Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications

Abstract

The purpose of this paper is to provide a new, unified and complete study for uniform dichotomy and exponential dichotomy on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over R+ which are invariant under translations. After that, we obtain a classification of the main classes of function spaces over R+, in order to deduce necessary and sufficient conditions for the existence of exponential dichotomy, emphasizing on the main technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years.Mathematics Subject Classification (2010). Primary 34D09; Secondary 34D05.

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

“…We recall that on the one hand, it is of interest to establish the structure of the potential candidates for projections and, on the other hand, one should investigate whether the families of projections are unique or not. The studies on these matters were developed from various perspectives (see Aulbach and Kalkbrenner [1], Aulbach, Minh and Zabreiko [2], Barreira, Dragičević and Valls [5], Battelli, Franca and Palmer [9], Chow and Yi [19], Chow and Leiva [20,21], Dragičević, Sasu and Sasu [26], Kloeden and Rasmussen [34], Megan, Sasu and Sasu [36][37][38], Minh, Räbiger and Schnaubelt [41], Minh and Huy [42], Palmer [44,45], P ötzsche [51], Pliss and Sell [50], Sacker and Sell [55], Sasu [56], Sasu, Babut ¸ia and Sasu [63], Sasu and Sasu [57][58][59][60][61][62][64][65][66][67][68], Zhou, Lu and Zhang [72], Zhou and Zhang [73,75]). We stress that depending on the method, sometimes it is important to establish the structure of the (initial) stable subspace(s) (see Sasu [56], Sasu and Sasu [57,62,65], Sasu, Babut ¸ia and Sasu [63]), in some cases it is necessary to study the projections uniqueness (see Battelli, Franca and Palmer…”

Admissibility and polynomial dichotomies for evolution families

“…Since then various authors obtained valuable contributions to this line of the research. For the results dealing with characterizations of uniform exponential behavior in terms of appropriate admissibility properties, we refer to the works of Huy [15], Latushkin, Randolph and Schnaubelt [16], Van Minh, Räbiger and Schnaubelt [22], Van Minh and Huy [23], Preda, Pogan and Preda [28,29] as well as Sasu and Sasu [30,31,32,33,34]. For contributions dealing with various concepts of nonuniform exponential behavior, we refer to [4,5,17,21,27,35,36] and references therein.…”

“…The case of exponential dichotomies on the half-line was first considered (in the infinitedimensional case) in [32]. For more recent results, we refer to the works of Huy [14], Latushkin, Randolph and Schnaubelt [15], Preda, Pogan and Preda [26,27] as well as Sasu and Sasu [29,30,31]. For results dealing with various flavours of nonuniform behaviour, we refer to [1,2,17,20,25,28,33,34] and references therein.…”

(h, k)-Dichotomy and Lyapunov Type Norms

“…Dichotomies have been the subject of extensive research over the last years, leading to exciting new results. For more details, we refer the reader [10], [13], [18]. A natural generalization of both the uniform and nonuniform, exponential and polynomial dichotomy is successfully modeled by the concept of (h, k)-dichotomy, where h, k are growth rates (nondecreasing functions that go to infinity).…”