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Shadowing and hyperbolicity for linear delay difference equations

Part of: Smooth dynamical systems: general theory Topological dynamics

Published online by Cambridge University Press:  09 December 2024

Lucas Backes ,
Davor Dragičević  and
Mihály Pituk
Lucas Backes
Affiliation:
Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil (lucas.backes@ufrgs.br)
Davor Dragičević*
Affiliation:
Faculty of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia (ddragicevic@math.uniri.hr) (corresponding author)
Mihály Pituk
Affiliation:
Department of Mathematics, University of Pannonia, Egyetem út 10, 8200 Veszprém, Hungary; HUN–REN–ELTE Numerical Analysis and Large Networks Research Group, Budapest, Hungary (pituk.mihaly@mik.uni-pannon.hu)
*
*Corresponding author.
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Abstract

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this article, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a) for non-autonomous equations with finite delays and uniformly bounded compact coefficient operators in Banach spaces and (b) for Volterra difference equations with infinite delay in finite dimensional spaces.

Keywords

delay difference equationexponential dichotomyhyperbolicityshadowingPerron property

MSC classification

Primary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
Secondary: 37B99: None of the above, but in this section

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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