# zbMATH — the first resource for mathematics

Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type. (English) Zbl 0728.34019
Summary: This paper is devoted to obtaining natural existence and uniqueness theorems for the fourth-order boundary value problem $$d^ 4u/dx^ 4+f(x,u(x),u'(x),u''(x))=e(x)$$, $$0<x<1$$, $$u(0)=u(1)=0$$, $$u'''(0)- hu''(0)=0$$, $$u'''(1)+ku''(1)=0$$, $$h\geq 0$$, $$k\geq 0$$, $$h+k>0$$, using degree theoretic methods for any given $$e\in L^ 1[0,1]$$. The function f: [0,1]$$\times {\mathbb{R}}\times {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}$$ is not required to be bounded on [0,1]$$\times {\mathbb{R}}\times {\mathbb{R}}\times {\mathbb{R}}$$ and satisfies conditions that are natural to the boundary value problem.

##### MSC:
 34B24 Sturm-Liouville theory 34B15 Nonlinear boundary value problems for ordinary differential equations