Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Vysotskaya I. A., Strukova I. I. The research of some classes of almost periodic at infinity functions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 1, pp. 4-14. DOI: 10.18500/1816-9791-2021-21-1-4-14, EDN: NNJJXS

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2021
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Russian
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Article
UDC: 
517.98
EDN: 
NNJJXS

The research of some classes of almost periodic at infinity functions

Autors: 
Vysotskaya Irina A., Federal State Official Military Educational Institution of Higher Education “Military Educational and Scientific Centre of the Air Force N. E. Zhukovsky and Y. A. Gagarin Air Force Academy”
Strukova Irina Igorevna, Voronezh State University
Abstract: 

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions. The first definition (approximating) is based on the approximation theorem. In the classical version, for almost periodic functions, they are represented as uniform closures of trigonometric polynomials. In our case, the Fourier coefficients are slowly varying at infinity functions. The second definition, which is an analogue of G. Bohr’s definition of an almost periodic function, is based on the concept of an $\varepsilon$-period. The third definition meets S. Bochner’s criterion for the almost periodicity of functions. The fourth definition is given in terms of factor space. With the help of the results of the theory of almost periodic vectors in Banach modules those four definitions are proved to be equivalent. In addition, it was proved that the introduced spaces of slowly varying and almost periodic at infinity functions with respect to different subspaces of functions vanishing at infinity coincide with the spaces of ordinary slowly varying and almost periodic at infinity functions, respectively. The feasibility of consideration of these functions is due to the fact that the solutions of some important classes of differential and difference equations are almost periodic at infinity. We consider differential equations whose right-hand side is a function vanishing at infinity and obtain necessary and sufficient conditions for their bounded solutions to be almost periodic at infinity functions. We also study an asymptotic representation of the solutions.

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Received: 
05.11.2019
Accepted: 
15.01.2020
Published: 
01.03.2021
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