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

In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form yΔ(t)=f(t,y(t)),yΔ(t)=f(t,y(t)),where f:T×Rn→Rn,f:T×Rn→Rn, and TT is a time scale. For a given set Ω⊂T×Rn,Ω⊂T×Rn, we formulate conditions for function f which guarantee that at least one solution y of the above system stays in Ω. Unlike previous papers the set Ω is considered in more general form, i.e., the time section Ωt   is an arbitrary closed bounded set homeomorphic to the disk (for every t∈Tt∈T) and the boundary ∂TΩ∂TΩ 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.