DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

WEIGHTED EXPONENTIAL TRICHOTOMY OF LINEAR
DIFFERENCE EQUATIONS
CLAUDIO VIDAL AND CLAUDIO CUEVAS
We introduce the weighted exponential trichotomy notion to difference equation and we
study the behavior in the future and the past of the solutions for linear system.
Copyright © 2006 C. Vidal and C. Cuevas. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The notion of dichotomy for a linear system of differential equations has gained prominence
since the appearance of two fundamental books: Dalietzkii and Krein [2], and
Massera and Schäffer [6]. These were followed by the important book of Coppel [1]who
synthesized and improved the results that existed in the literature up to 1978.
Two generalizations of dichotomy in differential equations have been introduced: the
first by Sacker and Sell [10] called (S-S) trichotomy and the second by Elaydi and Hájek
[3] called (E-H)-trichotomy. But it was not until 1990 that the notions of dichotomy
and trichotomy were extended to nonlinear difference equations by Papaschinopoulos in
[8] and by Elaydi and Janglajew in [4]. Pinto in [9] introduced a generalized notion of
dichotomies, called (h,k)-dichotomies, which contains the usual notion of ordinary or
exponential dichotomies.
In this paper we introduce a new notion of trichotomy, which is very useful in order
to study the asymptotic behavior for linear system of difference equations
x(n +1)= A(n)x(n), n ∈ Z, (1.1)
in the nonhomogeneous linear case. It consists essentially in taking into account the above
concepts of (E-H)-trichotomies and (h,k)-dichotomies; that is, we introduce both concepts
in only one, such trichotomies will be called weighted exponential trichotomy. Our
main purpose in this work is to extend the study of dichotomy and trichotomy in linear
ordinary difference equations and to study the asymptotic behavior of the solutions of
both the future and the past under the existence of weighted exponential trichotomy.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1077–1086