VARIATIONAL INEQUALITIES AND THE POSITIVITY OF MONOTONE MAPPINGS

Abstract

A necessary and sufficient condition for the existence of solutions of variational inequalities involving hemicontinuous mappings in a reflexive Banach space is given. For application of the criteria in the case of a Hilbert space a generalization of the Fredholm alternative is proved. Applications to partial differential equations with boundary restrictions in the form of inequalities are discussed. Breakdown of the positivity condition and lost of monotonicity are investigated.