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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 267572, 9 pages
Boundedness of Solutions for a Class of Second-Order Periodic Systems
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.
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  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.  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.
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 . The proof of Theorem 1 is based on a small twist theorem due to Ortega . 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.
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
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:
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.
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 tothen 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.
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 , 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.
Lemma 10. The functionsatisfies the following estimates:
Moreover, by the implicit function theorem, there exists a functionsuch that Since
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.
The proof is similar to the corresponding one in .
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 .
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 , 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
Since,we have Let Then
2.5. Proof of Theorem 2
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