In this paper, we study the following second-order periodic system:
where has a singularity. Under some assumptions on the and by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.
Keywords:boundedness of solutions; singularity; small twist theorem
1 Introduction and main result
In 1991, Levi  considered the following equation:
where satisfies 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 nonlinearity can guarantee the twist condition of KAM theorem. Later, several authors improved the Levi’s result; we refer to [2-4] and the references therein.
Recently, Capietto, Dambrosio and Liu  studied the following equation:
with is a π-periodic function and , where , and ν is a positive integer. Under the Lazer-Leach assumption that
they prove the boundedness of solutions and the existence of quasi-periodic solution by KAM theorem. It is the first time that the equation of the boundedness of all solution is treated in case of a singular potential.
We observe that in (1.2) is smooth and bounded, so a natural question is to find sufficient conditions on such that all solutions of (1.2) are bounded when is unbounded. The purpose of this paper is to deal with this problem.
where is a π-periodic function,
We suppose Lazer-Leach assumption hold:
Our main result is the following theorem.
Theorem 1Under the assumptions (1.5) and (1.6), all the solutions of (1.4) are bounded.
The main idea of our proof is acquired from . The proof of Theorem 1 is based on a small twist theorem due to Ortega . It mainly consists of two steps. The first one is to transform (1.4) into a perturbation of integrable Hamilton system. The second one is to show that Poincaré map of the equivalent system satisfies Ortega’s twist theorem, then some desired result can be obtained.
Moreover, we have the following theorem on solutions of Aubry-Mather type.
Theorem 2Assume that satisfies (1.6); then, there is an such that, for any , tequation (1.4) has a solution of the Mather type with rotation numberω. More precisely:
Case 1: is rational. The solutions , are independent periodic solutions of periodicqπ; moreover, in this case,
Case 2: ωis irrational. The solution is either a usual quasi-periodic solution or a generalized one.
We will apply Aubry-Mather theory, more precisely, the theorem in , to prove this theorem.
2 Proof of theorem
2.1 Action-angle variables and some estimates
Observe that (1.4) 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 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 . It is easy to see that
By direct computation, we get
We then have
Similar in estimating in , we have the estimation of functions and .
Lemma 1We have
where , . Note that here and below we always useC, or to indicate some constants.
Remark 1 It follows from the definitions of , and Lemma 1 that
Thus the time period is dominated by when h is sufficiently large. From the relation between and , we know is dominated by when h is sufficiently large.
Remark 2 It also follow from the definition of , , and Remark 1 that
Remark 3 Note that is the inverse function of . By Remark 2, we have
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 point .
We define the well-know map by
which is symplectic since
From the above discussion, we can easily get
In the new variables , the system (2.1) is
In order to estimate , we need the following lemma.
Lemma 2 [, Lemma 2.2]
ForIsufficient large and , the following estimates hold:
2.2 New action and angle variables
Now we are concerned with the Hamiltonian system (2.5) with 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.
From (2.6) and Lemma 1, we have
So, we assume that I can be written as
where R satisfies . Recalling that is the inverse function of , we have
which implies that
As a consequence, R is implicitly defined by
Lemma 3The function satisfies the following estimates:
Proof Case . By (2.8), Lemma 2 and noticing that as , we have
Case . Derivative both sides of (2.8) with respect to H, we have
By Remark 2, Lemma 2 and the estimate of R, we have
We suppose that
holds where . We will prove (2.9) also holds where , .
By direct calculation, we have
where , , .
by Lemma 1 and (2.9), when , we have
When , we have
By (2.11) and (2.12), we have
By (2.13), we have
By (2.10), (2.14) and Lemma 2, we have (2.9) holds where . Thus, we prove Lemma 3. □
Analogously, one may obtain, by a direct but cumbersome commutation, the following estimates.
Lemma 4The function satisfies the following estimates:
Moreover, by the implicit function theorem, there exists a function such that
for , we have
For , we have
By Lemmas 1 and 4, we have the estimates on .
For concision, in the estimates and the calculation below, we only consider the case , since the case have the similar result.
Lemma 5 for .
For the estimates of , we need the estimates on . By Lemmas 1 and 5, noticing that , we have the following lemma.
Lemma 6 for .
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 (2.15) is changed into the form
which is also Hamiltonian system with the new Hamiltonian function
Obviously, if , the solution of (2.16) with the initial date is defined in the interval and . So the Poincaré map of (2.16) is well defined in the domain .
Lemma 7 [, Lemma 5.1]
The Poincaré map of (2.16) has 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. Let us first recall the theorem in .
Lemma 8 (Ortega’s theorem)
Let be a finite cylinder with universal cover . The coordinate in is denoted by . Consider a map
We assume that the map has the intersection property. Suppose that , is a lift of and it has the form
whereNis 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 constantϵis independent ofδ.
We make the ansatz that the solution of (2.16) with the initial condition is of the form
Then the Poincaré map of (2.16) is
The functions and satisfy
where , . By Lemmas 4, 6 and 7, we know that
Hence, for , we may choose ϵ sufficiently small such that
Moreover, we can prove that
Lemma 9The following estimates hold:
By Lemma 2 and (2.25), we have
Take the derivative with respect to in the both sides of , we have
Using Lemma 2 and noticing , we have
Analogously, one may obtain, by a direct but cumbersome commutation that
which means that
The estimates for follow from a similar argument, we omit it here. Thus, Lemma 9 is proved. □
Now we turn to give an asymptotic expression of Poincaré map of (2.15), that is, we study the behavior of the functions and at as . In order to estimate and , we need introduce the following definition and lemma. Let
Proof This lemma was proved in , so we omit the details. □
For estimate and , we need the estimates of x and .
We recall that when , we have
When , by the definition of θ, we have
which yields that
Now we can give the estimates of and .
Lemma 11The following estimates hold true:
Proof Firstly, we consider . By Lemmas 2, 6 and (2.23), we have
Now we consider .
Thus, Lemma 11 is proved. □
2.4 Proof of Theorem 1
Then there are two functions and such that the Poincaré map of (2.16), given by (2.22), is of the form
Since , , we have
The other assumptions of Ortega’s theorem are easily verified. Hence, there is an invariant curve of P in the annulus which imply that the boundedness of our original equation (1.4). Then Theorem 1 is proved.
2.5 Proof of Theorem 2
We apply Aubry-Mather theory. By Theorem B in  and the monotone twist property of the Poincaré map P guaranteed by . It is straightforward to check that Theorem 2 is correct.
Remark 4 In , the authors study the multiplicity of positive periodic solutions of singular Duffing equations
where satisfies the semilinear condition at infinity and the time map satisfies an oscillation condition, and prove that the given equation possesses infinitely many positive 2π-periodic solutions by using the Poincaré-Birkhoff theorem. By the methods and techniques in , we can also prove the existence of 2π-periodic solutions of (1.4) where satisfies the sublinear condition.
The author declares that they have no competing interests.
Thanks are given to referees whose comments and suggestions were very helpful for revising our paper.
Kupper, T, You, J: Existence of quasiperiodic solutions and Littlewood’s boundedness problem of Duffing equations with subquadratic potentials. Nonlinear Anal.. 35, 549–559 (1999). Publisher Full Text
Li, X: Boundedness of solutions for Duffing’s equations with semilinear potentials. J. Differ. Equ.. 176, 248–268 (2001). Publisher Full Text
Wang, Y: Boundedness for sublinear Duffing equations with time-dependent potential. J. Differ. Equ.. 247, 104–118 (2009). Publisher Full Text
Capietto, A, Dambrosio, W, Liu, B: On the boundedness of solutions to a nonlinear singular oscillator. Z. Angew. Math. Phys.. 60(6), 1007–1034 (2009). Publisher Full Text
Liu, B: Quasi-periodic solutions of forced isochronous oscillators at resonance. J. Differ. Equ.. 246, 3471–3495 (2009). Publisher Full Text
Ortega, R: Boundedness in a piecewise linear oscillator and a variant of the small twist theorem. Proc. Lond. Math. Soc.. 79, 381–413 (1999). Publisher Full Text
Pei, ML: Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations. J. Differ. Equ.. 113, 106–127 (1994). Publisher Full Text
Wang, Z, Ma, T: Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities. Nonlinearity. 25, 279–307 (2012). Publisher Full Text