On the periodic solutions of discontinuous piecewise differential systems

Motivated by problems coming from different areas of the applied science we study the periodic solutions of the following differential system $$x'(t)=F_0(t,x)+\varepsilon F_1(t,x)+\varepsilon^2 R(t,x,\varepsilon),$$ when \(F_0\), \(F_1\), and \(R\) are discontinuous piecewise functions, and \(\varepsilon\) is a small parameter. It is assumed that the manifold \(\mathbb{Z}\) of all periodic solutions of the unperturbed system \(x'=F_0(t,x)\) has dimension \(n\) or smaller then \(n\). The averaging theory is one of the best tools to attack this problem. This theory is completely developed when \(F_0\), \(F_1\) and \(R\) are continuous functions, and also when \(F_0=0\) for a class of discontinuous differential systems. Nevertheless does not exist the averaging theory for studying the periodic solutions of discontinuous differential system when \(F_0\neq0\). In this paper we develop this theory for a big class of discontinuous differential systems.