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

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

WoS Article

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.

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.

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

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

GABOR, G.; RUSZKOWSKI, S.; VÍTOVEC, J.

2016

02.06.2015

0096-3003

APPLIED MATHEMATICS AND COMPUTATION

265

6

United States of America

358

369

12

http://www.sciencedirect.com/science/article/pii/S009630031500644X