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 ))+ϕ(x) = 0. Here f is a Carathéodory function on [0,T] × (0,∞) × 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 ϕ 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.  2003 Elsevier Science (USA). All rights reserved