Maximum and anti-maximum principles for three point SBVPs and nonlinear three point SBVPs

Abstract

We prove maximum and anti-maximum principle for the following differential inequalities,

$$\begin{aligned}&-(x y'(x))'-\lambda x y(x)\ge 0 ,\quad 0<x<1,\\&y'(0)=0,\quad y(1)-\delta y(\eta )\ge 0, \end{aligned}$$

where $\delta >0$ and $0<\eta <1$ and use it to examine the existence of solutions of the following class of nonlinear three point singular boundary value problems (SBVPs)

$$\begin{aligned}&-y''(x)-\frac{1}{x}y'(x)=f(x,y),\quad 0<x<1,\\&y'(0)=0,\quad y(1)=\delta y(\eta ). \end{aligned}$$

We use monotone iterative technique in the presence of upper and lower solutions which can be arranged in one way (well order) or the other (reverse order) and prove new existence theorems.