Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.