# 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.

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.

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

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.

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.

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.

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

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

## References

- Agarwal RP, Deng S, Zhang W:
**Generalization of a retarded Gronwall-like inequality and its applications.***Applied Mathematics and Computation*2005,**165**(3):599–612. 10.1016/j.amc.2004.04.067MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Kim Y-H, Sen SK:
**New discrete Halanay inequalities: stability of difference equations.***Communications in Applied Analysis*2008,**12**(1):83–90.MathSciNetMATHGoogle Scholar - Agarwal RP, Wong PJY:
*Advanced Topics in Difference Equations, Mathematics and Its Applications*.*Volume 404*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:viii+507.View ArticleGoogle Scholar - Baĭnov D, Simeonov P:
*Integral Inequalities and Applications, Mathematics and Its Applications*.*Volume 57*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.View ArticleMATHGoogle Scholar - Bay NS, Phat VN:
**Stability analysis of nonlinear retarded difference equations in Banach spaces.***Computers & Mathematics with Applications*2003,**45**(6–9):951–960.MathSciNetView ArticleMATHGoogle Scholar - Cooke KL, Ivanov AF:
**On the discretization of a delay differential equation.***Journal of Difference Equations and Applications*2000,**6**(1):105–119. 10.1080/10236190008808216MathSciNetView ArticleMATHGoogle Scholar - Gopalsamy K:
*Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications*.*Volume 74*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+501.View ArticleMATHGoogle Scholar - Liz E, Trofimchuk S:
**Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality.***Journal of Mathematical Analysis and Applications*2000,**248**(2):625–644. 10.1006/jmaa.2000.6947MathSciNetView ArticleMATHGoogle Scholar - Liz E, Ferreiro JB:
**A note on the global stability of generalized difference equations.***Applied Mathematics Letters*2002,**15**(6):655–659. 10.1016/S0893-9659(02)00024-1MathSciNetView ArticleMATHGoogle Scholar - Liz E, Ivanov AF, Ferreiro JB:
**Discrete Halanay-type inequalities and applications.***Nonlinear Analysis: Theory, Methods & Applications*2003,**55**(6):669–678. 10.1016/j.na.2003.07.013MathSciNetView ArticleMATHGoogle Scholar - Niamsup P, Phat VN:
**Asymptotic stability of nonlinear control systems described by difference equations with multiple delays.***Electronic Journal of Differential Equations*2000,**2000**(11):1–17.MathSciNetMATHGoogle Scholar - Udpin S, Niamsup P: New discrete type inequalities and global stability of nonlinear difference equations. to appear in Applied Mathematics Letters to appear in Applied Mathematics LettersGoogle Scholar
- Mohamad S, Gopalsamy K:
**Continuous and discrete Halanay-type inequalities.***Bulletin of the Australian Mathematical Society*2000,**61**(3):371–385. 10.1017/S0004972700022413MathSciNetView ArticleMATHGoogle Scholar - Pinto M, Trofimchuk S:
**Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima.***Proceedings of the Royal Society of Edinburgh. Section A*2000,**130**(5):1103–1118. 10.1017/S0308210500000597MathSciNetView ArticleMATHGoogle Scholar - Halanay A:
*Differential Equations: Stability, Oscillations, Time Lags*. Academic Press, New York, NY, USA; 1966:xii+528.MATHGoogle Scholar

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