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.
Find such that
The problem of type (1.1) is called the system of nonlinear mixed implicit equilibrium problems.
We denote by SMIE the set of all solutions of the problem (1.1).
If where is a maximal monotone mapping for then the problem (1.1) becomes the following problem.
Find such that
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 .
If for all , where is a real valued function for each Then the problem (1.1) reduces to the following problem.
Find such that
For each , let be a nonlinear mapping and fixed positive real numbers. If and for all , then the problem (1.3) reduces to the following problem.
Find such that
If for all , where is the indicator function of defined by
then the problem (1.4) reduces to the following problem.
Find such that
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].
If , and is a univariate mapping, then the problem (1.6) reduces to the following problem.
Find such that
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 ).
A real valued bifunction is said to be:
Lemma 1.3 (Combettes and Hirstoaga ).
Let be a nonempty closed convex subset of and be a bifunction of into satisfying the following conditions:
(C1) is monotone and upper hemicontinuous;
(C2) is convex and lower semi-continuous for all .
Then is a single-valued mapping.
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.
Let be a set-valued mapping.
Example 1.7 (Huang et al. ).
where is the Yosida approximation of and we recover the classical concepts.
Using the idea as in Huang et al. , we have the following result.
This implies that is a nonexpansive mapping. This completes the proof.
Now, for solving the problem (1.1), we consider the following equation: let and be fixed positive real numbers. Find such that
The proof directly follows from the definitions of and .
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).
Then is a singleton.
By the condition (2.3), we have , which implies that is a contraction mapping. Hence, by Banach contraction principle, there exists a unique such that This completes the proof.
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)
For any , compute approximate solution given by the iterative schemes:
where is a sequence of real numbers such that and
In order to consider the convergence theorem of the sequences generated by the algorithm (MTA), we need the following lemma.
where with and . Then
Now, we prove the convergence theorem for a solution for the problem (1.1).
If all the conditions of the Theorem 2.2 hold, then the sequence in generated by the algorithm (3.1) converges strongly to the unique solution for the problem (1.1).
Thus the sequence in converges strongly to a solution for the problem (1.1). This completes the proof.
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 ).
If implies that , then the iterative procedure is said to be -stable or stable with respect to .
where is a sequence defined in (3.1). If is defined as in (2.1), then the iterative procedure generated by (3.1) is -stable.
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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