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.