Admissibility and polynomial dichotomies for evolution families

Abstract

For an arbitrary noninvertible evolution family on the half-line and for ρ : [0, ∞) → [0, ∞) in a large class of rate functions, we consider the notion of a ρ-dichotomy with respect to a family of norms and characterize it in terms of two admissibility conditions. In particular, our results are applicable to exponential as well as polynomial dichotomies with respect to a family of norms. As a nontrivial application of our work, we establish the robustness of general nonuniform dichotomies.

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

“…The relevance of these results stem from the fact that the notion of polynomial dichotomy with respect to a sequence of norms includes both the notions of polynomial dichotomy and respectively of nonuniform polynomial dichotomy as particular cases. 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]).…”

“…On the other hand, it is relatively easy to construct broad classes of nonautonomous dynamics which admit a splitting into stable and unstable directions, but with non-exponential rates in describing stability and instability. Among many meaningful and useful concepts used to describe this type of asymptotic behaviors, we particularly mention those of polynomial type (see [7,8,10,11,[15][16][17]24,25,31,32,40,53,54,69] and the references therein). The polynomial stability was studied in the autonomous case by Bátkai, Engel, Pr üss and Schnaubelt in [8] and later, from a different perspective, by Hai for evolution families in [31] and for skewevolution semiflows in [32].…”

“…As mentioned above, the majority of the studies regarding the polynomial behaviors were devoted to Datko and Barbashin type criteria (see [15-17, 32, 40, 53, 54] and the references therein), while the admissibility type methods have proved to be a challenging topic that required distinct approaches and completely different techniques (see [24,25,31] and the presentations therein). In order to further develop studies in this direction, based on new admissibility properties among other topics, it would be necessary to establish new connections between polynomial dichotomies and other dichotomy concepts for which the admissibility methods are more advanced.…”

"Admissibility and Polynomial Dichotomy of Discrete Nonautonomous Systems"

“…For all that, as we pointed out in [24], in certain situations, a dynamical system may exhibit a splitting of the state space into (closed, invariant) stable and unstable subspaces, but with non-exponential rates in describing stability and instability. In this framework, some of the most representative asymptotic behaviors, which are not of an exponential nature, are those of polynomial type (see [5,[7][8][9]18,19,24,29,30,51,52] and the references therein). Thus, we emphasize that, in the case of the dichotomic behaviors, in contrast with the concepts of exponential dichotomy, in the notions of polynomial dichotomy the rates of contraction and expansion are of polynomial type (see [5, 7-9, 18, 19, 24, 51]).…”

Admissibility and generalized nonuniform dichotomies for discrete dynamics

“…Among the long list of works, several results were obtained in [1,2,6,9,12] for general nonuniform behavior and in [8,10,11] for nonuniform polynomial behavior. In [18], Dragicevic used the notion of admissibility to establish the existence of nonuniform polynomial dichotomies in the discrete time setting (for corresponding results in continuous-time see [17]). Our work was motivated by [18,4].…”