On Riemann integrability and almost periodic functions

Raouf Doss

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Let $f (t, x)$ be a vector valued function almost periodic in $t$ uniformly for $x$ , and let $m o d (f) = L_{1} \oplus L_{2}$ be its frequency module. We say that an almost periodic solution $x (t)$ of the system $\dot{x} = f (t, x), t \in ℝ, x \in D \subset ℝ^{n}$ is irregular with respect to $L_{2}$ (or partially irregular) if $(m o d (x) + L_{1}) \cap L_{2} = {0}$ . Suppose that $f (t, x) = A (t) x + X (t, x),$ where $A (t)$ is an almost periodic $(n \times n)$ -matrix and $m o d (A) \cap m o d (X) = {0} .$ We consider the existence problem for almost periodic irregular with respect to $m o d (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient...