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)=atu′(t)+λf(t,u(t),u′(t)), subject to periodic boundary conditions, where a>0a>0 is a given constant, λ>0λ>0 is a parameter, and the nonlinearity f(t,x,y)f(t,x,y) satisfies the local Carathéodory conditions on [0,T]×R×R[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.