### 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.

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.

#### Keywords

Singular nonlinear boundary value problems; positive solutions; periodic solutions

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