Existence Results for System of Variational Inequality Problems with Semimonotone Operators

Somyot Plubtieng,Kamonrat Sombut

doi:10.1155/2010/251510

Abstract

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 finite-dimensional 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.

Highlights

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, let, K be a nonempty, closed, and convex subset of a Hilbert space H and let A : K → H be a mapping
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

Summary

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. Fang and Huang 20 introduced two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, they obtained the existence of solutions for variational-like inequalities with relaxed monotone mappings in reflexive Banach spaces. In 2002, Kassay et al introduced and studied Minty and Stampacchia variational inequality systems by the Kakutani-Fan-Glicksberg fixed point theorem. Zhao and Xia 24 introduced and established some existence results for systems of vector variational-like inequalities in Banach spaces by using the Kakutani-Fan-Glicksberg fixed point theorem. By using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of generalized variational inequality for semimonotone and finite dimensional continuous in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results of Chen 19 and many others

Preliminaries

The Existence of the System of Generalized Variational Inequality