Positive solutions of singular problems with sign changing Carathéodory nonlinearities depending on x′

We consider the singular boundary value problem for the differential equation x″+f(t,x,x′)=0 with the boundary conditions x(0)=0, w(x(T),x′(T))+?(x)=0. Here f is a Carathéodory function on which may by singular at the value x=0 of the phase variable x and f may change sign, w is a continuous function, and ? is a continuous nondecreasing functional on C⁰(0,T]). The existence of positive solutions on (0,T] in the classes AC¹(0,T]) and C⁰(0,T])∩AC¹_loc((0,T]) is considered. Existence results are proved by combining the method of lower and upper functions with Leray-Schauder degree theory.