Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

Abstract

New existence results are presented for the singular second-order nonlinear boundary value problems u″ + g(t)f(u) = 0, 0 < t < 1, αu(0) − βu′ (0) = 0, γu(1) + δu′(1) = 0 under the conditions $0 \leqslant f_0^ + < M_1 ,m_1 < f_\infty ^ - \leqslant \infty {\text{ }}or{\text{ 0}} \leqslant f_\infty ^ + < M_1 ,m_1 < f_0^ - \leqslant \infty$, where $f_0^ + = \overline {\lim } _{u \to 0} f\left( u \right)/u,f_\infty ^ - = \underline {\lim } _{u \to \infty } f\left( u \right)/u,f_0^ - = \underline {\lim } _{u \to 0} f\left( u \right)/u,f_\infty ^ + = \overline {\lim } _{u \to \infty } f\left( u \right)/u,$ g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.