Higher order averaging theory for finding periodic solutions via Brouwer degree

摘要

In this paper we deal with nonlinear differential systems of the form x'(t) =k ∑i=0 ε~i F~i (t, x) + ε~(k+1)R(t, x, ε), where F_i: R×D → R~n for i = 0, 1, . . ., k, and R: R×D×(?ε_0, ε_0) → R~n are continuous functions, and T -periodic in the first variable, D being an open subset of R~n, and ε a small parameter. For such differential systems, which do not need to be of class C~1, under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in ε. Some applications are also performed.