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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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