Mathematics – Dynamical Systems
Scientific paper
2009-06-17
Mathematics
Dynamical Systems
19 pages. To appear in Journal of Difference Equations and Applications. To appear in Journal of Difference Equations and Appl
Scientific paper
The Bohl-Perron result on exponential dichotomy for a linear difference equation $$ x(n+1)-x(n) + \sum_{l=1}^m a_l(n)x(h_l(n))=0, h_l(n)\leq n, $$ 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.
Berezansky Leonid
Braverman Elena
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