A spectral notion for dynamic equations on time scales
type: Item
instance of: scholarly article
title: A spectral notion for dynamic equations on time scales
zbMATH Open document ID: 1013.34050
DOI: 10.1016/S0377-0427(01)00451-4
author: S. Siegmund
published in: Journal of Computational and Applied Mathematics
publication date: 2002-06-24T00:00:00Z
review text: The author considers the linear \(N\)-dimensional system of dynamic equations \[ x^\Delta= A(t) x\tag{1} \] on a time scale \(\mathbb{T}\) with an \(rd\)-continuous and regressive matrix-function \(A: \mathbb{T}\to \mathbb{R}^{N\times N}\), \(N\in\mathbb{N}= \{1,2,\ldots\}\). It is assumed that \(\mathbb{T}\) is unbounded to the right and either bounded to the left (one sided-time) or unbounded to the left (two-sided time), and \(\phi_A(t,s)\) is the evolution operator of (1). The notion of invariant vector bundle of (1) is defined to be a nonempty set \(W\subset\mathbb{T}\times \mathbb{R}^N\) with the properties: (a) for every \(\tau\in\mathbb{T}\) the fiber \(W(\tau)= \{\xi\in\mathbb{R}^N: (\tau,\xi)\in W\}\) is a linear subspace of \(\mathbb{R}^N\), (b) it is invariant, i.e., \((\tau,\xi)\in W\Rightarrow (t,\phi_A(t,\tau)\xi)\in W\) for all \(t\in\mathbb{T}\). An invariant projector of (1) is also defined to be a function \(P:\mathbb{T}\to \mathbb{R}^{N\times N}\) of projection \(P(t)\), \(t\in\mathbb{T}\), such that \[ P(t)\phi_A(t, s)= \phi_A(t,s) P(s)\quad\text{for }t,s\in\mathbb{T}. \] For \(\gamma\in \mathbb{R}\), it is said that (1) admits an exponential dichotomy with growth rate \(\gamma\) (\(\gamma\)-ED) if there exist an invariant projector \(P:\mathbb{T}\to \mathbb{R}^{N\times N}\) and constants \(K\geq 1\) and \(\alpha> 0\) such that, for \(t,s\in\mathbb{T}\), the dichotomy estimates \[ \|\phi_A(t,s)P(s)\|\leq ke^{(\gamma-\alpha)(t-s)}\quad\text{for }t\geq s\qquad\text{and} \] \[ \|\phi_A(t, s)(I- P(s)\|\leq ke^{(\gamma+\alpha)(t-s)}\quad\text{for }t\leq s, \] hold. The dichotomy spectrum of (1) is defined to be the set \[ \Sigma(A)= \{\gamma\in \mathbb{R}: x^\Delta= A(t) x\text{ admits no }\gamma\text{-ED}\} \] and the resolvent set \(\rho(A)= \mathbb{R}\setminus \Sigma(A)\) is its complement. The main result is the spectral theorem, which states that the dichotomy spectrum \(\Sigma(A)\) of (1) is the disjoint union of \(n\) closed intervals (called …
reviewed by: Marian Ivanovici
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34N05
zbMATH DE Number: 1758484
zbMATH Keywords: linear dynamic equations
zbMATH Keywords: time scales
zbMATH Keywords: exponential dichotomy
zbMATH Keywords: dichotomy spectrum
zbMATH Keywords: qualitative theory
MaRDI profile type: Publication
cites work: Q4033029
cites work: Q4650838
cites work: The dichotomy spectrum for noninvertible systems of linear difference equations
cites work: Q2723195
cites work: Dichotomies in stability theory
cites work: Analysis on measure chains - a unified approach to continuous and discrete calculus
cites work: Q4716262
cites work: A spectral theory for linear differential systems
cites work: Dichotomy spectrum for nonautonomous differential equations
full work available at URL: s0377-0427(01)00451-4
OpenAlex ID: W1985951228
Wikidata QID: Q127953754
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description: scientific article; zbMATH DE number 1758484