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Construction of almost periodic functions with given properties. (English) Zbl 1221.42011
The author introduces almost periodic functions in pseudometric spaces using a trivial extension of the Bohr concept, where the modulus is replaced by the distance. Necessary and sufficient conditions for a continuous function with values in a Banach space to be almost periodic may be no longer valid for continuous functions in general metric spaces. For the approximation condition, it is seen that the completeness of the space of values is necessary. The way one can construct almost periodic functions with prescribed properties in a pseudometric space is mentioned. The author also considers the set of systems of the form ${x}^{\text{'}}=A\left(t\right)x$, where $A$ is almost periodic and all matrices $A\left(t\right)$, $t\in ℝ$, are skew-Hermitian, with the uniform topology of matrix functions $A$ on the real axis. Using a certain method for constructing almost periodic functions, it is proved that, in any neighbourhood of a system of the form ${x}^{\text{'}}=A\left(t\right)x$, there exists a system which does not possess an almost periodic solution other than the trivial one, not only with a fundamental matrix which is not almost periodic.
##### MSC:
 42A75 Classical almost periodic functions, mean periodic functions