Periodic solutions of ordinary differential equations with bounded nonlinearities

Abstract

In this article we discuss the existence and non-existence of forced $T$-periodic solutions to ordinary differential equations of the form $u^{\prime \prime }+g(u)=e(t)$. The results concern equations with bounded nonlinear terms $g$ satisfying $g(s)> 0$ (or $g(s)< 0$) for all real numbers $s$, and $g(\pm \infty )=0$. Variational and topological methods are employed.