# Integrodifferential Inequality for Stability of Singularly Perturbed Impulsive Delay Integrodifferential Equations

- Danhua He
^{1}and - Liguang Xu
^{2}Email author

**2009**:369185

**DOI: **10.1155/2009/369185

© D. He and L. Xu. 2009

**Received: **24 February 2009

**Accepted: **11 June 2009

**Published: **16 June 2009

## Abstract

The exponential stability of singularly perturbed impulsive delay integrodifferential equations (SPIDIDEs) is concerned. By establishing an impulsive delay integrodifferential inequality (IDIDI), some sufficient conditions ensuring the exponentially stable of any solution of SPIDIDEs for sufficiently small are obtained. A numerical example shows the effectiveness of our theoretical results.

## 1. Introduction

Integrodifferential equations (IDEs) arise from many areas of science (from physics, biology, medicine, etc.), which have extensive scientific backgrounds and realistic mathematical models, and hence have been emerging as an important area of investigation in recent years, see [1–6]. Correspondingly, the stability of impulsive delay integrodifferential equations has been studied quite well, for example, [7–9]. However, besides delay and impulsive effects, singular perturbation likewise exists in a wide models for physiological processes or diseases [10]. And many good results on the stability of singularly perturbed delay differential equations have been reported, see, for example, [11–14]. Therefore, it is necessary to consider delay, impulse and singular perturbation on the stability of integrodifferential equations. However, to the best of our knowledge, there are no results on the problems of the exponential stability of solutions for SPIDIDEs due to some theoretical and technical difficulties. Based on this, this article is devoted to the discussion of this problem.

Applying differential inequalities, in [14–17], authors investigated the stability of impulsive differential equations. In [14], Zhu et al. established a delay differential inequality with impulsive initial conditions and derived some sufficient conditions ensuring the exponential stability of solutions for the singular perturbed impulsive delay differential equations (SPIDDEs). In this paper, we will improve the inequality established in [14] such that it is effective for SPIDIDEs. By establishing an IDIDI, some sufficient conditions ensuring the exponential stability of any solution of SPIDIDEs for sufficiently small are obtained. The results extend and improve the earlier publications, and which will be shown by the Remarks 3.2 and 3.5 provided later. An example is given to illustrate the theory.

## 2. Preliminaries

Throughout this letter, unless otherwise specified, let be the space of -dimensional real column vectors and be the set of real matrices. . For or , means that each pair of corresponding elements of and satisfies the inequality " ( )". Especially, is called a nonnegative matrix if , and is called a positive vector if .

where , , , , , is a small parameter, and is a strictly increasing sequence such that .

Definition 2.1.

The solution of (2.2) is said to be exponentially stable for sufficiently small if there exist finite constant vectors and , which are independent of for some , and a constant such that for and for any initial perturbation satisfying . Here is the solution of (2.2) corresponding to the initial condition .

## 3. Main Results

In order to prove the main result in this paper, we first need the following technique lemma.

Lemma 3.1.

Proof.

the first inequality and the second inequality are from (3.2), the last inequality is because , , , .

Therefore, from the definition of , one can know that .

this contradiction shows that , so there at least exists a positive constant such that , that is, the definition of for (3.3) is reasonable.

which contradicts the inequality in (3.13), and so (3.12) holds for all . Letting , then (3.3) holds, and the proof is completed.

Remark 3.2.

If in Lemma 3.1, then we get [14, Lemma 1].

Theorem 3.3.

Assume that for and , further suppose the following

Then there exists a small such that the solution of (2.2) is exponentially stable for sufficiently small .

Proof.

where is determined by (3.21) and the positive constant vector is determined by (3.18).

so is monotonically decreasing with respect to the variable , which implies that is also monotonically decreasing with respect to the variable . So we can choose the in (3.21) satisfying the same monotonicity with , for example, , where . Hence we can deduce that there exists a small such that the solution of (2.2) is exponentially stable for sufficiently small . The proof is completed.

Remark 3.4.

Suppose that in Theorem 3.3, then we can easily get [14, Theorem 1]. In fact, " " of condition in [14, Theorem 1] ensure that the above (3.20) holds.

Remark 3.5.

If , , that is there have no impulses in (2.2), then by Theorem 3.3, we can obtain the following result.

Corollary 3.6.

Assume that for and , , further suppose that and hold. Then there exists a small such that the solution of (2.2) is exponentially stable for sufficiently small .

Remark 3.7.

From Lemma 3.1 and the proof of Theorem 3.3, it is obvious that the results obtained in this paper still hold for . So this type of exponential stability can obviously be applied to general impulsive delay integrodifferential equations.

Remark 3.8.

When and , the global exponential stability criteria for (2.2) have been established in [18] by utilizing the Lyapunov functional method. However, the additional assumption that is bounded is required in [18].

## 4. An Illustrative Example

In this section, we will give an example to illustrate the exponential stability of (2.2).

Example 4.1.

where , are constants, , , , .

Case 1.

So for a given , we can obtain the corresponding by (4.5). By the proof of Theorem 3.3, we know that is monotonically decreasing with respect to the variable , then there exists an such that for any , we have . Therefore, all the conditions of Theorem 3.3 are satisfied, we conclude that the solution of (4.1) is exponentially stable for sufficiently small .

Case 2.

Let and , then (4.1) becomes the singularly perturbed delay integrodifferential equations without impulses. So by Corollary 3.6, the solution of (4.1) is exponentially stable for sufficiently small .

Remark 4.2.

Obviously, the delay differential inequality which established in [14] is ineffective for studing the stability of SPIDIDEs (4.1).

## Declarations

### Acknowledgments

The authors would like to thank the editor and the referees for their very helpful suggestions. The work is supported by National Natural Science Foundation of China under Grant 10671133.

## Authors’ Affiliations

## References

- Hale J:
*Theory of Functional Differential Equations, Applied Mathematical Sciences*. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle 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 ArticleGoogle Scholar - Kolmanovskii VB, Myshkis A:
*Applied Theory of Functional Differential Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992.View ArticleGoogle Scholar - Gan S:
**Dissipativity of -methods for nonlinear Volterra delay-integro-differential equations.***Journal of Computational and Applied Mathematics*2007,**206**(2):898–907. 10.1016/j.cam.2006.08.030MathSciNetView ArticleMATHGoogle Scholar - Alsaedi A, Ahmad B:
**Existence and analytic approximation of solutions of duffing type nonlinear integro-differential equation with integral boundary conditions.***Journal of Inequalities and Applications*2009,**2009:**-19.Google Scholar - Brown BM, Kirby VG:
**A computational investigation of an integro-differential inequality with periodic potential.***Journal of Inequalities and Applications*2000,**5**(4):321–342.MathSciNetMATHGoogle Scholar - Xu L, Xu D:
**Exponential stability of nonlinear impulsive neutral integro-differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(9):2910–2923. 10.1016/j.na.2007.08.062MathSciNetView ArticleMATHGoogle Scholar - Akhmetov MU, Zafer A, Sejilova RD:
**The control of boundary value problems for quasilinear impulsive integro-differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2002,**48**(2):271–286. 10.1016/S0362-546X(00)00186-3MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Rodríguez-López R:
**New comparison results for impulsive integro-differential equations and applications.***Journal of Mathematical Analysis and Applications*2007,**328**(2):1343–1368. 10.1016/j.jmaa.2006.06.029MathSciNetView ArticleMATHGoogle Scholar - Mackey MC, Glass L:
**Oscillation and chaos in physiological control systems.***Science*1977,**197**(4300):287–289. 10.1126/science.267326View ArticleGoogle Scholar - Tian H:
**The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag.***Journal of Mathematical Analysis and Applications*2002,**270**(1):143–149. 10.1016/S0022-247X(02)00056-2MathSciNetView ArticleMATHGoogle Scholar - da Cruz JH, Táboas PZ:
**Periodic solutions and stability for a singularly perturbed linear delay differential equation.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(6):1657–1667. 10.1016/j.na.2006.08.004MathSciNetView ArticleMATHGoogle Scholar - Liu X, Shen X, Zhang Y:
**Exponential stability of singularly perturbed systems with time delay.***Applicable Analysis*2003,**82**(2):117–130. 10.1080/0003681031000063775MathSciNetView ArticleMATHGoogle Scholar - Zhu W, Xu D, Yang C:
**Exponential stability of singularly perturbed impulsive delay differential equations.***Journal of Mathematical Analysis and Applications*2007,**328**(2):1161–1172. 10.1016/j.jmaa.2006.05.082MathSciNetView ArticleMATHGoogle Scholar - Xu D, Yang Z, Yang Z:
**Exponential stability of nonlinear impulsive neutral differential equations with delays.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(5):1426–1439. 10.1016/j.na.2006.07.043MathSciNetView ArticleMATHGoogle Scholar - Guan Z-H:
**Impulsive inequalities and stability of large scale singular measure differential systems with unbounded delays.***Dynamics of Continuous, Discrete and Impulsive Systems for Theory and Applications*1997,**3**(1):113–130.Google Scholar - Xu D, Yang Z:
**Impulsive delay differential inequality and stability of neural networks.***Journal of Mathematical Analysis and Applications*2005,**305**(1):107–120. 10.1016/j.jmaa.2004.10.040MathSciNetView ArticleMATHGoogle Scholar - Huang Z-T, Yang Q-G, Luo X-S:
**Exponential stability of impulsive neural networks with time-varying delays.***Chaos, Solitons & Fractals*2008,**35**(4):770–780. 10.1016/j.chaos.2006.05.089View ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.