A VARIATIONAL INEQUALITY THEORY FOR CONSTRAINED PROBLEMS IN REFLEXIVE BANACH SPACES

Let X be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space X*, and let K be a nonempty, closed, and convex subset of X. Let T : X superset of D(T) -> 2(X)* be maximal monotone, let S : K -> 2(X)* be bounded and of type (S+), and let C : X superset of D(C) -> X* with D(T) boolean AND D(partial derivative phi) boolean AND K subset of D(C). Let phi : X -> (-infinity, infinity] be a proper, convex, and lower semicontinuous function. New existence theorems are proved for solvability of variational inequality problems of the type VIP(T + S + C, K, phi, f*) if C is compact and VIP(T + C, K, phi, f*) if T is of compact resolvent and C is bounded and continuous. Various improvements and generalizations of the existing results for T + S and phi are obtained. The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems.