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 equationNote that can be written as From , , we obtain . Thus .
Next, we show thatBy direct computation, we get Hence, we have Moreover, it is easy to check that .
Remark 2. If , then
Remark 3. ConsiderDefine 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 equationthen is a -periodic solution of (1).
If satisfies the differential equationthen is a -periodic solution of (1).
Proof. Note thatIf , andIf , andHenceIfthen If satisfies (17), we have Hence, is a -periodic solution of (1).
On the other hand, we haveIf , then andFrom the above, we see that if , then andWe haveIfthen 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 equationFrom 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 ; andfor any .
Then (1) has at least one -periodic solution with .
Proof. From , we obtain that there exists a constant with in . Consider the differential equationFrom 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 equationNote that hereLetting , By Theorem 7, (51) has infinitely many 2-periodic solutions. One such solution is illustrated in Figure 1.
Example 2. Consider the differential equationWe claim that (54) has at least two -periodic solutions. Note that hereLet , , , and . Then By Theorems 7 and 8, (54) has at least two -periodic solutions . One such solution is illustrated in Figure 2.
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).