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

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.