Nonlinear partial functional differential equations: Existence and stability
type: Item
instance of: scholarly article
title: Nonlinear partial functional differential equations: Existence and stability
zbMATH Open document ID: 0991.35098
author: T. Caraballo
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-02-19T00:00:00Z
reviewed by: Drumi Bainov
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 1688654
zbMATH Keywords: exponential stability
zbMATH Keywords: nonlinear functional differential equations
zbMATH Keywords: finite delay
zbMATH Keywords: nonlinear monotone and coercive operators
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: variational approach
MaRDI profile type: Publication
OpenAlex ID: W2049746289
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DOI: 10.1006/JMAA.2001.7545
review text: The author investigates the existence and uniqueness of the solutions of a class of nonlinear functional differential equations in Hilbert spaces with finite delay, NEWLINE\[NEWLINEdx(t)/dt= A\bigl(t,x(t) \bigr)+ f(t, x_t)\;(t>0),\quad x(t)= \psi(t)\;(t\in [-h,0]),NEWLINE\]NEWLINE where \(A(t,\cdot): V\to V'\) is a family of nonlinear monotone and coercive operators and \(f(t,\cdot):X\to H\) is Lipschitz continuous, \(V\) is a separable Banach space and \(H\) is a real separable Hilbert space such that \(V\hookrightarrow H\equiv H'\hookrightarrow V'\). Here \(V'\) is the dual of \(V\) and the injections are continuous and dense, \(x_t(s)= x(t+s)\) \((s\in [-h,0])\).NEWLINENEWLINENEWLINEThe main purpose of the paper is to establish existence and uniqueness results using a variational approach. Some stability criteria for the problem under consideration are studied by using a Razumikhin type argument. Several examples illustrating the theory are given.
description: scientific article; zbMATH DE number 1688654