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Additive groups connected with asymptotic stability of some differential equations

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Arpad Elbert
*

** Address.**
Mathematical Institute of the Hungarian Academy of Sciences,
Budapest, P.O.B. 127, H--1364, Hungary

** E-mail:**
elbert@math-inst.hu

**Abstract.**
The asymptotic behaviour of a Sturm-Liouville differential
equ\-ation 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 $s\to\infty$. 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 $S=\{0\}$, $S=
\mathbb Z$, $S=\mathbb D$ (i.e. the set of dyadic numbers), and $\mathbb Q\subset
S\subsetneqq \mathbb R$.

**AMS classification.**
34C10

**Key words.**
Asymptotic stability, additive groups, parameter dependence