Abstract and Applied Analysis

Volume 2013 (2013), Article ID 267572, 9 pages

http://dx.doi.org/10.1155/2013/267572

Research Article

Boundedness of Solutions for a Class of Second-Order Periodic Systems

Shunjun Jiang¹ and Yan Ding²

¹College of Sciences, Nanjing University of Technology, Nanjing 210009, China

²Overseas Education College, Nanjing University of Technology, Nanjing 210009, China

Received 26 June 2013; Accepted 4 November 2013

Academic Editor: Yuriy Rogovchenko

Copyright © 2013 Shunjun Jiang and Yan Ding. 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.

In this paper we study the following second-order periodic system: where has a singularity. Under some assumptions on the and by Ortega’ small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

1. Introduction and Main Result

In 1991, Levi [1] considered the following equation:

wheresatisfies some growth conditions and. The author reduced the system to a normal form and then applied Moser twist theorem to prove the existence of quasi-periodic solution and the boundedness of all solutions. This result relies on the fact that the nonlinearitycan guarantee the twist condition of KAM theorem. Later, several authors improved Levi's result; we refer to [2–4] and the references therein.

Recently, Capietto et al. [5] studied the following equation:

wheris a-periodic function and, where,  , andis a positive integer. Under the Lazer-Leach assumption that

they prove the boundedness of solutions and the existence of quasi-periodic solution by Moser twist theorem. It is the first time that the equation of the boundedness of all solutions is treated in case of a singular potential.

We observe thatin (2) is smooth and bounded, so a natural question is to find sufficient conditions onsuch that all solutions of (2) are bounded whenis unbounded. The purpose of this paper is to deal with this problem.

Motivated by the papers [1, 5, 6], we consider the following equation:

where

In order to state our main results, we give some notation and assumptions. Letbe some fixed constant. Let

(A1)Assume anduniformly in.(A2)uniformly inforwhereand(A3)We suppose Lazer-Leach assumption holds:

Our main result is the following theorem.

Theorem 1. Under the assumptions (A1)–(A3), all the solutions of (4) are defined for all, and for each solution, one has.

The main idea of our proof is acquired from [6]. The proof of Theorem 1 is based on a small twist theorem due to Ortega [7]. The hypotheses (A1)–(A3) of our theorem are used to prove that the Poincaré mapping of (4) satisfies the assumptions of Ortega's theorem.

Moreover, we have the following theorem on solutions of Mather type.

Theorem 2. Assume thatsatisfies (7); then, there is ansuch that, for any, (4) has a solutionof Mather type with rotation number. More precisely,

Case  ( is rational). The solutions,, are mutually unlinked periodic solution of periodic; moreover, in this case,

Case  2 ( is irrational). The solutionis either a usual quasi-periodic solution or a generalized one.

2. Proof of Theorem

2.1. Action-Angle Variables and Some Estimates

Observe that (4) is equivalent to the following Hamiltonian system:

with the Hamiltonian function

In order to introduce action and angle variables, we first consider the auxiliary autonomous equation:

which is an integrable Hamiltonian system with Hamiltonian function

The closed curvesare just the integral curves of (11).

Denote bythe time period of the integral curveof (11) defined byand bythe area enclosed by the closed curvefor everyLetbe such thatIt is easy to see that

By direct computation we get

so

We then have

where

We now give the estimates on the functionsand.

Lemma 3. One has

where,. Note that here and below one always uses ,, orto indicate some constants.

Proof. Now we estimate the first inequality. We choseas the new variable of integration; then we have

Sinceand, we haveBy direct computation, we have

and then we get

Whenandis sufficiently large, there exitssuch that, so we have

Since, we have

and then

Observing that there issuch thatwhenand, we have

By (22)–(25) we have,.

The proof of the second inequality is similar to the first one, so we only give a brief proof.

We chooseas the new variable of integration, so we have

By direct computation, we have

By (27), we can easily get

where.

By a similar way to that used in estimating, we get

which means that

Thus Lemma 3 is proved.

Remark 4. It follows from the definitions of  ,and Lemma 3 that

Thus the time periodis dominated bywhenis sufficiently large. From the relation betweenand, we knowis dominated bywhenis sufficiently large.

Remark 5. It also follows from the definition of,,and Remark 4 that

Remark 6. Note thatis the inverse function of. By Remark 5, we have

We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function, whereis the part of the closed curveconnecting the point on the-axis and point.

We define the well-known mapby

which is symplectic since

From the above discussion, we can easily get

In the new variables, the system (9) becomes

where

In order to estimate, we need the estimate on the function.

Lemma 7. Forsufficiently large and, the following estimates hold:

The Lemma was first proved in [1], and later [5] gives a different proof; [8] using the method of induction hypothesis also gives another proof. So, for concision, we omit the proof.

2.2. New Action and Angle Variables

Now we are concerned with the Hamiltonian system (37) with Hamiltonian functiongiven by (38). Note that

This means that if one can solvefrom (38) as a function of(usingandas parameters), then

is also a Hamiltonian system with Hamiltonian functionand now the action, angle, and time variables are,, and.

From (38) and Lemma 3, we have

So we assume thatcan be written as

wheresatisfiesRecalling thatis the inverse function of, we have

which implies that

As a consequence,is implicitly defined by

For the estimates of, we need the following lemmas.

Lemma 8. Let and be continuously differentiable for, whereis an interval of. If

(1)as, uniformly with respect to,(2)as, uniformly with respect to

then one has as, uniformly with respect to.

Proof. For any, there exits, such that if, we have

LetThen by Lagrangian differential mean value theorem, it follows that, for all, we have

Moreover, there exists a constantsuch that

By condition (A1), there exists; we have.

Thus

Lemma 9. Under the assumptions (A1) and (A2), the following results hold:

(1)uniformly in, forand, wherefor, andfor.(2), forand, wherefor.

Proof. Result (1) is similar to Lemmain [9], so we omit the proof.

For result (2), we first prove that, for and ,

Forand, by result (1), we have that

uniformly in. It follows that

This means that (51) holds forFor, combining this with result (1) and the result of (51) for, we have that (51) still holds. Inductively we can prove that (51) holds for all

Obviously, it follows that

Using Lemma 8 and the first result (1) for, it follows that

for. In a similar way to the proof of (51), we have

whereThus we proved Lemma 9.

Now we give the estimates ofBy Lemma 9, in a similar way to that for Lemma 2.3 in [5], we have the following lemma.

Lemma 10. The functionsatisfies the following estimates:

Moreover, by the implicit function theorem, there exists a functionsuch that

Since

by Lemmas 3 and 10, we have the estimates on.

Lemma 11. Consider the following:

For the estimate of, we need the estimate onBy Lemma 3 and noticing that, we have the following lemma.

Lemma 12. Consider the following:

Now the new Hamiltonian functionis written in the form

The system (41) is of the form

Introduce a new action variableand a parameterby. Then,Under this transformation, the system (63) is changed into the form

which is also Hamiltonian system with the new Hamiltonian function

Obviously, if, the solutionof (64) with the initial dateis defined in the intervaland. So the Poincaré map of (64) is well defined in the domain.

Lemma 13 (see [6] Lemma). The Poincaré map of (64) has intersection property.

The proof is similar to the corresponding one in [6].

For convenience we introduce the notationand. We say a functionifis smooth inand, for,

for some constantwhich is independent of the arguments,,, and .

Similarly, we sayifis smooth inand, for,

uniformly in.

2.3. Poincaré Map and Twist Theorems

We will use Ortega' small twist theorem to prove that the Pioncaré maphas an invariant closed curve, ifis sufficiently small. Let us first recall the theorem in [7].

Lemma 14 (Ortega’s Theorem). Letbe a finite cylinder with universal coverThe coordinate inis denoted byConsider a map

One assumes that the map has the intersection property. Suppose that,is a lift ofand it has the form

whereis an integer and is a parameter. The functions,  , , andsatisfy

In addition, one assumes that there is a functionsatisfying

Moreover, suppose that there are two numbers andsuch thatand

where

Then there existandsuch that, ifand

the mappinghas an invariant curve in. The constantis independent of.

We make the ansatz that the solution of (64) with the initial conditionis of the form

Then, the Poincaré map of (64) is

The functionsandsatisfy

whereandBy Lemmas 9, 11, and 12, we know that

Hence, for, we may choosesufficiently small such that

Moreover we can prove that

In a similar way to that used for estimating, by direct calculation we have the following lemma.

Lemma 15. The following estimates hold:

Now we give an asymptotic expression of Poincaré map of (63); that is, we study the behavior of the functionsandatasIn order to estimateand, we need to introduce the following definition and lemma. Let

where.

Lemma 16. Consider the following:

Proof. This Lemma was proved in [5], so we omit the details.

For estimateand, we need the estimates ofand.

We recall that, when, we have

When, by the definition of, we have

which yields that

Now we can give the estimates ofand.

Lemma 17. The following estimates hold true:

for

Proof. Firstly we considerBy Lemmas 11, 12, and 15 and (77), we have

By result (2) of Lemma 9, aswhich meanswe have

By the measure of, we have

By (90) and (91), we have

Now we considerBy Lemmas 11, 12, and 15 and (77), we have

By result (2) of Lemma 9, as, we have

By the measure of, we have

By (94) and (95), we have

Thus Lemma 17 is proved.

2.4. Proof of Theorem 1

Let

Then there are two functions and such that the Poincaré map of (64), given by (76) of the form

where

Since,we have

Let

Then

The other assumptions of Ortega’s theorem are easily verified. Hence, there is an invariant curve ofin the annuluswhich implies the boundedness of our original equation (4). Then Theorem 1 is proved.

2.5. Proof of Theorem 2

We apply Aubry-Mather theory. By Theorem B in [10] and the monotone twist property of the Pioncarémapguaranteed by, it is straightforward to check that Theorem 2 is correct.

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