TITLE:
New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations
AUTHORS:
Limin Zhang, Chaofeng Zhang
KEYWORDS:
Nonautonomous Difference Equations, New Asymptotical Stability Theorem, New Uniformly Asymptotical Stability Theorem, Liapunovs Direct Method
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.10,
June
7,
2016
ABSTRACT: 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.