Datko criteria for uniform instability in Banach spaces

Abstract

The main objective of this paper is to present some necessary and sufficient conditions of Datko type for the uniform exponential and uniform polynomial instability concepts for evolution operators in Banach spaces.

Some Datko and Barbashin type characterizations for the uniform \(h\)-instability of evolution families

“…Corollary 3.8 ( [6]). Let U = {U (t, s)} t≥s≥0 be an evolution family with uniform polynomial decay.…”

“…After the path-breaking research of Barbashin and Datko, through the usage of Datko-Pazy theorem, admissibility, Banach function spaces, Lyapunov functions, discrete-time methods and other ways, there has been numerous extensions and generalizations devoted to this area (see [2,6,9,12,13,14,15,16,17,18,22,23,24,27,30,31] and the references therein). For instance, in [12] Dragičević obtained some continuous and discrete versions of Datko type theorem for the concept of the exponential stability in average.…”

“…For instance, in [12] Dragičević obtained some continuous and discrete versions of Datko type theorem for the concept of the exponential stability in average. In [6] and [23], the Datko and Barbashin type integral characterizations for uniform exponential instability of evolution operators were studied by Boruga and Megan. Also, a significant contribution in this direction was made by Bento et al in [5] who obtained an interesting Datko type characterization for nonuniform µ-dichotomy.…”

“…Motivated by the recent work of Boruga, Megan and Toth ( [8]), in this paper, we introduce the concept of uniform h-instability for evolution families which is an extension of classical concepts of uniform exponential instability and uniform polynomial instability. Our main objective is to give some necessary and sufficient conditions of Datko type and Barbashin type for the uniform h-instability concept of evolution families on Banach spaces, and variants for uniform h-instability of some well-known results in stability theory (Barbashin ( [1]), Datko ([10]), Boruga et al ([8])) and instability theory (Boruga and Megan, [6,23]) are obtained. We emphasize that the set of growth rates considered in this paper are different from that used in [5].…”

"On Uniform Instability with Growth Rates in Banach Spaces"

“…Among the most studied concepts in this field are the exponential instability ( [1], [5], [8], [9], [10], [12]), and the polynomial instability ( [2], [3], [6], [11]).…”