In this paper we consider the second-order three-point boundary value Problem~ \[\begin{cases} u''(t)+h(t)f(u)=0,~~\ \ \ t\in (0,1),\\[2ex] u'(0)=0, ~u(1)=\lambda u(\eta), \end{cases} \] where~$\eta\in[0,1)$,~$\lambda\in[0,1)$~is a parameter,~$f\in C( [0,\infty),[0,\infty))$~satisfies~$f(s)>0$~for $s>0$, and $h\in C( [0,1],[0,\infty))$~is not identically zero on any subinterval of [0,1]. We give information on the interesting problem as to what happens to the norms of positive solutions as $\lambda$ varies in $[0,1)$ under the conditions of~$f_{0}=0,~f_{\infty}=\infty$.~The proof of main result is based upon the fixed point index theory on cone and connectivity properties of the solution set.
