Periodic solutions of second order superlinear singular dynamical systems (Q983678)

The purpose of the paper under review is to study the following system of differential equations \[ \ddot{x}+k^2x=f(t,x)+e(t), \] where \(0<k<\pi/T\), the functions \(f:(\mathbb{R}/T\mathbb{Z})\times\mathbb{R}^N\setminus\{0\}\to\mathbb{R}_+^N\) and \(e:\mathbb{R}/T\mathbb{Z}\to\mathbb{R}^N\) are continuous, while \(x:\mathbb{R}/T\mathbb{Z}\to\mathbb{R}^N\) is of class \({\mathcal C}^2\). The authors establish the existence of solutions \(x\) which are nontrivial (i.e., \(x(t)\neq 0\) for all \(t\)) provided that the system has a singularity of repulsive type at \(x=0\) (i.e., there exists \(v\in\mathbb{R}_+^N\) such that \(\lim_{x\to 0,\langle v,x\rangle\geq 0}\langle v,f(t,x)\rangle=+\infty\)) and \(f\) satisfies some superlinear conditions. The proof is based on a well-known fixed point theorem for completely continuous operators.