Singular mixed boundary value problem

We study singular boundary value problems with mixed boundary conditions of the form u + f (t,u,u ) = 0, u (0) = 0, u(T)= 0, where [0,T] ⊂ R, D = (0,∞)×(−∞, 0), f is a nonnegative function and satisfies the Carathéodory conditions on (0,T ) × D. Here, f can have a time singularity at t = 0 and/or t = T and a space singularity at x = 0 and/or y = 0. We present conditions for the existence of solutions positive on [0,T ) and having continuous first derivatives on [0,T ]. © 2005 Elsevier Inc. All rights reserved.