Multivalued evolution equations with nonlocal initial conditions in Banach spaces

We prove the existence of integral solutions to the nonlocal Cauchy problem u'(t) is an element of -Au(t) + F(t,u(t)), 0 <= t <= T; u(0) = g(u) in a Banach space X, where A : D (A) subset of X -> X is rn-accretive and such that -A generates a compact semigroup, F : [0, T] x X -> 2(x) has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and g : C ( [0, T]; <(D(A))over bar>) -> <(D(A))over bar>. The case when A depends on time is also considered.