Time scale extensions of a theorem of Wintner on systems with asymptotic equilibrium

Abstract

We consider quasilinear dynamic systems of the form

where 𝕋 is a time scale, and provide extensions of a theorem of Wintner on systems with asymptotic equilibrium to arbitrary time scales. More specifically, we give sufficient conditions for the asymptotic equilibrium of the above system in the sense that for any given constant vector c, there is a solution satisfying

Our results are new for difference equations, q-difference equations and many other time scale systems even though their analogous for differential equations have been known for some time.

Keywords:

• asymptotic equilibrium
• Schauder fixed point theorem
• dynamic systems
• time scale

Notes

^†This work is a part of PhD dissertation of R. Mert at Middle East Technical University.

1. [email protected]

Additional information

Notes on contributors

A. Zafer

1