Approximation of Solutions of Singular Second Order Boundary Value Problems

Abstract

The boundary value problems \[ ({\text{P}}_1 )\begin{array}{*{20}c} {y'' + \frac{{n - 1}}{x}y' + f(x,y) = 0,} \\ {y'(0) = y(1) = 0,} \\ \end{array} \] and \[ \begin{gathered} y'' + f(x,y) = 0, \hfill \\ ({\text{P}}_2 )\quad \alpha y(0) - \beta y'(0) = 0, \hfill \\ \gamma y(1) + \delta y'(1) = 0 \hfill \\ \end{gathered} \] with the function $f(x,y)$ satisfying conditions that allow for singularities to be present are studied with the view of obtaining general existence, uniqueness, and approximation of positive solutions. Furthermore, the behavior of solutions near $x = 1$ is described for the special case $f(x,y) = a(x)y^{ - p} $.