On the robustness of a concept of dichotomy with different growth rates for linear discrete-time systems in Banach spaces

Crai, Violeta

doi:10.1109/saci.2016.7507353

Cited by 3 publications

(1 citation statement)

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“…A natural generalization of both exponential and polynomial dichotomy is successfully modeled by the concept of (h, k)− dichotomy introduced by M. Pinto [20] for invertible difference equations. Two years later M. Megan ([17]) developed his research and lately the concept was intensively studied in its various forms: uniform and nonuniform, strong and weak [5,11]. In this paper we consider the general and more realistic case of a non-invertible dynamics as well as of a nonuniform (h, k)− dichotomy (this concept is a generalization of a dichotomy concept studied by L. Barreira and C. Valls in [7]).…”

Section: Introductionmentioning

confidence: 99%

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

Crai

Aldescu

2019

Springer Proceedings in Mathematics &Amp; Statistics

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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].

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