Topological degree of solution mappings in functional and ordinary differential equations
type: Item
instance of: scholarly article
title: Topological degree of solution mappings in functional and ordinary differential equations
zbMATH Open document ID: 0898.34063
author: Takashi Kaminogo
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-10-15T00:00:00Z
review text: Initial value problems for functional- differential equations \[ x'(t)=f(t,x_t), \quad x_0=\phi , \tag{1} \] with \(x_t(s)=x(t+s)\) for \(s\in [-r,0]\) and \(\phi \in C[-r,0]\), are considered. Function \(f\) is assumed to satisfy the following condition: there exist two positive constants \(a\) and \(b\) such that \(| f(t,\psi)| \leq a+b| \psi | \) for \((t,\psi)\in [0,T]\times C[-r,0]\). This assumption ensures that every solution to (1) is extendable to be continuous up to \(t=T\). The mapping \(\Gamma =\{x_T: x\) is a solution to (1) defined on \([-r,T]\}\) is multivalued and acts from \(C[-r,0]\) to \(C[-r,0]\). A topological degree for the mapping \(\Gamma\) is defined. It is shown that in case \(r=0\) this definition is equivalent to the definition of the topological degree for ordinary differential equations, which was introduced formerly by the author.
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 34C99
zbMATH DE Number: 1140486
zbMATH Keywords: topological degree
zbMATH Keywords: functional-differential equations
reviewed by: Alexandra Rodkina
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225059
OpenAlex ID: W1989913944
cites work: Q5185568
cites work: Introduction to functional differential equations
cites work: Kneser's property and boundary value problems for some retarded functional differential equations
cites work: Q3955689
cites work: Q4208360
cites work: Q4143788
cites work: Q3706042
DOI: 10.2748/TMJ/1178225059
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description: scientific article; zbMATH DE number 1140486