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Boundedness of solutions of forced isochronous oscillators with singularity at resonance
Advances in Difference Equations volume 2014, Article number: 55 (2014)
In this paper we study the second-order periodic system: where has a singularity. Under some assumptions on the V, F, g, and p, by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.
1 Introduction and main result
The question of boundedness of all solutions of
initiated by Littlewood , has been the subject of numerous studies. Morris  has shown that when , where is continuous, all solutions are bounded. Levi  considered (1.1), where the potential V satisfies a superquadratic growth and has a singularity. The author reached a similar conclusion as in . In , Liu has proved boundedness of all the solutions of the following equation with an asymmetric nonlinearity
where , , . For this case, all the solutions of the unperturbed equation
have a common period , which means that (1.2) is isochronous.
In , Bonheure et al. study the following equation, which is a perturbation of the isochronous oscillator:
where , defined on , has a repulsive singularity and satisfies
m is a positive integer, . The perturbation satisfies the Lazer-Landsman condition
where the function is smooth and bounded, , and . The authors assume that all solutions of the unperturbed equation
are , which means that (1.4) is an isochronous oscillator with period . The authors proved the existence of 2π-periodic solutions of (1.3). We refer for more details on the isochronous system to  and the references therein.
Recently, Liu  studied the quasi-periodic solutions and boundedness of all solutions of the isochronous oscillators equation (1.3) where has a singularity, the nonlinearity is a bounded perturbation, and is 2π-periodic. Moreover, the following assumptions hold:
, for and , , and is defined on .
is smooth in and the limit exists. Furthermore, the following estimates hold: for each there is a constant such that
The positive function V is smooth and, for ,
where is a positive constant.
For , let ; the function Φ satisfies
for every positive integer k. By the results in , the auxiliary autonomous system is an isochronous system with period .
The function g is bounded on the interval and for . Moreover, the following inequalities hold:
If limit exists and , the following condition of Lazer-Landesman type holds:
Liu first reduced the system to a normal form and then applied a variant of Moser’s twist theorem of invariant curves to prove the existence of quasi-periodic solution and the boundedness of all solutions. This result relies on the fact that the nonlinearity can guarantee the twist condition of the KAM theorem. The assumptions (5) and (6) satisfy the Lazer-Landesman condition which plays a key role in the boundedness problem. In fact, it has been shown by Alonso and Ortega  that, when the Lazer-Landesman condition cannot be satisfied, the solutions with large initial conditions are unbounded either in the past or in the future.
We observe that in [5, 6], the perturbation is smooth and bounded, so a natural question is to find sufficient conditions on and such that all solutions of (1.3) are bounded when the perturbation is unbounded. The purpose of this paper is to deal with this problem. We can refer to more papers on the Littlewood Problem on unbounded perturbation such as [8, 9].
where . We suppose that (1)-(4) hold; moreover, we have
(7′) We mention a condition similar to the Lazer-Landesman condition: .
Our main result is the following theorem.
Theorem 1 Suppose the assumptions (1)-(4) and (5′)-(7′) hold, then (1.5) has infinitely many quasi-periodic solutions and all the solutions of (1.5) are bounded.
The main idea of our proof is acquired from . The proof of Theorem 1 is based on the small twist theorem due to Ortega . It mainly consists of two steps. The first one is to transform (1.5) into a perturbation a integrable Hamilton system. The second one is to show that the Poincaré map of the equivalent system satisfies Ortega’s twist theorem, then the desired result can be obtained.
2 The proof of the theorem
2.1 Action-angle variables
Observe that (1.5) 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 curves are just the integral curves of (2.2).
Denote by the time period of the integral curve of (2.2) defined by and by I the area enclosed by the closed curve for every . Let be such that . Then by assumption (1), we have
It is easy to see that
Let us denote
We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function , where C is the part of the closed curve connecting the point on the y-axis and the point .
We define the well-known map by
which is symplectic, since
From the above discussion, we easily get
In the new variables , the system (2.1) becomes
2.2 New action and angle variables
Now we are concerned with the Hamiltonian system (2.5) with the Hamiltonian function given by (2.6). Note that
This means that if one can solve I from (2.6) as a function of H (θ and t as parameters), then
is also a Hamiltonian system with Hamiltonian function I and now the action, angle, and time variables are H, t, and θ.
Lemma 1 There is a constant C such that
By (5′), (2.8), and Lemma 1, we can directly get the following lemma.
Lemma 2 The following estimates hold:
Using (2.6), we have
Hence, by the implicit function theorem, there exists a function such that
The function is defined implicitly by
Now we give the estimates of . By a direct computation, we have the following.
Lemma 3 for .
Proof Case . By (2.9), Lemma 2 and noticing that as , we have
Case . Taking derivatives on both sides of (2.9) with respect to H, we have
By Lemma 2, we have
We suppose that
holds where . We will prove that (2.10) also holds when , .
By direct calculation, we have
where , , .
By (2.10), we have
By (2.11), (2.12), and Lemma 2, we have
then we proved that (2.10) holds when . Thus we proved Lemma 3. □
By similarity to the proofs of Lemmas 2 and 3, we have the following.
Lemma 4 The following estimates hold:
Remark 1 By the proofs and estimates of R, , , we can easily see that is the main (twist) term of R and is the small term of R.
Now the new Hamiltonian function is written in the form
The system (2.7) is of the form
Introduce a new action variable and a parameter by . Then . Under this transformation, the system (2.13) is changed into the form
which is also a Hamiltonian system, with the new Hamiltonian function
Obviously, if , the solution of (2.14) with the initial date is defined in the interval and . So the Poincaré map of (2.14) is well defined in the domain .
Lemma 5 ([, Lemma 5.1])
The Poincaré map of (2.14) has the intersection property.
The proof is similar to the corresponding one in .
For convenience we introduce the notation and . We say a function if f is smooth in and for ,
for some constant which is independent of the arguments t, ρ, θ, ϵ.
Similarly, we say if f is smooth in and for ,
uniformly in .
2.3 Poincaré map and twist theorems
We will use Ortega’s small twist theorem to prove that the Poincaré map P has an invariant closed curve, if ϵ is sufficiently small. The earlier result was due to Moser [11–13]. Later, Orgeta  improved the previous results. Let us first recall the theorem in .
Lemma 6 (Ortega’s Theorem)
Let be a finite cylinder with universal cover . The coordinate in is denoted by . Consider the map
We assume that the map has the intersection property. Suppose that , is a lift of and it has the form
where N is an integer, is a parameter. The functions , , , and satisfy
In addition, we assume that there is a function satisfying
Moreover, suppose that there are two numbers and such that and
Then there exist and such that, if and
the mapping has an invariant curve in . The constant ϵ is independent of δ.
We make the ansatz that the solution of (2.14) with the initial condition is of the form
Then the Poincaré map of (2.14) is
The functions and satisfy
where , . By Lemmas 1 and 3, we know that
Moreover we can prove that
Lemma 7 The following estimates hold:
Proof We only consider , since the case can be proved similarly.
By (6′), Lemma 2 and (2.21), we have
Taking the derivative with respect to in both sides of , we have
Similar to the estimates of and noticing , we have
Now we will estimate , . By direct calculation, we have
where , , , , . For estimating , we need the estimates of and .
We firstly give the estimates of . By direct calculation, we get
where , , .
Since , we have
By (2.25) and (2.26), we have
We now give the estimates of . By (6′), (2.26), (2.27), and Lemma 1, we have
We suppose that the following inequality holds:
By direct calculation, we have
where , . By the estimates of Ψ (2.26), the estimates of (2.27) and (2.28), (2.29), we have
Since we have the estimates of Ψ, , and , now we can give the estimates of . By (2.26), (2.27), (2.30), and (2.24), we have the estimate of Δ:
which means that
The estimates for follow from a similar argument, and we omit it here. Thus Lemma 7 is proved. □
Now we turn to giving an asymptotic expression of the Poincaré map of (2.13), that is, we study the behavior of the functions and at as .
In order to estimate and , we introduce the following definition and lemma. Let be the subset of the interval such that , and satisfies , . By (2.3) and (2.4), similar to the computation in , there is a function such that . By the definition of and , we have
Moreover we will use the following lemma.
Lemma 8 ([, Lemma 4.1])
For , the function x has the following expression:
Now we can give the estimates of and .
Proof Firstly we consider . By Lemmas 4, 7, and (2.21), we have
By Lemma 1, we know that when
By (2.32) and (6′), we have
By Lemma 8, we have
Now we consider :
By (6′), Lemmas 4 and 7, and noticing that
By Lemmas 1, 8, and (2.32)
Then there are two functions and such that the Poincaré map of (2.14), given by (2.20), is of the form
Note that ,
The other assumptions of Ortega’s theorem are easily verified. Hence, there is an invariant curve of P in the annulus , which implies the boundedness of our original equation (1.5). Then Theorem 1 is proved. □
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Thanks are given to referees whose comments and suggestions were very helpful for revising our paper. This work is supported by National Natural Science Foundation of China (Grant Nos. 60973140, 61170276, 61373135, 60873231), Major Natural Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No. 12KJA520003), the Innovation Project for postgraduate cultivation of Jiangsu Province Grant No. CXLX11-0415, the research project of Jiangsu Province Grant No. BY2013011, the Natural Science Foundation of Jiangsu Province Grant No. BK20130096.
The authors declare that they have no competing interests.
The article is a joint work of two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.