- Research Article
- Open Access
A New Singular Impulsive Delay Differential Inequality and Its Application
© Z. Ma and X.Wang. 2009
- Received: 10 January 2009
- Accepted: 5 March 2009
- Published: 8 March 2009
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.
- Positive Constant
- Differential System
- Integral Inequality
- Nonnegative Matrix
- Differential Inequality
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 , 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.
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 .
For convenience, we introduce the following conditions.
which contradicts the second inequality in (3.12).
The proof is complete.
So we can get [7, Lemma 1] when we choose , , in Theorem 3.1.
Suppose that and in Theorem 3.1, then we can get [10, Theorem 3.1].
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 .
Throughout this section, we suppose the following.
This implies that the conclusion of the theorem holds.
Suppose that in , and , then we can get Theorems 1 and 2 in .
If then (4.1) becomes the nonautonomous neutral neural networks without impulses, we can get Theorem 4.1 in .
The following illustrative example will demonstrate the effectiveness of our results.
Now, we discuss the asymptotical behavior of the system (5.1) as follows.
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).
- 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
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