On the asymptotic behaviour of Volterra difference equations

Abstract

We consider the asymptotic behaviour of solutions of the difference equations of the form in a Banach space X, where n = 0,1,2,…; A, B(n) are linear bounded operators in X. Our method of study is based on the concept of spectrum of a unilateral sequence. The obtained results on asymptotic stability and almost periodicity are stated in terms of spectral properties of the equation and its solutions. To this end, a relation between the Z-transform and spectrum of a unilateral sequence is established. The main results extend previous results.

Keywords:

• Volterra equation
• convolution type
• asymptotic almost periodicity
• stability
• instability
• Katznelson–Tzafriri theorem

2000 Mathematics Subject Classification::

• Primary: 47D06
• Secondary: 47A35
• 39A11

Acknowledgements

The author would like to thank the anonymous referees and the editor for carefully reading the manuscript, and for remarks and suggestions to improve the previous version of the paper.