Existence Results for System of Variational Inequality Problems with Semimonotone Operators
- PDF / 145,523 Bytes
- 15 Pages / 600.05 x 792 pts Page_size
- 98 Downloads / 191 Views
Research Article Existence Results for System of Variational Inequality Problems with Semimonotone Operators Somyot Plubtieng and Kamonrat Sombut Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Somyot Plubtieng, [email protected] Received 14 July 2010; Accepted 29 October 2010 Academic Editor: Jozef Bana´s ´ Copyright q 2010 S. Plubtieng and K. Sombut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the system of variational inequality problems for semimonotone operators in reflexive Banach space. Using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of variational inequality problems for semimonotone with finitedimensional continuous operators in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results for variational inequality problems studied in recent years.
1. Introduction Let E be a Banach space, let E∗ be the dual space of E, and let ·, · denote the duality pairing of E∗ and E. If E is a Hilbert space and K is a nonempty, closed, and convex subset of E, then let, K be a nonempty, closed, and convex subset of a Hilbert space H and let A : K → H be a mapping. The classical variational inequality problem, denoted by VIPA, K, is to find x∗ ∈ K such that Ax∗ , z − x∗ ≥ 0
1.1
for all z ∈ K. The variational inequality problem VIP has been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as optimization theory, game theory, economic equilibrium, mechanics. In the last four decades, since the time of the celebrated Hartman Stampacchia theorem see 1, 2, solution existence of variational inequality and other related problems has become a basic research topic which continues to attract attention of researchers in applied mathematics see, e.g., 3–14 and the
2
Journal of Inequalities and Applications
references therein. Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the current interest in functional analysis. It is natural to consider a unified approach to these different problems; see, for example, 10, 11. Let K be a nonempty, closed, and convex subset of E and let A, B : K → E∗ be single valued. For the system of generalized variational inequality problem SGVIP, find x∗ , y∗ ∈ K × K such that Ay∗ , z − x∗ ≥ 0, ∗ Bx , z − y∗ ≥ 0,
∀z ∈ K, ∀z ∈ K.
1.2
There are many kinds of mappings in the literature of recent years; see, for example, 12, 13, 15–18. In 1999, Chen 19 introduced the concept of semimonotonicity for a single valued mapping, which occurred in the study of nonlinear partial differential equations of divergence type. Recently, Fang and Huang 20 introduced two classes
Data Loading...
