Abstract
We study limit periodic and almost periodic homogeneous linear difference systems. The coefficient matrices of the considered systems are taken from a given commutative group. We mention a condition on the group which ensures that, by arbitrarily small changes, the considered systems can be transformed to new systems, which do not possess any almost periodic solution other than the trivial one. The elements of the coefficient matrices are taken from an infinite field with an absolute value.
Highlights
In this paper, for a commutative group X of square matrices over a field, we analyse the homogeneous linear difference systems xk+1 = Akxk, k ∈ Z, (1.1)where {Ak}k∈Z ⊆ X
One of the main results of [18] says that the systems with non-almost periodic solutions form a dense subset of the space of all unitary systems
In papers [9, 24], the limit periodic systems of the form (1.1) are investigated, where matrices Ak are taken from a commutative group or from a bounded group
Summary
One of the main results of [18] says that the systems with non-almost periodic solutions form a dense subset of the space of all unitary systems. Transformable and strongly transformable groups of matrices are introduced Based on this concept, the above-mentioned result of [18] is generalized for other matrix groups. In papers [9, 24], the limit periodic systems of the form (1.1) are investigated, where matrices Ak are taken from a commutative group or from a bounded group. The properties of limit periodic homogeneous linear difference systems with respect to their almost periodic solutions are mentioned, e.g., in [9, 24].
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