"Admissibility and Polynomial Dichotomy of Discrete Nonautonomous Systems"

Abstract

"We give new admissibility criteria for dichotomic behaviours of discrete nonautonomous sys- tems, in infinite dimensional spaces. First, we present admissibility conditions for uniform and exponential dichotomy. Next, our study is focused on polynomial dichotomy, providing new characterizations for this no- tion by means of some double admissibilities. We obtain two categories of criteria for polynomial dichotomy, based on input-output conditions imposed to some suitable systems such that, for each one, the input sequences belong to certain $l^p$ -spaces and the outputs are bounded. We point out the importance of the assumptions re- garding the complementarity of the stable subspaces at the initial time and we also discuss the relevance of the concept of solvability (unique or not) in the admissibility criteria for polynomial dichotomies on the half-line. All the results are obtained in the general case, without any additional hypotheses on the systems coefficients and without assuming any growth type properties for the associated propagators. Furthermore, as an applica- tion of the admissibility results we establish a robustness property of the polynomial dichotomy under small perturbations"

"On Polynomial Dichotomies of Discrete Nonautonomous Systems on the Half-Line"

“…A first class of methods in which we will apply the central results of the present paper will be devoted to admissibility criteria for polynomial dichotomies and will be provided in [27], based on a new and different approach compared with the one considered in [24].…”

On some growth concepts for dichotomic behaviors of evolution operators

“…The most recent result on this line of research was obtained by D. Dragičević, A.L. Sasu and B. Sasu in [13] where the authors give some new admissibility conditions for uniform and exponential dichotomy and they also provide new characterizations for polynomial dichotomy by means of some double admissibilities. Moreover, the same authors in [12] obtain for the first time a characterization of polynomial dichotomy with respect to a sequence of norms in terms of exponential dichotomy.…”

On uniform polynomial splitting of variational nonautonomous difference equations in Banach spaces

“…Together with the exponential concepts, polynomial ones have been considered. Various results have been obtained in [2], [3], [9], [10], [11]. In this line, the present work considers the concept of polynomial splitting for a cocycle over a discrete semiflow.…”