Liapunov functionals and periodicity in integral equations
type: Item
instance of: scholarly article
title: Liapunov functionals and periodicity in integral equations
zbMATH Open document ID: 0808.45009
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1994-10-16T00:00:00Z
review text: A Lyapunov functional is constructed for the scalar nonconvolution integral equation \[ x(t) = a(t) - \int^ t_{-\infty} D(t,s) g \bigl( s,x (s) \bigr) ds \quad (t \geq t_ 0). \] The kernel \(D\) is of the type \(D(t,s) = B(t,s) - Q(t,s)\), where \(B_ s (t,s) \geq 0\), \(B_{st} (t,s) \leq 0\), and \(Q(t,s)\) is a ``perturbation term.'' First a basic inequality is proved, and then this inequality is used to derive a number of asymptotic results concerning boundedness and some additional asymptotic properties of \(x\), such as \(\int^ \infty_{t_ 0} x(t) g(t,x(t)) dt < \infty\) and \(x(t) \to 0\) as \(t \to \infty\). The existence of periodic solutions is also studied, and some extensions to systems of linear equations are included.
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45M15
zbMATH DE Number: 650897
zbMATH Keywords: Lyapunov functional
zbMATH Keywords: nonconvolution integral equation
zbMATH Keywords: boundedness
zbMATH Keywords: asymptotic properties
zbMATH Keywords: periodic solutions
zbMATH Keywords: systems
author: T. A. Burton
reviewed by: Olof Staffans
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225758
OpenAlex ID: W2040663233
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DOI: 10.2748/TMJ/1178225758
description: scientific article; zbMATH DE number 650897