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

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.