Existence of a positive solution for a third-order three point boundary value problem

Abstract

By applying the Krasnoselskii fixed point theorem in cones and the fixedpoint index theory, we study the existence of positive solutions of the nonlinear third-order three point boundary value problem$u^{‴}(t)+a(t)f(t,u(t))=0$, $tın(0,1)$;$u^{\prime }(0)=u^{\prime }(1)=\alpha u(\eta )$, $u(0)=\beta u(\eta )$,where $\alpha$, $\beta$ and $\eta$ are constants with $\alpha ın[0,\frac{1}{\eta })$,and $0<\eta <1$. The results obtained heregeneralize the work of Torres [Positive solution for a third-order three pointboundary value problem, Electronic J. Diff. Equ. 2013 (2013), 147, 1–11].