Positive solutions of nonlinear multi-point boundary value problems

Abstract

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where $\xi _i \in (0,1)$ with $0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,$ and $a_i,b_i$ satisfy   $a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,$ and $\sum _{i=1}^{m-2} b_i <1.$ By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$