Existence theorems for perturbed variational inequalities in Banach spaces

Abstract

The aim of this paper is to establish existence and boundedness theorems for perturbed variational inequalities defined by a set-valued mapping without any kind of monotonicity in Banach spaces. The first result is shown that if a coercivity condition holds, then the solution set of a variational inequality perturbed along a direction is nonempty and uniformly bounded. Second, by employing the Minty variational inequalities perturbed by a nonlinear mapping without monotonicity, we prove the boundedness result for the corresponding perturbed variational inequalities under a kind of coercivity condition.