Publication detail

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

GABOR, G. RUSZKOWSKI, S. VÍTOVEC, J.

Original Title

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

English Title

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

Type

journal article

Language

en

Original Abstract

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.

English abstract

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.

Keywords

Time scale; Dynamic system; Non-autonomous system; Difference equation; Asymptotic behavior of solution; Retract method

RIV year

2015

Released

02.06.2015

Pages from

358

Pages to

369

Pages count

12

URL

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"
}
```