ABSTRACT
We begin with some notations needed later. If X is a Banach space, D is a closed subset in X and I is an interval, Cb(I;X) denotes the space of all bounded and continuous functions from I to X , equipped with the sup-norm ‖ · ‖Cb(I;X), while Cb(I;D) denotes the closed subset in Cb(I;X) consisting of all elements u ∈ Cb(I;X) satisfying u(t) ∈ D for each t ∈ I. Further, C([ a, b ];X) stands for the space of all continuous functions from [ a, b ] to X endowed with the sup-norm ‖ · ‖C([ a,b ];X) and C([ a, b ];D) is the closed subset of C([ a, b ];X) containing all u ∈ C([ a, b ];X) with u(t) ∈ D for each t ∈ [ a, b ]. If τ ≥ 0, X = C([−τ, 0 ];X). If σ ∈ R, u ∈ C([σ − τ,+∞);X) and t ∈ [σ,+∞), then ut ∈ X is defined by
ut(s) = u(t+ s)
for each s ∈ [−τ, 0 ]. If τ = 0, X identifies with X and ut identifies with u(t).
