`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 909387, 17 pageshttp://dx.doi.org/10.1155/2012/909387`
Research Article

The Existence and Uniqueness of Periodic Solutions for Some Nonlinear th-Order Differential Equations

1College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, China
2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

Received 15 December 2011; Accepted 19 March 2012

Academic Editor: Gabriel Turinici

Copyright © 2012 Qiyuan Zhou and Shuhua Gong. 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

The nonlinear th-order differential equations are considered. By using inequality techniques and coincidence degree theory, some criteria are obtained to guarantee the existence and uniqueness of -periodic solutions for the equations. The obtained results are also valid and new for the problem discussed in the previous literature. Moreover, two illustrative examples are provided to illustrate the effectiveness of our results.

1. Introduction

In applied science, some practical problems are associated with the periodic solutions for nonlinear high-order differential equations, such as nonlinear oscillations [1, 2], electronic theory [3], biological model, and other models [46]. In particular, during the past thirty years, there has been a great amount of work on the existence and uniqueness of periodic solutions for the th-order nonlinear differential equation where and are continuous functions, is -periodic with respect to , is -periodic in the first variable, and are constants. Many of these results can be found in [711] and the references cited therein. Among the known results, we find that the assumption is continuous, and there are positive constants and such that is employed, and it plays an important role in the proofs of these known results (see, e.g., [911]). Recently, under some spectral conditions of linear differential operator, Li [12, 13] discussed the existence and uniqueness of -periodic solutions of nonlinear differential equations.

However, to the best of our knowledge, there exist few results for the existence and uniqueness of periodic solutions of (1.1) without and the spectral conditions of linear differential operator. Thus, in this case, it is worth to study the problem of existence and uniqueness of periodic solutions of th-order nonlinear differential equation (1.1).

The purpose of this paper is to investigate the existence and uniqueness of -periodic solutions of (1.1). By using some inequality techniques and Mawhin’s continuation theorem, we establish some sufficient conditions for the existence and uniqueness of -periodic solutions of (1.1) when and the spectral conditions are avoided. Moreover, two illustrative examples are given in Section 4.

2. Preliminary Results

Let us introduce some notations. We will use to denote the empty set. For , we denote by the Banach space endowed with the norm where, for a function , we have that For , we will denote Now, let be a continuous function, -periodic with respect to the first variable, and consider the th-order differential equation

Lemma 2.1 (see [14]). Assume that the following conditions hold.(i)There exists such that, for each , one has that any possible -periodic solution of the problem satisfies the priori estimation .(ii)The continuous function defined by satisfies .Then, (2.5) has at least one -periodic solution such that .

From Lemma 2.2 in [15] and the proof of inequality (10) in [7, pp 3402], one obtains the following.

Lemma 2.2. Let . Suppose that there exists a constant such that then

Lemma 2.3. For any , one has that

Proof. Lemma 2.3 is a direct consequence of the Wirtinger inequality, and see [16, 17] for its proof.

By the same approach used in the proof of Lemma  3 of [7], we have the following.

Lemma 2.4. For any , one has that

Lemma 2.5. Let be an even number, , and Assume that one of the following conditions is satisfied: for, there exists a nonnegative constant such that where, then (1.1) has at most one -periodic solution.

Proof. Suppose that and are two -periodic solutions of (1.1). Set . Then, we obtain Integrating (2.15) from 0 to , it results that Therefore, in view of integral mean value theorem, it follows that there exists a constant such that Since is a strictly monotone function in , (2.17) implies that Then, from (2.9), we have Multiplying (2.15) by and then integrating it from 0 to , it follows that Now suppose that (or ) holds, and we will consider two cases as follows.
Case  i. If holds, (2.10) and (2.20) yield that which, together with (2.18), implies that Hence, (1.1) has at most one -periodic solution.Case  ii. If holds, using (2.9), (2.10), (2.19), and (2.20), we obtain that From (2.18) and , (2.23) yield that Therefore, (1.1) has at most one -periodic solution. The proof of Lemma 2.5 is now complete.

Similar to the proof of Lemma 2.5, one can prove the following result.

Lemma 2.6. Let be an odd number, , and Assume that one of the following conditions is satisfied: , there exists a nonnegative constant such that where , then (1.1) has at most one -periodic solution.

Lemma 2.7. Let be an even number, , and Assume that one of the following conditions is satisfied:for,there exists a nonnegative constant such that where, then (1.1) has at most one -periodic solution.

Proof. Multiplying (2.15) by and then integrating it from 0 to , yields that Now the proof proceeds in the same way as in Lemma 2.5.

Similar to the proof of Lemma 2.7, we can prove the following results.

Lemma 2.8. Let be an odd number, , and Assume that one of the following conditions is satisfied:, there exists a nonnegative constant such that where , then (1.1) has at most one -periodic solution.

3. Main Results

Theorem 3.1. Let be an even number and . Assume that one of the following conditions is satisfied: let hold, and there exists a nonnegative constant such that there exist nonnegative constants and such that holds, then (1.1) has a unique -periodic solution.

Proof. From Lemma 2.5, together with (or ), it is easy to see that (1.1) has at most one -periodic solution. Thus, to prove Theorem 3.1, it suffices to show that (1.1) has at least one -periodic solution. To do this, we shall use Lemma 2.1 with the nonlinearity given by
For , we consider the th-order differential equation Let us show that (i) in Lemma 2.1 is satisfied, which means that there exists such that any possible -periodic solution of (3.4) is such that Let and let be a possible -periodic solution of (3.4). In what follows, denotes a fixed constant independent of and . Integrating (3.4) from 0 to , it results that which together with (or ) implies that Hence, from (2.9), we have that In view of (2.10), (3.8) implies that It follows that
On the other hand, multiplying (3.4) by and integrating from 0 to , it follows that Now suppose that (or ) holds, and we will consider two cases as follows.
Case 1. If holds, using (2.10), (3.10), and (3.11), we have which imply that there exists a positive constant satisfying Case 2. If holds, using (2.9), (2.10), (3.10), and (3.11), we obtain which together with yield that (3.13) holds.
Using (3.9) and (3.13), it follows that there exists such that
Now, we shall estimate , multiplying (3.4) by and integrating from 0 to , we have
Using (2.10), (3.15), and (3.16), we have which imply that there exists a positive constant satisfying This implies the existence of a constant such that (3.5) holds.
Now, to show that (ii) in Lemma 2.1 is satisfied, it suffices to remark that Hence, from (or ) and , it results that . Then, using Lemma 2.1, it follows that (3.4) has at least one -periodic solution satisfying (3.5). This completes the proof.

Similar to the proof of Theorem 3.1, from Lemma 2.6, one can prove the following results.

Theorem 3.2. Let be an odd number and . Assume that one of the following conditions is satisfied: let hold, and there exists a nonnegative constant such thatthere exist nonnegative constants and such that holds, then (1.1) has a unique -periodic solution.

Theorem 3.3. Let be an even number and . Assume that one of the following conditions is satisfied: let hold, and there exists a nonnegative constant such that there exist nonnegative constants and such that holds, then (1.1) has a unique -periodic solution.

Proof. From Lemma 2.7, together with (or ), it is easy to see that (1.1) has at most one -periodic solution. Thus, to prove Theorem 3.3, it suffices to show that (1.1) has at least one -periodic solution.
Multiplying (3.4) by and integrating from 0 to , it follows that Then, from (3.24), by using similar arguments in proof of (3.15) and (3.18), we can obtain that there exists a constant such that In view of (3.4), (3.25) and (3.26) yield that which together with (2.11) and (3.25) implies the existence of a constant such that (3.5) holds.
Now the proof proceeds in the same way as in Theorem 3.1.

Similar to the proof of Theorem 3.3, from Lemma 2.8, we obtain the following.

Theorem 3.4. Let be an odd number and . Assume that one of the following conditions is satisfied: let hold, and there exists a nonnegative constant such that there exist nonnegative constants and such that holds, then (1.1) has a unique -periodic solution.

Remark 3.5. If and satisfies the following condition: there exist and such that, for any continuous -periodic function , we have Moreover, one of conditions holds. Then, by using the methods similarly to those used in Theorem 3.1, one may also establish the results similar to those in Theorems 3.13.4.

4. Examples and Remarks

Example 4.1. Let be three continuous, strictly positive, and -periodic functions, and let be continuous, -periodic, then the fourth-order differential equation has a unique -periodic solution. For the proof, it suffices to remark that the function satisfies where is sufficiently large. Hence, and satisfy , and the result follows from Theorem 3.1.

Example 4.2. Let be continuous, strictly positive, and -periodic, let be constants, and let be continuous, -periodic, then the five-order differential equations have a unique -periodic solution. For the proof, it suffices to remark that the function with , and satisfies . Hence, the result follows from Theorem 3.3.

Remark 4.3. Since in Examples 4.1 and 4.2 does not satisfy , the main results in [911] and the references therein cannot be applicable to (4.1)–(4.3) to obtain the existence and uniqueness of 2-periodic solutions. Moreover, all the results in this present paper avoid the spectral conditions in [12, 13]. This implies that the results of this paper are new, and they complement previously known results.

Acknowledgments

The authors would like to express their sincere appreciation to the anonymous referee for the valuable comments which have led to an improvement in the presentation of the paper. This work was supported by the Construct Program of the Key Discipline in Hunan Province (Mechanical Design and Theory), the Key Project of Chinese Ministry of Education (Grant no. 210 151), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of PR China (Grant no. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grants nos. 11C0916, 11C0915, 11C1186), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grants nos. Y6110436, Y12A010059), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).

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