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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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