  Consider a differential system of the form $$ x'=F_0(t,x)+\sum_{i=1}^k
\varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon), $$ where
$F_i:\mathbb{S}^1 \times D \to \mathbb{R}^m$ and $R:\mathbb{S}^1 \times D
\times (-\varepsilon_0,\varepsilon_0) \to \mathbb{R}^m$ are piecewise $C^{k+1}$
functions and $T$-periodic in the variable $t$. Assuming that the unperturbed
system $x'=F_0(t,x)$ has a $d$-dimensional submanifold of periodic solutions
with $d<m$, we use the Lyapunov-Schmidt reduction and the averaging theory to
study the existence of isolated $T$-periodic solutions of the above
differential system.
