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.
@article{107953,
author = {Stan\v ek, Svatoslav},
title = {An almost-periodicity criterion for solutions of the oscillatory differential equation $y''=q(t)y$ and its applications},
journal = {Archivum Mathematicum},
volume = {041},
year = {2005},
pages = {229-241},
zbl = {1117.34043},
mrnumber = {2164672},
language = {en},
url = {http://dml.mathdoc.fr/item/107953}
}
Staněk, Svatoslav. An almost-periodicity criterion for solutions of the oscillatory differential equation $y''=q(t)y$ and its applications. Archivum Mathematicum, Tome 041 (2005) pp. 229-241. http://gdmltest.u-ga.fr/item/107953/
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