Abstract and Applied Analysis

Volume 2013 (2013), Article ID 132619, 8 pages

http://dx.doi.org/10.1155/2013/132619

Research Article

New Results on Impulsive Functional Differential Equations with Infinite Delays

Jie Yang¹ and Bing Xie²

¹School of Mathematical Sciences, Shandong Normal University, Ji'nan 250014, China

²Department of Mathematics, Shandong University, Weihai, Shandong 264209, China

Received 28 January 2013; Accepted 6 April 2013

Academic Editor: Jinde Cao

Copyright © 2013 Jie Yang and Bing Xie. 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.

We investigate the stability for a class of impulsive functional differential equations with infinite delays by using Lyapunov functions and Razumikhin-technique. Some new Razumikhin-type theorems on stability are obtained, which shows that impulses do contribute to the system’s stability behavior. An example is also given to illustrate the importance of our results.

1. Introduction

Impulsive differential equations have attracted the interest of many researchers in recent years. It arises naturally from a wide variety of applications such as orbital transfer of satellite, ecosystems management, and threshold theory in biology. There has been a significant development in the theory of impulsive differential equations in the past several years ago, and various interesting results have been reported; see [1–4]. Recently, systems with impulses and time delay have received significant attention [5–16]. In fact, the system stability and convergence properties are strongly affected by time delays, which are often encountered in many industrial and natural processes due to measurement and computational delays, transmission, and transport lags. In [5, 6, 8], the authors considered the stability of impulsive differential equations with finite delay and got some results. In [7], by using Lyapunov functions and Razumikhin technique, some Razumikhin-type theorems on stability are obtained for a class of impulsive functional differential equations with infinite-delay. However, not much has been developed in the direction of the stability theory of impulsive functional differential systems, especially for infinite delays impulsive functional differential systems. As we know, there are a number of difficulties that one must face in developing the corresponding theory of impulsive functional differential systems with infinite-delay; for example, the interval is not compact, and the images of a solution map of closed and bounded sets in space may not be compact. Therefore, it is an interesting and complicated problem to study the stability theory for impulsive functional differential systems with infinite delays.

In the present paper, we will consider the stability of impulsive infinite-delay differential equations by using Lyapunov functions and the Razumikhin technique, we get some new results. The effect of delay and impulses which do contribute to the equations’s stability properties will be shown in this paper.

The rest of this paper is organized as follows. In Section 2, we introduce some notations and definitions. In Section 3, we get some criteria for uniform stability and uniform asymptotic stability for impulsive infinite-delay differential equations, and an example is given to illustrate our results. Finally, concluding remarks are given in Section 4.

2. Preliminaries

Let denote the set of real numbers, the set of nonnegative real numbers, and the n-dimensional real space equipped with the Euclidean norm . For any , let where or be a Volterra-type functional. In the case when , the interval is understood to be replaced by .

We consider the impulsive functional differential equations

where the impulse times satisfy and denotes the right-hand derivative of . is an open set in , where is continuous everywhere except at finite number of points , at which and exist and . For each .

For any , there exists a such that implies that , where .

Define . For , the norm of is defined by . For any, let .

For any given , the initial condition for system (1) is given by

where .

We assume that the solution for the initial problems, (1)-(2) does exist and is unique which will be written in the form ; see [4, 10]. Since , then is a solution of (1)-(2), which is called the trivial solution. In this paper, we always assume that the solution of (1)-(2) can be continued to from the right of .

For convenience, we also have the following classes in later sections:

  and for ;  and for and is nondecreasing in ; ;  .

We introduce some definitions as follows.

Definition 1 (see [4]). The function belongs to class if

() is continuous on each of the sets and exists;() is locally Lipschitzian in and .

Definition 2 (see [4]). Let , for any , the upper right-hand Dini derivative of along the solution of (1)-(2) is defined by

Similarly, we can define ,. If , then , where is any of the four derivatives.

For , the upper right-hand Dini derivative of along the solution of (1)-(2) is defined by

Similarly, we can define . If , then these are simply the second derivative of .

Definition 3 (see [4]). Assume to be the solution of (1)-(2) through . Then, the zero solution of (1)-(2) is said to be

(1)uniformly stable, if for any and , there exists a such that implies .(2)uniformly asymptotically stable, if it is uniformly stable, and there exists a such that for any, there is a such that implies .

3. Main Results

Theorem 4. Assume that there exist functions , and constants , such that the following conditions hold:

(i); (ii)for any and , if  , , then

where for any ;(iii)for all ,

Also, for all ,

where satisfies ;(iv)there exist constants such that the following inequalities hold:

where .

Then, the zero solution of (1)-(2) is uniformly asymptotically stable.

Proof. Condition () implies that for . So let and be continuous, strictly increasing functions satisfying for all . Then

We first show uniform stability.

For any , one may choose a such that . Let be a solution of (1)-(2) through . For any , we will prove that .

For convenience, let . Suppose that . First, for , we have

So, .

Next, we claim that

Suppose on the contrary that there exists some such that . Since , we can define . Thus, , and . Also, from (10) we obtain

On the other hand, note that and in view of (10), we can define ; it is obvious that and for . Therefore, combining (12), we have for

that is,

By assumption (), (), we have

However, we also have

which is a contradiction. So, (11) holds.

Hence, implies that . Thus, .

On the other hand, from condition (), we note for ,

Hence, we obtain . particularly, . In view of (10), we get

Next, we claim that

Suppose on the contrary that there exists some such that . Then applying exactly the same argument as in the proof of (11) yields our desired contradiction.

By induction hypothesis, we may prove, in general, that for ,

In view of condition , we obtain that . Therefore, we have proved that the solutions of (1)-(2) are uniformly stable.

Next, we claim that they are uniformly asymptotically stable. Since the zero solution of (1)-(2) is uniformly stable, for any given constant , then there exists such that implies that .

For any , let

From condition , we get that there exists a such that for ,

Choose a positive integer satisfying

and define , we will prove that implies .

First, we prove that there exists such that

Suppose on the contrary that for all ,

Let , from (17), we get

In general, combining (22), we deduce that

which is a contradiction. So, (24) holds.

Suppose . Furthermore, we can prove that for

Suppose this assertion is false, then there exists some such that . Since , so define

then and . Note that

thus, we can define

then and for .

Hence, we get for

which implies that for

Thus, .

By assumption, (, we have for ,

However, we also have

which is a contradiction. So, (28) holds.

On the other hand, it is easy to prove that the functions are nonincreasing for in view of condition for any .

Hence, the following inequalities hold: for ,

Next, we claim that

Or else, then ; from (17), we get

Considering (36), it holds that

which is a contradiction and (37) holds.

Next, we can prove that for

Suppose that this assertion is false, then there exists some such that . Then applying exactly the same argument as in the proof of (24) and (28) yields our desired contradiction. Here, we omit it.

By induction hypothesis, we may prove, for ,

that is,

Hence, we obtain

Next, we prove that there exists such that

Suppose that for all ,

Using the same argument as in the proof of (24), we get

where .

This is a contradiction. So, (44) holds.

Suppose . Furthermore, we claim that for )

Suppose on the contrary, that there exists some such that . We define

since in view of (44). Thus, and . Note that

furthermore, we can define

then and for .

Hence, we get for

considering the definition of and (43), we get for

Thus, .

Using assumptions , we have

However,

giving us a contradiction. So, (47) holds.

Next, we claim that

whose arguments are the same as was employed in the proof of (36), (37). there we omit it.

Repeating this process, it is easy to check that

By induction hypothesis, we have

Let , then for ,

Therefore, we arrive at . The proof of Theorem 4 is complete.

Corollary 5. Assume that there exist functions , and constants , such that the following conditions hold:

(i)  ;(ii)for any and , if  , then

where for any ;(iii)for all ,

Also, for all ,

(iv)there exist constants such that the following inequalities hold:

where .

Then the zero solution of (1)-(2) is uniformly stable.

Theorem 4 has a dual result when is nonincreasing on . Here, we only give the results whose proof is very similar to the proof of Theorem 4.

Theorem 6. Assume that there exist functions , and constants , such that the following conditions hold:

(i); (ii)for any and , if  , then

where for any ;(iii)for all ,

Also, for all ,

where satisfies ;(iv)there exist constants such that the following inequalities hold:

where .

Then the zero solution of (1)-(2) is uniformly asymptotically stable.

Example 7. Consider the impulsive delay differential equations:

where . For any given , we always suppose that (68) has and only has positive solutions, and assume without loss of generality that is a solutions of (68) through . Suppose that there exists such that the following inequalities hold:

where . Then, the zero solution of (68) is uniformly asymptotically stable.

In fact, let then implies that . Thus, for

where .

In view of condition (69), we note

So, condition in Corollary 5 holds.

On the other hand, we have for

in view of condition . Also, considering to be a positive solution of (68), we get

where .

Note that

in view of (69). So, the zero solution of (68) is uniformly stable by Corollary 5.

Furthermore, choose (positive constants), which implies that . On the other hand, since , implying that , then

By Theorem 4, we obtain that if (69) holds, then the zero solution of (68) is uniformly asymptotically stable.

Remark 8. In fact, is a positive solution of (68) through in the absence of impulses. It is obvious that the solution is unstable. However, the solution is uniformly asymptotically stable under proper impulses effect, which shows that impulses do contribute to the system’s stability behavior.

4. Conclusion

In this work, we have considered the stability of impulsive infinite-delay differential systems. By using Lyapunov functions and the Razumikhin technique, we have obtained some new results. We can see that impulses and delay do contribute to the system’s stability behavior.

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