Publication detail

# Positive solutions of nonlinear delayed differential equations with impulses

DIBLÍK, J.

Original Title

Positive solutions of nonlinear delayed differential equations with impulses

English Title

Positive solutions of nonlinear delayed differential equations with impulses

Type

journal article in Web of Science

Language

en

Original Abstract

The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations $$\dot y(t)=-f(t,y_t),\,\,\,t\ge t_0.$$ It is assumed that $f\colon [t_0,\infty)\times {\cal C} \mapsto {\mathbb{R}}$ is a~continuous mapping satisfying a~local Lipschitz condition with respect to the second argument and ${\cal C}:={C}([-r,0],\mathbb{R})$, $r>0$ is the Banach space of conti\-nu\-ous functions. The problem is solved of the existence of positive solutions if the equation is subjected to impulses $y(t_s^+)=b_sy(t_s)$, $s=1,2,\dots$, where $t_0\le t_1< t_2<\dots$ and $b_s>0$, $s=1,2,\dots\,\,$. A criterion for the existence of positive solutions on $[t_0-r,\infty)$ is proved and their upper estimates are given. Relations to previous results are discussed as well.

English abstract

The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations $$\dot y(t)=-f(t,y_t),\,\,\,t\ge t_0.$$ It is assumed that $f\colon [t_0,\infty)\times {\cal C} \mapsto {\mathbb{R}}$ is a~continuous mapping satisfying a~local Lipschitz condition with respect to the second argument and ${\cal C}:={C}([-r,0],\mathbb{R})$, $r>0$ is the Banach space of conti\-nu\-ous functions. The problem is solved of the existence of positive solutions if the equation is subjected to impulses $y(t_s^+)=b_sy(t_s)$, $s=1,2,\dots$, where $t_0\le t_1< t_2<\dots$ and $b_s>0$, $s=1,2,\dots\,\,$. A criterion for the existence of positive solutions on $[t_0-r,\infty)$ is proved and their upper estimates are given. Relations to previous results are discussed as well.

Keywords

Positive solution; large time behavior; delayed differential equation; impulse.

Released

12.04.2017

Publisher

Elsevier

ISBN

0893-9659

Periodical

APPLIED MATHEMATICS LETTERS

Year of study

72

Number

10

State

US

Pages from

16

Pages to

22

Pages count

7

URL

Documents

BibTex


@article{BUT137191,
author="Josef {Diblík}",
title="Positive solutions of nonlinear delayed differential equations with impulses",
annote="The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations $$\dot y(t)=-f(t,y_t),\,\,\,t\ge t_0.$$ It is assumed that $f\colon [t_0,\infty)\times {\cal C} \mapsto {\mathbb{R}}$ is a~continuous mapping satisfying a~local Lipschitz condition with respect to the second argument
and ${\cal C}:={C}([-r,0],\mathbb{R})$, $r>0$ is the Banach space of conti\-nu\-ous functions. The problem is solved of the existence of positive solutions if the equation is subjected to impulses $y(t_s^+)=b_sy(t_s)$, $s=1,2,\dots$,
where $t_0\le t_1< t_2<\dots$ and $b_s>0$, $s=1,2,\dots\,\,$. A criterion for the existence of positive solutions on $[t_0-r,\infty)$ is proved and their upper estimates are given. Relations to previous results are discussed as well.",
}