Admissibility and generalized nonuniform dichotomies for discrete dynamics

Abstract

We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth.

"Admissibility and Polynomial Dichotomy of Discrete Nonautonomous Systems"

“…The second category of methods is often stronger (and requires combined arguments of functional analysis and control) as in some cases it provides not only the robustness property, but also a radius within which the "size" of the perturbation should fit (see [10,11,32,58,63,66] and the references therein). We note that, for the special case of nonuniform polynomial dichotomic behaviors, recent robustness results have been obtained in [18,19,65].…”

“…Other notions of polynomial stabilities, instability and expansiveness were explored by Hai in [29,30]. The importance of the polynomial behaviors is certified even more by the recent studies on generalized dichotomies (see Silva [65] and the references therein). It should be mentioned here that, over more than a decade, various studies on the polynomial behaviors of dynamical systems both in nonautonomous and variational case were coordinated by Megan (see [9,51,52] and the references therein).…”

“…Among the most important methods in this area we mention the input-output techniques, or the so-called admissibility methods, which proved their effectiveness particularly in studying dichotomies and provided interesting answers to open problems regarding nonautonomous dynamics (see Aulbach and Minh [1], Barreira, Dragičević and Valls [2][3][4], Chicone and Latushkin [11], Chow and Leiva [12], Dragičević [17][18][19], Dragičević, Zhang and Zhou [25], Elaydi and Janglajew [27], Huy and Minh [33], Megan, Sasu and Sasu [38][39][40], Minh, Räbiger and Schnaubelt [42], Palmer [44][45][46], Pliss and Sell [50], Sasu and Sasu [55,59,[61][62][63], Sasu [56], Sasu [57,58], Sasu, Babut ¸ia and Sasu [60], Silva [65], Zhang [68], Zhou and Zhang [70], Zhou, Lu and Zhang [71]). Even though technically these methods trace back to the pioneering works of Perron [47] and Li [35], for the historical origins of the admissibility notions and related methods we refer to the landmark works of Massera and Schäffer [36,37] and Coffman and Schäffer [14] as well as to the books of Daleckȋi and Kreȋn [16], Coppel [15] and Henry [31]…”

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

“…For similar results in the case of continuous time dynamics, we refer to Dragičević [25]. Moreover, the polynomial dichotomies and their properties are also relevant in the context of generalized dichotomies (see Silva [71]). In this framework, over the last decade, many studies initiated and coordinated by Megan were devoted to polynomial behaviors (among which we mention polynomial stabilities, polynomial instabilities, polynomial dichotomies).…”