Periodic solutions of singular second order differential equations: Upper and lower functions

In this paper, we continue the study of the periodic problem for the second-order equation u′′+f(u)u′+g(u)=h(t,u)u′′+f(u)u′+g(u)=h(t,u), where hh is a Carathéodory function and f,gf,g are continuous functions on (0,+∞)(0,+∞) which may have singularities at zero. Both attractive and repulsive singularities are considered. The method relies on a novel technique of construction of lower and upper functions. As an application, we obtain new sufficient conditions for the existence of periodic solutions to the Rayleigh–Plesset equation.