Abstract: In this paper, the existence of positive solutions of the second-order three-point
boundary value problem $-u''(t)=b(t)f(u(t))$ for all $t\in[0,1]$ subject to $u'(0)=0$, $u(1)={\alpha}u({\eta})$ is studied, where $\alpha, \eta\in(0,1)$ are given, $f\in C\big([0,\infty),[0,\infty)\big)$, $b\in C\big([0,1],[0,\infty)\big)$ and there exists $t_0\in[0,1]$ such that $b(t_0)>0$. The problem is transformed into the Hammerstein's integral equation with its corresponding Green's funtion. By applying the fixed point index theory, authors obtain the optimal sufficient conditions for the existence of single and multiple positive solutions of the above mentioned problem concerning the first eigenvalue of the relevant linear problem.

Key words: Positive solutions, Cone, Fixed point index, First eigenvalue