Periodic solutions of measure functional differential equations
type: Item
instance of: scholarly article
title: Periodic solutions of measure functional differential equations
zbMATH Open document ID: 1481.45010
author: S. M. Afonso
author: Márcia R. da Silva
author: E. M. Bonotto
published in: Journal of Differential Equations
publication date: 2021-12-22T00:00:00Z
review text: Let \(f,g \colon \mathbb{R} \times G_T^- (\mathbb{R}, \mathbb{R}^n) \to \mathbb{R}^n\) be \(T\)-periodic functions in the first variable, \(T >0\) and \(G_T^- (\mathbb{R}, \mathbb{R}^n)\) denote the space of all \(T\)-periodic regulated left-continuous functions. For \(x \in G_T^- (\mathbb{R}, \mathbb{R}^n)\) and \(t \in \mathbb{R}\) the symbol \(x_t\) denotes the function from \(G_T^- (\mathbb{R}, \mathbb{R}^n)\) given as \(x_t(\theta) = x(t+ \theta)\), \(\theta \in \mathbb{R}\). The authors study the existence of regulated \(T\)-periodic solutions to the following measure functional equation \[ x(t) = x(0) + \int_0^t f(s,x_s)ds + \int_0^t g(s,x_s)du(s), \] where \(u \colon \mathbb{R} \to \mathbb{R}\) is a left-continuous regulated function and the integrals in the right-hand side are taken in the Perron and Perron-Stieltjes sense, respectively. As the main tool, the topological transversality theory which is a version of the topological degree theory (see, e.g., [\textit{A. Granas} and \textit{J. Dugundji}, Fixed point theory. New York, NY: Springer (2003; Zbl 1025.47002)]) is used. The obtained results are applied to a periodic problem for a class of impulsive functional differential equations.
Mathematics Subject Classification ID: 45M15
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 34K45
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 7450436
zbMATH Keywords: measure functional differential equations
zbMATH Keywords: periodic solutions
zbMATH Keywords: impulsive differential equations
zbMATH Keywords: topological transversality
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
full work available at URL: j.jde.2021.11.031
OpenAlex ID: W3215729229
cites work: Q4523439
cites work: Periodic solutions of abstract differential equations with infinite delay
cites work: On Optimal Controls for Measure Delay-Differential Equations
cites work: Existence and stability of measure differential equations
cites work: Basic results for functional differential and dynamic equations involving impulses
cites work: Regulated functions with values in Banach space
cites work: Coincidence degree and periodic solutions of neutral equations
cites work: Q3999786
cites work: Q4064496
cites work: Q3845518
cites work: Leray-Schauder degree for semilinear Fredholm maps and periodic boundary value problems of neutral equations
cites work: The existence of periodic solution of impulsive functional differential equations with infinite delay
cites work: Periodic solutions of nonlinear functional differential equations
cites work: Differential systems involving impulses
cites work: Optimal Control Theory for Nonlinear Vector Differential Equations Containing Measures
cites work: Q4023958
cites work: Q4368810
cites work: An Abstract Measure Differential Equation
cites work: Nonlinear integrodifferential equations and a priori bounds on periodic solutions
cites work: Topological transversality and periodic solutions of neutral functional differential equations
cites work: Q4258069
DOI: 10.1016/J.JDE.2021.11.031
Recommended article: Q4808146
Recommended article: Q5487048
Recommended article: Periodic boundary value problems for functional differential equations with impulses
Recommended article: Periodic boundary value problem for the first order functional differential equations with impulses
Recommended article: Q3015305
Recommended article: A survey and some new results on the existence of solutions of PBVPs for first order functional differential equations
Recommended article: Q5295770
Recommended article: Q4231518
Recommended article: Existence of solutions for second order impulsive differential equations
Recommended article: Periodic solutions to periodically perturbed functional-differential equations
description: scientific article; zbMATH DE number 7450436