Exponential trichotomy and p-admissibility for evolution families on the real line

Abstract

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1, ∞) and the pair (C b (R, X ), C c (R, X )) is uniformly p-admissible for an evolution family U = {U (t, s)} t≥s then U is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C b (R, X ), C c (R, X )) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C b (R, X ), L p (R, X )) for an evolution family U = {U (t, s)} t≥s is equivalent with its uniform exponential dichotomy.

Pairs of Function Spaces and Exponential Dichotomy on the Real Line

“…If one analyzes the dichotomous properties of differential equations, then it is easily seen that there are some main technical differences between the case of evolution families 2 Advances in Difference Equations on the half-line see 5, 9, 10 and the case of evolution families on the real line see [11][12][13][14][15][16] , which require a distinct analysis for each case. For instance, when one determines sufficient conditions for the existence of exponential dichotomy on the half-line, an important hypothesis is that the initial stable subspace is closed and complemented see, e.g., 5, Theorem 4.3 or 9, Theorem 3.3 .…”

Integral Equations on Function Spaces and Dichotomy on the Real Line

“…A different dichotomy concept is characterized by Preda, Pogan and Preda in [19], under the assumption that there exists a dichotomy projection family compatible with U (see (i)⇔(v)). An approach which generalizes the above equivalences (see (i)⇔(vi)) was given in [25], treating both discrete and integral case. The investigation was completed in [26], where the author deduced the equivalence (i)⇔(vii), based on the homologous discrete-time result.…”

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

“…In the study of the exponential dichotomy, there are two crucial differences between the real line and the half-line. The first one is that on the real line, if an evolution family is exponentially dichotomic with respect to a family of projections then the projections family is uniquely determined (see [6], [11], [21], [22], [25] and the references therein). In contrast, on the half-line an evolution family may be exponential dichotomic with respect to an infinite class of projection families (see e.g.…”

“…These arguments and also many others led to distinct studies concerning these two cases: the real line on the one side (see e.g. [6], [11], [21], [22], [25]) and the half-line on the other side (see e.g. [10], [14], [15], [19], [23], [24]).…”

“…L. Sasu and B. Sasu I EOT Remark 5.9. Using different arguments, the above corollary was recently proved in [23] (see Theorem 3.3).…”