The existence of mild solutions to the non-convex Cauchy problem
is investigated. Here A is the infinitesimal generator of a C0-semigroup in a reflexive and separable Banach space , F is a Pompeiu–Hausdorff lower semicontinuous multifunction whose values are closed convex and bounded sets with non-empty interior contained in , 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.