Solutions of Sturm-Liouville type multi-point boundary value problems for higher-order differential equations

The existence of solutions of the following multi-point boundary value problem $\left\{ \begin{gathered} x^{(n)} (t) = f(t,x(t),x^\prime (t),...,x^{(n - 2)} (t)) + r(t),0 < t < 1, \\ x^{(i)} (\xi _i ) = 0 for i = 0,1,... ,n - 3, ( * ) \\ \alpha x^{(n - 2)} (0) = \beta x^{(n - 1)} (0) = \gamma x^{(n - 1)} (1) + \tau x^{(n - 1)} (1) = 0 \\ \end{gathered} \right.$ is studied. Sufficient conditions for the existence of at least one solution of BVP(*) are established. It is of interest that the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we don’t apply the Green’s functions of the corresponding problem and the method to obtain a priori bounds of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.