An existence theorem for generalized variational inequalities with discontinuous and pseudomonotone operators

Abstract

This paper gives a solution existence theorem for a generalized variational inequality problem with an operator which is defined on an infinite dimensional space, which is C-pseudomonotone in the sense of Inoan and Kolumbn [D. Inoan, J. Kolumbn, On pseudomonotone set-valued mappings, Nonlinear Analysis 68 (2008) 4753], but which may not be upper semicontinuous on finite dimensional subspaces. The proof of the theorem provides a new technique which reduces infinite variational inequality problems to finite ones. Two examples are given and analyzed to illustrate the theorem. Moreover, an example is presented to show that the C-pseudomonotonicity of the operator cannot be omitted in the theorem. © 2010 Elsevier Ltd. All rights reserved.