Positive solutions of nonlinear delayed differential equations with impulses

Positive solutions of nonlinear delayed differential equations with impulses

WoS Article

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.

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.

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

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

DIBLÍK, J.

2018

12.04.2017

Elsevier

0893-9659

Applied Mathematics Letters

72

10

United States of America

16

22

7

https://doi.org/10.1016/j.aml.2017.04.004