- Research Article
- Open Access
Strong Convergence Theorems for Countable Lipschitzian Mappings and Its Applications in Equilibrium and Optimization Problems
© L. Yang and Y. Su. 2009
- Received: 21 October 2008
- Accepted: 5 March 2009
- Published: 17 March 2009
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Further, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the results of W. Nilsrakoo and S. Saejung (2008) and some others in some respects.
- Variational Inequality
- Mathematical Programming
- Hybrid Method
- Equilibrium Problem
- Nonexpansive Mapping
where denotes the metric projection from onto a closed convex subset of . The iterative method (1.3) is said to be hybrid method or method. In recent years, the hybrid method (1.3) has been modified by many authors for other nonlinear operators [2, 5–10].
In 2008, Nilsrakoo and Saejung  used the hybrid method to obtain a strong convergence theorem for countable Lipschitzian mappings as follows.
Nilsrakoo and Saejung also apply the aforementioned result to obtain an applied result for equilibrium problems.
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the Nilsrakoo and Saejung results in some respects.
The following lemma is well known.
Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence of -Lipschitzian mappings from into itself with . is said to satisfy the (SU) condition, if the following conditions hold:
The results (1)–(3) are easy to prove.
The following theorem directly follows from Theorem 2.1.
The set of solutions of (3.1) is denoted by . Given a mapping , let , for all . Then, if and only if , for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (3.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [12–16].
We need the following lemmas for the proof of our main results.
Now, we prove the following lemma which is very important for the main results of this section.
Now, we study a kind of optimization problem by using the aforementioned results of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
It is obvious that , where denotes the set of solutions of equilibrium Problem (3.25). In addition, it is easy to see that satisfies the conditions (A1)–(A4) in Section 2. Therefore, from Theorem 3.5, we can obtain the following theorem.
Remark 3.7 ..
It is easy to see that this paper hassome new methods and results than the results of Nilsrakoo and Saejung :
(1)proposed a modified hybrid iterative scheme, so that the new simple method of proof has been used;
(4)give an application for optimization problem;
This project is supported by the National Natural Science Foundation of China under grant(10771050).
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