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An almost-periodicity criterion for solutions of the
oscillatory differential equation $y''=q(t)y$ and its applications

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*Svatopluk Stanek*

**Address.**

Department of Mathematical Analysis, Faculty of
Science, Palacky University,
Tomkova 40, 779 00 Olomouc, Czech Republic

**E-mail. ** stanek@inf.upol.cz

**Abstract.**

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

**AMSclassification. **34C27, 34A30.

**Keywords. **Linear second-order differential equation,
Appell equation, Kummer equation,
uniformly almost-periodic solution, bounded solution, phase.