Journal of Function Spaces
Volume 2018 (2018), Article ID 8254758, 9 pages
https://doi.org/10.1155/2018/8254758

Research Article

A New Approach to the Existence of Quasiperiodic Solutions for Second-Order Asymmetric -Laplacian Differential Equations

Xiaoming Wang¹ and Lixia Wang²

¹School of Mathematics & Computer Science, Shangrao Normal University, Shangrao 334001, China
²School of Sciences, Tianjin Chengjian University, Tianjin 300384, China

Correspondence should be addressed to Xiaoming Wang; wxmsuda03@163.com

Received 24 April 2018; Accepted 17 May 2018; Published 19 June 2018

Academic Editor: Liguang Wang

Copyright © 2018 Xiaoming Wang and Lixia Wang. 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.

For and , we propose a new estimate approach to study the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order asymmetric -Laplacian differential equations where and are two positive constants satisfying with , is a continuous function, -periodic in the first argument and continuously differentiable in the second one, , , and . Using the Aubry-Mather theorem given by Pei, we obtain the existence of Aubry-Mather sets and quasiperiodic solutions under some reasonable conditions. Particularly, the advantage of our approach is that it not only gives a simpler estimation procedure, but also weakens the smoothness assumption on the function in the existing literature.

1. Introduction

In recent years, all kinds of nonlinear dynamic behavior, such as the existence of positive solutions [1–16] and sign-changing solutions [17, 18], the existence and uniqueness of solutions [19–25], the existence and multiplicity results [26–30], and the existence of unbounded solutions[31, 32], have been widely investigated for some nonlinear ordinary differential equations and partial differential equations due to the application in many fields such as physics, mechanics, and the engineering technique fields. In the present paper, we deal with the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order differential equations with a -Laplacian and an asymmetric nonlinear term

where , , and and are positive constants satisfying

where is a continuous function, -periodic in the first argument and continuously differentiable in the second one, where . Since the pioneering works of Aubry [33] and Mather [34], the existence of Aubry-Mather sets and quasiperiodic solutions for a variety of differential equations, such as Hamiltonian systems [35–41], and reversible systems [42–44] had been widely investigated due to the application in many fields such as one-dimensional crystal model of solid state physics, differential geometry, and dynamical systems (see [45, 46]).

If , then and (1) reduces to the following piecewise linear equation:

and (2) becomes

The first result is due to Capietto and Liu [38], who proved that the existence of Aubry-Mather sets and quasiperiodic solutions of (3) for some in (4), provided that , , and the perturbation term satisfies some growth conditions. Recently, this result was extended to a much weaker smoothness nonlinearity . In [41], by using the Aubry-Mather theorem generalized by Pei [37], the present author [41] studied the existence of Aubry-Mather sets and quasiperiodic solutions of (3), under the condition that in (4) and can be allowed to be either a bounded function or an unbounded function, which differs from above existing results.

In [39], (3) has been generalized to the following -Laplacian-like nonlinear differential equation:

where , and are positive constants satisfying (2) with , , and is a -periodic function. They considered the existence of Aubry-Mather sets and quasiperiodic solutions of (5) when satisfies some further approximate properties at infinity. We notice that in [39], to overcome the barriers of weak smoothness, they made use of exchange of the role of time and angle variables skills and showed the existence of Aubry-Mather sets and quasiperiodic solutions by employing a version of Aubry-Mather theorem obtained by Pei [37]. Moreover, the results in [39] need the smoothness requirement of the perturbation function at least to smooth in .

Now a natural question to ask is whether the smoothness of the function in (5) is further reduced; we can also obtain the same results as [13]. In this paper, we will deal with this interesting problem and answer this question in the form of Theorem 1 with more general case (1) than that of (5). Because of the presence of weak smoothness nonlinearity, the methods of seeking the existence of Aubry-Mather sets and quasiperiodic solutions for problems as [38, 39] do not seem to be applicable to (1). This phenomenon provokes some mathematical difficulties, which make the study of (1) particularly interesting. Our approach here is mainly based on the direct proof of the Poincaré map of the transformed system satisfying monotone twist property and is developed from the present author (see the recent papers [41, 44]) but is more subtle than the ones in [38–40]. More efforts have to be made to estimate the monotone twist property for the Poincaré map of the transformed system, but the procedure is a little simpler than those in [38–40]. One important advantage of our approach is that it does not require any high smoothness assumptions on function . Our results improve and generalize some results of the previous studies [39, 41] to some extent.

The main result of this paper is the following theorem.

Theorem 1. Suppose that (2) holds. Moreover, satisfies the following conditions:
The limit is

There exist constants , such that

Then there exists , such that, for any , (1) possesses an Aubry-Mather type solution with rotation number ; that is,
(i) if is rational and , the solutions , are mutually unlinked periodic solutions of period ;
(ii) if is irrational, the solution is either a usual quasiperiodic solution or a generalized one.

Remark 2. A solution is called generalized quasiperiodic one if the closed set

is Denjoy’s minimal set (see its definition in [47]).

Remark 3. Using the rule of L’Hospital to condition , it can easily be seen that

Remark 4. We noticed that the perturbations and in [39] need to be bounded. But from and of this paper, it is easy to verify that the perturbation can be either a bounded function or an unbounded function. For example, we can set to be a bounded function or an unbounded function for when and in Theorem 1. Moreover, positive constant satisfying in [39] has been extended to the case in this paper. Thus, our situation is more general than the results obtained in [39] for .

Remark 5. If , let us point out that the results in Theorem 1 have covered the conclusions obtained by Wang [41]. Besides, the estimation process in this paper is much more meticulous than that in [41] since the -Laplacian of a function , with , is no longer linear. Therefore, the results obtained in this paper are natural generalizations and refinements of the results obtained in [41].

The main idea of our proof is acquired from [39, 41]. The proof of Theorem 1 is based on an Aubry-Mather theorem due to Pei [37]. The rest of this manuscript is as follows. In Section 2, we introduce some action-angle variables transformation to transform system (1) into an equivalent integral Hamiltonian system and then present some growth properties on the corresponding action and angle variables functions. In Section 3, we provide some crucial estimates by some lemmas which say that the Poincaré mapping of the new system is monotone twist around the infinity. At last, Section 4 gives the proof of Theorem 1 by using Pei’s Aubry-Mather theorem [37].

2. Preliminaries

2.1. The Action and Angle Variables

Let be the solution of

satisfying the initial condition Then it follows from [48] that is a -periodic odd function with , for , and for Moreover, for and can be implicitly given by

Introducing a new variable , then (9) is equivalent to the planar system

where is the conjugate exponent of Letting be the unique solution of (11) satisfying , then the functions and are much similar to cosine and sine. It follows from [49] that and are -periodic, and for , iff , and iff Moreover, and , and

Now we consider (1). Set in (1); then (1) can be rewritten as a planar system

where is the conjugate exponent of .

Lemma 6. For and for any , , the solution

of (12) satisfying the initial condition is unique and exists on the whole -axis.

Proof. The proof of uniqueness can be established similarly to the proof of Proposition 2 in [50]; the global existence result can be acquired similarly to Lemma 3.1 in [51].

Let be the solution of the following homogeneous system:

Then, by using (2) and direct computation, one obtains the following.

Lemma 7. (i) Both and are -periodic functions, and can be given by

(ii) and .
(iii)

Now we introduce an action-angle variables transformation by the mapping , where defined by the formula

where is a constant. This transformation is said to be a generalized symplectic transformation because its Jacobian is equal to .

2.2. Some Properties on Action and Angle Variables Functions

Under the transformation and using Lemma 7 (iii), (12) is changed into

where

We notice that the relation between (17) and (12) is that if are the solutions of (17) with the initial value condition , then

and

are the solutions of (12) with initial data By Lemma 6, (17) has a unique solution for and . Moreover, this solution has continuous derivatives with respect to initial data and .

For notional convenience, hereinafter, we write instead of , , respectively.

Firstly, by some simple calculations, we have the following.

Lemma 8. (i) , .
(ii)

Now we are concerned with the growth estimates with regard to and .

Lemma 9. The limit

holds uniformly on .

Proof. In view of and (16), there exist constants , such that

Then, by the Gronwall inequality, one has

for all
So, by (22), as uniformly for .

According to (22), it is easy to see the following.

Corollary 10. and , there exist constants and , such that

when .

Lemma 11. and , there exists constant , such that

if .

Proof. Since holds, then, for every , there exists , such that

if and . Hence,

Thus, by using action-angle variables transformation (16) and Lemma 9, there exists such that if .
For if , we assume that , where , and then

So, by (16), Lemma 8 (ii), and Lemma 9, there exists a constant , such that if .
If we choose , then implies
Exploiting the same arguments, one can show that the inequality on the right side of (i) holds.

3. Twist Property and Proof of Theorem 1

Let the Poincaré mapping of equation (17) be

In order to apply the Aubry-Mather theorem developed by Pei [37], we only need to show that the Poincaré mapping is a monotone twist map around the infinity; that is, it is enough to show if In the following we are going to give its detailed proofs by some lemmas.

Similarly, for notional convenience, hereinafter, we also write instead of , , respectively.

Lemma 12. The following convergences hold uniformly on :

(i), as .(ii), as .(iii); ; , as .

Proof. If and hold, then to each there corresponds a positive number , such that

and

when and , where is a constant given in (16).
Denote , .
(i) By action-angle variables transformation (16) and Lemma 8 (ii) and , we have

Then, given , choosing so that , by using Corollary 10, provided

we have

Since is arbitrary, the proof of (i) is complete.
For (ii), observe that , and, combining (16) and Lemma 8 (ii), one has

Then, given , choosing so that , by using Corollary 10, provided

we have

where Since is arbitrary, (ii) is proved.
(iii) Set , , , In view of (16), Lemma 7 (ii), and , we can get

Then, given , choosing so that , by using Corollary 10, provided

we have

Since is arbitrary, the proof of (iii) is finished.

, set

As a result of Lemma 9, Corollary 10, and Lemma 12, we have the following.

Lemma 13. , the following conclusions hold:

(i), as .(ii), as .(iii), as .

Let us consider the variational equation of (17) with respect to the initial value One can verify that

Lemma 14. For all , , one has

(i);(ii);(iii)

Proof. From variational equations (41) and Lemma 13, one has

and here we have used and .
Hence, for all, , we have and . Thus, (i) and (ii) are proved.
To prove (iii), we consider the variational equation of (17) about ; one can get

By using a similar argument in (ii), we can also show that , , as . This completes the proof of Lemma 14.

Next, we will develop an estimate of upper bound and lower bound for

Lemma 15. Let satisfy .
(i) If for all , then there exists a constant , such that .
(ii) If for all , then there exists a constant , such that . Moreover, if , then

Proof. (i) If , take Then

Writing , we have .
(ii) If , with condition , it is easy to know that when and when . Hence, and

Therefore, setting , we obtain The proof is complete.

Let with . To estimate that the integral of on is smaller than the integral of on , we need the following lemma.

Lemma 16. Let be as in Theorem 1. Define Then there exist , such that

for all .

Proof. According to Lemma 11, we see that if and only if
By the action-angle variables transformation (16), it is not difficult to verify that there exists such that when Therefore, by using Corollary 10, we know that there exist , such that

for all . Thus, we prove Lemma 16.

The next lemma gives the estimates of for .

Lemma 17. For , one gets

Proof. The following results immediately from Corollary 10, Lemma 15, and Lemma 16:

So, if , we have

4. Proof of Theorem 1

Now we start to give the proof of Theorem 1.

Proof of Theorem 1. Based on Lemma 17 and the Aubry-Mather theorem [37], we can see that the Poincaré map of system (17) is a monotone twist map when At last, using similar arguments as in [37], we may broaden the Poincaré map to a new map which is a whole monotone twist homeomorphism on the cylinder and agree with on with a fixed constant . Hence, the existence of Aubry-Mather sets of is ensured by the Aubry-Mather theorem due to Pei [37]. Moreover, for some small , all those Aubry-Mather sets with rotation number lie in the domain . Therefore, they happen to be the Aubry-Mather sets of the Poincaré map of . From the above discussions, we have showed the existence of Aubry-Mather sets; this implies that (1) has an Aubry-Mather type solution with rotation number . This completes the proof of Theorem 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11461056), the Youth Natural Science Foundation of Jiangxi (Grant no. 20132BAB211008), and the Natural Science Foundation of Jiangxi Provincial Department of Education (Grant no. GJJ170926).

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