Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter E. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in epsilon). A result for third order averaging method is also given. As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given. (C) 2003 Elsevier SAS. All rights reserved.