Properties of the solution map

Abstract

In the study of retarded functional differential equations, the space of initial functions is preassigned, but the space in which one considers the trajectories is not. To be more specific, if x(σ, ϕ, f) is a solution of a RFDE(f) through (σ, ϕ) should the solution map be considered as the map x(σ, •‚ f)(t): C → ℝ ⁿ or the map T(t, σ):C → C defined in Section 2.5 as T(t, σ)ϕ = x _t (σ,ϕ, f)?