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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 246343, 18 pages
Research Article

Boundedness of Solutions for a Class of Sublinear Reversible Oscillators with Periodic Forcing

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

Received 3 February 2013; Accepted 30 April 2013

Academic Editor: Wenchang Sun

Copyright © 2013 Tingting Zhang and Jianguo Si. 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.


We study the boundedness of all solutions for the following differential equation where are odd functions, is an even function, are smooth -periodic functions, is a nonzero constant, and is a small parameter. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, the existence of Aubry-Mather sets is obtained as well.

1. Introduction

It is well known that the longtime behavior for periodically forced planar systems can be very intricate. For example, there are equations having unbounded solutions but with infinitely many zeros and with nearby unbounded solutions having randomly prescribed number of zeros and also periodic solutions; see [1]. In contrast to such unbounded phenomenon Littlewood [2] suggested to study the boundedness of all the solutions of the following differential equation: in the following two cases:(i)superlinear case: as ;(ii)sublinear case: and as . Later, one calls this subject as Littlewood boundedness problem.

The first result in superlinear case is obtained by Morris [3], who showed that all solutions of are bounded, where . Later, a series results in superlinear case were obtained by several authors, see [413] and references therein. However, in general, it is harder to study the Lagrange stability of sublinear systems since smoothness of sublinear term is insufficient. There are only a few works in sublinear case so far. In 1999, Küpper and You [14] proved the first result in the study of the equation where and . Later, Liu [15] proved the same result for more general equation where satisfying the sublinear condition (ii) and some inequalities, and . In 2004, Ortega and Verzini [16] studied the boundedness of (4) in a special case with the variational method. In 2009, Wang [17] gave a sufficient and necessary condition for the boundedness of all solutions for sublinear equation where .

As is widely known, there is a deep similarity between reversible and Hamiltonian dynamics. Many fundamental results of the Hamiltonian systems possess reversible counterparts. On boundedness problem for sublinear reversible systems, the first results were obtained by Li [18], later, Yang [19], in the study of a sublinear reversible systems Recently, Wang [20] gave a sufficient and necessary condition for the boundedness of all solutions of the differential equation with , .

By the discussions about the sublinear Hamiltonian equation (1.3) in [17] motivations, we will study the boundedness of all solutions for a sublinear reversible system like where and . Furthermore, we also show that (8) has solutions of Mather type. The results obtained in [1820] can be regarded as corollary of result of this paper.

Remark 1. Using the method of this paper we also can consider the more general equation provided of adding suitable conditions for . For convenience, we only consider the case .

Remark 2. Adding the perturbation term will lead to a new difficulty for estimating appeared in (86). Fortunately, we can easily verify that is bounded by a constant (see in the proof of Lemma 12).

Throughout this paper, we denote two universal positive constants without regarding their values by and , and suppose that the following conditions hold:(A1), and , and are odd, is even, and , are both -periodic functions, ;(A2) there is some positive constant such that the inequalities are satisfied for and all , where .

We decompose as , where is the average of and has zero mean value. That is and . If we write that , then it is easy to see that and have the same sign when with .

Now we state the main results of this paper.

Theorem 3. Assume that and - hold. Then there exists an such that for any , every solution of (8) is bounded if and only if .

Theorem 4. Under the conditions of Theorem 3, there is an such that, for any , (8) has a solution of Mather type with rotation number . More precisely:(i)if is rational, the solutions , , are periodic solutions of period q; moreover, in this case (ii)if is irrational, the solution is either a usual quasi-periodic solution or a generalized one.
We recall that a solution is called generalized quasi-periodic if the closed set is a Denjoys minimal set.

2. Reversible Systems and Action-Angle Variables

In this section, we will assume that and . Firstly, we consider (8) which is equivalent to the following system: where . Then we can obtain that (13) is reversible with respect to the transformation by (A1).

Lemma 5. There exists a -invariant diffeomorphism such that (13) is transformed into the following system: where .

Proof. Introduce a transformation : where will be determined later. Under this transformation, the system (13) is transformed into a new system as follows: Now, we define the function by Since , so we can obtain . Then the new system can be expressed as in (14) by direct computation.
It is easy to know that by (A1), then we can obtain that the transformation is a -invariant diffeomorphism.

Let us consider the auxiliary system which is a time-independent Hamiltonian system with Hamiltonian

It is easy to see that , , . Note that each level line is a close orbit of system (18), hence, all the solutions of (18) are periodic with period tending to zero as tends to infinity.

Assume that is the solution of (18) with initial conditions , and let be the minimal period. We can find that and satisfy(i), ;(ii), ;(iii), ;(iv); (v); (vi), ;(vii).

Then we introduce the transformation which is where . It is easy to see that by . Since , this transformation is invariant with respect to the involutions and , and we can find that the mapping is a generalized canonical transformation by (iv). In fact, where .

Under the transformation , the system (18) is transformed into the simpler form where .

The original system (13) is transformed into the system where

Let Clearly, is odd in and is even in by the definitions of and . Thus, by the evenness of and the oddness of and we have This implies that system (24) is reversible with respect to the involutions .

Lemma 6. For , the following inequalities hold:(1), (2), (3), (4), (5), (6), (7),
where , , and .

Proof. (1) It is easy to know that is a sum of terms of the form where . Meanwhile, is a sum terms of the form Hence, we obtain by the assumptions on and the definitions of and .
(2) From the expression of , we have
We can find that where .
(3) From the expression of , we have
(4) From the expression of , we can obtain that
(5) From the definition of , we have
(6) From the definition of , we can obtain Hence, we can know that where .
(7) From the expression of , we have

For , we define the domain

Lemma 7. There exists a -invariant diffeomorphism : such that for some . Under this transformation, (24) is transformed into the system where with

Proof. Define a transformation by By we get Let , . The system (24) is transformed into (41).

Lemma 8. For large enough, the following conclusions hold:(i), (ii).

Proof. In view of we obtain By , we have for large enough. Hence, is uniquely determined by the contraction mapping principle. Moreover, , for some , as a consequence of the implicit function theorem and
Above all, if , from (47) and (49), we get
We note that and the right side hand is sum of the term where , , . The highest order term in is the one with , namely, . We move the part to the left hand side of (52). Since for large enough, this also provides immediately a bound on . The rest part .
Now, we proceed inductively by assuming that for the estimates hold and we wish to conclude that the same estimate holds for .
Indeed, if , we have by This proves (i) of Lemma 8.
Now we check (ii). In fact, since we have From (47), we have for sufficiently large and therefore we obtain .

By the estimates in Lemma 6, we can prove the following inequalities.

Lemma 9. For , the following inequalities hold:(1), (2), (3), (4), (5), (6), (7).

Proof. (1) From the estimates (1) and (5) of Lemmas 6 and 8, it follows that (2) Since we can prove that Their proofs are similar to the proofs in (1).
Next, we check the last part of   . We get by the estimate in Lemma 6 and the definition of .
(3) It is clearly by (3) in Lemma 6.
(4) It is clearly by (4) in Lemmas 6 and 8.
(5) We have that From the last inequalities and (5) in Lemma 6, we obtain (6) Since we just have to prove that In fact, so we have proved (6).
(7) We have by (7) in Lemma 6.

3. The Proof of Boundedness

In this section, all the solutions of (8) which are bounded will be proved via the KAM theory for reversible systems developed by Sevryuk [21] or Moser [22, 23] if .

We define the functions , , , , , , and as Then system (41) is equivalent to the following system:

In addition, one can verify that system (70) is reversible with respect to involution .

Then some estimates on the functions and are given.

Lemma 10. The following inequalities hold:(1), (2), (3), (4), (5), (6), (7), for .

Proof. (1) It is clear.
(2) Note that , and it follows that as .
Moreover, we also have So From (72) and (74), it is easy to see that (3) We have By (72), and , we have