Schäffer spaces and uniform exponential stability of linear skew-product semiflows

Abstract

We study the exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers. Our main tool is the admissibility of a pair of the so-called Schäffer spaces. This characterization is a very general one, it includes as particular cases many interesting situations among them we can mention some results due to Clark, Datko, Latushkin, van Minh, Montgomery-Smith, Randolph, Räbiger, Schnaubelt.

Schäffer spaces and exponential dichotomy for evolutionary processes

“…In contrast to this "philosophy," the present paper shows that we can characterize the exponential dichotomy in terms of the admissibility of some suitable pairs of spaces in a direct way, without the so-called evolution semigroup, see also [14][15][16][17]. But this is not the first aim of our paper.…”

“…The main purpose of this paper is to continue the line of research in [14][15][16][17] and to give conditions general on the relationship between "input space" and on the "output space" so that the Perron type of result still holds in the general case of evolutionary processes. Example 3.1 shows that the condition (♦) in Theorem 3.1 cannot be dropped, showing that our conditions on the "input space" and on the "output space" are as general as possible.…”

Perron type theorems for skew-evolution semiflows

“…The theory of skew-product semiflows is well represented in the literature by the papers due to C. Chicone Characterization for the asymptotic properties of skew-product semiflows were obtained by C. Preda, P. Preda, A. Petre ( [16]) and P. Preda, A. Pogan, C. Preda in [17].…”

Nonuniform Exponential Dichotomy for Evolution Families on the Real Line

“…Preda et al [23] deal with the study of the uniform exponential stability for Schäffer spaces. In 2010, in [1], Barreira and Valls, using appropriate adapted norms (which can be seen as Lyapunov norms) mentioned the connection between a nonuniform exponentially stable evolution process and the admissibility of their associated L p spaces, denoted by L p (X), where p ∈ [1, ∞].…”