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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 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.

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 [

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 [

Consider the following nonautonomous difference system

where

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.

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

(i)

(ii) the sequence

(iii)

Then, there exists a positive integer sequence

Proof. We first prove that for arbitrary constant

Suppose that this conclusion of inequality (2.1) does not hold, then there exist

By the continuity of

Note that

to the exists of

Denote

for each positive integer

Lemma 2 Assume that there exists a function

(i)

(ii) the sequence

(iii)

Then, for each fixed r

Proof. We first prove that for arbitrary constants

The case of

trary

ment below inequality (2.2), there exists a positive integer sequence

Let

where

where

If

If

Inequalities (2.7) and (2.8) imply that

Since

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

(i)

(ii) the sequence

(iii)

Then, there exists a positive integer sequence

Proof. Let us first prove

Suppose that this is not true. Then there exist a constant c > 0 and a strictly increasing integer sequence

such that

The result of (2.11) implies the boundedness of

uniformly continuous on the same domain. And as shown above, we obtain

On the other hand, by Lemma 2.2, there exists a sequence

From (2.12) and (2.13) we easily get (2.10). The proof of Lemma 2.3 is complete .

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

Definition 1 A function

Definition 2 The function

for all

Definition 3 Let

(i) Stable if given

(ii) Attracting if there exists

formly attracting if the choice of

(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

(i)

(ii)

(iii)

(iv)

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 [

tion

By condition (ii) we know that

From condition (iii) we know that

According to the definition of function

Now we prove

Suppose that (3.3) is not true. Then there exist a constant

On the other hand, by (3.2) there is an integer j such that

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

Theorem 2 Provided that the hypotheses of Theorem 1 are satisfied, the zero solution of system (1.1) is uniformly asymptotically stable if positive definite function

Proof. Since

this is not true, then there exists a

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

In this section, we provide an example to illustrate the feasibility of our results.

Example 4.1. Consider the following difference equations

where

f is C^{1} with respect to

Moreover,

For

Now, we calculate

Then we get

Theorem (3.1). Denote

Then,

Thus condition (iv) of Theorem (3.1) is fulfilled. The zero solution of Example 4.1 is asymptotical stable. Inequation (4.2) implies that

We also can utilize Polar coordinate transformation to prove the above conclusion. Let

The square of the first equation adding the square of the second equation in system (4.3) yields

Denote

original system (4.1) is asymptotical stable and uniformly asymptotically stable. This confirm the correctness of utilizing Theorem 3.1 and Theorem 3.2 to judge Example 4.1.

This work was supported by the National Natural Science Foundation of China (Grant No.31170338), the General Project of Educational Commission in Sichuan Province (Grant No.16ZB0357) and the Major Project of Sichuan University of Arts and Science (Grant No.2014Z005Z).

Limin Zhang,Chaofeng Zhang,1 1, (2016) New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations. Applied Mathematics,07,1023-1031. doi: 10.4236/am.2016.710089