Higher order averaging theory for finding periodic solutions via Brouwer degree

In this paper we deal with nonlinear differential systems of the form x^′(t)=∑_i=0^kεⁱF_i(t,x)+ε^k+1R(t,x,ε), where F_i:R×D→Rⁿ for i = 0, 1, …, k, and R:R×D×(−ε₀,ε₀)→Rⁿ are continuous functions, and T-periodic in the first variable, D being an open subset of Rⁿ, and ε a small parameter. For such differential systems, which do not need to be of class C¹, under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in ε. Some applications are also performed. (paper)