Additive groups connected with asymptotic stability of some differential equations

Árpád Elbert

Archivum Mathematicum (1998)

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

Abstract

top
The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient λ 2 q ( s ) , s [ s 0 , ) is investigated, where λ and q ( s ) is a nondecreasing step function tending to as s . Let S denote the set of those λ ’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 = { 0 } , S = , S = 𝔻 (i.e. the set of dyadic numbers), and S .

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.