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Almost periodic Hill equations with bounded solutions. (Russian) Zbl 0825.34004
It is shown that in the case, when \(p\in C^2(\mathbb{R},\mathbb{R})\) is almost periodic and for any \(\bar p\in \bar\Sigma_p\) all the solutions of the equation \(\ddot x=\bar p(t)x\) are bounded, there exists for any \(\varepsilon > 0\) such \(p_ \varepsilon \in C(\mathbb{R},\mathbb{R})\) that \(\sup_ t\| p(t)-p_ \varepsilon(t)\| < \varepsilon\); \(p_\varepsilon\) is almost periodic, the condition of inclusion of the Fourier moduli is fulfilled \(M_{p_ \eta}\subset M_p\), and all solutions of the equation \(\ddot x=p_\varepsilon(t)x\) are almost periodic.
MSC:
34A30 Linear ordinary differential equations and systems, general
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34D99 Stability theory for ordinary differential equations
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