Continuity, compactness, fixed points, and integral equations

An integral equation, x(t) = a(t) − ∫ t −∞ D(t, s)g(x(s))ds with a(t) bounded, is studied by means of a Liapunov functional. There results an a priori bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer type. It is a very flexible fixed point theorem which enables us to show that the solution inherits properties of a(t), including periodic or almost periodic solutions in a Banach space.