On the System of Nonlinear Mixed Implicit Equilibrium Problems in Hilbert Spaces
© Y. J. Cho and N. Petrot. 2010
Received: 22 December 2009
Accepted: 10 January 2010
Published: 17 January 2010
We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems (SMIE) in Hilbert spaces. The algorithm for finding a solution of the problem (SMIE) is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.
1. Introduction and Preliminaries
Let be a real Hilbert space whose inner product and norm are denoted by and respectively. Let be given two bi-functions satisfying for all and . Let be a nonlinear mapping. Let be a nonempty closed convex subset of . In this paper, we consider the following problem.
The problem of type (1.1) is called the system of nonlinear mixed implicit equilibrium problems.
which is called the system of variational inclusion problems. In particular, when and the problem (1.2) is reduced to the problem, so-called the generalized variational inclusion problem, which was studied by Kazmi and Bhat .
then the problem (1.4) reduces to the following problem.
which is called the system of nonlinear variational inequalities problems. Some corresponding results to the problem (1.6) were studied by Agarwal et al. , Chang et al. , Cho et al. , J. K. Kim and D. S. Kim,  and Verma [8, 9].
which is known as the classical variational inequality introduced and studied by Stampacchia  in 1964. This shows that a number of classes of variational inequalities and related optimization problems can be obtained as special cases of the system (1.1) of mixed equilibrium problems.
Inspired and motivated by the recent research going on in this area, in this paper, we use the Wiener-Hopf equations and the Yosida approximation notion to suggest and prove the existence and uniqueness of solutions for the problem (1.1). We also discuss the convergence criteria and stability of the iterative algorithm. The results presented in this paper improve and generalize many known results in the literature.
In the sequel, we need the following basic concepts and lemmas.
Definition 1.1 (Blum and Oettli ).
Lemma 1.3 (Combettes and Hirstoaga ).
Definition 1.4 is an extension of the Yosida approximation notion introduced in . The existence and uniqueness of the solution of the problem (1.15) follow from Lemma 1.3.
Example 1.7 (Huang et al. ).
Using the idea as in Huang et al. , we have the following result.
In this paper, we are interested in the following class of nonlinear mappings.
2. Existence of Solutions of the Problem (1.1)
In this section, we give an existence theorem of solutions for the problem (1.1). Firstly, in view of Lemma 1.9, we can obtain the following, which is an important tool, immediately.
Now, we are in position to prove the existence theorem of solutions for the problem (1.1).
3. Convergence and Stability Analysis
In view of Lemma 2.1, for the fixed point formulation of the problem (2.1), we suggest the following iterative algorithm.
3.1. Mann Type Perturbed Iterative Algorithm (MTA)
In order to consider the convergence theorem of the sequences generated by the algorithm (MTA), we need the following lemma.
Now, we prove the convergence theorem for a solution for the problem (1.1).
3.2. Stability of the Algorithm (MTA)
Consider the following definition as an extension of the concept of stability of the iterative procedure given by Harder and Hicks .
Definition 3.3 (Kazmi and Khan ).
This completes the proof.
The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050). The second author was supported by the Commission on Higher Education and the Thailand Research Fund (project no. MRG5180178).
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