Functions without exceptional family of elements and the solvability of variational inequalities on unbounded sets

Abstract

In this paper we prove an alternative existence theorem for variational inequalities defined on an unbounded set in a Hilbert space. This theorem is based on the concept of exceptional family of elements (EFE) for a mapping and on the concept of $(0,k)$-epi mapping which is similar to the topological degree. We show that when a $k$-set field is without (EFE) then the variational inequality has a solution. Based on this result we present several classes of mappings without (EFE).