Two-Point Boundary Value Problems for Essentially Singular Nonlinear Second-Order Differential Equations

We establish new tests for the solvability and unique solvability of two-point boundary value problems for ordinary second-order differential equations with nonintegrable singularities in the time variable. In particular, we describe a set of functions f:]a, b[×ℝ → ℝ such that the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int\limits_a^b {{{\left( {t - a} \right)}^\ell }{{\left( {b - t} \right)}^\ell }|\left( {t,x} \right)|dt = + \infty } $$\end{document} is satisfied for arbitrary x ∈ ℝ and ℓ > 0, but nevertheless, 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}$$u'' = f\left( {t,u} \right);\,\,u\left( {a + } \right) = 0,\,u\left( {b - } \right) = 0$$\end{document} has a unique solution.