Polynomial expansiveness and admissibility of weighted Lebesgue spaces

Abstract

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"Generalized exponential behavior on the half-line via evolution semigroups"

“…one may step over the polynomial behavior (see [10,11,18]). Furthermore, (2.3) shows that if in particular the function ω : R + → R + is non-decreasing, then any ω-exponentially stable, ω-exponentially expansive or ω-exponentially dichotomic evolution family, with constants ν > 0 and N ≥ 1, admits the corresponding uniform exponential behavior with new constants N e νω(t0)t0 and νω(t 0 ) for any fixed t 0 > 0.…”

"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]).…”

“…In the case of continuous-time nonautonomous systems, Dragičević introduced in [19] the notion of polynomial dichotomy with respect to a family of norms and characterized it in terms of admissibility relative to an integral equation. 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).…”

“…Despite the advances made in the admissibility theory so far, the studies on the polynomial behaviors via admissibility methods represented a challenging topic that required new and different approaches compared with those previously used in the literature (see Dragičević [18,19], Hai [29,30]). For instance, it should be noted that in the first studies devoted to admissibilities for polynomial dichotomies (see [18,19]) the structures of the input-output systems were different from the classic ones.…”

“…Consequently, the admissibility notions were distinct compared with those generally used when exploring a dichotomy of a nonautonomous system -see the approaches in [18] compared with those in [39,55,56,61] and respectively the method (and the integral equation considered) in [19] versus the admissibility concepts (and the corresponding integral equations) employed in [38,42,59,60]. Similarly, nontrivial technical changes must be done when using admissibility conditions to study other polynomial behaviors of nonautonomous systems (such as polynomial stability, polynomial expansiveness), as it can be seen by comparing the approaches in [29,30] with the methods (and the input-output systems) considered in [1,21,40,42]. In this context, for the forthcoming studies, the natural question arises whether one can use the "classic structure" for the input-output systems when exploring a polynomial behavior and, if so, which would be the new requirements regarding the input or output spaces.…”