Viability for Semilinear Differential Inclusions via the Weak Sequential Tangency Condition

AbstractLet X be a reflexive and separable Banach space, A:D(A)⊂X→X the generator of a C0-semigroup S(t):X→X,t≥0,D a locally weakly sequentially closed subset in X, and F:D→2X 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 ξ∈D there exists at least one mild solution u ofdudtt∈Aut+Futsatisfying u(0)=ξ is the so-called “weak sequential tangency condition” below.(WSTC) For each ξ∈D there exists y∈F(ξ) and two sequences (tn)n∈Nin R+and (pn)n∈Nin X such that tn→0,pn→0 and satisfying S(tn)ξ+tn(y+pn)∈D