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An almost-periodicity criterion for solutions of the oscillatory differential equation \(y''=q(t)y\) and its applications. (English) Zbl 1117.34043
The author deals with the problem of uniform almost-periodicity of bounded solutions of the second order linear differential equation \[ y''=q(t)y\tag{*} \] with a uniformly almost-periodic function \(q\). The main tool is the theory of phase functions of (*) introduced by O. Borůvka [Linear differential transformations of the second order. (London: The English Universities Press Ltd. XVI) (1971; Zbl 0222.34002)]. The nonhomogeneous equation associated with (*) is considered as well. A typical result is the following statement:
Let \(q\) be a uniformly almost-periodic function and let \(\alpha =\alpha (t)\) be its phase function. Then all solutions of (*) are uniformly almost-periodic if and only if \(\alpha (t) = at +\varphi (t)\), where \(a\in \mathbb R\) and \(\varphi \) is a function such that \(\varphi ^{(i)}\), \(i=0, 1\), is almost periodic, and \(\inf \{| a+\varphi '(t)| , t\in \mathbb R\} >0\).
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
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