 Research Article
 Open Access
 Published:
A New Singular Impulsive Delay Differential Inequality and Its Application
Journal of Inequalities and Applications volume 2009, Article number: 461757 (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.
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 timevarying 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 .
denotes the space of continuous mappings from the topological space to the topological space . In particular, let , where is a constant.
denotes the space of piecewise continuous functions with at most countable discontinuous points and at this points are right continuous. Especially, let . Furthermore, put .
, where denotes the derivative of . In particular, let .
is a positive integrable function and satisfies and .
For , or , we define And we introduce the following norm, respectively,
For any , we define the following norm:
For an matrix defined in [23], we denote
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.
Let the dimensional diagonal matrix satisfy
Let be an matrix, where and satisfies
Theorem 3.1.
Assume the conditions and hold. Let and be a solution of the following singular delay differential inequality with the initial conditions :
where , and , , . Then
provided that the initial conditions satisfy
where , and the positive number satisfies the following inequality:
Proof.
By the conditions () and the definition of matrix, there is a constant vector such that , exists and .
By using continuity, we obtain that there must exist a positive constant satisfying the inequality (3.6), that is,
Denote by
It follows from (3.3) and (3.5) that
In the following, we will prove that for any positive constant ,
Let
If inequality (3.10) is not true, then is a nonempty set and there must exist some integer such that .
If , by and the inequality (3.5), we can get
By using , (3.3), (3.7), (3.12), (3.13), and , we obtain that
Since , we have by . Then (3.14) becomes
which contradicts the second inequality in (3.12).
If , then by and . From the inequality (3.5), we can get
By using , (3.3), (3.7), (3.16), and , we obtain that
This is a contradiction. Thus the inequality (3.10) holds. Therefore, letting in (3.10), we have
The proof is complete.
Remark 3.2.
In order to overcome the difficulty that in (3.3) may be discontinuous, we introduce the notation which is different from the notation in [7]. However, when is continuous in , we have
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
The singular impulsive delay differential inequality obtained in Section 3 can be widely applied to study the dynamics of impulsive neutral differential equations. To illustrate the theory, we consider the following nonautonomous impulsive neutral neural networks with delays
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 .
The initial condition for (4.1) is given by
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.
The set is called a global attracting set of (4.1), if for any initial value , the solution converges to as . That is,
where , for .
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
There exists nonnegative matrix , such that for all , and
.
Denote by
and let be an matrix, and .
There exists a constant such that
where , and the scalar is determined by the inequality
where , and
Theorem 4.3.
Assume that hold. Then is a global attracting set of (4.1).
Proof.
Denote . Let be the sign function. For , define
Calculating the upper right derivative along system (4.1). From (4.1), and we have
On the other hand, from (4.1) and , we have
Let
then from (4.12)–(4.14) and , we have
By the conditions and the definition of matrix, we may choose a vector such that
By using continuity, we obtain that there must be a positive constant satisfying the inequality (4.10). Let and , then . Since , denote
then . From the property of cone, we have, .
For the initial conditions , , where and (no loss of generality, we assume , and , we can get
Then (4.18) yield
Let , and . Thus, all conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we have
Suppose that for all , the inequalities
hold, where .
From (4.21), , and , we can get
Since , we have
On the other hand, it follows from that
Then from (4.21)–(4.24), we have
which together with (4.22) yields that
Then, it follows from (4.21) and (4.26) that
Using Theorem 3.1 again, we have
By mathematical induction, we can conclude that
Noticing that , by , we can use (4.29) to conclude that
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.
Consider nonlinear impulsive neutral neural networks:
with
where , , , .
The parameters of conditions are as follows:
It is easy to prove that is an matrix and
Let , then and . Let which satisfies the inequality
Now, we discuss the asymptotical behavior of the system (5.1) as follows.

(i)
If , then
(5.6)
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).
References
Walter W: Differential and Integral Inequalities. Springer, New York, NY, USA; 1970:x+352.
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.
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.
Xu DY: Integrodifferential equations and delay integral inequalities. Tohoku Mathematical Journal 1992,44(3):365–378. 10.2748/tmj/1178227303
Wang L, Xu DY: Global exponential stability of Hopfield reactiondiffusion neural networks with timevarying delays. Science in China. Series F 2003,46(6):466–474. 10.1360/02yf0146
Huang YM, Xu DY, Yang ZG: Dissipativity and periodic attractor for nonautonomous neural networks with timevarying delays. Neurocomputing 2007,70(16–18):2953–2958.
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.2004.10.040
Xu DY, Zhu W, Long S: Global exponential stability of impulsive integrodifferential equation. Nonlinear Analysis: Theory, Methods & Applications 2006,64(12):2805–2816. 10.1016/j.na.2005.09.020
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.2006.05.072
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.2006.07.043
Driver RD: Ordinary and Delay Differential Equations, Applied Mathematical Sciences. Volume 20. Springer, New York, NY, USA; 1977:ix+501.
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.
Halanay A: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York, NY, USA; 1966:xii+528.
Amemiya T: Delayindependent stabilization of linear systems. International Journal of Control 1983,37(5):1071–1079. 10.1080/00207178308933029
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.2000.6947
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/1113247567
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/S0022247X(02)000562
Lu KN, Xu DY, Yang ZC: Global attraction and stability for CohenGrossberg neural networks with delays. Neural Networks 2006,19(10):1538–1549. 10.1016/j.neunet.2006.07.006
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/S0362546X(00)001115
Xu DY, Zhao HY: Invariant and attracting sets of Hopfield neural networks with delay. International Journal of Systems Science 2001,32(7):863–866.
Zhao H: Invariant set and attractor of nonautonomous functional differential systems. Journal of Mathematical Analysis and Applications 2003,282(2):437–443. 10.1016/S0022247X(02)003700
Guo Q, Wang X, Ma Z: Dissipativity of nonautonomous neutral neural networks with timevarying delays. Far East Journal of Mathematical Sciences 2008,29(1):89–100.
Berman A, Plemmons RJ: Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematic. Academic Press, New York, NY, USA; 1979:xviii+316.
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ma, Z., Wang, X. A New Singular Impulsive Delay Differential Inequality and Its Application. J Inequal Appl 2009, 461757 (2009). https://doi.org/10.1155/2009/461757
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/461757
Keywords
 Positive Constant
 Differential System
 Integral Inequality
 Nonnegative Matrix
 Differential Inequality