Singular mixed boundary value problem

We study singular boundary value problems with mixed boundary conditions of the formu″+f(t,u,u′)=0,u′(0)=0,u(T)=0, where [0,T]⊂R[0,T]⊂R, D=(0,∞)×(−∞,0)D=(0,∞)×(−∞,0), f   is a nonnegative function and satisfies the Carathéodory conditions on (0,T)×D(0,T)×D. Here, f   can have a time singularity at t=0t=0 and/or t=Tt=T and a space singularity at x=0x=0 and/or y=0y=0. We present conditions for the existence of solutions positive on [0,T)[0,T) and having continuous first derivatives on [0,T][0,T].