We show the existence of positive
solutions and the global behavior of positive solutions of
the nonlinear multi-point boundary value problem
$u''+f(t,u)=0$,
$u(0)=0$, $u(1)=\alpha u(\eta)$,
where $\eta \in (0,1)$. This is achieved by the Fixed-Point Index and
Global Continuation Principle of Leray-Schauder.

The boundary condition reduces to the Dirichlet boundary condition
$u(0)=0$, $u(1)=0$,
if $\alpha=0$, and to the Robin boundary condition
$u(0)=0$, $u'(1)=0$
if $\alpha=1$ and $\eta$ approaches $1$.