Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 735758, 7 pages
http://dx.doi.org/10.1155/2015/735758

Research Article

Multiplicity of Periodic Solutions for Third-Order Nonlinear Differential Equations

Xuxin Yang,¹ Weibing Wang,² and Dingyang Lv¹

¹Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China
²Department of Mathematics, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

Received 29 November 2014; Revised 16 March 2015; Accepted 20 March 2015

Academic Editor: Alexander Klimenko

Copyright © 2015 Xuxin Yang 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.

We study the existence of periodic solutions for third-order nonlinear differential equations. The method of proof relies on Schauder’s fixed point theorem applied in a novel way, where the original equation is transformed into second-order integrodifferential equation through a linear integral operator. Finally, examples are presented to illustrate applications of the main results.

1. Introduction

Questions on the existence and the multiplicity of periodic solutions are important topics in qualitative analysis of differential equations. Much work related to periodic solutions for second-order differential equations has been done by using various theorems and methods of nonlinear functional analysis; see [1–10] and references therein. In this paper, we investigate existence of periodic solutions of the following differential equation:

where , is -periodic in , and . Third-order differential equations of the above type arise, for example, in various fields of agriculture, biology, economics, and physics [11–14]. Questions related to this class of differential equations have recently attracted considerable attention from the researcher community; see, for example, [15–19].

A naive idea in study of higher-order (in particular third-order) differential equations is to translate the equation into a first-order system of differential equations by defining . This method works well, if the task is to show the existence of periodic solutions. However, it does not obviously lead to existence proofs for positive periodic solutions, since the condition of positivity for the higher-order equation is different from the natural positivity condition () for the corresponding system.

Another frequently used approach is to transform the third-order equation into a corresponding integral equation and to establish the existence of positive periodic solutions based on a fixed point theorem in cones. In order to follow this path, one needs an explicit representation of Green’s function for corresponding ordinary equation; see [20, 21]. In [20], Agarwal gave the explicit Green function for the th-order and th-order differential equations. Futhermore, Anderson has studied Green’s function for a third-order boundary value problem in [21].

It should be noted that (1) includes many important models. For example, it arises in many fields of science and technology, such as physics, mechanics, and engineering. We refer the reader to [22–27] for recent results on such models.

The main purpose of this paper is to show the existence of periodic solutions of (1) by means of Schauder’s fixed point theorem. The method of proof used in this paper is based on a simple but novel idea, and it consists of two steps as follows.

(1)The first step is to transform the original equation into a second-order integrodifferential equation through a linear integral operator.(2)The second step is to apply Schauder’s fixed point theorem.

After the above steps, the existence of a single periodic solution for (1) has been established under suitable behavior of functions and on some closed set. In addition, some information on the location of a periodic solution is obtained, leading to results on the multiplicity of solutions. To our belief, neither this method nor similar results can be found in the literature.

The paper is organized as follows. In Section 2 we introduce a lemma, which is crucial in proving the main results. Section 3 is devoted to the existence results on solutions of (1). In Section 4 we give some examples to illustrate potential applications of the main results.

2. Preliminaries

Let with the norm ; then is a Banach space.

Let , , and consider the following two differential equations:

Lemma 1. Assume that . Then (2) has a unique -periodic solution satisfying

and (3) has a unique -periodic solution satisfying

where

Proof. We only consider (2). Let , be two -periodic solutions of (2). Then, is -periodic solution of the differential equation

Note that can be written as

From , , we obtain . Thus .
Next, we show that

By direct computation, we get

Hence, we have

Moreover, it is easy to check that .

Remark 2. If , then

Remark 3. Consider

Define two operators by

From Lemma 1, one can easily check that are compact, increasing operators for .

Remark 4. If is a constant function, then

for .
Next, we define two operators and on by

where is a constant. For any and .

Lemma 5. If satisfies the differential equation

then is a -periodic solution of (1).
If satisfies the differential equation

then is a -periodic solution of (1).

Proof. Note that

If , and

If ,   and

Hence

If

then

If satisfies (17), we have

Hence, is a -periodic solution of (1).
On the other hand, we have

If , then and

From the above, we see that if , then and

We have

If

then

If satisfies (18), we have

Hence, is a -periodic solution of (1).

Lemma 6 (see [28]). Let be a Banach space so that is closed and convex. Assume that is a completely continuous operator. Then has a fixed point in .

3. Main Results

The following theorems are the main results of this paper.

Theorem 7. Assume that there exist constants such that

and in ;  and

for any .
Then (1) has at least one -periodic solution with .

Proof. From , we obtain that there exists a constant with in . Consider the differential equation

From Lemma 1 we see that if is a solution of (34), then satisfies

where is composition of defined by and the operator is defined by

Set . For all , we have . Put ; then for . Thus

for any . Hence, for all ,

Using , we obtain that, for all ,

Since is an increasing operator, we obtain that, for ,

By Remark 4, we have

for ; that is, .
Also, from the facts that is completely continuous and is continuous it follows that is a continuous and compact map. By Lemma 6, has at least one fixed point in and is periodic solution of (1). The proof is complete.

Theorem 8. Assume that there exist constants such that

  and in ;  and

for any .
Then (1) has at least one -periodic solution with .

Proof. From , we obtain that there exists a constant with in . Consider the differential equation

From Lemma 1, if is a solution of (43), satisfies

where is composition of and defined as , and the operator is defined as

Set , where with in . We set ; then ; we have

for any . Hence, for all , we have

Using , we obtain that, for all ,

Since is an increasing operator, we obtain that, for ,

By Remark 4, we have

for ; that is, .
Also, from the facts that is completely continuous and is continuous, it follows that is a continuous and compact map. By Lemma 6, has at least one fixed point in . The proof is complete.

4. Some Examples

In this section, two examples are provided to highlight potential applications of the results obtained in the previous section.

Example 1. Consider the differential equation

Note that here

Letting ,

By Theorem 7, (51) has infinitely many 2-periodic solutions. One such solution is illustrated in Figure 1.

Figure 1: A numerical approximation to a -periodic solution to (51). The approximation is obtained by using the built-in  NDsolve  function of Mathematica 10.

Example 2. Consider the differential equation

We claim that (54) has at least two -periodic solutions. Note that here

Let , , , and . Then

By Theorems 7 and 8, (54) has at least two -periodic solutions . One such solution is illustrated in Figure 2.

Figure 2: A numerical approximation of a -periodic solution to (54). The approximation is obtained by using the built-in  NDsolve  function of Mathematica 10.

Conflict of Interests

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

Acknowledgments

This work is supported by the NNSF of China (11171085), Hunan Provincial Natural Science Foundation of China (14JJ7083), and a Key Project supported by Scientific Research Fund of Hunan Provincial Education Department (14A028) and supported by Scientific Research Fund of Hunan Provincial Education Department (14C0253).

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