Positive Solutions of Nonlinear Three-Point Integral Boundary-Value Problems for Second-Order Differential Equations

We investigate the existence of positive solutions to the nonlinear second-order three-point integral boundary value problem \label{eq-1} {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, 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. We show the existence of at least one positive solution if $f$ is either superlinear or sublinear by applying Krasnoselskii's fixed point theorem in cones.