A Viability Result in the Upper Semicontinuous Case

We prove the existence of solutions of a differential inclusion u′ ∈ F(t, u) in a separable Banach space X with constraint u(t) ∈ D(t). F is globally measurable, weakly upper semicontinuous with respect to u and takes convex, weakly compact values. D is upper semicontinuous from the left, and, for every r > 0, the sets D(t) ∩ r B are compact. F and D fulfil a well-known tangential condition, which is expressed by means of the Bouligand cone.