Geodesic
Additive groups connected with asymptotic stability of some differential equations
Archivum mathematicum, Tome 34 (1998) no. 1, pp. 49-58

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 
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     author = {Elbert, \'Arp\'ad},
     title = {Additive groups connected with asymptotic stability of some differential equations},
     journal = {Archivum mathematicum},
     pages = {49--58},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {1998},
     mrnumber = {1629656},
     zbl = {0917.34022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1998__34_1_a5/}
}
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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/