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 $$x(n+1)-x(n)+\sum _{l=1}^m{a}_l(n)x(h_l(n))=0,h_l(n)\le n,$$ 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 {\em 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.

19 pages. To appear in Journal of Difference Equations and Applications. To appear in Journal of Difference Equations and Applications