Periodic solutions to singular second order differential equations: the repulsive case

Abstract

This paper is devoted to study the existence of periodic solutions to the second-order differential equation $u''+f(u)u'+g(u)=h(t,u)$, where $h$ is a Carathéodory function and $f,g$ are continuous functions on $(0,\infty)$ which may have singularities at zero. The repulsive case is considered. By using Schaefer's fixed point theorem, new conditions for existence of periodic solutions are obtained. Such conditions are compared with those existent in the related literature and applied to the Rayleigh-Plesset equation, a physical model for the oscillations of a spherical bubble in a liquid under the influence of a periodic acoustic field. Such a model has been the main motivation of this work.