Positive Solutions of a Three-Point Boundary Value Problem

Abstract:

      We study the existence of positive solutions of the three-point boundary value problem $$u''+g(t)f(u)=0,\ \ t\in(0,\ 1),$$ $$ u'(0)=0,\qquad u(1)=\alpha u(\eta), $$ where $\eta\in(0,1)$, and $\alpha \in \mathbb{R}$ with $0<\alpha<1$. The existence of positive solutions is studied by means of fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The results here essentially extend and improve the main result in [1].