Existence and roughness of nonuniform \((h,k,\mu,\nu)\)-trichotomy for nonautonomous differential equations
type: Item
instance of: scholarly article
title: Existence and roughness of nonuniform \((h,k,\mu,\nu)\)-trichotomy for nonautonomous differential equations
zbMATH Open document ID: 1414.34044
DOI: 10.1216/RMJ-2018-48-8-2751
author: Meng Fan
author: Chunmei Zhang
author: Jimin Zhang
published in: Rocky Mountain Journal of Mathematics
publication date: 2019-01-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1546138830
review text: The property of exponential trichotomy for a nonautonomous system of linear differential equations \[ \dot{x}=A(t)x \tag{1} \] on a Banach space $\mathcal{X}$ has been introduced by R. J. Sacker and G. Sell for the special case $\mathcal{X}=\mathbb{R}^{n}$. This property has been generalized in several ways. In particular, this article introduces the property of nonuniform $(h,k,\mu,\nu)$-trichotomy on $\mathbb{R}^{+}$ in the following way: \par There exists projections $P(t),Q(t),R(t): \mathcal{X}\to \mathcal{X}$ such that \[ P(t)+Q(t)+R(t)=\mathrm{Id} \] and the evolution operator $T(t,s)$ associated to (1) satisfies the following properties: \begin{itemize} \item[(i)] For any $t,s\in \mathbb{R}^{+}$ it follows that: \[ \begin{aligned} T(t,s)P(s)&=P(t)T(t,s)\\ T(t,s)Q(s)&=Q(t)T(t,s)\\ T(t,s)R(s)&=R(t)T(t,s). \end{aligned} \] \item[(ii)] There exist constants $a\leq 0 < b$, $c\leq 0<d$, $\varepsilon \geq 0$ and $K_{1}\geq 0$ such that, for any $t\geq s \geq 0$, \[ \begin{aligned} |T(t,s)P(s)|&\leq K_{1}\left(\frac{h_{1}(t)}{h_{1}(s)}\right)^{a}\mu_{1}^{\varepsilon}(s),\\ |T(t,s)R(s)|&\leq K_{1}\left(\frac{h_{2}(t)}{h_{2}(s)}\right)^{b}\mu_{2}^{\varepsilon}(s), \end{aligned} \] and, for any $0\leq t \leq s$, \[ \begin{aligned} |T(t,s)Q(s)|&\leq K_{1}\left(\frac{k_{1}(s)}{k_{1}(t)}\right)^{c}\nu_{1}^{\varepsilon}(s),\\ |T(t,s)R(s)|&\leq K_{1}\left(\frac{k_{2}(s)}{k_{2}(t)}\right)^{d}\nu_{2}^{\varepsilon}(s), \end{aligned} \] where the functions $h_{i},k_{i},\mu_{i},\nu_{i}$ ($i=1,2$) are in the set \[ \triangle=\left\{ u: \mathbb{R}^{+}\to [1,+\infty) \mid u\text{ is increasing}, u(0)=1\text{ and }\lim_{t\to +\infty}u(t)=+\infty\right\}. \] \end{itemize} In order to introduce the first result, the authors focus its attention on finite dimensional systems (1), where the linear operator $A(t)$ is of the type \[ A(t)=\left[\begin{matrix} A_{1}(t) & 0 & 0 \\ 0 & A_{2}(t) & 0 \\ 0 & 0 & A_{3}(t) \end{matrix}\right], \tag{2} \] which leads to the subsystems \[ x_{1}'=A_{1}…
reviewed by: Gonzalo Robledo
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 6999283
zbMATH Keywords: nonautonomous differential equations
zbMATH Keywords: trichotomy
zbMATH Keywords: $(H,K,L)$ Lyapunov exponent
zbMATH Keywords: roughness
MaRDI profile type: Publication
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OpenAlex ID: W2908131848
description: scientific article; zbMATH DE number 6999283