Lyapunov pairs for continuous perturbations of nonlinear evolutions
type: Item
instance of: scholarly article
title: Lyapunov pairs for continuous perturbations of nonlinear evolutions
zbMATH Open document ID: 1173.37009
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-06-24T00:00:00Z
review text: For the abstract Cauchy problem \[ y'(t)\in Ay(t) + F(y(t))\;,\;y(0)=\xi \] with \(A:D(A)\subseteq X\mapsto X\) a (possibly) multivalued \(m\)-dissipative operator, \(X\) being a real Banach space, \(\emptyset\neq M\subset \overline{D(A)}\) and \(F:M\mapsto X\) a function. The pair \(V,g:X\mapsto R\) is a Liapunov pair of the above equation if \(\mathrm{dom} (V)\subseteq M\) and the dissipativeness inequality holds \[ V(y(t)) + \int_0^t g(y(\tau))d\tau \leq V(\xi)\;,\;0\leq t\leq T \] The necessary and sufficient condition for \((V,g)\) to be a Liapunov pair is \[ \underline{D}^AV(\xi)F(\xi)+g(\xi)\leq 0\;,\;\xi\in \mathrm{dom}(V) \] where \(\underline{D}^AV(\xi)(v)\) is the \(A\)-contingent derivative of \(V\) at \(\xi\in \mathrm{dom}(V)\) in the direction \(v\in X\).
reviewed by: Vladimir Răsvan
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 93D30
zbMATH DE Number: 5569395
zbMATH Keywords: Lyapunov pair
zbMATH Keywords: \(m\)-dissipative operator
zbMATH Keywords: viability
zbMATH Keywords: contingent derivative
author: Ovidiu Cârjă
author: A. I. Lazu
MaRDI profile type: Publication
full work available at URL: j.na.2008.11.022
OpenAlex ID: W2047337165
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DOI: 10.1016/J.NA.2008.11.022
description: scientific article; zbMATH DE number 5569395