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Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

It has been observed that the solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero (see, e.g., the recent paper [

One such type consists of impulsive differential equations which are important in simulation of processes with jump conditions (see, e.g., [

By a solution of (

A solution of (

We will establish dichotomous criteria that guarantee solutions of (

For each

One has

In the next section, we state four theorems to ensure that every solution of (

The main results of the paper are as follows.

Assume that the conditions (A)–(C) hold. Suppose further that there exists a positive integer

Assume that the conditions (A)–(C) hold. Suppose further that there exists a positive integer

Assume that the conditions (A)–(C) hold and that

Assume that the conditions (A)–(C) hold and that

Before giving proofs, we first illustrate our theorems by several examples.

Consider the equation

Consider the equation

Consider the equation

Note that the ordinary differential equation

has a nonnegative solution

To prove our theorems, we need the following lemmas.

Assume the following.

For

Suppose that conditions (A)–(C) hold and

If there exists some

If there exists some

First of all, we will prove that (a) is true. Without loss of generality, we may assume that

Next, we will prove that (b) is true. Without loss of generality, we may assume that

We may prove in similar manners the following statements.

If we replace the condition (a) in Lemma

If we replace the condition (b) in Lemma

Suppose that conditions (A)–(C) hold and

Without loss of generality, we may assume that

Suppose that

there exists

We now turn to the proof of Theorem

Suppose that (a) holds. Then we see that the conditions (

Suppose that (b) holds. Let

Next, we give the proof of Theorem

Suppose that (a) holds. Note that

Similarly, for

By induction, for each

so that

That is,

Suppose (b) holds. Without loss of generality, we may assume that

In particular,

Similarly, for

By induction, we know that

By Lemma

It follows that

We now give the proof of Theorem

Suppose that (a) holds. Note that

Similarly, for

By induction, for any

So

If (b) holds, let

Finally, we give the proof of Theorem

Suppose that (a) holds. We may easily see that the conditions (

In particular,

Similarly, we have for

In particular,

By induction, we obtain for any

In particular,

Similarly, we have for

In particular,

By induction, we obtain for any

Suppose that (b) holds. Let

This research is supported by the Natural Science Foundation of Guang Dong of China under Grant 9151008002000012. The authors would also like to thank the reviewers for their comments and corrections of their mistakes in the original version of this paper.