Article

 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: http://hdl.handle.net/10338.dmlcz/107632 . 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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