Averaging method for neutral differential equations in finite dimension (Q1958988)

This paper deals with the existence of periodic solutions for a class of neutral differential equations \[ y'(\tau)=\Phi(\frac{\tau}{\varepsilon}, y(\tau-h(\varepsilon)),\quad y'(\tau-h(\varepsilon)), \varepsilon),\quad \tau\in \mathbb{R}, \] where \(\Phi\) is a continuous map on \({\mathbb R}\times {\mathbb R}^{N}\times {\mathbb R}^{N}\times [0, 1]\), \(T\)-periodic in its first variable, Lipschitz in its third variable and \(h\) is an arbitrary map from \([0, 1]\) to \([0, \infty)\). Such equations with high periodic frequency term \(\tau/\varepsilon\) concern in physical applications the field of high frequency phenomena. A theorem on the existence of a periodic solution is proved by using the topological degree theory combined with the averaging method. An example is given to show the necessity of an auxiliary condition in the main theorem.