Periodic BVPs in ODEs with time singularities

In this paper we show the existence of solutions to a nonlinear singular second order ordinary differential equation, u(t) = a t u(t) + λf(t, u(t), u(t)), subject to periodic boundary conditions, where a > 0 is a given constant, λ > 0 is a parameter, and the nonlinearity f(t, x, y) satisfies the local Carathéodory conditions on [0, T ] × R × R. 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.