Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

Abstract

AbstractThe existence of mild solutions to the non-convex Cauchy problemis investigated. Here A is the infinitesimal generator of a C0-semigroup in a reflexive and separable Banach space $\mathbb{E}$, F is a Pompeiu–Hausdorff lower semicontinuous multifunction whose values are closed convex and bounded sets with non-empty interior contained in $\mathbb{E}$, and ∂F(t, x(t)) denotes the boundary of F(t, x(t)). Our approach is based on the Baire category method, with appropriate modifications which are actually necessary because, under our assumptions, the underlying metric space that naturally enters in the Baire method, i.e. the solution set of the convexified Cauchy problem (CF), can fail to be a complete metric space.