Existence and Uniqueness of Periodic Solutions for Second Order Differential Equations

Abstract

We study some second order ordinary differential equations. We establish the existence and uniqueness in some appropriate function space. By using Schauder’s fixed-point theorem, new results on the existence and uniqueness of periodic solutions are obtained.

1. Introduction

In this paper, we are concerned with the existence and uniqueness of periodic solution for the nonlinear equation where is continuous, periodic in with period , and with respect to .

Because of wide interests in physics and engineering, periodic solutions of second order differential equations have been investigated by many authors. We refer the reader to [1–7] and the references cited therein.

The purpose of this paper is to study the existence and uniqueness of periodic solution in some appropriate function space . To be precise, we first derive a result of existence and uniqueness, when the nonlinearity is a function with respect to . Then, a similar result concerning the existence of periodic solutions is obtained, when the nonlinearity is not a function.

Throughout this paper we use the following assumptions.(A1)There exist two continuous functions and such that, for all ,  where means and .(A2)There exists a positive constant such that, for all ,

We now present our main results of this paper.

Theorem 1. Let assumptions (A1) and (A2) hold. Then (1) has a unique -periodic solution.

Remark 2. We point out that the condition in (A1) is necessary. Let us take the following equation, for example: where is a constant and . The equation has no periodic solution.

Remark 3. Consider the following example: We can easily check that assumptions (A1) and (A2) hold. By Theorem 1, the equation has a unique -periodic solution.

We also consider the case when the right-hand side of the equation is only continuous. In this case, we establish the existence of periodic solutions for the differential equation where , are continuous and periodic in with period . We assume the following.(A3)There exist continuous functions and and a positive constant such that  for all .

Then, we have the following result.

Theorem 4. Let (A3) hold. Then (6) has at least one -periodic solution.

Remark 5. Consider the following example: By Theorem 4, the equation has at least one -periodic solution.

To establish the main results, we introduce some appropriate function space. By using Schauder’s fixed-point theorem, the existence and uniqueness of periodic solutions are obtained. We know from the anonymous referee that the proof of this paper can be simplified, by using the results in [8]. The proof of this paper can be seen as an application of Schauder’s fixed-point theorem.

2. Preliminary

In this section, We first introduce the function space, in which we will obtain the periodic solution for the problem. Then, some preliminary lemma is introduced, which is valuable for the proof of our main results.

Let be the space of continuously differentiable -periodic functions, with norm given by It is well known that is a Banach space.

We now introduce the following lemma.

Lemma 6. and are -periodic functions. If , for all , then the equation has a unique -periodic solution .

Proof. Assume there exists a -periodic solution , . Since , is not a constant. Otherwise we will get a contradiction by substituting into (10). Then we claim that there exist and , , such that Now we prove it.
Case 1. has zero points. Assume that ; then, we have because the initial value problem of (10) has a unique solution. If , then we let ; if , by the periodic condition, there exists , , such that So . Also by the periodic condition there exists , , such that So we can find a such that . Thus, under Case 1 we verify (11) holds.
Case 2. has no zero point. Then has a constant sign. Suppose (if not we can consider the -periodic solution , . Since is periodic and is not a constant, there exists such that , . So we can find a , , . Then we prove that (11) holds.
Multiplying both sides of (10) by and integrating between and we get which leads to a contradiction. This completes the proof of Lemma 6.

Remark 7. We can also prove the following. If , , , a.e. , then the following periodic boundary value problem, also has a unique solution .

3. Proof of the Main Results

Proof of Theorem 1. First prove the uniqueness. Assume that and are two -periodic solutions of (1). Setting we get From assumption (A1) we know Hence, by Lemma 6, .
We next prove the existence. Rewrite (1) in the following form:
From Lemma 6, for each , the equation has a unique -periodic solution because the corresponding homogeneous equation only has trivial -periodic solution .
We define operator for given ,   is the unique -periodic solution of (20). Then the existence of the -periodic solution is equivalent to the existence of fixed point of in the space . We will prove that is continuous and compact, and is a bounded subset of .
Proof of Continuity. For any convergent sequence , satisfying , let . Then We claim that is a bounded sequence in . If not, we can find a subsequence of (for convenience we also use the same notations) such that . Let . Then , , So . Since is bounded and is bounded and equicontinuous sequence of functions. Furthermore, by is bounded and equicontinuous sequence of functions. By Ascoli-Arzelà Theorem, and contain a uniformly convergent subsequence, respectively, (for convenience we also use the same notations) such that where the notation “” means uniform convergence. Obviously, , . From (23) and (24), we obtain Let . From (25) and (27), we get Hence By Lemma 6, we have , which is in contradiction with , so is a bounded sequence. Then, by (22), we know is bounded, so and are bounded and equicontinuous sequences of functions. By Ascoli-Arzelà Theorem, and contain a uniformly convergent subsequence, respectively, (for convenience we also use the same notations) such that We know When , from (31), we obtain By the uniqueness we know ;  thus, operator is continuous.
Proof of Compactness. For each bounded set , we claim that is bounded in . If not, by an analogous manner as above we will reach a contradiction. For every , is defined by (20). Since , , , and are all bounded, then . Proceeding as proof of continuity we conclude that and are bounded and equicontinuous sequences of functions. By the Ascoli-Arzelà Theorem, is a compact operator.
We claim that is bounded in . If not, there exist , , such that . Let . Then (22) holds. Take . Then , , and (23), (24), (25), and (27) hold. By the above proof we know and are bounded and equicontinuous sequences of functions and contain a uniformly convergent subsequence, respectively, (also use the same notations) such that Sequences and are bounded sequences in . By the weakly sequential compactness of space, both of them have a weakly convergent subsequence (also use the same notations) such that in , where “” means weak convergence. Obviously,
When , from (25) and (27), for a.e. , we have Then
By Remark 3, we obtain , which is in contradiction with . Then there exists a constant such that , .
Assume . By Schauder’s fixed-point theorem, has at least one fixed point. This completes the proof of Theorem 1.

Proof of Theorem 4. For each , from Lemma 6, we know the equation has a unique -periodic solution because the corresponding homogeneous equation only has trivial -periodic solution .
We define , for each given    is the unique -periodic solution of (38). Hence the existence of the periodic solutions is equivalent to the existence of fixed points of in the Banach space .
Proceeding as the proof of Theorem 1 we can prove that is a compact continuous operator and is a bounded subset of .
Then there exists a constant such that , for all . Let . By Schauder’s fixed-point theorem, has at least one fixed point. This completes the proof of Theorem 4.

4. Another Simple Proof

Actually, by the anonymous referee, we know that the proof of the theorem can be much simplified if we use the theorem in [8]. In fact, for equation where is continuous and -periodic in , Theorem 2.2 of [8] implies the following propositions.

Lemma 8. Let there exist continuous and -periodic in the first argument functions such that are satisfied on , where and are continuous -periodic functions. Let, moreover, for any continuous -periodic functions , satisfying the equation have no nontrivial -periodic solution. Then equation has at least one -periodic solution.

Lemma 9. Let the function in the last arguments have continuous partial derivatives, satisfying where are continuous -periodic functions. Let, moreover, for any -periodic , satisfying (41), (42) has no nontrivial -periodic solution. Then, the equation has a unique -periodic solution.

On the basis of these theorems, we can prove the theorem by checking the conditions in the previous lemma. By Lemma 6, the conditions above can be easily proved.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (Grant no. 11301209). The author would like to thank the anonymous referee for the valuable comments and suggestions on the paper, especially for the considerations in Section 4. The author would like to thank Professor Yong Li for his helpful instruction and valuable suggestions.