- Research Article
- Open Access

# A New Singular Impulsive Delay Differential Inequality and Its Application

- Zhixia Ma
^{1}and - Xiaohu Wang
^{2}Email author

**2009**:461757

https://doi.org/10.1155/2009/461757

© Z. Ma and X.Wang. 2009

**Received:**10 January 2009**Accepted:**5 March 2009**Published:**8 March 2009

## Abstract

A new singular impulsive delay differential inequality is established. Using this inequality, the invariant and attracting sets for impulsive neutral neural networks with delays are obtained. Our results can extend and improve earlier publications.

## Keywords

- Positive Constant
- Differential System
- Integral Inequality
- Nonnegative Matrix
- Differential Inequality

## 1. Introduction

It is well known that inequality technique is an important tool for investigating dynamical behavior of differential equation. The significance of differential and integral inequalities in the qualitative investigation of various classes of functional equations has been fully illustrated during the last 40 years [1–3]. Various inequalities have been established such as the delay integral inequality in [4], the differential inequalities in [5, 6], the impulsive differential inequalities in [7–10], Halanay inequalities in [11–13], and generalized Halanay inequalities in [14–17]. By using the technique of inequality, the invariant and attracting sets for differential systems have been studied by many authors [9, 18–21].

However, the inequalities mentioned above are ineffective for studying the invariant and attracting sets of impulsive nonautonomous neutral neural networks with time-varying delays. On the basis of this, this article is devoted to the discussion of this problem.

Motivated by the above discussions, in this paper, a new singular impulsive delay differential inequality is established. Applying this equality and using the methods in [10, 22], some sufficient conditions ensuring the invariant set and the global attracting set for a class of neutral neural networks system with impulsive effects are obtained.

## 2. Preliminaries

Throughout the paper, means -dimensional unit matrix, the set of real numbers, the set of positive integers, and . means that each pair of corresponding elements of and satisfies the inequality " ( )". Especially, is called a nonnegative matrix if .

It is a cone without conical surface in . We call it an " -cone".

## 3. Singular Impulsive Delay Differential Inequality

For convenience, we introduce the following conditions.

Theorem 3.1.

Proof.

By the conditions ( ) and the definition of -matrix, there is a constant vector such that , exists and .

If inequality (3.10) is not true, then is a nonempty set and there must exist some integer such that .

which contradicts the second inequality in (3.12).

The proof is complete.

Remark 3.2.

So we can get [7, Lemma 1] when we choose , , in Theorem 3.1.

Remark 3.3.

Suppose that and in Theorem 3.1, then we can get [10, Theorem 3.1].

## 4. Applications

where is the neural state vector; , , are the interconnection matrices representing the weighting coefficients of the neurons; are activation functions; are transmission delays; denotes the external inputs at time . represents impulsive perturbations; the fixed moments of time satisfy .

We always assume that for any , (4.1) has at least one solution through ( ), denoted by or (simply or if no confusion should occur).

Definition 4.1.

The set is called a positive invariant set of (4.1), if for any initial value , we have the solution for .

Definition 4.2.

Throughout this section, we suppose the following.

are continuous. Moreover, and .

There exist nonnegative matrices , , , and a constant such that

There exist nonnegative matrices such that

There exist nonnegative matrices , , such that for all the activating functions and satisfy

.

Theorem 4.3.

Assume that hold. Then is a global attracting set of (4.1).

Proof.

Denote . Let be the sign function. For , define

then . From the property of -cone, we have, .

This implies that the conclusion of the theorem holds.

By using Theorem 4.3 with , we can obtain a positive invariant set of (4.1), and the proof is similar to that of Theorem 4.3.

Theorem 4.4.

Assume that with hold. Then is a positive invariant set and also a global attracting set of (4.1).

Remark 4.5.

Suppose that in , and , then we can get Theorems 1 and 2 in [9].

Remark 4.6.

If then (4.1) becomes the nonautonomous neutral neural networks without impulses, we can get Theorem 4.1 in [22].

## 5. Illustrative Example

The following illustrative example will demonstrate the effectiveness of our results.

Example 5.1.

Now, we discuss the asymptotical behavior of the system (5.1) as follows.

Thus , , , and . Clearly, all conditions of Theorem 4.3 are satisfied, by Theorem 4.3, is a global attracting set of (5.1).

(ii)If , then . By Theorem 4.4, is a positive invariant set of (5.1).

## Declarations

### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10671133 and the Scientific Research Fund of Sichuan Provincial Education Department (08ZA044).

## Authors’ Affiliations

## References

- Walter W:
*Differential and Integral Inequalities*. Springer, New York, NY, USA; 1970:x+352.View ArticleGoogle Scholar - Lakshmikantham V, Leela S:
*Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations, Mathematics in Science and Engineering*.*Volume 55*. Academic Press, New York, NY, USA; 1969:ix+390.MATHGoogle Scholar - Lakshmikantham V, Leela S:
*Differential and Integral Inequalities: Theory and Applications. Vol. II: Functional, Partial, Abstract, and Complex Differential Equations, Mathematics in Science and Engineering*.*Volume 55*. Academic Press, New York, NY, USA; 1969:ix+319.MATHGoogle Scholar - Xu DY:
**Integro-differential equations and delay integral inequalities.***Tohoku Mathematical Journal*1992,**44**(3):365–378. 10.2748/tmj/1178227303MathSciNetView ArticleMATHGoogle Scholar - Wang L, Xu DY:
**Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays.***Science in China. Series F*2003,**46**(6):466–474. 10.1360/02yf0146MathSciNetView ArticleMATHGoogle Scholar - Huang YM, Xu DY, Yang ZG:
**Dissipativity and periodic attractor for non-autonomous neural networks with time-varying delays.***Neurocomputing*2007,**70**(16–18):2953–2958.View ArticleGoogle Scholar - Xu DY, 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 - Xu DY, Zhu W, Long S:
**Global exponential stability of impulsive integro-differential equation.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(12):2805–2816. 10.1016/j.na.2005.09.020MathSciNetView ArticleMATHGoogle Scholar - Xu DY, Yang Z:
**Attracting and invariant sets for a class of impulsive functional differential equations.***Journal of Mathematical Analysis and Applications*2007,**329**(2):1036–1044. 10.1016/j.jmaa.2006.05.072MathSciNetView ArticleMATHGoogle Scholar - Xu DY, 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 - Driver RD:
*Ordinary and Delay Differential Equations, Applied Mathematical Sciences*.*Volume 20*. Springer, New York, NY, USA; 1977:ix+501.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 ArticleMATHGoogle Scholar - Halanay A:
*Differential Equations: Stability, Oscillations, Time Lags*. Academic Press, New York, NY, USA; 1966:xii+528.MATHGoogle Scholar - Amemiya T:
**Delay-independent stabilization of linear systems.***International Journal of Control*1983,**37**(5):1071–1079. 10.1080/00207178308933029MathSciNetView 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 - Ivanov A, Liz E, Trofimchuk S:
**Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima.***Tohoku Mathematical Journal*2002,**54**(2):277–295. 10.2748/tmj/1113247567MathSciNetView ArticleMATHGoogle 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 - Lu KN, Xu DY, Yang ZC:
**Global attraction and stability for Cohen-Grossberg neural networks with delays.***Neural Networks*2006,**19**(10):1538–1549. 10.1016/j.neunet.2006.07.006View ArticleMATHGoogle Scholar - Xu DY, Li S, Zhou X, Pu Z:
**Invariant set and stable region of a class of partial differential equations with time delays.***Nonlinear Analysis: Real World Applications*2001,**2**(2):161–169. 10.1016/S0362-546X(00)00111-5MathSciNetView ArticleMATHGoogle Scholar - Xu DY, Zhao H-Y:
**Invariant and attracting sets of Hopfield neural networks with delay.***International Journal of Systems Science*2001,**32**(7):863–866.MathSciNetView ArticleMATHGoogle Scholar - Zhao H:
**Invariant set and attractor of nonautonomous functional differential systems.***Journal of Mathematical Analysis and Applications*2003,**282**(2):437–443. 10.1016/S0022-247X(02)00370-0MathSciNetView ArticleMATHGoogle Scholar - Guo Q, Wang X, Ma Z:
**Dissipativity of non-autonomous neutral neural networks with time-varying delays.***Far East Journal of Mathematical Sciences*2008,**29**(1):89–100.MathSciNetMATHGoogle Scholar - Berman A, Plemmons RJ:
*Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematic*. Academic Press, New York, NY, USA; 1979:xviii+316.Google 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.