Weighted exponential trichotomy of linear difference equations. (English) Zbl 1155.39004

The authors introduce the notion of weighted exponential trichotomy for the linear ordinary difference equations
\[ x(n+1)=A(n)x(n), \quad n\in\mathbb Z, \tag \(*\) \]
where \(A(n)\) is an \(m\times m\) invertible matrix defined on \(\mathbb Z\), and present several illustrative examples of this new class of trichotomy. The main result of the paper gives a complete description of the space \(S\) of all the solutions with their asymptotic behavior of the system \((*)\) possessing weighted exponential trichotomy. This extends some of the results of S. Elaydi and K. Janglajew [J. Difference Equ. Appl. 3, No. 5–6, 417–448 (1998; Zbl 0914.39013)] and G. Papaschinopoulos [Appl. Anal. 40, No. 2–3, 89–109 (1991; Zbl 0687.39003)]. The final part of the paper is devoted to the study of the nonhomogeneous linear difference systems.


39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations