A theorem of Ważewski and dynamic equations on time scales

Abstract

An important and well known extension of the direct method of Lyapunov functions is a simple topological principle, in the geometrical theory of ordinary differential equations known as Ważewski's principle. It is much less known that there is an old variant of this principle for difference equations stated by Coffman. In this paper, we combine both methods to a method usable for dynamic systems on time scales and discuss some first results.

Ważewski theorem on time scales with a set of egress points that is not the whole boundary

“…The results on the Ważewski topological principle for dynamic equations on time scales are also not satisfactory yet. In fact, the only cases explored are the ones where the set of constraints is negatively invariant (see [9,10]) so the Ważewski idea has not been exploited enough.…”

“…: differential inclusions (see, e.g., [2] or [3] and references therein), retarded functional differential equations (see [4] and also [2] for some explanations of this development), difference equations (e.g. [5][6][7][8]) or, recently, dynamic equations on time scales (e.g., [9,10]). This last area of research has been intensively developed since the 1990's as a unification and generalization of the theory of difference equations and differential equations, and has found applications in many mathematical models in biology and physics, where discrete and continuous dynamics have to be studied simultaneously.…”

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

“…We show that, for a given set ⊂ T × R n and a function f satisfying some conditions, there exists at least one solution y = y(t ) of system (1) such that (t, y(t)) ∈ for every t ∈ T.…”

“…In this paper we investigate an asymptotic behavior of solutions of systems of n dynamic equations on time scales of the form y (t) = f (t, y(t )), (1) where f : T × R n → R n , and T is a time scale. We show that, for a given set ⊂ T × R n and a function f satisfying some conditions, there exists at least one solution y = y(t ) of system (1) such that (t, y(t)) ∈ for every t ∈ T.…”

“…[4][5][6]. Finally and recently, it was used for dynamic equations on time scales, see e.g., [1,[7][8][9][10]12].…”