Individual stability and instability for evolutionary processes
type: Item
instance of: scholarly article
title: Individual stability and instability for evolutionary processes
zbMATH Open document ID: 1399.34174
author: Juan-Miguel Gracia
published in: Acta Mathematica Hungarica
publication date: 2018-02-13T00:00:00Z
review text: The authors consider an evolutionary process defined by a linear bounded operator \(\Phi(t,t_0)\), \(t\geq t_0\), on a Banach space \(X\). They are interested to study the convergence to zero or the divergence of a single trajectory \(\Phi(t,t_0)x\), when \(t\to +\infty\). Under some technical assumptions, the main result states that if there exists \(k(t_0)\) such that \[ \left( \int_t^\infty | | \Phi(\tau,t_0)x| | ^p\, d\tau \right)^{1/p}\leq k(t_0) | | \Phi(t,t_0)x| | \] for each \(t\geq t_0\), then the trajectory converges to zero with exponential decay. A similar result is given for the divergence of the trajectory \(\Phi(t,t_0)x\).
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 6837897
zbMATH Keywords: evolutionary process
zbMATH Keywords: convergence of a trajectory
zbMATH Keywords: divergence of a trajectory
reviewed by: Andrea Bacciotti
MaRDI profile type: Publication
full work available at URL: s10474-017-0754-y
OpenAlex ID: W2754501809
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cites work: Q3972482
cites work: On the Applicability of Lyapunov’s Theorem in Hilbert Space
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: On uniform N-equistability
cites work: On uniform exponential stability for skew-evolution semiflows on Banach spaces
DOI: 10.1007/S10474-017-0754-Y
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description: scientific article; zbMATH DE number 6837897