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In this paper, we study the existence of positive solutions of a three-pointintegral boundary value problem (BVP) for the following second-orderdifferential 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, $0<{\eta}<1$, $0<{\alpha}< \frac{1}{{\eta}}$. Byusing the properties of the Green's function and Krasnoselskii's fixed pointtheorem on cones, the eigenvalue intervals of the nonlinear boundary valueproblem are considered, some sufficient conditions for the existence of atleast one positive solutions are established.

Positive solutions of a nonlinear three-point eigenvalue problem with integral boundary conditions