Viability for Semilinear Differential Inclusions via the Weak Sequential Tangency Condition

Abstract

Let X be a reflexive and separable Banach space, A: D( A) ⊂ X → X the generator of a C 0-semigroup S( t) : 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 above a necessary and sufficient condition in order that for each ξ ∈ D there exists at least one mild solution u of du dt t ∈ Au t + F u t satisfying u(0) = ξ is the so-called “weak sequential tangency condition” below. ( W S T C ) For each ξ ∈ D there exists y ∈ F(ξ) and two sequences ( t n ) n ∈ N in R + and ( p n ) n ∈ N in X such that t n → 0, p n → 0 and satisfying S( t n )ξ + t n ( y + p n ) ∈ D.