Global continuation for bounded solutions of ordinary differential equations
type: Item
instance of: scholarly article
title: Global continuation for bounded solutions of ordinary differential equations
zbMATH Open document ID: 0802.34038
DOI: 10.12775/TMNA.1993.031
published in: Topological Methods in Nonlinear Analysis
publication date: 1994-12-18T00:00:00Z
review text: The author considers a parameter dependent family of differential equations of the form \((*)\) \(dx/dt= \mu F(x,t,\mu)\), where \(F\) is a continuous function of \((x,t,\mu)\in D\times \mathbb{R}\times [0,1]\), and gives some sufficient conditions which guarantee the existence of bounded solutions (i.e., \(\| x\|:= \sup_{t\in \mathbb{R}} | x(t)|<\infty\)) when \(F(x,t,\mu)\) is almost periodic in \(t\in \mathbb{R}\). Here, \(D\) stands for an open set in \(\mathbb{R}^ m\). The result is stated in the form of a theorem whose proof, which is rather long, is based upon applying Conley index theory to a family of skew-product flows associated with \((*)\). As a particular application he derives the sufficient conditions for the equation \(dx/dt= F(x,t)\) to have bounded solutions. Here \(F: \mathbb{R}^ m\times \mathbb{R}\to\mathbb{R}^ m\) is continuous and uniformly almost periodic in \(t\in \mathbb{R}\).
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C27
zbMATH DE Number: 591506
zbMATH Keywords: parameter dependent family of differential equations
zbMATH Keywords: bounded solutions
zbMATH Keywords: almost periodic
zbMATH Keywords: Conley index
zbMATH Keywords: family of skew-product flows
author: James Robert jun. Ward
reviewed by: Mithat Idemen
MaRDI profile type: Publication
full work available at URL: tmna.1993.031
OpenAlex ID: W2602072880
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description: scientific article; zbMATH DE number 591506