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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 494067, 7 pages
http://dx.doi.org/10.1155/2013/494067
Research Article

The Asymptotic Behavior for a Class of Impulsive Delay Differential Equations

Department of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 27 January 2013; Accepted 21 March 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Zhichun Yang. 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.

Abstract

This paper is concerned with asymptotical behavior for a class of impulsive delay differential equations. The new criteria for determining attracting sets and attracting basin of the impulsive system are obtained by developing the properties of quasi-invariant sets. Examples and numerical simulations are given to illustrate the effectiveness of our results. In addition, we show that the impulsive effects may play a key role to these asymptotical properties even though the solutions of corresponding nonimpulsive systems are unbounded.

1. Introduction

Impulsive delay differential equations have attracted increasing interests since time delays and impulsive effects commonly exist in many fields such as population dynamics, automatic control, drug administration, and communication networks [14]. In past two decades, its asymptotical behaviors such as stability and attractivity of the equilibrium point or periodical solutions have been deeply studied for impulsive functional differential equations (see, [518]).

However, under impulsive perturbation, the solutions may not be attracted to an equilibrium point or periodical trajectory but to some bounded region. In this case, it is interesting to investigate the attracting set and attracting basin, that is, the region attracting the solutions and the range in which initial values vary when remaining the attractivity for impulsive delay differential equations. In [19], Xu and Yang first give the method to estimate global attracting set and invariant set for impulsive delayed systems by developing delayed differential inequalities. The techniques are further developed to study global attractivity for some complex impulsive systems such as impulsive neutral differential equations [20, 21] and impulsive stochastic systems [22]. But the techniques and methods given in the existing publications are invalid for determining locally attracting set and attracting basin for impulsive delay differential equations.

In this paper, our objective is to mainly discuss the asymptotical behavior on (locally) attracting set and its attracting basin for a class of impulsive delay differential equations. Based on the quasi-invariant properties, we estimate the existence range of attracting set and attracting basin of the impulsive delay systems by solving algebraic equations and employing differential inequality technique. Examples are given to illustrate the effectiveness of our method and show that the asymptotic behavior of the impulsive systems may be different from one of the corresponding continuous systems.

2. Preliminaries

Let be the set of all positive integers, the space of -dimensional real column vectors, and the set of real matrices. For or , means that each pair of corresponding elements of and satisfies the inequality “().” , , and denotes an unit matrix.

Let and be the fixed points with (called impulsive moments).

denotes the space of continuous mappings from the topological space to the topological space . Let especially.

Morever, for , exists for , for all but at most a finite number of points . is a space of piecewise right-hand continuous functions which is a nature extension of the phrase space .

We define is continuous at , and exist, , for .

For , , or , we define where and is an norm in .

In this paper, we will consider a impulsive delay differential equations: where denotes the right-hand derivative of , , ,  = diag,,  = diag, , and the limit exists, , and is defined by .

A function is called to be a solution of (2) through (), if as , and satisfies (2) with the initial condition Throughout the paper, we always assume that for any , system (2) has at least one solution through (), denoted by or (simply and if no confusion should occur), where , .

In this paper, we need the following definitions involving attracting set, attracting basin, the quasi-invariant set of impulsive systems, and monotonous vector functions.

Definition 1. The set is called to be an attracting set of (2), and is called an attraction basin of , if for any initial value , the solution converges to as . That is, where dist dist, dist, for .

Definition 2. The set is called to be a positive quasi-invariant set of (2), if there is a positive diagonal matrix  = diag such that for any initial value , the solutions satisfy , for . When (identity matrix) especially, the set is called positively invariant.

Definition 3. Let . The vector function is called to be monotonically nondecreasing in , if for any , implies .

3. Main Results

In this paper, we always make the following assumptions. ) There exist nonnegative constants such that , for .() for and , where the vector function is continuous and monotonically nondecreasing in .() , for , and , where the vector function is continuous and monotonically nondecreasing in .

To obtain attractivity, we first give the quasi-invariant properties of (2).

Theorem 4. Assume that in addition to , there is a vector such that where  = ,  = , , are defined by Then, the set is a positive quasi-invariant set of (2). When especially, is a positive invariant set of (2).

Proof. Let be a solution of (2) through (). It is easily verified that the following formula for the variation of parameters is valid: where is the Cauchy matrix of linear impulsive system According to the representation of the Cauchy matrix (see page 74 [2]), Since , for , we obtain the following estimate: In terms of the definition of and , By (7) and (11) and the assumptions () and (), then Since and  = , we have From the strict inequality (5), there is an enough small number such that In the following, we will prove that implies Otherwise, from the piecewise continuity of , there must be an integer and such that By using (12), (13), (14), (17), , and the monotonicity of , we can get This contradicts (16), and so (15) holds. Letting , from (15), we have for any (i.e., ), Therefore, the set is a positive quasi-invariant set of (2). When especially, is a positive invariant set of (2). The proof is complete.

Based on the obtained quasi-invariant set, we have the following

Theorem 5. Let Assume that all conditions in Theorem 4 hold. Define Then, is an attracting set of (2) and is the attracting basin of .

Proof. From (5) and the definitions of the above sets, then , , , , . Obviously, and are nonempty, and so the definitions of the sets of and are valid. For any , there is a satisfying . According to Theorem 4, we obtain That is, Then, for any given , there is a such that In light of  = diag, for the above and , we can find an enough large such that Using (12), (13), (22), (24), and (25), we have for , This implies that Letting , then That is, and . Thus, From the definition of and , dist as . The proof is complete.

From the above theorems, we can obtain sufficient conditions ensuring global attractivity and stability in the following corollaries.

Corollary 6. Assume that ()–() hold with If the spectral radius then is a positive quasi-invariant set of (2), and is a global attracting set of (2).

Proof. Since and , we directly calculate Without loss of generality, we assume that . Since , exists and (see [23]), and so . For any , we take in Theorem 4 and verify the condition (5): According to Theorem 4, when , we deduce that is a positive quasi-invariant set of (2). Furthermore, by (33), From the arbitrariness of , we obtain . Moreover, It follows from Theorem 5 that is a global attracting set of (2) and is also a global attracting set due to . The proof is complete.

Corollary 7. Assume that all conditions in Corollary 6 hold with . Then, the zero solution of (2) is globally asymptotically stable.

4. Illustrative Examples

The following illustrative examples will demonstrate the effectiveness of our results and also show the different asymptotical behaviors between the impulsive system and the corresponding continuous system.

Example 8. Consider a scalar nonlinear impulsive delay system According to Theorems 4 and 5, we have , , , , , , , , and so Thus, is an attracting set of (36), and is an attracting basin of . However, solutions of the corresponding continuous system (i.e., in (36)) may be unbounded. Taking the initial condition , , Figure 1 shows the different asymptotic behavior between the solution of (36) with no impulse and one with impulses.

fig1
Figure 1: The trajectory of (36) with: (a) no impulse (i.e., ) and (b) impulses.

Example 9. Consider a 2-dimensional impulsive delay system According to Corollary 6, we have ,  = diag,  = diag,  = diag,  = diag, , , , where and so . Therefore, is a positive quasi-invariant set of (38), and is a global attracting set of (38). Figure 2 shows the asymptotic properties of solutions of (38) under the different initial conditions.

494067.fig.002
Figure 2: Global attracting set of (38).

Acknowledgments

This work is supported partially by the National Natural Science Foundation of China under Grant nos. 10971240, 61263020, and 61004042, the Key Project of Chinese Education Ministry under Grant no. 212138, the Natural Science Foundation of Chongqing under Grant CQ CSTC 2011BB0117, and the Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ120630.

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