#### Abstract

We study the oscillation and asymptotic behavior of third-order nonlinear delay differential equation with piecewise constant argument of the form . We establish several sufficient conditions which insure that any solution of this equation oscillates or converges to zero. Some examples are given to illustrate the importance of our results.

#### 1. Introduction

Let denote the greatest-integer function. Consider the following third-order nonlinear delay differential equation with piecewise constant argument: where are continuous on with , is continuously differentiable on with . We will show that every solution of (1.1) oscillates or converges to zero, provided appropriate conditions are imposed.

Throughout this paper, we assume that and that there exist functions and such that (i) is continuous on with ,(ii) is continuously differentiable and nondecreasing on , for , (iii), (iv) and is not identically zero in any subinterval of .

The delay functional differential equations provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon its past history. In recent years, the oscillation theory and asymptotic behavior of delay functional differential equations and their applications have been and still are receiving intensive attention. In fact, in the last few years several monographs and hundreds of research papers have been written, see, for example [1–4]. In particular case, determining oscillation criteria for second-order delay differential equations has received a great deal of attention, while the study of oscillation and asymptotic behavior of the third-order delay differential equations has received considerably less attention in the literature.

The delay differential equations with piecewise continuous arguments can be looked as a special kind of delay functional differential equations. Since the delays of such equations are discontinuous, it need to be investigated individually. As is shown in [5], the solutions of differential equations with piecewise continuous arguments are determined by a finite set of initial data, rather than by an initial function as in the case of general functional differential equations. Moreover, the strong interest in such equations is motivated by the fact that they represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equation.

In the last few decades, there has been increasing interest in obtaining sufficient conditions for the oscillatory of solutions of different classes of the first-order differential equations with piecewise constant arguments, see [6–9] and the references therein. It is found that the presence of piecewise constant arguments plays an important role in the oscillation of the solution. For instant, all solution of are oscillatory (see [6]). But the corresponding ordinary differential equation has nonoscillatory solution .

However, there are few results about the oscillation of higher order equations. As mentioned in [5, 10], there are reasons for investigating the higher order equations with piecewise constant arguments. For example, suppose a moving particle with time variable mass is subjected to a restoring controller which acts at sampled time , then the second law of motion asserts that

In [10], the authors study a slightly more general second-order delay differential equations of the form: Two sufficient conditions which insure that any solution of (1.3) oscillates are obtained.

However, as far as we know, there are not works studying the oscillation and asymptotic behavior of third-order delay differential equations with piecewise constant argument. Motivated by this fact, in the present paper, we will investigate the oscillatory and asymptotic behavior of a certain class of third-order equation (1.1) with damping. The main ideas we used here are based on the paper [3, 4, 10].

The rest of this paper is organized as follows. In Section 2 we give the definition of the solution of (1.1) and establish some lemmas which are useful in the proof of our main results. Section 3 is devoted to the presentation of several sufficient conditions for the oscillation of (1.1). In Section 4, several examples are given to illustrate the importance of our results.

#### 2. Definitions and Preliminary Lemmas

Similar to [11], we give the following definition.

*Definition 2.1. *A solution of (1.1) on is a function that satisfies the conditions:(i) is continuous on ,(ii) is differentiable at each point , with the possible exception of the points , where one-sided derivatives exist,(iii) Equation (1.1) is satisfied on each interval with .

Our attention is restricted to those solutions of (1.1) which exist on the half line and satisfy sup for any . We make a standing hypothesis that (1.1) does possess such solutions. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. Equation (1.1) itself is called oscillatory if all its solutions are oscillatory.

*Remark 2.2. *If is a solution of (1.1), then is a solution of the equation
where . It is easy to check that and , where with . Thus, concerning nonoscillatory solutions of (1.1), we can restrict our attention only to the positive ones.

For the sake of brevity, we denote the following two operators
Thus (1.1) can be written as

Similar to [12], we give the following definition.

*Definition 2.3. *Let be a solution of (1.1). We say that has property on , if for every .

It is worth pointing out here that if has property on , then by (1.1), is eventually nonpositive.

Define the functions

We assume that

To obtain our main results we need the following lemmas.

Lemma 2.4. * Suppose that
**
is nonoscillatory. If is a nonoscillatory solution of (1.1), then does not change sign eventually.*

*Proof. *Suppose that is a nonoscillatory solution of (1.1) on . Without loss of generality, we may assume that for . Let be a nonoscillatory solution of (2.7). We will firstly consider the case that is eventually negative, that is, there exists a constant such that for . By (1.1) and (2.7), it is easy to see that

Suppose to the contrary that has arbitrarily large zeros, then there exist consecutive zeros of , , and , such that and , . If there exists an integer such that
see Figure 1(a), then integrating (2.8) we find

Note that , it follows from that
which is contrary to (2.10).

If there exists an integer such that , see Figure 1(b), then integrating (2.8) directly from to , we also get a contradiction.

Next we consider the case that is eventually positive. Let be the constant such that for . It follows from (1.1) and (2.7) that,
where with . Suppose to the contrary that has arbitrarily large zeros, then there exist consecutive zeros of , , and , such that and . Using the argument as above, we also arrive at a contradiction. Thus the proof is complete.

**(a)**

**(b)**

*Remark 2.5. *Lemma 2.4 shows that, the oscillatory or asymptotic behavior of (1.1) is linked to nonoscillation of the second-order homogeneous equation (2.7). The source of this interesting phenomenon can be explained as follows.

In (1.1), if we let , then verifies the following equation:
Let be any solution of (1.1). If is nonoscillatory, then and thus is eventually of one sign (see (2.13)). Hence, by the comparison method (as was shown by the proof of Lemma 2.4), the assumption that (2.7) is nonoscillatory guarantees that
is nonoscillatory. This fact means that, under the hypotheses of Lemma 2.4, any solution of (1.1) is either oscillatory or is monotone.

Lemma 2.6. *Suppose that assumption (2.6) is satisfied and that is a nonoscillatory solution of (1.1) such that for every . Then has property on for some . *

*Proof. *Assume without loss of generality that for . We assert firstly that for any integer ,. If this is not true, then there exists an integer such that . By (1.1), we obtain for , that

This implies that , and hence
Using induction, we obtain that . Thus
Integrating this inequality from to , we find
Letting , it follows from (2.18) and the continuity of that
Consequently,
This contradicts that . Therefore, for any integer . By the continuity and monotonicity of , we get that , . Finally, since , we conclude that is eventually positive.

This completes the proof.

Lemma 2.7. *Let be a solution of (1.1) such that , where is a constant. If
**
then for any arbitrary constant , there exists an integer such that for any , we have
*

*Proof. *Let us assume that is eventually positive. The case when is eventually negative can be similarly dealt with. For any constant , it follows from (2.21) and that, there exists an integer such that for . By (1.1), we have
where . Since and is nondecreasing, it follows from (2.23) that

Integrating (2.24) from to , where , we have

If there exists an integer such that , then (2.22) follows from (2.25) directly. Otherwise, there exist an integer such that
Using (2.25) and the continuity of , we have that
Thus we conclude that (2.22) holds for any .

#### 3. Main Results

In this section we present some sufficient conditions which guarantee that every solution of (1.1) oscillates or converges to zero. For convenience, we let

Theorem 3.1. *Suppose that*(i)*(2.7) is nonoscillatory,*(ii)*(2.5) and (2.6) are satisfied, and for every ,
*(iii)*one of the following two conditions holds:
or
**Then any solution of (1.1) is oscillatory or satisfies as . *

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for . By Lemma 2.4, there exists a constant such that one of the following cases holds:

We will firstly show that Case 1 is impossible. Indeed, if this case holds, then it follows from Lemma 2.6 that has property on for some . We define
Clearly, , . It follows from (1.1) that
where . Noting that
we get from (3.8) that
Denoted by
then we get from (3.10) that
Integrating (3.12), we have
Using the increasing property of , it follows that
Thus
where . For fixed , by letting , we get from (3.2) that for large integer , which is contrary to the fact that .

Therefore, we only need to consider Case 2: , , . We assert that
Suppose to the contrary that . For any arbitrary constant , it follows from Lemma 2.7 that, there exists an integer such that () and
where .

Now assume that (3.3) holds. Letting in (3.17), we have
where is a constant. By this inequality and (3.3), there exists a negative constant and a positive constant such that for . Integrating from to twice, we have
Thus we get from (2.5) that for large , a contradiction.

Next, assume that (3.4) holds. We distinguish the following three subcases: (i) for all large , (ii) for all large , (iii) changes sign for arbitrarily large .

Case is equivalent to that for all large . Since , we conclude that is eventually negative, a contradiction.

In case , we get from (2.22) that
where is a constant such that on . Letting , we obtain that
Integrating (3.21) from to , using , it turns out that

Integrating again from to , we find
By the condition (3.4), we have that for all large . This is a contradiction.

Finally, in case , we let be the sequence of zeros of such that . By choosing in (2.22), we get
Let , it follows that

Integrating this inequality twice yields that
which also leads to a contradiction.

Therefore, in Case 2 we conclude that (3.16) holds. This completes the proof of the theorem.

From the conclusion of Theorem 3.1, we get the following corollary.

Corollary 3.2. *Suppose that (2.5), (2.6) hold and that (2.7) is nonoscillatory. Assume further that for large . If , then any solution of (1.1) is oscillatory or satisfies as . *

*Proof. *Since , it follows that
Thus (3.3) is satisfied.

On the other hand, with the condition we get (3.2) immediately. Consequently, the expected conclusion follows from Theorem 3.1 directly.

We next consider the following equation which is different from (1.1): or the more general equation with where is a quotient of odd integers. We note that the result of Theorem 3.1 is not applicable. In what follows we give an oscillation criteria for (3.29).

Let Suppose that and not identically zero in any subinterval of for any . Moreover, we need the following conditions: or

Theorem 3.3. * Suppose that*(i)*(2.7) is nonoscillatory,*(ii)*(2.5) and (2.6) are satisfied, and for every ,
*(iii)*for any , one of (3.32) and (3.33) is satisfied.**Then any solution of (3.29) is oscillatory or satisfies as . *

*Proof. *From the proof of Theorem 3.1, it suffices to verify in Case 2 that there exists a positive constant such that
for any , where is the integer such that , .

By (3.29), we obtain that
, where . If , then it follows from this inequality that
while , we have
Let if and if , it turns out for the both cases that
The rest of the proof is exactly the same as in Theorem 3.1 and hence is omitted.

The following result follows from Theorem 3.3 directly.

Corollary 3.4. *Suppose that (2.5), (2.6) hold and that (2.7) is nonoscillatory. Assume further that for large . If , then any solution of (3.29) is oscillatory or satisfies as . *

*Remark 3.5. * In the literature dealing with the third-order delay differential equation, the Riccati transformation , where is the delay and is a differentiable positive function, is used widely, see [3, 4] and the references therein. However, in our paper we find that the transformation (see (3.7)) plays the same role as the more general one
In fact, if we replace (3.7) by (3.40) in the proof of Theorem 3.1, then it yields that
Similarly to Theorem 3.1, we need the following condition (instead of (3.2))
to obtain a contradiction. Noting that
we conclude that (3.42) is equivalent to (3.2).

Therefore, we cannot get a more general result by using the “more general” transformation (3.40). This is different from the theory of function differential equation with the continuous delay.

#### 4. Examples

In this section, we give several examples to illustrate our main results. For the convenience of readers, let us firstly recall the famous lemma of Kneser [13]. Consider the following second-order ordinary differential equation: where is a locally integrable function of . Kneser [13] shows that (4.1) is nonoscillatory if and is oscillatory if , where is an any arbitrary positive constant.

*Example 4.1. *Consider the third-order delay differential equation with piecewise constant argument
where is a quotient of odd integers. Clearly, . Let , , then . By the results of [13], is nonoscillatory. A simple calculation shows that for . Therefore, it is easy to see from Corollary 3.4 that any solution of (4.2) is either oscillatory or converges to zero.

*Example 4.2. * Consider the third-order delay differential equation with piecewise constant argument
where . Let , then . It is easy to see that the Euler equation is nonoscillatory, hence is nonoscillatory. A simple calculation leads to
Thus
as .

On the other hand,
which yields that
where are positive constants. Hence

Obviously, the other conditions of Theorem 3.1 are also satisfied. Hence we conclude that any solution of (4.3) is either oscillatory or converges to zero.

*Example 4.3. * Consider the third-order delay differential equation with piecewise constant argument
Here . Let , then . Since the equation can be reduced to , we conclude that the former is nonoscillatory. It is easy to see that and . Therefore, By Corollary 3.4, any solution of (4.9) is either oscillatory or converges to zero.

*Example 4.4. *Consider the third-order delay differential equation with piecewise constant argument
where is a quotient of odd integers is a integrable function such that
Let , . We will show that
is nonoscillatory. We introduce the change of variables:
which transforms (4.12) into
Note that
and that
we obtain that
Thus by Kneser [13], it follows that (4.14) (and hence (4.12)) is nonoscillatory.

Let , then it is easy to find that . Therefore, by Corollary 3.4, any solution of (4.10) is either oscillatory or converges to zero.

#### Acknowledgments

The authors would like to thank the referee very much for his valuable comments and suggestions. Liang was supported by the NSF of China (no. 11201086) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant 2012LYM_0087).