Author(s)

Limin Zhang^1,2, Chaofeng Zhang¹

¹School of Mathematics and Finance-Economics, Sichuan University of Arts and Science, Dazhou, China.
²School of Mathematics, Sichuan University, Chengdu, China.

New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous difference equations relies on the existence of a positive definite Liapunov function that has an indefinitely small upper bound and whose variation along a given nonautonomous difference equations is negative definite. In this paper, we consider the case that the Liapunov function is only positive definite and its variation is semi-negative definite. At these weaker conditions, we put forward a new asymptotical stability theorem of nonautonomous difference equations by adding to extra conditions on the variation. After that, in addition to the hypotheses of our new asymptotical stability theorem, we obtain a new uniformly asymptotical stability theorem of nonautonomous difference equations provided that the Liapunov function has an indefinitely small upper bound. Example is given to verify our results in the last.

Nonautonomous Difference Equations, New Asymptotical Stability Theorem, New Uniformly Asymptotical Stability Theorem, Liapunovs Direct Method

Zhang, L. and Zhang, C. (2016) New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations. Applied Mathematics, 7, 1023-1031. doi: 10.4236/am.2016.710089.