Asymptotic summation of perturbed linear difference systems in critical case

Abstract

We propose an asymptotic summation method for certain class of linear difference systems. Both the ideas of the centre manifold theory and the averaging method are used to construct the asymptotics for solutions. We illustrate the asymptotic summation method by constructing the asymptotics for solutions of certain higher order scalar difference equation with an oscillatory decreasing coefficient.

Keywords:

• Centre manifold
• method of averaging
• oscillatory decreasing functions
• asymptotic behaviour
• discrete delay equation

AMS CLASSIFICATIONS:

• 39A06
• 39A10
• 39A30

Acknowledgments

The author is grateful to the referee for some useful comments and remarks that helped to improve the text.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was carried out within the framework of the State Programme of the Ministry of Science and Higher Education of the Russian Federation, project No. 1.13560.2019/13.1.