Invariant sets for a class of perturbed differential equations of retarded type

LetX be a Banach space and leta, b, q be real numbers such thata,q>0. Denote byD a locally closed subset ofX. A necessary and sufficient condition for the existence of a mild solutionu∈C([a−q, b ₁],X),a ₁<b, to the differential equationdu(t)/dt=Au(t)+f(t, u _t), such thatu:[a,b ₁]→D, u _a=ϕ is given. The linear operatorA is the generator of aC ₀ semigroupT(t), t≧0, withT(t) compact fort>0,f: [a, b)×C([−q,0],D _λ)→X is continuous and ϕ∈C([−q,0],D _λ) with ϕ(0)∈D. D _λ is a neighbourhood ofD. Applications to parabolic partial differential equations with retarded argument are given.