Geodesic
Additive groups connected with asymptotic stability of some differential equations
Archivum mathematicum, Tome 34 (1998) no. 1, pp. 49-58 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\mathbb Q\subset S\subsetneqq \mathbb R$.
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 $\mathbb Q\subset S\subsetneqq \mathbb R$.
Classification : 34B24, 34C10, 34D05, 34M99
Keywords: Asymptotic stability; additive groups; parameter dependence 
@article{ARM_1998_34_1_a5,
     author = {Elbert, \'Arp\'ad},
     title = {Additive groups connected with asymptotic stability of some differential equations},
     journal = {Archivum mathematicum},
     pages = {49--58},
     year = {1998},
     volume = {34},
     number = {1},
     mrnumber = {1629656},
     zbl = {0917.34022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1998_34_1_a5/}
}
TY  - JOUR
AU  - Elbert, Árpád
TI  - Additive groups connected with asymptotic stability of some differential equations
JO  - Archivum mathematicum
PY  - 1998
SP  - 49
EP  - 58
VL  - 34
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ARM_1998_34_1_a5/
LA  - en
ID  - ARM_1998_34_1_a5
ER  - 
%0 Journal Article
%A Elbert, Árpád
%T Additive groups connected with asymptotic stability of some differential equations
%J Archivum mathematicum
%D 1998
%P 49-58
%V 34
%N 1
%U http://geodesic.mathdoc.fr/item/ARM_1998_34_1_a5/
%G en
%F ARM_1998_34_1_a5
Elbert, Árpád. Additive groups connected with asymptotic stability of some differential equations. Archivum mathematicum, Tome 34 (1998) no. 1, pp. 49-58. http://geodesic.mathdoc.fr/item/ARM_1998_34_1_a5/

[1] F. V. Atkinson: A stability problem with algebraic aspects. Proc. Roy. Soc. Edinburgh, Sect. A 78 (1977/78), 299–314. | MR

[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

[3] Á. Elbert: On asymptotic stability of some Sturm-Liouville differential equations. General Seminars of Mathematics (University of Patras) 22–23 (1997), 57–66.