Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: An averaging approach (Q1012387)

Consider the system of differential equations \[ {dx\over dt}=\varepsilon f(t,x),\tag{1} \] where \(f: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous and \(JT\)-periodic in the first variable, \(\varepsilon> 0\) is a small parameter. The goal of the authors is to establish the existence of \(T\)-periodic solutions of (1) by means of the averaged system \[ {dy\over dt}=\varepsilon f_0(x) \] and the topological degree of \(f_0\) with respect to some compact epi-Lipschitz subset \(k\) of \(\mathbb{R}^n\). Moreover, the authors the existence of \(T\)-periodic solutions of the system \[ {dx\over dt}= \varepsilon\varphi(t, x)+ \psi(t,x), \] where \(\varphi,\psi: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) are continuously differentiable and \(T\)-periodic in \(t\), by means of the linearized system and topological degree.