An Almost-periodicity Criterion for Solutions of the Oscillatory Differential Equation $y''=q(t)y$ and its Applications

摘要

The linear differential equation (q) : y 00 = q(t)y with the uniformly almost-periodic function q is considered. Necessary and sufficient conditions which guarantee that all bounded (on 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 00 = q(t)y + f(t) are considered. The results are applied to the Appell and Kummer differential equations.