Boundedness and persistence of delay differential equations with mixed nonlinearity

Leonid Berezansky; Elena Braverman

doi:10.1016/j.amc.2016.01.015

Boundedness and persistence of delay differential equations with mixed nonlinearity

Leonid Berezansky, Elena Braverman

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

For a nonlinear equation with several variable delays x(t)=_k=1^mf_k(t,x(h₁(t)),⋯,x(h_l(t)))-g(t,x(t)), where the functions f_k increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on [0, ∞), as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.

Original language	English
Pages (from-to)	154-169
Number of pages	16
Journal	Applied Mathematics and Computation
Volume	279
DOIs	https://doi.org/10.1016/j.amc.2016.01.015
State	Published - 10 Apr 2016

Keywords

A global positive solution
Mackey-Glass equation
Nonlinear delay differential equations
Persistent
Population dynamics models
permanent and unbounded solutions

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.amc.2016.01.015

Cite this

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title = "Boundedness and persistence of delay differential equations with mixed nonlinearity",

abstract = "For a nonlinear equation with several variable delays x(t)=k=1mfk(t,x(h1(t)),⋯,x(hl(t)))-g(t,x(t)), where the functions fk increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on [0, ∞), as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.",

keywords = "A global positive solution, Mackey-Glass equation, Nonlinear delay differential equations, Persistent, Population dynamics models, permanent and unbounded solutions",

author = "Leonid Berezansky and Elena Braverman",

note = "Funding Information: L. Berezansky was partially supported by Israeli Ministry of Absorption, E. Braverman was partially supported by the NSERC research grant RGPIN-2015-05976 . The authors are grateful to the reviewers whose thoughtful comments significantly contributed to the presentation of the results of the paper. Publisher Copyright: {\textcopyright} 2016 Elsevier Inc. All rights reserved.",

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doi = "10.1016/j.amc.2016.01.015",

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AU - Berezansky, Leonid

AU - Braverman, Elena

N1 - Funding Information: L. Berezansky was partially supported by Israeli Ministry of Absorption, E. Braverman was partially supported by the NSERC research grant RGPIN-2015-05976 . The authors are grateful to the reviewers whose thoughtful comments significantly contributed to the presentation of the results of the paper. Publisher Copyright: © 2016 Elsevier Inc. All rights reserved.

PY - 2016/4/10

Y1 - 2016/4/10

N2 - For a nonlinear equation with several variable delays x(t)=k=1mfk(t,x(h1(t)),⋯,x(hl(t)))-g(t,x(t)), where the functions fk increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on [0, ∞), as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.

AB - For a nonlinear equation with several variable delays x(t)=k=1mfk(t,x(h1(t)),⋯,x(hl(t)))-g(t,x(t)), where the functions fk increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on [0, ∞), as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.

KW - A global positive solution

KW - Mackey-Glass equation

KW - Nonlinear delay differential equations

KW - Persistent

KW - Population dynamics models

KW - permanent and unbounded solutions

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JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -

Boundedness and persistence of delay differential equations with mixed nonlinearity

Abstract

Keywords

ASJC Scopus subject areas

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