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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
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