Existence of positive solutions for a three-point integral boundary-value problem

In this paper, by using the Krasnosel'skii's fixed-point theorem, we study the existence of at least one or two positive solutions to the three-point integral boundary value problem {equation*} \label{eq-1} {gathered} {u^{\prime \prime}}(t)+a(t)f(u(t))=0,\ 0<t<T, u(0)={\beta}u(\eta),\ u(T)={\alpha}\int_{0}^{\eta}u(s)ds, {gathered} {equation*} where $0<{\eta}<T$, $0<{\alpha}< \frac{2T}{{\eta}^{2}}$, $0\leq{\beta}<\frac{2T-\alpha\eta^{2}}{\alpha\eta^{2}-2\eta+2T}$ are given constants.