MinSup Problems with Quasi-equilibrium Constraints and Viscosity Solutions

MinSup problems with constraints described by quasi-equilibrium problems are considered in Banach spaces. The solutions set of such problems may be empty even in very good situations, so the aim of this paper is twofold. First, we determine appropriate regularizations which allow to asymptotically reach the value of the original problem. Then, among these regularizations we identify those which allow to bypass the lack of exact solutions to these problems by a suitable concept of viscosity solution whose existence is then proved under reasonable assumptions.