Nonlinear three point Singular BVPs : A Classification

Abstract

We analyze the existence of unique solutions of the following class of nonlinear three point singular boundary value problems (SBVPs), \begin{eqnarray*}\label{NL-Singular-P} &&-(x^α y'(x))'= x^αf(x,y),\quad 0<x<1,\\ &&y'(0)=0,\quad y(1)=δy(η), \end{eqnarray*} where $\delta >0$, $0<\eta <1$ and $\alpha \ge 1$. This study shows some novel observations regarding the nature of the solution of the nonlinear three point SBVPs. We observe that when $sup\left(\partial f/\partial y\right)>0$ for $\alpha \in {\cup }_{n\in \mathbb{N}}\left(4n-1,4n+1\right)$ or $\alpha \in \{1,5,9,\cdots \}$ reverse ordered case occur. When $sup\left(\partial f/\partial y\right)>0$ for $\alpha \in {\cup }_{n\in \mathbb{N}}\left(4n-3,4n-1\right)$ or $\alpha \in \{3,7,11,\cdots \}$ and when $sup\left(\partial f/\partial y\right)<0$ for all $\alpha \ge 1$ well order case occur.

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