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.
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 ).
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]).
As an application of our main result, we solve a multiobjective optimization problem.
We need the following lemmas for our main results.
Lemma 2.1 (see ).
Lemma 2.2 (see [10, L mma 3.1(b)]).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Then the following holds:
The following lemma is an immediate consequence of an inner product.
3. Main Results
First, we prove some lemmas as follows.
This completes the proof.
This completes the proof.
Next we prove the main results of this paper.
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.
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.
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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