Computational fixed-point theory for differential delay equations with multiple time lags
type: Item
instance of: scholarly article
title: Computational fixed-point theory for differential delay equations with multiple time lags
zbMATH Open document ID: 1247.34119
author: Gábor Kiss
author: Jean-Philippe Lessard
published in: Journal of Differential Equations
publication date: 2012-03-07T00:00:00Z
review text: The authors consider the problem of existence of periodic solutions of differential delay equations with multiple time lags. Using an original and elaborated construction they find an equivalent fixed point operator equation. An appropriate notion of radii polynomials is defined, which is used in the hypothesis of some result that prove the unique solvability of the fixed point operator equation mentioned before. The contraction mapping theorem is used at this step. For a certain class of differential delay equations with multiple time lags the authors provide an explicit construction of the radii polynomials. For this, one needs to find numerical approximations of periodic solutions using Newton's method applied on a finite dimensional projection. At the end, two applications are provided. The coexistence of three periodic solutions for a given delay equation with two time lags is obtained. The second application provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags.
reviewed by: Adriana Buică
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6012780
zbMATH Keywords: delay equation
zbMATH Keywords: multiple time lags
zbMATH Keywords: Fourier expansion
zbMATH Keywords: contraction mapping theorem
zbMATH Keywords: Newton-type operator
zbMATH Keywords: radii polynomials
describes a project that uses: INTLAB
MaRDI profile type: Publication
full work available at URL: j.jde.2011.11.020
OpenAlex ID: W2003645160
cites work: A further generalization of the Schauder fixed point theorem
cites work: The existence of periodic solutions of \(f'(x)=-\alpha f(x-1)(1+f(x))\)
cites work: The fixed point index and asymptotic fixed point theorems for $k$-set-contractions
cites work: Periodic solutions for functional differential equations with multiple state-dependent time lags
cites work: Some global results for nonlinear eigenvalue problems
cites work: A global bifurcation theorem with applications to functional differential equations
cites work: A Hopf Global Bifurcation Theorem for Retarded Functional Differential Equations
cites work: Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs
cites work: Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift–Hohenberg Equation
cites work: Validated Continuation for Equilibria of PDEs
cites work: Periodic solutions of analytic functional differential equations are analytic
cites work: INFINITE DIMENSIONAL KRAWCZYK OPERATOR FOR FINDING PERIODIC ORBITS OF DISCRETE DYNAMICAL SYSTEMS
cites work: Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation
cites work: NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
cites work: Computer-assisted enclosure methods for elliptic differential equations
cites work: A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem
cites work: Differential-delay equations with two time lags
cites work: Global dynamics of delay differential equations
cites work: Periodic solutions of delay equations with several fixed delays
cites work: Validated continuation over large parameter ranges for equilibria of PDEs
cites work: Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation
cites work: Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation
cites work: Q1110275
cites work: Q4949347
cites work: Stability implications of delay distribution for first-order and second-order systems
cites work: Delay equations. Functional-, complex-, and nonlinear analysis
DOI: 10.1016/J.JDE.2011.11.020
Recommended article: Q3149553
Recommended article: A new approach to numerical solution of fixed-point problems and its application to delay differential equations
Recommended article: Computation of double Hopf points for delay differential equations
Recommended article: Q3583417
Recommended article: Computation, Continuation and Bifurcation Analysis of Periodic Solutions of Delay Differential Equations
Recommended article: Q3560398
Recommended article: Computing stability of differential equations with bounded distributed delays
Recommended article: Computation of Asymptotic Stability for a Class of Partial Differential Equations with Delay
Recommended article: A new fixed point algorithm for finding the solution of a delay differential equation
Recommended article: Q3698186
description: scientific article; zbMATH DE number 6012780