Summary: We show the existence of solutions to a nonlinear singular second order ordinary differential equation,
subject to periodic boundary conditions, where is a given constant, is a parameter, and the nonlinearity satisfies the local Carathéodory conditions on . Here, we study the case that a well-ordered pair of lower and upper functions does not exist and therefore the underlying problem cannot be treated by well-known standard techniques. Instead, we assume the existence of constant lower and upper functions having opposite order. Analytical results are illustrated by means of numerical experiments.