Abstract

In this paper we consider Aronszajn’s-type topological characterization (or compact Rδ property) of the set of solutions to the following integral equation x(t) = V  t, x(θ1(t)), Rt 0 F t, s, x(θ2(s)), Rs 0 r(s, τ )x(θ3(τ ))dτ
ds + Rt 0 K(t, s)g(s, x(θ4(s)))ds where t ∈ [0, ∞); θi : [0, ∞) → [0, ∞), i = 1, 2, 3, 4 ; K : [0, ∞) × [0, ∞) → L(E, E); V : [0, ∞) × E × E → E; F : [0, ∞) × [0, ∞) × E × E → E; r : [0, ∞) × [0, ∞) → R; g : [0, ∞) × E → E; E is a real Banach space with norm |.|; L(E, E) the Banach space of continuous linear operators with domain E and range in E.