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Title: Additive groups connected with asymptotic stability of some differential equations (English)
Author: Elbert, Árpád
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 34
Issue: 1
Year: 1998
Pages: 49-58
Summary lang: English
Category: math
Summary: The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient $\lambda ^2q(s),\ s\in [s_0,\infty )$ is investigated, where $\lambda \in \mathbb R$ and $q(s)$ is a nondecreasing step function tending to $\infty $ as $s\rightarrow \infty $. Let $S$ denote the set of those $\lambda $’s for which the corresponding differential equation has a solution not tending to 0. It is proved that $S$ is an additive group. Four examples are given with $S=\lbrace 0\rbrace $, $S= \mathbb Z$, $S=\mathbb D$ (i.e. the set of dyadic numbers), and $\mathbb Q\subset S\subsetneqq \mathbb R$. (English)
Keyword: Asymptotic stability
Keyword: additive groups
Keyword: parameter dependence
MSC: 34B24
MSC: 34C10
MSC: 34D05
MSC: 34M99
idZBL: Zbl 0917.34022
idMR: MR1629656
Date available: 2009-02-17T10:10:14Z
Last updated: 2012-05-10
Stable URL:
Reference: [1] F. V. Atkinson: A stability problem with algebraic aspects.Proc. Roy. Soc. Edinburgh, Sect. A 78 (1977/78), 299–314. MR 0492522
Reference: [2] Á. Elbert: Stability of some difference equations.Advances in Difference Equations: Proceedings of the Second International Conference on Difference Equations and Applications (held in Veszprém, Hungary, 7–11 August 1995), Gordon and Breach Science Publishers, eds. Saber Elaydi, István Győri and Gerasimos Ladas, 1997, 155–178. MR 1638535
Reference: [3] Á. Elbert: On asymptotic stability of some Sturm-Liouville differential equations.General Seminars of Mathematics (University of Patras) 22–23 (1997), 57–66.


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