Construction of almost periodic sequences with given properties

Abstract

We define almost periodic sequences with values in a pseudometric space X and we modify the Bochner definition of almost periodicity so that it remains equivalent with the Bohr definition. Then, we present one (easily modifiable) method for constructing almost periodic sequences in X. Using this method, we find almost periodic homogeneous linear difference systems that do not have any non-trivial almost periodic solution. We treat this problem in a general setting where we suppose that entries of matrices in linear systems belong to a ring with a unit.