| Contents |
| We deal with nonlinear differential systems of the
form
\begin{equation*}
x'(t)=\sum_{i=0}^k\varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon),
\end{equation*}
where $F_i:\mathbb{R}\times D\rightarrow\mathbb{R}^n$ for $i=0,1,\cdots,k$, and
$R:\mathbb{R}\times D\times(-\varepsilon_0,\varepsilon_0)\rightarrow\mathbb{R}^n$ are continuous
functions, $T$--periodic in the first variable and Lipschitz in the second variable, being $D$ an
open subset of $\mathbb{R}^n$, and $\varepsilon$ a small parameter. For such
differential systems, which do not need to be of class $\mathcal{C}^1$, under
convenient assumptions we extend the averaging theory for computing
their periodic solutions to $k$--th order in $\varepsilon$. The main tool used is the Brouwer degree theory for finite dimensional spaces. |
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