Variational inequalities and complementarity problems

We shall consider variational inequalities for multivalued mappings to unify mathematical programming problems and extended fixed point problems.Let X and Y be two real separated topological vector spaces with a given bilinear form ·, · of Y× X into the reals R .Let T be a multivalued mapping from its domain D(T)⊂ X to subsets of $Y,$ f a function from X to R .Under these conditions a solution of a variational inequality is the following; x₀∈ $D(T)$ and w₀∈ T(x₀) such that w₀, x-x₀ f(x₀)-f(x) for all x∈ D(T) .When $D(T)$ is a cone, variational inequalities are related to complementarity problems.Variational inequalities in infinite dimensional spaces were studied by Browder [1], Karamardian [5] and others.Karamardian [5] also considered complementarity problems, for which we also refer to Mor\'e [6].In this paper we shall give two existence theorems.Using them, we shall solve variational inequalities for multivalued mappings on closed convex subsets in topological vector spaces.Then the results are applied to complementarity problems.