that is, for all ;
(A4)For each , is convex and lower semicontinuous.
is the metric projection of onto and denotes nearest point in from .
The variational inequality problem is denoted by .
respectively. It is well known that if is strongly monotone and Lipschitzian on , then has a unique solution. An important problem is how to find a solution of . Recently, there are many results to solve the (see, e.g., [10–14]).
Lemma 2.5 (see below) shows that, for each , is firmly nonexpansive and hence nonexpansive and . Suppose that is a -Lipschitzian and -strong monotone operator and let . Assume that .
As an application of our main result, we solve a multiobjective optimization problem.
for all and with .
Let denote the identity operator of and let be a sequence in a Hilbert space and . Throughout this paper, denotes that strongly converges to and denotes that weakly converges to .
We need the following lemmas for our main results.
Lemma 2.1 (see ).
Lemma 2.2 (see [10, L mma 3.1(b)]).
Then for all , where .
If , Lemma 2.2 still holds.
Lemma 2.3 (see ).
If for all , where , then is a bounded sequence.
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Then the following holds:
(1) is single-valued;
(4) is closed and convex.
The following lemma is an immediate consequence of an inner product.
3. Main Results
First, we prove some lemmas as follows.
The sequence generated by (1.10) is bounded.
which shows that is bounded. This completes the proof.
Therefore, by Lemma 2.3, we have . This completes the proof.
then for each .
This completes the proof.
then for all
This completes the proof.
Next we prove the main results of this paper.
Then the sequence generated by (1.10) converges strongly to an element in , which is the unique solution of the variational inequality .
Therefore, by applying Lemma 2.3 to (3.35), we conclude that the sequence strongly converges to a point .
In order to prove the uniqueness of solution of the , we assume that is another solution of . Similarly, we can conclude that converges strongly to a point . Hence , that is, is the unique solution of . This completes the proof.
As direct consequences of Theorem 3.5, we obtain the following corollaries.
then the sequence converges strongly to an element .
Put and for each in Theorem 3.5. Then we know that is -Lipschitzian and -strongly monotone, and . Therefore, by Theorem 3.5, we conclude the desired result.
Put for each and . Set in Theorem 3.5. Then, by (2.6), we have . Therefore, by Theorem 3.5, we conclude the desired result.
Recently, many authors have studied the iteration sequences for infinite family of nonexpansive mappings. But our iterative sequence (1.10) is very different from others because we do not use -mapping generated by the infinite family of nonexpansive mappings and we have no any restriction with the infinite family of nonlinear mappings.
We do not use Suzuki's lemma  for obtaining the result that . However, many authors have used Suzuki's lemma  for obtaining the result that in the process of studying the similar algorithms. For example, see [5, 19, 20] and so on.
where and are both convex and lower semicontinuous functions defined on a nonempty closed and convex subset of of a Hilbert space . We denote by the set of solutions of (4.1) and assume that .
Obviously, if we find a solution , then one must have .
By Corollary 3.6, we know that the sequence converges strongly to a solution , which is a solution of the multiobjective optimization problem (4.1).
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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