Positive solutions of singular problems with sign changing Carathéodory nonlinearities depending on x (cid:1) ✩

We consider the singular boundary value problem for the differential equation x (cid:1)(cid:1) + f (t,x,x (cid:1) ) = 0 with the boundary conditions x( 0 ) = 0, w(x(T ),x (cid:1) (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 C 0 ( [ 0 ,T ] ) . The existence of positive solutions on ( 0 ,T ] in the classes AC 1 ( [ 0 ,T ] ) and C 0 ( [ 0 ,T ] ) ∩ AC 1loc (( 0 ,T ] ) is considered. Existence results are proved by combining the method of lower and upper functions with Leray–Schauder degree theory.