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

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.


Introduction
Difference equations usually describe the evolution of certain phenomena over the course of time.These equations occur in biology, economics, psychology, sociology, and other fields.In addition, difference equations also 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 ( ) . 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 ( ) , n V n x which has indefinitely small upper bound and whose variation ( ) along the solution of system (1.1) is negative definite.Sometimes it is not easy to determine the positive definite Liapunov function for a given equations in applications.If we further require that the function has indefinitely small upper bound besides its negative definite variation, the work would become more difficult to do.In this paper, we weaken the Liapunov function to positive definite and also weaken the negative definite variation to semi-negative definite on orbits of Equations (1.1), then we put forward a new Liapunov asymptotical stability theorem for difference Equations (1.1) by adding to extra conditions on the variation.Subsequently, 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.

Some Lemmas
In this section, we introduce the following lemmas, which play a key role in obtaining our results.

Lemma 1 Suppose that there exists a function ( )
, n g n x satisfying the following conditions: , and Then, there exists a positive integer sequence { } i n with i n → +∞ as i → +∞ such that ( ) Proof.We first prove that for arbitrary constant 0 a > there exists a sufficient large integer 1 N ∈  for every positive integer , and Then, for each fixed r ( ) , there exists a positive integer sequence Proof.We first prove that for arbitrary constants 0 a > there exists a sufficient large integer 2 N ∈  such that for every The case of 1 r = is proved by (2.1) in the proof of Lemma 2.1.Suppose that inequality (2.4) holds in the case of 1 r − ( ) but is not true in the case of r.Then there exist constants 0 a >  such that for arbitrary N ′ ∈  there exists a positive integer ( ) Similarly to the statement below inequality (2.2), there exists a positive integer sequence     denote the maximum integer not exceeding x and P + ∈   denote a constant.Same as above, without loss of generality, we only consider the case By the discrete analogue fundamental theorem of calculus [10], we get where 0,1, 2, , 1 2 where , 1, 2, , 2 2 2 , 0 2 , from inequality (2.5), we obtain , 0 2 Since ( ) . This leads to a contradiction because of the inductive assumption for (2.4) in the case of 1 r − .Therefore, the conclusion of (2.4) is proved.
Similarly to the second part of the proof of Lemma 2.1, for each r ( ) According to Lemma 2.2 we prove the following result.

Lemma 3
Assume that there exists a function ( ) , n g n x satisfying the following conditions: is uniformly continuous with respect to the second argument x, , and Then, there exists a positive integer sequence { }

New Asymptotical Stability and Uniformly Asymptotical Stability Theorems
In this section, we propose and prove the new asymptotical stability and uniformly asymptotical stability theorems of system (1.1).First of all, we introduce a special class of function and then give the definition of positive definite function.Subsequently, we introduce the various stability notions of the equilibrium point * x of system (1.1).These are very useful for obtaining our results besides the above Lemmas.Definition 1 A function φ is said to be class of K if it is continuous in [ ) 0, H , strictly increasing, and ( ) Definition 2 The function ( ) formly attracting if the choice of µ is independent of 0 n .(iii) Asymptotically stable if it is stable and attracting, and uniformly asymptotically stable if it is uniformly stable and uniformly attracting.
Theorem 1 Consider nonautonomous difference Equations (1.1), where : with respect to the second argument x and satisfies ( ) , 0 f n o = .Suppose that there exists a , : , , , , , where the function φ defined by Definition 1.
Then the zero solution of system (1.1) is asymptotically stable.Proof.By conditions (i) and (ii), the origin of system (1.1) is stable according to the references [12] According to the definition of function On the other hand, by (3.2) there is an integer j such that ( ) with respect to the second argument and ( ) Clear, l j m n > for sufficiently large l such that ( ) which contradicts to the definition of v given by (3.4).Therefore, (3.3) is proved.According to Definition 3, we obtain that the zero solution of system (1.1) is asymptotically stable.
In addition to the hypotheses of Theorem 1, we can obtain that the zero solution of system (1.1) is uniformly asymptotically stable if ( ) has an indefinitely small upper bound as in the classical Liapunov asymptotical stability theorem of nonautonomous difference equations.
Theorem 2 Provided that the hypotheses of Theorem 1 are satisfied, the zero solution of system (1. Then we obtain that ( ) ( ) ( ) ( ) ( ) ( ) ( ) This is a contradiction.Since all the conditions of Theorem 1 are satisfied, the zero solution of system (1.1) is asymptotically stable.Therefore, for the above ε , ( ) δ ε , there exists ( )

Example
In this section, we provide an example to illustrate the feasibility of our results.
where ( ) ( ) ( ) ( ) n n V n u w ∆ ≤ , then the zero solution of system (1.1) is stable.At the same condition, we also get ,    , , , ,

.≥
The equilibrium point * x of system (1.1) is said to be: , uniformly stable if δ may be chosen in dependent of 0 . Then, by the definition of positive definite ( )

Example 4 . 1 .
Consider the following difference equations we only need to verify the example whether satisfies condition (iv) of Theorem (3.1).Denote ( )

3 )
Thus condition (iv) of Theorem (3.1) is fulfilled.The zero solution of Example 4.1 is asymptotical stable.The square of the first equation adding the square of the second equation in system (4Definition 3, we obtain the zero solution of the original system (4.1) is asymptotical stable and asymptotically stable.This confirm the correctness of utilizing Theorem 3.1 and Theorem 3.2 to judge Example 4.1. 4 − + 10) i n → +∞ as i → +∞ and( ) * * , .