On properties of systems of linear difference equations
by
Ali Mahmud Ateiwi
Department of Mathematics and Statistics, Faculty of Science, Al-Hussein Bin Talal University
Coauthors: Iryna Volodymyrivna Komashynska and Hussam Rabbaia

Consider a system of liner difference equations with variable coefficients

( 1)

where n = 0, 1, 2, …, xn is a vector from the Euclidean the space Rd and An is a d d matrixes of coefficient.

We assume that E + An is nondegenerate for all n 0 So system ( 1 ) has the unique solution.

We study the problem of reduction of the system ( 1 ) to a system with constant coefficients.

Yn+1 = yn + Byn , ( 2)

Where B is a constant matrix.

Definition 1. A linear difference system is called reducible if there exists a lyapunov transformation that reduces it to a system with constant coefficients ( 2).

We establish reducibility conditions.

Theorem1. the linear difference system ( 1) is reducible if and only if a certain fundamental matrix Xn of it is representable in the form

( 3)

Where E is the d d identity matrix and B is a certain constant d d matrixes.

Reference

[1 ] Yu. A. Mitropol’skii, A. M. Samoilenko, and D. I. Martynyuk, systems of evolution Equations with Periodic and Quasiperiodic coefficients [ in Russian], Naukova Dumka, Kiev ( 1984).

[ 2 ] D. I. Martynyuk and N. A. Perestyuk, “ Reducibility of linear systems of difference equations with smooth right- hand side, “ vychisl. Prikl. Mat., Issue 27, 34-40 ( 1990).

[3] D. I. Martynyuk and N. A. Perestyuk, On Reducibility of linear systems of difference equations with quasiperiodic coefficient“ vychisl. Prikl. Mat., Issue 29, 116-127 ( 1992).

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Date received: February 7, 2007