The existence of positive solutions of a second order differential equation of the form

(formula here)

with the separated boundary conditions: αz(0) − βz′(0) = 0 and γz(1)+δz′(1) = 0 has proved to be important in physics and applied mathematics. For example, the Thomas–Fermi equation, where f = z3/2 and g = t−1/2 (see [12, 13, 24]), so g has a singularity at 0, was developed in studies of atomic structures (see for example, [24]) and atomic calculations [6]. The separated boundary conditions are obtained from the usual Thomas–Fermi boundary conditions by a change of variable and a normalization (see [22, 24]). The generalized Emden–Fowler equation, where f = zp, p > 0 and g is continuous (see [24, 28]) arises in the fields of gas dynamics, nuclear physics, chemically reacting systems [28] and in the study of multipole toroidal plasmas [4]. In most of these applications, the physical interest lies in the existence and uniqueness of positive solutions.