Multiple positive solutions for a nonlinear three-point integral boundary-value problem

We investigate the existence of positive solutions to the nonlinear second-order three-point integral boundary value problem \begin{equation*} \label{eq-1} \begin{gathered} {u^{\prime \prime}}(t)+f(t, u(t))=0,\ 0<t<T, \\ u(0)={\beta}u(\eta),\ u(T)={\alpha}\int_{0}^{\eta}u(s)ds, \end{gathered} \end{equation*} where $0<{\eta}<T$, $0<{\alpha}< \frac{2T}{{\eta}^{2}}$, $0<{\beta}<\frac{2T-\alpha\eta^{2}}{\alpha\eta^{2}-2\eta+2T}$ are given constants. We establish the existence of at least three positive solutions by using the Leggett-Williams fixed-point theorem.