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.
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 .
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.
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.
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 .
We always assume that for any , (4.1) has at least one solution through ( ), denoted by or (simply or if no confusion should occur).
The set is called a positive invariant set of (4.1), if for any initial value , we have the solution for .
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
and let be an -matrix, and .
Assume that hold. Then is a global attracting set of (4.1).
Denote . Let be the sign function. For , define
then . From the property of -cone, we have, .
hold, where .
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.
Assume that with hold. Then is a positive invariant set and also a global attracting set of (4.1).
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 .
5. Illustrative Example
The following illustrative example will demonstrate the effectiveness of our results.
where , , , .
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).
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).
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