Lyapunov functions and boundedness for differential and delay equations
type: Item
instance of: scholarly article
title: Lyapunov functions and boundedness for differential and delay equations
zbMATH Open document ID: 0653.34049
published in: Hiroshima Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: In the study of ordinary and functional differential equations by means of Lyapunov's direct method, it frequently happens that the Lyapunov function fails to be radially unbounded and the derivative fails to be strictly negative definite at the same time. The classic example is given by the system \(\{x'=y\), \(y'=-q(x,y)y-g(x)\}\) in which q(x,y)\(\geq 0\), \(xg(x)>0\) if \(x\neq 0\). The Lyapunov function \(V=y^ 2+2\int^{x}_{0}g(s)ds\) yields \(V'=-2q(x,y)y^ 2.\) Thus, if \(\int^{x}_{0}g(s)ds\) is bounded, then the described situation obtains along the x-axis. In this paper it is shown how to convert the problem into a differential inequality yielding bounded solutions. Several examples are given for both ordinary and functional differential equations.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 4065480
zbMATH Keywords: functional differential equations
zbMATH Keywords: Lyapunov function
author: T. A. Burton
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206129727
OpenAlex ID: W1499486256
DOI: 10.32917/HMJ/1206129727
description: scientific article; zbMATH DE number 4065480