Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Abstract

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game);The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X).In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset;the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X).If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).