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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u'' = f(t,u) + g(t,u)u';{\text{ }}u(a + ) = 0,{\text{ }}u(b - ) = 0$$ \end{document} is considered, where the functions f and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g:{\text{ }}\left] {a,b} \right[ \times \left] {0, + \infty } \right[ \to \mathbb{R}$$ \end{document} satisfy the local Carathéodory 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.