Existence of solutions for fourth order differential equation with four-point boundary conditions
The authors study the fourth order differential equation \[ u^{(4)}(t)=f(t,u(t)), \; t \in (0,1), \] subject to the boundary conditions \[ \begin{gathered} u(0)=0,\quad au''(\xi_1)-bu'''(\xi_1)=0,\\ u(1)=0,\quad cu''(\xi_2)-du'''(\xi_2)=0, \end{gathered} \] where \(0\leq \xi_1 \leq \xi_2\leq 1\). They prove that this boundary value problem (BVP) has a solution. The main ingredient of the proof is a nonlinear alternative theorem of Leray-Schauder-type. The authors also give some remarks on a paper by \textit{S. Chen}, \textit{W. Ni} and \textit{C. Wang} [ Appl. Math. Lett. 19, 161-168 (2006; Zbl 1096.34009)], where this particular BVP has been studied previously.
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