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

Svatopluk Stanek

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


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.