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We establish a new comparison principle for impulsive differential systems with time delay. Then, using this comparison principle, we obtain some sufficient conditions for several stabilities of impulsive delay differential equations. Finally, we present an example to show the effectiveness of our results.

The impulsive functional differential systems provide very important mathematical models for many real phenomena and processes in the field of natural sciences and technology [

In view of the importance of comparison principles in the qualitative analysis for differential equations, in this paper we establish a new comparison principle for impulsive delay differential systems. As an application, we use it to deal with the stability of impulsive functional differential equations.

The rest of this paper is organized as follows. In Section

Let

Consider the following impulsive functional differential equations:

Define

In this paper, we suppose that there exists a unique solution of system (

We now give some useful notations and definitions that will be used in the sequel.

A function

Let

Assume

stable, if, for any

attractive, if, for any

asymptotically stable if

exponentially stable; assume

In the proof of our main results we will use the following lemma.

Let

In this section, we will establish a general comparison principle for the impulsive delay differential system (

Assume that

Since

Assume that the conditions in Lemma

Then

Let

For simplicity, let

First, we will prove

If (

Consequently, condition (i) yields

When

Next, we will prove

By the above proof, it easily follows that

If (

This implies that

Consequently, condition (i) yields

By induction, we can obtain

This means that

If

Next, we give some special cases of Theorem

Assume that

In particular, let

If

Assume that

Then

From Corollary

Next, we will apply the comparison result to establish some stability criteria of system (

Assume that the conditions in Theorem

We establish asymptotical stability. First, we prove that the trivial solution of system (

Now, we prove that the trivial solution of system (

For any given

Assume that the conditions in Theorem

Since the trivial solution of (

In this section, we will give an example to illustrate the effectiveness of our results.

Consider the following impulsive delay differential equations:

The trivial solution of system (

Choose

If

In [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by National Science Foundation of China under Grant (no. 61201430) and Scientific and Technological Program of Huangdao District (no. 2014-1-28).