Viability for Semilinear Differential Inclusions via the Weak Sequential Tangency Condition

Let X be a reflexive and separable Banach space, A:DŽA. X X the generator of a C0-semigroup SŽt.: X X, t 0, D a locally weakly sequentially 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 abo e a necessary and sufficient condition in order that for each D there exists at least one mild solution u of du Žt. AuŽt. FŽuŽt.. satisfying uŽ0. is the so-called ‘‘weak sequential tan- dt gency condition’’ below. ŽW STC.For each D there exists y FŽ .and two sequencesŽtn.n in andŽpn.n in X such that tn 0, pn 0 and satisfying SŽtn. tnŽy pn. D.