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.
Highlights
Difference equations usually describe the evolution of certain phenomena over the course of time
Liapunov introduced the method for investigating the stability of nonlinear differential equations. He put forward Liapunov stability theorem, Liapunov asymptotical stability theorem and Liapunov unstable theorem, which have been known as the fundamental theorems of stability
We weaken the Liapunov function to positive definite and weaken the negative definite variation to semi-negative definite on orbits of Equations (1.1), we put forward a new Liapunov asymptotical stability theorem for difference Equations (1.1) by adding to extra conditions on the variation
Summary
Difference equations usually describe the evolution of certain phenomena over the course of time. As shown in [12] [13], using Liapunov’s direct method to study the asymptotical stability of the zero solution of system (1.1) relies on the existence of a positive definite Liapunov function V (n, xn ) which has indefinitely small upper bound and whose variation ∆V (n, xn ) along the solution of system (1.1) is negative definite. We weaken the Liapunov function to positive definite and weaken the negative definite variation to semi-negative definite on orbits of Equations (1.1), we put forward a new Liapunov asymptotical stability theorem for difference Equations (1.1) by adding to extra conditions on the variation. Provided that all the conditions of our new asymptotical stability theorem are satisfied, we obtain a new uniformly asymptotical stability theorem of nonautonomous difference equations if the Liapunov function has an indefinitely small upper bound
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