Positive solutions of four-point boundary value problem for fourth order ordinary differential equation (Q611753)

The boundary value problem \[ \begin{aligned} &y^{(4)}(t)-f(t,y(t),y''(t))=0,\quad 0\leq t\leq 1,\\ & y(0)=y(1)=0,\\ &ay''(\xi_1)-by'''(\xi_1)=0,\qquad cy''(\xi_2)+dy'''(\xi_2)=0\end{aligned} \] is studied, where \(0\leq \xi_1<\xi_2\leq 1\). Some sufficient conditions guaranteeing the existence of a positive solution to the above four-point boundary value problem are obtained by using the Krasnoselskii fixed point theorem.