Weak and generic bang-bang properties for continuous evolution inclusions and Baire's method

We consider evolution inclusions, in a separable and reflexive Banach space , of the form and , where A is the infinitesimal generator of a C (0)-semigroup, F is a continuous and bounded multifunction defined on with values F(t, x) in the space of all closed convex and bounded subsets of with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C (0)-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).