Existence and The Structure of The Solution Set for Hemivariational Inequalities Under Hybrid-Kind Monotonicity on Unbounded Domains

Abstract

We study the hemivariational inequality HVI( ) under -monotonicity, on closed convex subsets  in a reflexive Banach space. First, we give a concrete coercivity condition and use it to prove existence of a solution when  is unbounded. The proof truncates  into bounded balls, solves the problem on each ball, and shows the solutions stay away from the boundary once the ball is large enough. Second, we study the solution set . We show every solution lies in one fixed ball, so  is bounded. We show  is weakly closed, hence weakly compact. Third, under a strong monotonicity hypothesis,  is a single point, and compactness is then trivial. We show why norm-compactness of  does not follow from our hypotheses when  has more than one point, and we record this as an open problem.

2020 Mathematics Subject Classification: 49J40, 47H04, 47J20, 46B10.