- Open Access
Boundedness of solutions for a class of second-order differential equation with singularity
© Jiang; licensee Springer. 2013
- Received: 14 September 2012
- Accepted: 22 March 2013
- Published: 10 April 2013
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.
- boundedness of solutions
- small twist theorem
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.
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.
Our main result is the following theorem.
Theorem 1 Under 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 2 Assume that satisfies (1.6); then, there is an such that, for any , t equation (1.4) has a solution of the Mather type with rotation number ω. More precisely:
Case 2: ω is 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.
2.1 Action-angle variables and some estimates
The closed curves are just the integral curves of (2.2).
Similar in estimating in , we have the estimation of functions and .
where , . Note that here and below we always use C, or to indicate some constants.
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.
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 .
In order to estimate , we need the following lemma.
Lemma 2 [, Lemma 2.2]
2.2 New action and angle variables
is also a Hamiltonian system with Hamiltonian function I and now the action, angle and time variables are H, t and θ.
holds where . We will prove (2.9) also holds where , .
where , , .
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.
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 .
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 some constant which is independent of the arguments t, ρ, θ, ϵ.
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)
the mapping has an invariant curve in . The constant ϵ is independent of δ.
The estimates for follow from a similar argument, we omit it here. Thus, Lemma 9 is proved. □
Proof This lemma was proved in , so we omit the details. □
For estimate and , we need the estimates of x and .
Now we can give the estimates of and .
Thus, Lemma 11 is proved. □
2.4 Proof of Theorem 1
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.
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.
Thanks are given to referees whose comments and suggestions were very helpful for revising our paper.
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