Positive solutions of singular problems with sign changing Caratheodory 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)) + phi (x) = 0. Here f is a Carathdodory function on [0, T] x (0, infinity) x R 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 phi is a continuous nondecreasing functional on C-0 ([0, T]). The existence of positive solutions on (0, T] in the classes AC(1) ([0, T]) and C-0([0, T]) boolean AND AC(loc)(1)((0, T]) is considered. Existence results are proved by combining the method of lower and upper functions with Leray-Schauder degree theory. (C) 2003 Elsevier Science (USA). All rights reserved.