Positive solutions of three-point boundary value problems for n-th order differential equations

Abstract

In this paper, we establish the existence and non-existence results of positive solutions for the (n-1,1) three-point boundary value problems consisting of the equation $$ u^{(n)}+\lambda a(t)f(u(t))=0,\;\;\;t\in (0,1) $$ and one of the following boundary value conditions $$ u(1)=\beta u(\eta ),\;\;u^{(i)}(0)=0\;\hbox{ for }\;i=1,2,\cdots,n-1 $$ and $$ u^{(n-1)}(1)=\beta u^{(n-1)}(\eta ),\;u^{(i)}(0)=0\;\hbox{ for }\;i=0,1,\cdot,n-2, $$ where $\eta \in [0,1)$, $\beta\in [0,1)$ and $a:\;(0,1)\rightarrow R$ may change sign. $f(0)>0$, $\lambda >0$ is a parameter. Our approach is based on the Leray-Schauder fixed point Theorem. This paper is motivated by Eloe and Henderson [6].