In this paper we consider Aronszajn’s-type topological charac- terization (or compact $R_{\delta }$ property) of the set of solutions to the following integral equation $$x(t)=V(t,x({\theta }_1(t)),{\int }_0^{t}F(t,s,x({\theta }_2(s)),{\int }_0^{t}r(s,\tau )x({\theta }_3(\tau ))d\tau ))+{\int }_0^{t}K(t,s)g(s,x({\theta }_4(s)))ds,$$ where $t\in [0,\mathrm{\infty });$ ${\theta }_i:[0,\mathrm{\infty })\to [0,\mathrm{\infty }),$ $i=1,2,3,4;$ $K:[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to L(E,E);$ $V:[0,\mathrm{\infty })\times E\times E\to E$; $F:[0,\mathrm{\infty })\times [0,\mathrm{\infty })\times E\times E\to E$; $r:[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to R$; $g:[0,\mathrm{\infty })\times E\to E$; $E$ is a real Banach space with norm $|.|$; $L(E,E)$ the Banach space of continuous linear operators with domain $E$ and range in $E$.