# Exponential Stability of Difference Equations with Several Delays: Recursive Approach

- Leonid Berezansky
^{1}and - Elena Braverman
^{2}Email author

**2009**:104310

https://doi.org/10.1155/2009/104310

© L. Berezansky and E. Braverman. 2009

**Received: **8 January 2009

**Accepted: **23 April 2009

**Published: **5 May 2009

## Abstract

We obtain new explicit exponential stability results for difference equations with several variable delays and variable coefficients. Several known results, such as Clark's asymptotic stability criterion, are generalized and extended to a new class of equations.

## Keywords

## 1. Introduction and Preliminaries

where is an integer for any , is an integer, . Stability of (1.1) and relevant nonlinear equations has been an intensively developed area during the last two decades.

Let us compare stability methods for delay differential equations and delay difference equations. Many of the methods previously used for differential equations have also been applied to difference equations. However, there are at least two methods which are specific for difference equations. The first approach is reducing a solution of a delay difference equation to the values of a solution of a delay differential equation with piecewise constant arguments at integer points. The second method is based on a recursive form of difference equations and is described in detail later. In this paper we obtain new stability results based on the recursive solution representation.

Definition 1.1.

*exponentially stable*if there exist constants , such that for every solution of (1.1) and (1.2) the inequality

holds for all , where do not depend on .

Equation (1.1) is *stable* if for any
there exists
such that
implies
,
; if
does not depend on
, then (1.1) is *uniformly stable*.

Equation (1.1) is *attractive* if for any
a solution tends to zero
. It is *asymptotically stable* if it is both stable and attractive.

One of the methods to establish stability of difference equations is based on a recursive form of these equations; see the monographs [1, 2]. The following result was also obtained by this method.

for every solution of (1.4), where . In particular, the zero solution of (1.4) is globally exponentially stable.

Similar argument leads to the following result.

Lemma 1.3 ([4]).

Recently several results on exponential stability of high-order difference equations appeared where the results are based on the recursive representations; see, for example, [5, 6]. In particular, in [6, Corollary ] contains the following statement.

Lemma 1.4 ([6]).

is globally exponentially stable.

In the present paper we obtain some new stability results for (1.1) with several variable delays. In contrast to many other stability tests, we consider the case when the sum of coefficients or some of its subsum is allowed to be in the interval , not just . We illustrate our results with several examples.

## 2. Main Results

Now we can proceed to the main results of this paper. Let us note that any sum where the lower index exceeds the upper index is assumed to vanish.

Theorem 2.1.

Then (1.1) is exponentially stable.

Proof.

By Lemma 1.2, (1.1) is exponentially stable.

We can reformulate Theorem 2.1 in the following equivalent form.

Theorem 2.2.

Then (1.1) is exponentially stable.

Assuming first and then we obtain the following two corollaries for an equation with a nondelay term.

Corollary 2.3.

is exponentially stable.

Example 2.4.

with an arbitrary bounded delay : for some integer and any . Since , then by Corollary 2.3 this equation is exponentially stable. Lemma 1.4 is formulated for a constant delay, some other tests do not apply since .

Corollary 2.5.

Then (2.5) is exponentially stable.

Example 2.6.

is exponentially stable, since and .

Now let us assume that all coefficients are proportional. Such equations arise as linear approximations of nonlinear difference equations in mathematical biology. Then a straightforward computation leads to the following result.

Corollary 2.7.

Then (1.1) is exponentially stable.

Assuming constant coefficients and we obtain the following corollary.

Corollary 2.8.

Then (1.1) is exponentially stable.

Remark 2.9.

Corollary 2.8 for the case was obtained in Proposition of [7].

Choosing , , we obtain Parts and of Corollary 2.10, respectively.

Corollary 2.10.

- (1)
- (2)

Then (2.11) is exponentially stable.

For and we obtain Parts , , and , respectively.

Corollary 2.11.

- (1)
- (2)
- (3)
- (4)

Then (2.12) is exponentially stable.

Theorem 2.1 and its corollaries imply new explicit conditions of exponential stability for autonomous difference equations with several delays, as well as a new justification for known ones.

where . Choosing we immediately obtain the following stability test.

Corollary 2.12.

Let . Then (2.13) is exponentially stable.

Remark 2.13.

This result is well known; see, for example, [8] for as well as some results for autonomous equations below. We presented it just to illustrate our method.

Further, Theorem 2.1 and Corollaries 2.8 and 2.11 can be reformulated for (2.13) as follows.

Corollary 2.14.

Then (2.13) is exponentially stable.

Corollary 2.15.

If , then (2.13) is exponentially stable.

Corollary 2.16.

Suppose that at least one of the following conditions holds

Then (2.15) is exponentially stable.

which is a different form of (1.1).

We recall that we assume for all delays in this paper.

Theorem 2.17.

Then (2.16) is exponentially stable.

Proof.

Thus by Lemma 1.2 , where , , for , so (2.16) is exponentially stable.

Theorem 2.18.

Then (2.16) is exponentially stable.

Proof.

then the reference to Lemma 1.2 completes the proof.

## 3. Discussion and Examples

once the delays are bounded: , .

The following example outlines the sharpness of the condition that the delays are bounded in Theorems 2.1, 2.2, 2.17, and 2.18.

Example 3.1.

satisfies all assumptions of Theorems 2.1, 2.2, 2.17, and 2.18 but the boundedness of the delay. Since the solution with tends to 1 as , then the zero solution of (3.3) is neither asymptotically nor exponentially stable. Here even the condition is not satisfied.

Example 3.2.

for any , , so for any ; the equation is asymptotically stable. Since the solution decay is not faster than , then the equation is not exponentially stable.

Next, let us compare Corollary 2.10 and Theorem of [6] which is also Lemma 1.4 of the present paper.

Example 3.3.

Then, (1.12) is exponentially stable for any by Corollary 2.10, Part ( ; here , .

which is not satisfied for even , so Lemma 1.4 cannot be applied to deduce exponential stability.

Example 3.4.

for odd , which gives the intervals and , respectively. Finally, Lemma 1.4 implies exponential stability for .

In addition to Theorems 2.1, 2.2, 2.17, and 2.18, let us review some other known stability conditions for equations with several delays. For comparison, we will cite the following two results.

is globally asymptotically stable.

Theorem 3 B ([15]).

Then (1.1) is asymptotically stable.

Let us note that unlike Theorems A and B we do not assume that coefficients are either nonnegative (as in Theorem A) or constant (as in Theorem B). Further, let us compare our stability tests with known results, including Theorems A and B.

Example 3.5.

Theorem B is applicable to equations with constant coefficients only.

## Declarations

### Acknowledgments

The authors are very grateful to the referee for valuable comments and remarks. This paper is partially supported by Israeli Ministry of Absorption and by the NSERC Research Grant.

## Authors’ Affiliations

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