  In this paper, we study the existence of positive solutions of a three-point
integral boundary value problem (BVP) for the following second-order
differential equation \begin{equation*} \begin{gathered} {u^{\prime \prime
}}(t)+\lambda a(t)f(u(t))=0,\ \ 0<t<1, \\ u^{\prime}(0)=0, \
u(1)={\alpha}\int_{0}^{\eta}u(s)ds, \end{gathered} \end{equation*} where
$\lambda>0$ is a parameter, $0<{\eta}<1$, $0<{\alpha}< \frac{1}{{\eta}}$. By
using the properties of the Green's function and Krasnoselskii's fixed point
theorem on cones, the eigenvalue intervals of the nonlinear boundary value
problem are considered, some sufficient conditions for the existence of at
least one positive solutions are established.
