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An almost-periodicity criterion for solutions of the oscillatory differential equation $y''=q(t)y$ and its applications
Archivum mathematicum, Tome 41 (2005) no. 2, pp. 229-241 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The linear differential equation $(q):y''=q(t)y$ with the uniformly almost-periodic function $q$ is considered. Necessary and sufficient conditions which guarantee that all bounded (on $\mathbb{R}$) solutions of $(q)$ are uniformly almost-periodic functions are presented. The conditions are stated by a phase of $(q)$. Next, a class of equations of the type $(q)$ whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations $y''=q(t)y+f(t)$ are considered. The results are applied to the Appell and Kummer differential equations.
The linear differential equation $(q):y''=q(t)y$ with the uniformly almost-periodic function $q$ is considered. Necessary and sufficient conditions which guarantee that all bounded (on $\mathbb{R}$) solutions of $(q)$ are uniformly almost-periodic functions are presented. The conditions are stated by a phase of $(q)$. Next, a class of equations of the type $(q)$ whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations $y''=q(t)y+f(t)$ are considered. The results are applied to the Appell and Kummer differential equations.
Classification : 34C27
Keywords: linear second-order differential equation; Appell equation; Kummer equation; uniformly almost-periodic solution; bounded solution; phase 
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Staněk, Svatoslav. An almost-periodicity criterion for solutions of the oscillatory differential equation $y''=q(t)y$ and its applications. Archivum mathematicum, Tome 41 (2005) no. 2, pp. 229-241. http://geodesic.mathdoc.fr/item/ARM_2005_41_2_a10/

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