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Some Singular Nonlinear Boundary Value Problems

1991; Society for Industrial and Applied Mathematics; Volume: 22; Issue: 2 Linguagem: Inglês

10.1137/0522030

1095-7154

John V. Baxley,

Differential Equations and Numerical Methods

Two-point boundary value problems associated with the (possibly) singular nonlinear ordinary differential equation $y'' + g(x,y') + f(x,y) = 0$, $a \leqq x \leqq b$, are considered. The goal is to obtain rather general existence and uniqueness theorems for positive solutions. In the case of general separated linear boundary conditions, the results allow $f(x,y)$ to be singular as $y \to 0^ + $ and at the endpoints, with significant nonlinearity in both f and g. For the special condition $y'(a) = 0$, the results also allow g to be singular as $x \to a^ + $. In this way, the case $g(x,y) = ({{(N - 1)} / x})y'$, which arises when seeking radial solutions of $\nabla ^2 y = f(x,y)$, is included. The results extend previous theorems of Taliaferro and more recent theorems of Gatica, Waltman, et al.