Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity

The aim of this paper is to establish existence results for some variational-like inequality problems involving set-valued maps, in reflexive and nonreflexive Banach spaces. When the set K, in which we seek solutions, is compact and convex, we no dot impose any monotonicity assumptions on the set-valued map A, which appears in the formulation of the inequality problems. In the case when K is only bounded, closed, and convex, certain monotonicity assumptions are needed: We ask A to be relaxed η-α monotone for generalized variational-like inequalities and relaxed η-α quasimonotone for variational-like inequalities. We also provide sufficient conditions for the existence of solutions in the case when K is unbounded, closed, and convex.