Semigroups of translations and functional differential equations
type: Item
instance of: scholarly article
title: Semigroups of translations and functional differential equations
zbMATH Open document ID: 0592.34043
DOI: 10.1016/0022-247X(85)90252-5
author: S. Thaller
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: A retarded functional differential equation of the form \(\dot x(\)t)\(=f(t,x_ t)\), \(t\geq t_ 0\), \(x_{t_ 0}=\phi\) is studied and an appropriate Banach space of measurable functions \(\phi\) : (-\(\infty,0]\to {\mathbb{R}}^ n\) is introduced on which it is assumed that the equation has unique solutions which depend continuously on the initial data. Using the theory of nonlinear evolution operators, it is shown that the solution of the equation U(t,s)x for \(x\in X\) is an evolution operator. Finally it is shown that, under the conditions of a theorem of \textit{M. G. Crandall} and \textit{A. Pazy} [Isr. J. Math. 11, 57-94 (1972; Zbl 0249.34049)] this evolution operator is a translation.
Mathematics Subject Classification ID: 34K05
zbMATH DE Number: 3951046
zbMATH Keywords: retarded functional differential equation
zbMATH Keywords: Banach space
zbMATH Keywords: evolution operator
reviewed by: S. P. Banks
MaRDI profile type: Publication
cites work: Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces
cites work: Nonlinear evolution equations in Banach spaces
cites work: 20.—Functional Differential Equations and Non-linear Evolution Operators
cites work: On semigroups of nonlinear operators and the solution of the functional differential equation \(\dot x(t)=F(x_t)\)
cites work: Q4163814
cites work: Q5509729
cites work: Some considerations to the fundamental theory of infinite delay equations
cites work: Nonlinear semigroups of translations in Banach space generated by functional differential equations
cites work: Existence and continuous dependence for functional-differential equations with unbounded delay
cites work: Autonomous nonlinear functional differential equations and nonlinear semigroups
cites work: Functional differential equations and nonlinear semigroups in \(L^p\) spaces
cites work: Asymptotic Stability for Abstract Nonlinear Functional Differential Equations
cites work: Nonlinear nonautonomous functional differential equations in Lp spaces
description: scientific article; zbMATH DE number 3951046