A perron type theorem for positive solutions of functional differential equations

Research output: Contribution to journalArticle

Abstract

A nonlinear perturbation of a linear autonomous retarded functional differential equation is considered. According to a Perron type theorem, with the possible exception of small solutions the Lyapunov exponents of the solutions of the perturbed equation coincide with the real parts of the characteristic roots of the linear part. In this paper, we study those solutions which are positive in the sense that they lie in a given order cone in the phase space. The main result shows that if the Lyapunov exponent of a positive solution of the perturbed equation is finite, then it is a characteristic root of the unperturbed equation with a positive eigenfunction. As a corollary, a necessary and sufficient condition for the existence of a positive solution of a linear autonomous delay differential equation is obtained.

Original languageEnglish
Article number57
JournalElectronic Journal of Qualitative Theory of Differential Equations
Volume2018
DOIs
Publication statusPublished - Jan 1 2018

Fingerprint

Functional Differential Equations
Characteristic Roots
Positive Solution
Differential equations
Lyapunov Exponent
Theorem
Retarded Functional Differential Equations
Small Solutions
Nonlinear Perturbations
Delay Differential Equations
Exception
Eigenfunctions
Phase Space
Corollary
Cone
Eigenvalues and eigenfunctions
Cones
Necessary Conditions
Sufficient Conditions

Keywords

  • Cone
  • Functional differential equation
  • Lyapunov exponent
  • Perturbation
  • Positivity

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

@article{5e10c17c072f47ab9aac04a82880aed6,
title = "A perron type theorem for positive solutions of functional differential equations",
abstract = "A nonlinear perturbation of a linear autonomous retarded functional differential equation is considered. According to a Perron type theorem, with the possible exception of small solutions the Lyapunov exponents of the solutions of the perturbed equation coincide with the real parts of the characteristic roots of the linear part. In this paper, we study those solutions which are positive in the sense that they lie in a given order cone in the phase space. The main result shows that if the Lyapunov exponent of a positive solution of the perturbed equation is finite, then it is a characteristic root of the unperturbed equation with a positive eigenfunction. As a corollary, a necessary and sufficient condition for the existence of a positive solution of a linear autonomous delay differential equation is obtained.",
keywords = "Cone, Functional differential equation, Lyapunov exponent, Perturbation, Positivity",
author = "M. Pituk",
year = "2018",
month = "1",
day = "1",
doi = "10.14232/ejqtde.2018.1.57",
language = "English",
volume = "2018",
journal = "Electronic Journal of Qualitative Theory of Differential Equations",
issn = "1417-3875",
publisher = "University of Szeged",

}

TY - JOUR

T1 - A perron type theorem for positive solutions of functional differential equations

AU - Pituk, M.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - A nonlinear perturbation of a linear autonomous retarded functional differential equation is considered. According to a Perron type theorem, with the possible exception of small solutions the Lyapunov exponents of the solutions of the perturbed equation coincide with the real parts of the characteristic roots of the linear part. In this paper, we study those solutions which are positive in the sense that they lie in a given order cone in the phase space. The main result shows that if the Lyapunov exponent of a positive solution of the perturbed equation is finite, then it is a characteristic root of the unperturbed equation with a positive eigenfunction. As a corollary, a necessary and sufficient condition for the existence of a positive solution of a linear autonomous delay differential equation is obtained.

AB - A nonlinear perturbation of a linear autonomous retarded functional differential equation is considered. According to a Perron type theorem, with the possible exception of small solutions the Lyapunov exponents of the solutions of the perturbed equation coincide with the real parts of the characteristic roots of the linear part. In this paper, we study those solutions which are positive in the sense that they lie in a given order cone in the phase space. The main result shows that if the Lyapunov exponent of a positive solution of the perturbed equation is finite, then it is a characteristic root of the unperturbed equation with a positive eigenfunction. As a corollary, a necessary and sufficient condition for the existence of a positive solution of a linear autonomous delay differential equation is obtained.

KW - Cone

KW - Functional differential equation

KW - Lyapunov exponent

KW - Perturbation

KW - Positivity

UR - http://www.scopus.com/inward/record.url?scp=85050115529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050115529&partnerID=8YFLogxK

U2 - 10.14232/ejqtde.2018.1.57

DO - 10.14232/ejqtde.2018.1.57

M3 - Article

VL - 2018

JO - Electronic Journal of Qualitative Theory of Differential Equations

JF - Electronic Journal of Qualitative Theory of Differential Equations

SN - 1417-3875

M1 - 57

ER -