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

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 kolumbán [d. inoan, j. kolumbán, on pseudomonotone set-valued mappings, nonlinear analysis 68 (2008) 47–53], 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.