Additive groups connected with asymptotic stability of some differential equations

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 -