Periodic BVPs in ODEs with time singularities

Abstract: In this paper we show the existence of solutions to a nonlinear singular second order ordinary differential equation, 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

Neumann problems with time singularities

AbstractIn this paper we study the existence and uniqueness of solutions to a nonlinear Neumann problem for a scalar second order ordinary differential equation u″=atu′+f(t,u,u′), where a<0, and f(t,x,y) satisfies the local Carathéodory conditions on [0,T]×R×R