Detail publikace

# Positive solutions of nonlinear delayed differential equations with impulses

Originální název

Positive solutions of nonlinear delayed differential equations with impulses

Anglický název

Positive solutions of nonlinear delayed differential equations with impulses

Jazyk

en

Originální abstrakt

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.

Anglický abstrakt

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.

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.",
}