On a two-point boundary value problem for second order singular equations

The problem on the existence of a positive in the interval]a, b[ solution of the boundary value problem u" = f(t, u) + g(t, u) u'; u(a+) = 0, u(b-) = 0 is considered, where the functions f and g:]a, b[ x]0, +infinity[ --> R satisfy the local Caratheodory conditions. The possibility for the functions f and g to have singularities in the first argument (for t = a and t = b) and in the phase variable (for u = 0) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.