On (h, k)-Dichotomy of Linear Discrete-Time Systems in Banach Spaces

Abstract

The paper considers a general concept of dichotomy with different growth rates for linear discrete-time systems in Banach spaces. Characterizations in terms of Lyapunov type sequences of norms are given. The approach is illustrated by various examples. IntroductionThe notion of (uniform) exponential dichotomy for difference equations was introduced in the literature by T. Li [16] and plays a central role in the theory of dynamical system such as, for example, in the study of stable and unstable manifolds and in many aspects of the theory of stability. We note that the theory of exponential dichotomies and its applications are very much developed.Early results in the study of dichotomies for difference equations appeared in the paper of C.V. Coffman and J.J. Schaffer [9]. Later, in 1981, D. Henry included discrete dichotomies in his book [15]. These were followed by the monographs due to R.P. Agarwal [1], C. Pötzsche [23] and S. Elaydi [13] (deals with ordinary dichotomy).Lately, characterizations of the nonuniform exponential dichotomy for discrete linear systems can be found in the works of M. Megan, T. Ceauşu, A.L. Sasu, B. Sasu, L. Popa, M.G. Babuţia and colleagues (see [4,18,21]). In 2009 A. Bento and C. Silva introduced a new concept of dichotomy called polynomial dichotomy [8].

Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line

“…iii) When considering assumptions different and/or more general than those stated by (P2) we can obtain other dichotomies and we mention some as the (h, k)-dichotomies [13], the nonuniform exponential dichotomy [2,5], the (µ, ν)nonuniform dichotomies [6] and and the polynomial dichotomies [4].…”