New stability conditions for linear difference equations using Bohl–Perron type theorems

Abstract

The Bohl–Perron result on exponential dichotomy for a linear difference equation

states (under some natural conditions) that if all solutions of the non-homogeneous equation with a bounded right hand side are bounded, then the relevant homogeneous equation is exponentially stable. According to its corollary, if a given equation is close to an exponentially stable comparison equation (the norm of some operator is less than one), then the considered equation is exponentially stable. For a difference equation with several variable delays and coefficients we obtain new exponential stability tests using the above results, representation of solutions and comparison equations with a positive fundamental function.

Keywords:

• linear delay difference equations
• exponential stability
• positive fundamental function

AMS Subject Classification::

• 39A11
• 39A12

Acknowledgements

The authors are grateful to the referee for valuable comments and remarks. L. Berezansky was partially supported by the Israeli Ministry of Absorption and E. Braverman was partially supported by the NSERC Research Grant.