#### Abstract

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.

#### 1. Introduction

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 [1] in which a third order nonlinear delay differential equation with damping is considered). Such a dichotomy may yield useful information in real problems (see, e.g., [2] in which implications of this dichotomy are applied to the deflection of an elastic beam). Thus it is of interest to see whether similar dichotomies occur in different types of functional differential equations.

One such type consists of impulsive differential equations which are important in simulation of processes with jump conditions (see, e.g., [3–22]). But papers devoted to the study of asymptotic behaviors of third-order equations with impulses are quite rare. For this reason, we study here the third-order nonlinear differential equation with impulses of the form where , such that ,

for Here and are real functions and are real numbers.

By a solution of (1.1), we mean a real function defined on such that

(i) for (ii) and are continuous on for and exist, and for any (iii) satisfies at each pointA solution of (1.1) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it is said to be oscillatory.

We will establish dichotomous criteria that guarantee solutions of (1.1) that are either oscillatory or zero convergent based on combinations of the following conditions.

(A) is positive and continuous on is continuous on for , and , where is positive and continuous on and is differentiable in such that for .(B)For each is continuous in and there exist positive numbers such that for and (C)One hasIn the next section, we state four theorems to ensure that every solution of (1.1) either oscillates or tends to zero. Examples will also be given. Then in Section 3, we prove several preparatory lemmas. In the final section, proofs of our main theorems will be given.

#### 2. Main Results

The main results of the paper are as follows.

Theorem 2.1. *Assume that the conditions (A)–(C) hold. Suppose further that there exists a positive integer such that for **
Then every solution of (1.1) either oscillates or tends to zero.*

Theorem 2.2. *Assume that the conditions (A)–(C) hold. Suppose further that there exists a positive integer such that for , **
Then every solution of (1.1) either oscillates or tends to zero.*

Theorem 2.3. *Assume that the conditions (A)–(C) hold and that for any . Suppose further that there exists a positive integer such that for
**
Then every solution of (1.1) either oscillates or tends to zero.*

Theorem 2.4. *Assume that the conditions (A)–(C) hold and that for any . Suppose further that , is bounded, that
**
Then every solution of (1.1) either oscillates or tends to zero.*

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

*Example 2.5. *Consider the equation
where for ; , , , . It is not difficult to see that conditions (A)–(C) are satisfied. Furthermore,
Thus by Theorem 2.1, every solution of (2.9) either oscillates or tends to zero.

*Example 2.6. *Consider the equation
where , for ; , , and
Here, we do not assume that is bounded, monotonic, or differential. It is not difficult to see that conditions (A)–(C) are satisfied. Furthermore,
Thus by Theorem 2.2, every solution of (2.11) either oscillates or tends to zero.

*Example 2.7. *Consider the equation
where for ; , , ; It is not difficult to see that conditions (A)–(C) are satisfied. Furthermore,
Thus, by Theorem 2.2, every solution of (2.14) either oscillates or tends to zero.

Note that the ordinary differential equation

has a nonnegative solution as . This example shows that impulses play an important role in oscillatory and asymptotic behaviors of equations under perturbing impulses.

#### 3. Preparatory Lemmas

To prove our theorems, we need the following lemmas.

Lemma 3.1 (Lakshmikantham et al. [3]). *Assume the following. *(*)** and is left-continuous at .*(*)**For and ,** **
where and are real constants. Then for ,
*

Lemma 3.2. *Suppose that conditions (A)–(C) hold and is a solution of (1.1). One has the following statements. *(a)*If there exists some such that and for , then there exists some such that for .*(b)*If there exists some such that and for , then there exists some such that for .*

*Proof. *First of all, we will prove that (a) is true. Without loss of generality, we may assume that and for . We assert that there exists some such that for . If this is not true, then for any we have . Since is increasing on intervals of the form , we see that for . Since is increasing on intervals of the form , we see that for
that is,
In particular,
Similarly, for , we have
By induction, we know that for
From condition (B), we have
Set . Then from (3.7) and (3.8), we see that for
It follows from Lemma 3.1 that
That is,
Note that , and the second equality of condition (C) holds. Thus we get for all sufficiently large . The relation leads to a contradiction. Thus, there exists some such that and . Since is increasing on intervals of the form for thus for , we have
Similarly, for ,
We can easily prove that, for any positive integer and
Therefore, for . Thus, (a) is true.

Next, we will prove that (b) is true. Without loss of generality, we may assume that and for We assert that there exists some such that for . If this is not true, then for any we have . Since is increasing on intervals of the form , we see that for . By , , we have that is nondecreasing on . For , we have

In particular,
Similarly, for , we have
By induction, we know that for
From condition (B), we have
Set . Then from (3.18) and (3.19), we see that for
It follows from Lemma 3.1 that
That is,
Note that , and the first equality of condition (C) holds. Thus we get for all sufficiently large . The relation leads to a contradiction. So there exists some such that and . Then
Since we see that is strictly monotonically increasing on for For , we have
In particular,
Similarly, for , we have
By induction, we have for . Thus, we know that for . The proof of Lemma 3.2 is complete.

*Remark 3.3. *We may prove in similar manners the following statements.(*)If we replace the condition (a) in Lemma 3.2 “ and for ” with “ and for ”, then there exists some such that for .(**)If we replace the condition (b) in Lemma 3.2 “ and for ” with “ and for ”, then there exists some such that for *

Lemma 3.4. *Suppose that conditions (A)–(C) hold and is a solution of (1.1) such that for where . Then there exists such that either (a) , for or (b) , for *

*Proof. *Without loss of generality, we may assume that for . By (1.1) and condition (A), we have for
We assert that for any . If this is not true, then there exists some such that , so . Since is decreasing on for , we see that for
In particular,
Similarly, for , we have
In particular,
By induction, for any for we have
Hence, for By Remark 3.3(), there exists such that for by Remark 3.3(), we get for which is contrary to for . Hence, for any since is decreasing on for , therefore for It follows that is strictly increasing on for . Furthermore, note that , we see that if for any , then for . If there exists some such that , then for . The proof of Lemma 3.4 is complete.

Lemma 3.5 (see [12]). *Suppose that is continuous at and , it is left-continuous at and exists for Further assume that *(*)**there exists such that for *(*)** is nonincreasing (resp., nondecreasing) on for *(*)** is convergent.**Then exists and (resp., ).*

#### 4. Proofs of Main Theorems

We now turn to the proof of Theorem 2.1. Without loss of generality, we may assume that . If (1.1) has a nonoscillatory solution , we first assume that for . By (1.1) and the condition (A), for , we get

From the condition (B), we know that

By Lemma 3.4, there exists a such that either (a) , for or (b) , for

Suppose that (a) holds. Then we see that the conditions () and () of Lemma 3.5 are satisfied. Furthermore, note that and . Then we have

Since we obtain for any

By (4.3) and (4.4), we know that the sequence is bounded. Thus there exists such that It follows from the condition (B) that

From (4.5) and the fact that is convergent, we know that is convergent. Therefore, the condition () of Lemma 3.5 is also satisfied. By Lemma 3.5, we know that We assert that If then there exists such that for any . Note further that so we obtain for Let for By (4.1) and (4.2), we have

where From (4.6), (4.7), and Lemma 3.1, we get for

It is easy to see from (2.2) and (4.8) that for sufficiently large This is contrary to for Thus that is,

Suppose that (b) holds. Let for Then for . By (1.1) and the condition (A), we get, for

From the conditions (A), (B) and , we know that

From (4.9), (4.10), and Lemma 3.1, we get, for

It is easy to see from (2.2) and (4.11) that for sufficiently large This is contrary to for and hence we obtain a contradiction. Thus in case (b) must be oscillatory. The proof of Theorem 2.1 is complete.

Next, we give the proof of Theorem 2.2. Without loss of generality, we may assume that . If (1.1) has an eventually positive solution for By (1.1) and conditions (A) and (B), we have that (4.1) and (4.2) hold. By Lemma 3.4, there exists a such that either (a) , for or (b) , for

Suppose that (a) holds. Note that and for and each , is decreasing on ; we have for

Similarly, for we have

By induction, for each , we have

so that is decreasing on We know that is convergent as Let Then We assert that If then there exists such that for . Since then Let for Then By (4.1) and (4.2), we have that (4.6) and (4.7) hold. From (4.6), (4.7), and Lemma 3.1, we get for

That is,

It is easy to see from (4.16) that the following inequality holds:

Note that ; it follows from integrating (4.17) from to and by using the condition (B) that

It is easy to see from (2.4) and (4.18) that for sufficiently large This is contrary to for Thus that is,

Suppose (b) holds. Without loss of generality, we may assume that Then we see that . Since is nondecreasing on , for , we have

In particular,

Similarly, for , we have

By induction, we know that

That is, for Note that and From the condition (B), we have Since we have Let by (4.1) and (4.2), we have, for that

Similar to the proof of (4.17), we obtain

Let for Then . By (4.24) and the condition (B), and noting that , we have for

By Lemma 3.1, we get

It follows that

In view of (4.27), we have, for ,

It is easy to see from (2.4) and (4.28) that This is contrary to for Thus in case (b) must be oscillatory. The proof of Theorem 2.2 is complete.

We now give the proof of Theorem 2.3. Without loss of generality, we may assume that . If (1.1) has an eventually positive solution, for . By Lemma 3.4, there exists a such that either (a) , , or (b) , , holds.

Suppose that (a) holds. Note that since for and each is decreasing on then for we have

Similarly, for we have

By induction, for any for we have

So is decreasing and bounded on we know that is convergent as Let then We assert that If then there exists such that for . Since then By (1.1) and condition (A), we have for

From condition (B), and noting that we have

Let Then for By (4.32) and (4.33), we have for that

From (4.34), (4.35), and Lemma 3.1, we get, for that

It is easy to see from (2.6) and (4.36) that for sufficiently large This is contrary to for Thus that is,

If (b) holds, let for We see that for . By (1.1) and the condition (A), we get for

From the conditions (A) and (B), we know that

From (4.37), (4.38), and Lemma 3.1, we get for

It is easy to see from (2.6) and (4.39) that for sufficiently large This is contrary to for Thus in case (b) must be oscillatory. The proof of Theorem 2.3 is complete.

Finally, we give the proof of Theorem 2.4. Without loss of generality, we may assume that . If (1.1) has an eventually positive solution, for . By Lemma 3.4, there exists a such that either (a) , , or (b) , , holds.

Suppose that (a) holds. We may easily see that the conditions (), () of Lemma 3.5 are satisfied. Furthermore, since , then there exists some , such that for

In particular,

Similarly, we have for

In particular,

By induction, we obtain for any

Since is bounded and (4.44) holds, we know that is bounded. Thus there exists such that It follows from the condition (B) that

By (4.45), we know that is convergent. Therefore, the condition () of Lemma 3.5 is also satisfied. By Lemma 3.5, we know that We assert that If then there exists such that for . Since we have Since , , there exists some such that for

In particular,

Similarly, we have for

In particular,

By induction, we obtain for any

By and the condition (B), we know that is bounded, and from (4.50), we see that is bounded. There then exists such that Therefore, we have

By (1.1) and the condition (A), we have that (4.1) holds. Integrating (4.1) from to , it follows from (4.51) and for that

Note that is convergent. Thus it is easy to see from (2.8) and (4.52) that for sufficiently large This is contrary to for Thus that is,

Suppose that (b) holds. Let for We see that for . Similar to the proof of (4.39), we also obtain

It is easy to see from (2.8) and (4.53) that for sufficiently large This is contrary to for . Thus in case (b) must be oscillatory. The proof of Theorem 2.4 is complete.

#### Acknowledgments

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.