FUNCTION-SPACE FLOW INVARIANCE FOR FUNCTIONAL-DIFFERENTIAL EQUATIONS OF RETARDED TYPE

Let C denote the Banach space of continuous functions ϕ: [- r, 0] → ℝn, let Ω ∈ C be closed, and let f: [0, ∞) × Ω → ℝn be continuous. In this note we establish necessary and sufficient conditions for function space flow invariance for the functional differential equation: X′(t) = f(t, xt) for t ≥ 0, x0 = ϕ ∈ Ω. That is, for each ϕ ∈ Ω there exist b > 0 and a solution x: [- r, b] → ℝn such that xt ∈ Ω for each t ∈ [0, b]. © 1979 American Mathematical Society.