Existence of periodic solutions for differential equations with multiple delays under dichotomy condition

Using Krasnoselskii's fixed point theorem and dichotomy theory, we prove the existence of periodic solutions for differential equations with multiple delays of the form x' (t) + cx' (t - tau) = A(t)x(t) + f(t, x(t - alpha(1)(t)), ... , x(t - alpha(m)(t))), where the parameter c << 1 is a small perturbation for a delayed forced term. Moreover, we discuss the convergence of these solutions to a solution of the unperturbed problem as c -> 0.