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Asymptotic stability; additive groups; parameter dependence
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 $\mathbb Q\subset S\subsetneqq \mathbb R$.
[1] F. V. Atkinson: A stability problem with algebraic aspects. Proc. Roy. Soc. Edinburgh, Sect. A 78 (1977/78), 299–314. MR 0492522
[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 1638535
[3] Á. Elbert: On asymptotic stability of some Sturm-Liouville differential equations. General Seminars of Mathematics (University of Patras) 22–23 (1997), 57–66.
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