On the asymptotic stability of differential equations with ''maxima''
type: Item
instance of: scholarly article
title: On the asymptotic stability of differential equations with ''maxima''
zbMATH Open document ID: 0766.34054
DOI: 10.1007/BF02845076
author: Drumi Bainov
author: Hristo Voulov
publication date: 1992-09-27T00:00:00Z
review text: The scalar differential equation with maxima of the following form is considered: (1) \(\dot x(t)=F(t,x(t),\max\{x(s):s\in[g(t),t]\})\), where \(g(t)<t\), \(t\geq t_ 0\) and \(\dot x\) denotes the right derivative of the function \(x\). Effective sufficient as well as necessary and sufficient conditions for the asymptotic stability of the trivial solution are given. The following formulation gives an idea about the obtained results. Assumptions: \(H_ 1\). The function \(F:(t,x,y)\to R\) is continuous for \(t\in[t_ 0,\infty)=I\), \(x^ 2+y^ 2<Q\leq+\infty\) and satisfies \(xF(t,x,x)<0\) for \(x\neq 0\). \(H_ 2\). The function \(g\) is continuous, nondecreasing for \(t\in I\) and satisfies the inequality \(g(t)<t\). \(H_ 3\). For any \(\sigma\in I\) the initial value problem (1), \(x_ \sigma=0\) has a unique trivial solution. \(H_ 4\). There exists a function \(\ell:I\to I\), such that \(\ell(\sigma)\geq\sigma\) and \(L(x)\subset(\sigma,\ell(\sigma)\) for each solution \(x\) of the initial problem (1), \(x_ \sigma=\varphi\in B\). The definition of \(L(x)\) is clear from the following equalities: \(y(t)=\max\{x(s):s\in[g(t),t]\}\), \(D(x)=\{t\in(\sigma,\infty):x(t)<y(t)\}\), \(L(x)=\{t\in D(x):y(t)\neq x(g(t))\}\). The equation (2) \(\dot x(t)=F(t,x(t),\;x(g(t)))\) is associated to equation (1). Provided that conditions \(H_ 1\), \(H_ 2\), \(H_ 3\) and \(H_ 4\) hold and the trivial solution of equation (2) is stable, then the trivial solution of equation (1) is stable.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 60498
zbMATH Keywords: asymptotic stability
zbMATH Keywords: scalar differential equation with maxima
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02845076
OpenAlex ID: W1972456890
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 60498