We consider the practical stability of impulsive differential equations with infinite delay in terms of two measures. New stability criteria are established by employing Lyapunov functions and Razumikhin technique. Moreover, an example is given to illustrate the advantage of the obtained result.

1. Introduction

One of the trends in the stability theory of the solutions of differential equations is the so-called practical stability, which was introduced by LaSalle and Lefschetz [1]. This is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Fundamental results in this direction were obtained in [2]. In recent years the theory of practical stability and stability has been developed very intensively [37].

The theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modelling of many real world phenomena. Impulsive differential equations and impulsive functional differential equations have been intensively researched [820].

By employing the Razumikhin technique and Lyapunov functions, several stability criteria are established for general impulsive differential equations with finite delay [57, 14, 21]. Systems with infinite delay deserve study because they describe a kind of system present in the real world. For example, it is very useful in a predator-prey system. Therefore, it is an interesting and complicated problem to study the stability of impulsive functional differential systems with infinite delay. Usually, the Lyapunov functions are defined on whole components of system's state [1222]. In this paper, we divided the components of into several groups and correspondingly, we employ several Lyapunov functions , where for each . In this way, Lyapunov, functions are easier constructed, and the conditions ensuring the required stability are less restrictive. Furthermore, the stability results on impulsive finite delay differential equations considered in [4, 5] are generalized into the results on impulsive infinite delay differential equations in terms of two measures.

The work is organized as follows. In Section 2, we introduce some preliminary definitions which will be employed throughout the paper. In Section 3, based on Lyapunov functions and Razumikhin method, sufficient conditions for the uniformly practical stability in terms of two measures are given; an example is presented to illustrate the effectiveness of the approach.

2. Preliminaries

Consider the following impulsive infinite delay differential equations: where could be is a Volterra-type function. denotes the space of piecewise right continuous functions with the sup-norm ,    , for , and . The functions , are such that if and , then , where const..

The initial condition for system (2.1) is given by where , for .

We assume that a solution for the initial problem (2.1) and (2.2) does exist and is unique. Since , then is a solution of (2.1), which is called the zero solution. Let . For convenience, we define .

Definition 2.1. A continuous function is called a wedge function if and is (strictly) increasing.

Definition 2.2. For , and , for any , we define

Definition 2.3 (see[22]). Let . The impulsive functional differential 1 (2.1), (2.2) is said to be(S1) practically stable, if given with , we have implies for some ; (S2) uniformly practically stable if (S1) holds for every .

In what follows, we will split into several vectors, such that and . For convenience, we define . For , we adopt notation as for . Similarly, let is piecewise continuous and bounded}, and .

3. Main Results

In the sequence, we assume that is defined on for some . For simplicity, denote by , respectively, . Now we start with the case of . be the right-hand derivative of .

Theorem 3.1. For , let be continuous, for with some constants , and let be wedge functions. If there exist Lyapunov functions such that (i), where ; (ii)when , there holds if for ; when , there holds if for ;(iii), and ; (iv) are given, ; when , there holds , where are wedge functions, and is a solution of (2.1) and (2.2). Then the zero solution of (2.1) and (2.2) is uniformly practically stable with respect to .

Proof. Since , and , it follows that there exists some , such that and . Define a function for all
We claim first that for any In fact, if , then by (3.1) and condition (i), ; whereas, if , it also holds. On the other hand, the right-hand inequality in (3.2) is trivially valid.
Step 1. We aim to show that for each , Indeed, suppose and there exists some such that for . Then by (3.1), .

Case 1 .. If for some , then By (iii) .

Case 2 .. for any , and . Then if we have . Clearly, implies . If we have . Obviously, implies . In conclusion, , implies . So by (ii) we have .
If we arrive at the assertion that (3.3) is true for all . Otherwise, there exists a such that . When for some we have and . In this case, by (iii) we have . When for any , we set for .
By the similar analysis to Cases 1 and 2, we also have (3.3) when .
If then (3.3) holds for all . Otherwise, repeat the above argument to arrive at the assertion that (3.3) is valid for all . As for the case of for , the process is similar and thus omitted.
For any , we assume there is a unique solution of (2.1), (2.2) through . Furthermore, we denote
If , such that . By condition (iv), From the definition of , we have .
Let , we assume and , where .
Step 2. We aim to prove that .
First, for any , from Definition 2.2 and condition (iv), we know . Then by (3.2), for . Hence, .
Assume is the first impulse of all such that . Now we claim that
If it does not hold, then there is a such that and for . From (3.3) we have . It is a contradiction, so (3.6) holds.
Without loss of generality, we assume , then ; from inequality (3.6) and condition (iii) we have . Thus,
Similarly, with the process in proving (3.6) and (3.7), we have By simple induction, we can prove that, in general Taking this together with (3.2) and , we have Since , we have Therefore, by the definition of , we have . Thus the zero solution of (2.1), (2.2) with respect to is -uniformly practically stable.

Remark 3.2. Since in our result may be and the upper bound of the Lyapunov functions in our paper is improved by , the result we have obtained is more general than that in [47, 14] with or without finite delay; furthermore, we have divided the components of into several groups, correspondingly, several Lyapunov functions are employed, where for each . In this way, construction of the suitable Lyapunov functions is much easier than for as [4, 6, 7, 10]. In additional, compared with [9, 12] where the infinite delay was considered in the Lyapunov stability of differential equations, we obtain the uniformly practical stability in terms of two measures.

Now, we may develop the ideas behind Theorem 3.1 to obtain the following more general results.

Theorem 3.3. For , let be continuous, for with some constants , and let be wedge functions. If there also exist Lyapunov functions such that (i), where ;(ii)when , there holds if for ;(iii), and ;(iv) are given, ; when where are wedge functions, and is a solution of (2.1) and (2.2).Then the zero solution of (2.1) and (2.2) is -uniformly practically stable.

It suffices to mention a few points in the proofs of Theorem 3.3, the rest are the same as in the proofing of Theorem 3.1, thus, are omitted.

First, for , we define Second, instead of (3.2) we can claim that for any

Example 3.4. Consider the equation where , with . Let . and are all continuous functions.

We first assume that and , with , and . Without loss of generality, we may assume that the right-hand sides of (3.14) are defined on , then set and .

Let , then from the definition . For given , we assume implies that there exists a such that for any . Let ,   then ; furthermore, when , then for , so conditions (i) and (iv) in Theorem 3.1 are verified.

Moreover, when , that is, , and for , we have similarly, when and for , we also have . Thus, condition (ii) in Theorem 3.1 is satisfied and the zero solution of system (3.14) is -uniformly practically stable.

It is easy to see that if we put two variables in one Lyapunov function, then the arguments to get the desired stability conclusions would be much more complicated and the imposed conditions would be more restrictive. Furthermore, we extend the uniformly practically stable results to the infinite delay systems, and it is easy to see that the criteria in [310] are limited to judge the practical stability of Example 3.4.


This work was supported by the National Natural Science Foundation of China (nos. 11101373 and 61074011), the Natural Science Foundation of Zhejiang Province of China (no. Y6100007), and Zhejiang Innovation Project (no. T200905).