# Additive groups connected with asymptotic stability of some differential equations

Archivum Mathematicum (1998)

- Volume: 034, Issue: 1, page 49-58
- ISSN: 0044-8753

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topElbert, Árpád. "Additive groups connected with asymptotic stability of some differential equations." Archivum Mathematicum 034.1 (1998): 49-58. <http://eudml.org/doc/248218>.

@article{Elbert1998,

abstract = {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 $\mathbb \{Q\}\subset S\subsetneqq \mathbb \{R\}$.},

author = {Elbert, Árpád},

journal = {Archivum Mathematicum},

keywords = {Asymptotic stability; additive groups; parameter dependence; asymptotic stability; additive groups; parameter dependence},

language = {eng},

number = {1},

pages = {49-58},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Additive groups connected with asymptotic stability of some differential equations},

url = {http://eudml.org/doc/248218},

volume = {034},

year = {1998},

}

TY - JOUR

AU - Elbert, Árpád

TI - Additive groups connected with asymptotic stability of some differential equations

JO - Archivum Mathematicum

PY - 1998

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 034

IS - 1

SP - 49

EP - 58

AB - 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 $\mathbb {Q}\subset S\subsetneqq \mathbb {R}$.

LA - eng

KW - Asymptotic stability; additive groups; parameter dependence; asymptotic stability; additive groups; parameter dependence

UR - http://eudml.org/doc/248218

ER -

## References

top- F. V. Atkinson, A stability problem with algebraic aspects, Proc. Roy. Soc. Edinburgh, Sect. A 78 (1977/78), 299–314. (1977) MR0492522
- 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. (1995) MR1638535
- Á. Elbert, On asymptotic stability of some Sturm-Liouville differential equations, General Seminars of Mathematics (University of Patras) 22–23 (1997), 57–66. (1997)

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