Measure neutral functional differential equations as generalized ODEs
type: Item
instance of: scholarly article
title: Measure neutral functional differential equations as generalized ODEs
zbMATH Open document ID: 1417.34195
published in: Journal of Dynamics and Differential Equations
publication date: 2019-04-25T00:00:00Z
review text: The authors consider integral equations of the form \[ x(t)=x(t_0)+\int_{t_0}^t f(x_s,s)\,\mathrm{d}g(s)+\int_{-r}^0 \mathrm{d}_\theta[\mu(t,\theta)]x(t+\theta)-\int_{-r}^0 \mathrm{d}_\theta[\mu(t_0,\theta)]\varphi(\theta), \] for which they coin the term ``measure neutral functional differential equations''. The solution $x$ and the function $f$ take values in $\mathbb{R}^n$, $g$ is a real nondecreasing function, $\mu$ takes values in $\mathbb{R}^{n\times n}$ and is left-continuous with bounded variation with respect to the second variable, and $\varphi$ is the initial condition on $[-r,0]$. \par It is shown that under certain assumptions, the integral equations of the above-mentioned type represent a special case of Kurzweil's generalized ordinary differential equations. Hence, one can easily obtain some basic results on the local existence and uniqueness of solutions, as well as continuous dependence on parameters.
reviewed by: Antonín Slavík
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 7047664
zbMATH Keywords: generalized ordinary differential equations
zbMATH Keywords: measure neutral functional differential equations
zbMATH Keywords: Kurzweil-Henstock-Stieltjes integral
author: M. Federson
author: Miguel V. S. Frasson
author: Jaqueline Godoy Mesquita
author: Patricia H. Tacuri
MaRDI profile type: Publication
full work available at URL: s10884-018-9682-y
OpenAlex ID: W2829317630
cites work: Boundedness of solutions of retarded functional differential equations with variable impulses via generalized ordinary differential equations
cites work: Existence results for initial value problems for neutral functional differential equations
cites work: Differential-difference equations
cites work: On Optimal Controls for Measure Delay-Differential Equations
cites work: Existence and stability of measure differential equations
cites work: Q3081879
cites work: Generalized ODE approach to impulsive retarded functional differential equations
cites work: Topological dynamics of retarded functional differential equations.
cites work: Measure functional differential equations and functional dynamic equations on time scales
cites work: Continuous dependence on a parameter of solutions of generalized differential equations
cites work: Q5758403
cites work: A Class of Neutral Equations with the Fixed Point Property
cites work: Asymptotic behavior of neutral functional differential equations
cites work: Introduction to functional differential equations
cites work: A class of functional equations of neutral type
cites work: Periodic solutions of abstract neutral functional differential equations
cites work: Linear autonomous neutral functional differential equations
cites work: On a new class of abstract neutral differential equations
cites work: Q5568181
cites work: Q3901847
cites work: Generalized ordinary differential equations and continuous dependence on a parameter
cites work: Addition to my paper “Generalized ordinary differential equations and continuous dependence on a parameter”
cites work: A class of neutral functional differential equations
cites work: Q5568182
cites work: Differential systems involving impulses
cites work: Fundamental theorems for linear measure differential equations.
cites work: Optimal Control Theory for Nonlinear Vector Differential Equations Containing Measures
cites work: Q4023958
cites work: An Abstract Measure Differential Equation
cites work: Dynamic equations on time scales and generalized ordinary differential equations
cites work: Measure functional differential equations with infinite delay
cites work: Neutral equations and associated semigroups
cites work: Integrals and Banach spaces for finite order distributions
cites work: Regulated functions and the Perron-Stieltjes integral
Wikidata QID: Q129545209
DOI: 10.1007/S10884-018-9682-Y
description: scientific article; zbMATH DE number 7047664