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

Agraïments: FEDER-UNAB-10-4E-378. The second author is partially supported by a FAPESP-BRAZIL grant 2013/16492-0. The two authors are also supported by a CAPES CSF-PVE grant 88881.030454/ 2013-01.For m = 1, 2, 3, we consider differential systems of the form x0 = F0(t, x) +Xmi=1εiFi(t, x) + εm+1R(t, x, ε), where Fi: R × D → Rn, and R : R × D × (−ε0, ε0) → Rn are Cm+1 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 x0 = 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 (ε) > 0 is sufficiently small