Nonlinear three point Singular BVPs : A Classification

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^{\alpha} y'(x))'= x^{\alpha}f(x,y),\quad 0<x<1,\\ &&y'(0)=0,\quad y(1)=\delta y(\eta), \end{eqnarray*} where $\delta>0$, $0<\eta<1$ and $\alpha \geq 1$. 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 $\alpha\geq 1$ well order case occur.