Neumann problems with time singularities

In 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<0a<0, and f(t,x,y)f(t,x,y) satisfies the local Carathéodory conditions on [0,T]×R×R[0,T]×R×R.