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

summary: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