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DEPENDENCE OF EIGENVALUES OF SIXTH-ORDER BOUNDARY VALUE PROBLEMS ON THE BOUNDARY

Part of: Boundary value problems Special classes of linear operators

Published online by Cambridge University Press:  09 September 2014

SUQIN GE ,
WANYI WANG  and
QIUXIA YANG
SUQIN GE*
Affiliation:
School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, BaoTou 014010, PR China email 15647280518@163.com School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China
WANYI WANG
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China email wwy@imu.edu.cn
QIUXIA YANG
Affiliation:
College of Information and Management, Dezhou University, Dezhou 253023, PR China email yawenxuan@21cn.com
*
✉15647280518@163.com
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Abstract

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In this paper, we consider the dependence of eigenvalues of sixth-order boundary value problems on the boundary. We show that the eigenvalues depend not only continuously but also smoothly on boundary points, and that the derivative of the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$th eigenvalue as a function of an endpoint satisfies a first-order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher eigenvalues of such boundary value problems march off to plus infinity. This is also true for the first (that is, lowest) eigenvalue.

Keywords

sixth-order boundary value problemsboundary conditioneigenvalues

MSC classification

Secondary: 34B20: Weyl theory and its generalizations 34B24: Sturm-Liouville theory 47B25: Symmetric and selfadjoint operators (unbounded)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 457 - 468
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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