# Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations

- Ravi P. Agarwal
^{1}Email author, - Young-Ho Kim
^{2}and - S. K. Sen
^{1}

**2009**:535849

**DOI: **10.1155/2009/535849

© Ravi P. Agarwal et al. 2009

**Received: **7 December 2008

**Accepted: **21 January 2009

**Published: **5 February 2009

## Abstract

We derive new nonlinear discrete analogue of the continuous Halanay-type inequality. These inequalities can be used as basic tools in the study of the global asymptotic stability of the equilibrium of certain generalized difference equations.

## 1. Introduction

The investigation of stability of nonlinear difference equations with delays has attracted a lot of attention from many researchers such as Agarwal et al. [1–3], Baĭnov and Simeonov [4], Bay and Phat [5], Cooke and Ivanov [6], Gopalsamy [7], Liz et al. [8–10], Niamsup et al. [11, 12], Mohamad and Gopalsamy [13], Pinto and Trofimchuk [14], and references sited therein. In [15], Halanay proved an asymptotic formula for the solutions of a differential inequality involving the "maximum" functional and applied it in the stability theory of linear systems with delay. Such an inequality was called *Halanay inequality* in several works. Some generalizations as well as new applications can be found, for instance, in Agarwal et al. [2], Gopalsamy [7], Liz et al. [8–10], Niamsup et al. [11, 12], Mohamad and Gopalsamy [13], and Pinto and Trofimchuk [14]. In particular, in [2, 6, 10, 12, 13], the authors considered discrete Halanay-type inequalities to study some discrete version of functional differential equations.

In the following results of Liz et al. [10], authors showed that some discrete versions of these (maximum) inequalities can be applied to study the global asymptotic stability of a family of difference equations.

Theorem 1.

By a simple use of Theorem A, authors also demonstrated the validity of the following statement, namely, Theorem B.

Theorem 1.

If either

where can be calculated in the form established in Theorem A. As a consequence, the trivial solution of (1.6) is globally asymptotically stable.

The main aim of the present paper is to establish some new nonlinear retarded Halanay-type inequalities, which extend Theorem A, along with the derivation of new global stability conditions for nonlinear difference equations.

## 2. Halanay-Type Discrete Inequalities

In this section, we introduce new discrete inequalities which will be used to derive global stability conditions in the next section.

Theorem 2.1.

with

Proof.

where Since it is easy to prove by induction that if and for then and for all

This is equivalent to the existence of a solution of equation where is the polynomial defined by (2.6).

Now, in view of On the other hand, in view of (2.3). As a consequence, there exists such that Hence, is a solution of (2.9) with

For this value of , the pair is a solution of (2.8) for every Thus, choosing we have that , and for all

Hence, using the first part of the proof, we can conclude that , and for all

By the similar argument used in Theorem 2.1, we obtain the following result.

Theorem 2.2.

with

Proof.

Since it is easy to prove by induction that if and for then and for all

This is equivalent to the existence of a solution of equation where is the polynomial defined in (2.13).

Now, in view of , we have in case in case and in case

On the other hand, in view of (2.10). As a consequence, there exists such that Hence, is a solution of (2.15) with

For this value of , the pair is a solution of (2.14) for every Thus, choosing we have , and for all These imply , and for all Hence, using the first part of the proof, we can conclude that , and for all

Remark 2.3.

Therefore, in the case of positive sequences, the discrete inequality (2.4) is less conservative than the discrete Halanay-type inequality given by (2.16).

## 3. Global Stability of Difference Equations

where

Although, for every initial string the solution of (3.1) can be explicitly calculated by a recurrence formula similar to (2.2), it is in general difficult to investigate the asymptotic behavior of the solutions using that formula. The next result gives an asymptotic estimate by a simple use of the discrete Halanay inequality.

Theorem 3.1.

where and with If either

(a) and or

(b) and

with

As a consequence, the trivial solution of (3.1) is globally asymptotically stable.

Proof.

where and are chosen as in Theorem 3.1. This completes the proof of the theorem.

Next, we obtain new conditions for the asymptotic stability of (3.1) using inequality (3.13) instead of (3.3).

Corollary 3.2.

with

As a consequence, the trivial solution of (3.1) is globally asymptotically stable.

Similarly, using Theorem 2.2 instead of Theorem 2.1, we obtain the following result.

Theorem 3.3.

with

As a consequence, the trivial solution of (3.1) is globally asymptotically stable.

Remark 3.4.

Next, we study the asymptotic behavior of the solutions of (3.21). We can apply Theorem 3.1, Corollary 3.2, or Theorem 3.3 to obtain some relations between coefficients and that ensure the global asymptotic stability of the zero solution. Moreover, from Theorem 3.1 we know that if there exists such that for all and if either

(a) and or

(b) and

hold, then all solutions of (3.21) converge to zero.

## Declarations

### Acknowledgment

The authors thank the referees of this paper for their careful and insightful critique.

## Authors’ Affiliations

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