© 2014, Springer Basel. For m=1,2,3, we consider differential systems of the form (Formula Presented.) functions, and T -periodic in the first variable, being D an open subset of Rn, and ε a small parameter. For such system, we assume that the unperturbed system x′ = F0(t, x) has a k-dimensional manifold of periodic solutions with k ≤ n. We weaken the sufficient assumptions for studying the periodic solutions of the perturbed system when (Formula Presented.) is sufficiently small.