Detail publikace

# Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points

Originální název

Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points

Anglický název

Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points

Jazyk

en

Originální abstrakt

In this paper we study an asymptotic behaviour of solutions of nonlinear dynamic systems on time scales of the form $$y^{\Delta}(t)=f(t,y(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$, and $\mathbb{T}$ is a time scale. For a given set $\Omega\subset\mathbb{T}\times\R^{n}$, we formulate conditions for function $f$ which guarantee that at least one solution $y$ of the above system stays in $\Omega$. Unlike previous papers the set $\Omega$ is considered in more general form, i.e., the time section $\Omega_t$ is an arbitrary closed bounded set homeomorphic to the disk (for every $t\in\mathbb{T}$) and the boundary $\partial_\mathbb{T}\Omega$ does not contain only egress points. Thanks to this, we can investigate a substantially wider range of equations with various types of bounded solutions. A relevant example is considered. The results are new also for non-autonomous systems of difference equations and the systems of impulsive differential equations.

Anglický abstrakt

In this paper we study an asymptotic behaviour of solutions of nonlinear dynamic systems on time scales of the form $$y^{\Delta}(t)=f(t,y(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$, and $\mathbb{T}$ is a time scale. For a given set $\Omega\subset\mathbb{T}\times\R^{n}$, we formulate conditions for function $f$ which guarantee that at least one solution $y$ of the above system stays in $\Omega$. Unlike previous papers the set $\Omega$ is considered in more general form, i.e., the time section $\Omega_t$ is an arbitrary closed bounded set homeomorphic to the disk (for every $t\in\mathbb{T}$) and the boundary $\partial_\mathbb{T}\Omega$ does not contain only egress points. Thanks to this, we can investigate a substantially wider range of equations with various types of bounded solutions. A relevant example is considered. The results are new also for non-autonomous systems of difference equations and the systems of impulsive differential equations.

BibTex


@article{BUT114696,
author="Jiří {Vítovec} and Grzegorz {Gabor} and Sebastian {Ruszkowski}",
title="Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points",
annote="In this paper we study an asymptotic behaviour of solutions
of nonlinear dynamic systems on time scales of the form
$$y^{\Delta}(t)=f(t,y(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$, and $\mathbb{T}$ is a time scale.
For a given set
$\Omega\subset\mathbb{T}\times\R^{n}$, we formulate conditions for function $f$ which
guarantee that at least one solution $y$ of the above system stays in $\Omega$.  Unlike previous papers the set $\Omega$ is considered in more general form, i.e., the time section $\Omega_t$ is an arbitrary closed bounded set homeomorphic to the disk (for every $t\in\mathbb{T}$) and the boundary $\partial_\mathbb{T}\Omega$ does not contain only egress points. Thanks to this, we can investigate a substantially wider range of equations with various types of bounded solutions. A relevant example is considered.

The results are new also for non-autonomous systems of difference equations and the systems of impulsive differential equations.",
chapter="114696",
doi="10.1016/j.amc.2015.05.027",
howpublished="online",
number="6",
volume="265",
year="2015",
month="june",
pages="358--369",
type="journal article in Web of Science"
}