Abstract and Applied Analysis

Volume 2013 (2013), Article ID 454619, 12 pages

http://dx.doi.org/10.1155/2013/454619

Research Article

Antiperiodic Solutions for a Generalized High-Order -Laplacian Neutral Differential System with Delays in the Critical Case

Yongzhi Liao,¹ Tianwei Zhang,² and Yongkun Li³

¹School of Mathematics and Computer Science, Panzhihua University, Panzhihua, Sichuan 617000, China

²City College, Kunming University of Science and Technology, Kunming, Yunnan 650051, China

³Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 23 January 2013; Accepted 13 April 2013

Academic Editor: Anna Capietto

Copyright © 2013 Yongzhi Liao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By applying the method of coincidence degree, some criteria are established for the existence of antiperiodic solutions for a generalized high-order -Laplacian neutral differential system with delays , in the critical case . The results of this paper are completely new. Finally, an example is employed to illustrate our results.

1. Introduction

During the last twenty years, there have been quite a few results on the existence of periodic solutions for delay differential equations and neutral functional differential equations. We can see [1–7]. For example, the authors of [8–11] investigated the existence of periodic solutions for the following types of neutral functional differential equations:

respectively. But the condition of constant is required. For example, under the assumption , they obtain that , has a unique inverse defined by

and then

which was crucial to obtaining estimation of a priori bounds of periodic solutions in the noncritical case .

Under the critical case , the authors of [12–15] studied a first-order neutral differential equation

a Duffing differential equation of neutral type

a Rayleigh differential equation of neutral type

and a -Laplacian differential equation of neutral type

respectively.

In the past thirty years, there has been a great deal of work on the problem of the periodic solutions of high-order nonlinear differential equations, especially for the third-order and fourth-order differential equations which have been used to describe nonlinear oscillations [16–20], and fluid mechanical and nonlinear elastic mechanical phenomena [21–27]. In [28], Jin and Lu discussed the existence of periodic solutions of third-order -Laplacian equation with a deviating argument

Before continuing, by applying Mawhin’s continuation theorem of coincidence degree theory, the authors of [29] studied the existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

Arising from problems in applied sciences, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations as a special periodic solution and has been extensively studied by many authors during the past twenty years; see [30–44] and references therein. For example, antiperiodic trigonometric polynomials are important in the study of interpolation problems [45, 46], and antiperiodic wavelets are discussed in [47].

However, to the best of our knowledge, due to the neutral term and -Laplace operator term, the existence of antiperiodic solutions for (4)–(7) is very difficult to obtain by applying traditional researching methods. Therefore, to date, there are few papers to investigate the existence of antiperiodic solutions for (4)–(7).

Motivated by above statements, in this paper, we will apply the method of coincidence degree to study the existence of antiperiodic solutions for a generalized high-order -Laplacian neutral differential system with delays in the critical case

where , , , and , ; , , , and are -periodic functions; for any , is defined by ; are -periodic in their first arguments; , are constants; , , , and are nonnegative integers, , .

Throughout this paper, we will denote by the set of nonnegative integers and by the set of odd positive integers.

Let , , , , , , and in system (10), then system (10) is reformulated as

Furthermore, one can easily obtain the following.

(a) If ,

 then system (10) reduces to (4). (b) If ,

 then system (10) reduces to (5). (c) If ,

 then system (10) reduces to (6). (d) If ,

 then system (10) reduces to (7).

The main purpose of this paper is to establish sufficient conditions for the existence of -antiperiodic solutions to system (10) by using the method of coincidence degree.

The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, by using the method of coincidence degree, we establish sufficient conditions for the existence of -antiperiodic solutions to system (10). An illustrative example is given in Section 4.

2. Preliminaries

The following continuation theorem of coincidence degree is crucial in the arguments of our main results.

Lemma 1 (see [48]). Let be two Banach spaces; let be open bounded and symmetric with . Suppose that is a linear Fredholm operator of index zero with and is -compact. Further, one also assumes that

(H) , for all .

Then equation has at least one solution on .

Definition 2. Let be continuous. is said to be -antiperiodic on , if

We will adopt the following notations:

where is a -periodic function.

For the sake of convenience, we introduce the following assumptions.

There exist nonnegative constants , such that

 for any , . For all ,

In order to apply Lemma 1 to study the existence of antiperiodic solutions for system (10), we set

are two Banach spaces with the norms

respectively. Define

and two difference operators and as follows:

Then system (10) reduces to

where . Obviously, the existence of antiperiodic solutions to system (10) is equivalent to that of antiperiodic solutions to system (24). Thus, the problem of finding a -antiperiodic solution for system (10) reduces to finding one for system (24).

Define a linear operator by setting

and by setting

It is easy to see that

Thus, , and is a linear Fredholm operator of index zero.

Define the continuous projector and the averaging projector by

Hence and Ker . Denoting by the inverse of , we have

where

in which are decided by , where

, and are decided by , where

, and are decided by , where

, and are decided by , where

, and .

Clearly, and are continuous. Using the Arzela-Ascoli theorem, it is not difficult to show that are relatively compact for any open bounded set . Therefore, is -compact on for any open bounded set .

Lemma 3 (see [12]). Let , where and are coprime positive integers. Then

(1)if , is odd and is even, then

(2)if and is odd, then

(3)if and , then

Lemma 4 (see [13]). Suppose , for all . Then the following propositions are true.

(1)If , , where and are coprime positive integers with odd and even, then has a unique inverse satisfying . (2)If , , where and are coprime positive integers with odd, then has a unique inverse satisfying . (3)If , , then has a unique inverse satisfying .

Lemma 5 (see [12]). Let , where and are coprime positive integers. Then

(1)if , is odd and is even, then

(2)if and is odd, then

(3)if and , then

Lemma 6 (see [13]). Suppose for all . Then the following propositions are true.

(1)If , where and are coprime positive integers with odd and even, then has a unique inverse satisfying . (2)If , where and are coprime positive integers with odd, then has a unique inverse satisfying . (3)If , then has a unique inverse satisfying .

3. Main Result

In this section, we will study the existence of -antiperiodic solutions for system (10) in the critical case .

Theorem 7. Assume that - hold. Suppose further that , , where and are coprime positive integers with odd and even, then system (10) has at least one -antiperiodic solution, if one of the following conditions holds.

(1), where and are coprime positive integers with odd and even, and , where

(2), where and are coprime positive integers with odd, and , where

(3), and , where

 in which , and are constants defined by Lemma 3 or Lemma 5.

Proof. As the proof of other cases works almost exactly as the proof of case (1), we will prove case (1) only. Consider the operator equation

Then we have

where

Suppose that is an arbitrary -antiperiodic solution of system (45). Because is -antiperiodic, hence, we have

Then there exists a constant such that

Therefore, we have

for all . Combining the above two inequalities, we can get

Similar to (50), one can easily get

which yield

By a parallel argument to (47)–(52), we can also obtain

Since is -antiperiodic, similar to (47), there exists a constant such that . By a parallel argument to (50), we can obtain from (45), , (52), and (53) that

Namely,

Since is -antiperiodic, similar to (50), there exists a constant such that . By a parallel argument to (50), we can obtain from (45), , (52), and (53) that

Namely,

From (1) of Lemmas 4 and 6, one can obtain

That is,

From (54) and (55), we can get

With (57)–(62), we have

where

Let . Then we have from (63) that

in which

Since , (65) implies from (1) that there exists a positive constant such that ; that is,

which implies from (61) that there exist positive constants and such that

which implies from (57) and (59) that

Substituting (68), (69) into (52), (53), (54), and (55), we get

Let

Take

It is clear that satisfies all the requirements in Lemma 1 and condition is satisfied. In view of all the discussions above, we conclude from Lemma 1 that system (10) has at least one -antiperiodic solution. This completes the proof.

Similar to Theorem 7, we can easily obtain the following results.

Theorem 8. Assume that - hold. Suppose further that , where and are coprime positive integers with odd, then system (10) has at least one -antiperiodic solution, if one of the following conditions holds.

(1), where and are coprime positive integers with odd and even, and , where

(2), where and are coprime positive integers with odd, and , where

(3), and , where

 in which , and are constants defined by Lemma 3 or Lemma 5.

Theorem 9. Assume that - hold. Suppose further that , , then system (10) has at least one -antiperiodic solution, if one of the following conditions holds.

(1), where and are coprime positive integers with odd and even, and , where

(2), where and are coprime positive integers with odd, and , where

(3), and , where

 in which , and are constants defined by Lemma 3 or Lemma 5.

4. An Example

Example 1. Let , . Then the following third-order -Laplacian neutral differential system

has at least one -antiperiodic solution.

Proof. By calculation, , and . Therefore,

Hence,

which implies that case (1) in Theorem 7 holds. It is easy to verify that - hold and the result follows from Theorem 7. This completes the proof.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

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