Some new viability results for semilinear differential inclusions

Let X be a reflexive and separable Banach space, A : D(A) subset of X -> X the infinitesimal generator of a C-0-semigroup S(t) : X -> X, t >= 0, D a locally weakly closed subset in X and F : D -> 2(X) a nonempty, closed, convex and bounded valued mapping which is weakly-weakly upper semi-continuous. The main result of the paper is: Theorem. Under the general assumptions above a necessary and sufficient condition in order that for each xi is an element of D there exists at least one mild solution u of du/dt (t) is an element of Au(t) + F(u(t)) satisfying u(0) = xi is the so called "bounded w-tangency condition" below. (BwTC) There exists a locally bounded function M : D -> R+* enjoying the property that for each xi is an element of D there exists y is an element of F(xi) such that for each each delta > 0 and each weak neighborhood V of 0 there exist t is an element of (0; delta] and p is an element of V with parallel to p parallel to <= M(xi) and satisfying S(t)xi + t(y + p) is an element of D.