Bifurcations from families of periodic solutions in piecewise differential systems

Consider a differential system of the form $$x^{\mathrm{\prime }}=F_0(t,x)+\sum _{i=1}^k{\epsilon }^i{F}_i(t,x)+{\epsilon }^{k+1}R(t,x,\epsilon ),$$ 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$arepiecewise$C^{k+1}$functionsand$T$-periodicinthevariable$t$. Assuming that the unperturbed system $x^{\mathrm{\prime }}=F_0(t,x)$ 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.