Continuous selections of set of mild solutions of evolution inclusions

Abstract

We prove the existence of continuous selections of the set valued map ξ → S(ξ) where S(ξ) is the set of all mild solutions of the evolution inclusions of the form ẋ(t) ∈ A(t)x(t) + ∫t0 K(t, s) F(s, x(s))ds x(0) = ξ, t ∈ I = [0, T], where F is a lower semi continuous set valued map Lipchitzean with respect to x in a separate Banach space X, A is the infinitesimal generator of a C0-semi group of bounded linear operators from X to X, and K(t, s) is a continuous real valued function defined on I x I with t ≥ s for all t, s ∈ I and ξ ∈ X.