Krasnoselskii's fixed point theorem and stability
type: Item
instance of: scholarly article
title: Krasnoselskii's fixed point theorem and stability
zbMATH Open document ID: 1015.34046
DOI: 10.1016/S0362-546X(01)00111-0
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-07-29T00:00:00Z
review text: The authors present an application of the fixed-point theory in stability. They suggest a generalization of Krasnosel'skii's theorem on fixed-points of operators of the form \(A+B\), where \(A\) is completely continuous and \(B\) is contracting, and use their result to prove new theorems on the exponential stability of solutions to Cauchy problems. General theorems are applied to perturbed Liénard equations. One of the main results is as follows: Let \(M\) denote a closed convex nonempty subset of the Banach space \(U\) of bounded continuous functions \(\varphi: [0,\infty) \to\mathbb{R}^d\). Consider the Cauchy problem \[ x'=b(t,x) +a(t,x), \quad x(0)=x_0\in \mathbb{R}^d,\;t \in [0,\infty). \] Let \(b(t,x)\) be uniformly Lipschitz in \(x\) for \(t\in[0, \infty \)), \(x\in\mathbb{R}^d\). Let the operator \(A\) defined by \[ y\mapsto \int^t_0 a\bigl( s,y(s) \bigr)ds \] be continuous on \(M\) and the image \(A(M)\) of the set \(M\) be compact. Let the operator \(B\) defined by \[ y\mapsto \int^t_0 b\bigl(s,y(s) \bigr) ds \] be contracting on \(U\) with the constant \(\alpha<1\). Then if for each \(y\in M\) a unique solution \(x\) to \(x'=b(t,x)+ a(t,y)\), \(x(0)= x_0\), is in \(M\), then a solution to the Cauchy problem above is also in \(M\).
Mathematics Subject Classification ID: 34D30
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 1751610
zbMATH Keywords: exponential stability
zbMATH Keywords: solutions
zbMATH Keywords: Liénard equations
author: T. A. Burton
author: Tetsuo Furumochi
reviewed by: Dmitrii Rachinskii
MaRDI profile type: Publication
cites work: Q5822590
cites work: A generalization of a lemma of bellman and its application to uniqueness problems of differential equations
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cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q5330830
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full work available at URL: s0362-546x(01)00111-0
OpenAlex ID: W2023371347
description: scientific article; zbMATH DE number 1751610