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New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations

Limin Zhang, Chaofeng Zhang

2016
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Applied Mathematics
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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
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... y 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. 1024 appear in the study of discretization methods for differential equations. Realizing that most of the problems that arise in practice are nonlinear and mostly unsolvable, the qualitative behaviors of solutions without actually computing them are of vital importance in application process. The stability property of an equilibrium is the very important qualitative behavior for difference equations. The most powerful method for studying the stability property is Liapunov's second method or Liapunov's direct method. The main advantage of this method is that the stability can be obtained without any prior knowledge of the solutions. In 1892, the Russian mathematician A.M. Liapunov introduced the method for investigating the stability of nonlinear differential equations. According to the method, he put forward Liapunov stability theorem, Liapunov asymptotical stability theorem and Liapunov unstable theorem, which have been known as the fundamental theorems of stability. Utilizing these fundamental theorems of stability, many authors have investigated the stability of some specific differential systems [1]- [9] . We know that several results in the theory of difference equations have been obtained as more or less natural discrete analogues of corresponding results of differential equations, so Liapunov's direct method is much more useful for difference equations. Actually, some authors have utilized the methods for difference equations successfully [10]- [20] . Using the method, S. Elaydi [10] and J.P. Lasalle [11] gave the classical Liapunov stability theorem for autonomous difference equations. In [12] [13], the authors extended the technique to generalized nonautonomous difference equations and put forward the classical Liapunov stability theorem for nonautonomous difference equations. In [14]-[17], the direct approach was extended to some special delay difference systems to investigate the stability properties. In [18]-[20], how to construct Liapunov function for difference system or hybrid time-varying system was exploited. Consider the following nonautonomous difference system

doi:10.4236/am.2016.710089
fatcat:nzfof4ox2rebjjvxlyjy3hdaui