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 ) , where h is a Caratheodory function and f , g are continuous functions on ( 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.