Additive groups connected with asymptotic stability of some differential equations

Arpad Elbert

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


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