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

Abstract

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 \(\int\limits_a^b {{{\left( {t - a} \right)}^\ell }{{\left( {b - t} \right)}^\ell }|\left( {t,x} \right)|dt = + \infty } \) is satisfied for arbitrary x ∈ ℝ and ℓ > 0, but nevertheless, the boundary value problem \(u'' = f\left( {t,u} \right);\,\,u\left( {a + } \right) = 0,\,u\left( {b - } \right) = 0\) has a unique solution.