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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)
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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