## Existence of solutions for fourth order differential equation with four-point boundary conditions.(English)Zbl 1140.34308

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 S. Chen, W. Ni and C. Wang [ Appl. Math. Lett. 19, 161-168 (2006; Zbl 1096.34009)], where this particular BVP has been studied previously.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

Boundary value problem; fixed point; nonlinear alternative

Zbl 1096.34009
Full Text:

### References:

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