Title:

Additive groups connected with asymptotic stability of some differential equations (English) 
Author:

Elbert, Árpád 
Language:

English 
Journal:

Archivum Mathematicum 
ISSN:

00448753 (print) 
ISSN:

12125059 (online) 
Volume:

34 
Issue:

1 
Year:

1998 
Pages:

4958 
Summary lang:

English 
. 
Category:

math 
. 
Summary:

The asymptotic behaviour of a SturmLiouville 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:

20090217T10:10:14Z 
Last updated:

20120510 
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 SturmLiouville differential equations.General Seminars of Mathematics (University of Patras) 22–23 (1997), 57–66. 
. 