Improving the averaging theory for computing periodic solutions of the differential equations

For we consider differential systems of the form x' = F-0(t,x) + Sigma(m)(i=1) epsilon F-i(i)(t,x) + epsilon Rm+1(t,x,epsilon), where F-i: R x D -> R-n and R : R x D x(-epsilon(0), epsilon(0)) -> R-n are c(m+1) functions, and T-periodic in the first variable, being D an open subset of R-n, and epsilon a small parameter. For such system, we assume that the unperturbed system x' = F-0(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 vertical bar epsilon vertical bar > 0 is sufficiently small.