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Additive groups connected with asymptotic stability of some differential equations. (English) Zbl 0917.34022
The author investigates the differential equation $y''(s)+ \lambda ^2q(s)y(s)=0,\quad s\in (s_0,\infty), \tag{1}$ with $$\lambda \in \mathbb{R}$$, $$q(s)>0$$ is a step function tending to infinity. He denotes by $$S$$ the set of those $$\lambda$$’s for which the equation $$(1)$$ has a solution not tending to zero as $$s\to \infty$$ and proves that $$S$$ is an additive group.
The paper continues a similar problem studied by F. V. Atkinson [Proc. R. Soc. Edinb., Sect. A 78, 299-314 (1978; Zbl 0393.34023)] for the equation $y''(s)+\bigl(\lambda ^2q(s)+\lambda \sqrt {q(s)}g(s)\bigr)y(s)=0,\quad s\in (s_0,\infty), \tag{2}$ where $$q(s)>0$$ is continuous nondecreasing and $$\lim _{s\to \infty } q(s)=\infty$$.
The used method is the Prüfer transformation technique. In addition, four examples are given and some open problems are formulated.
Reviewer: Z.Došlá (Brno)
MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34M99 Ordinary differential equations in the complex domain 34D05 Asymptotic properties of solutions to ordinary differential equations
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