Some new viability results for semilinear differential inclusions

Abstract

Let X be a reflexive and separable Banach space, $ A : D(A) \subset X \rightarrow X $ the infinitesimal generator of a C 0-semigroup $ S(t) : X \rightarrow X, t \geq 0 $ , D a locally weakly closed subset in X and $ F : D \rightarrow 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 suffcient condition in order that for each $ \xi \in D $ there exists at least one mild solution u of¶¶ $ {du \over dt} (t) \in Au(t) + F(u(t)) $ ¶satisfying $ u(0) = \xi $ is the so called "bounded w-tangency condition" below. $ (Bw {\cal TC}) $ . There exists a locally bounded function $ {\cal M} : D \rightarrow {\bf R}^*_+ $ enjoying the property that for each $ \xi \in D $ there exists $ y \in F(\xi) $ such that for each $ \delta \gt 0 $ and each weak neighborhood V of 0 there exist $ t \in ({\rm 0},\delta ] $ and $ p \in V $ with $ \parallel p \parallel \leq {\cal M}(\xi) $ and satisfying¶¶ $ S(t)\xi + t(y + p) \in D. $