Lagrangian Stability of a Class of Second-Order Periodic Systems
© Shunjun Jiang et al. 2011
Received: 24 November 2010
Accepted: 5 January 2011
Published: 11 January 2011
1. Introduction and Main Result
are bounded for all time, that is, whether there are resonances that might cause the amplitude of the oscillations to increase without bound.
The first positive result of boundedness of solutions in the superlinear case (i.e., as ) was due to Morris . By means of KAM theorem, Morris proved that every solution of the differential equation (1.1) is bounded if , where is piecewise continuous and periodic. This result relies on the fact that the nonlinearity can guarantee the twist condition of KAM theorem. Later, several authors (see [3–5]) improved Morris's result and obtained similar result for a large class of superlinear function .
where as and is a -periodic function. After introducing new variables, the differential equation (1.3) can be changed into a Hamiltonian system. Under some suitable assumptions on and , by using a variant of Moser's small twist theorem  to the Pioncaré map, the author obtained the existence of quasi-periodic solutions and the boundedness of all solutions.
The above differential equation essentially possess Hamiltonian structure. It is well known that the Hamiltonian structure and reversible structure have many similar property. Especially, they have similar KAM theorem.
Then, the following conclusions hold true.
(ii)Every solution of (1.5) is bounded.
where , , and are constants. We want to generalize the result in  to a class of -Laplacian-type differential equations of the form (1.9). The main idea is similar to that in . We will assume that the functions and have some parities such that the differential system (1.9) still has a reversible structure. After some transformations, we change the systems (1.9) to a form of small perturbation of integrable reversible system. Then a KAM Theorem for reversible mapping can be applied to the Poincaré mapping of this nearly integrable reversible system and some desired result can be obtained.
Our main result is the following theorem.
Our main nonlinearity in (1.9) corresponds to in (1.5). Although it is more special than , it makes no essential difference of proof and can simplify our proof greatly. It is easy to see from the proof that this main nonlinearity is used to guarantee the small twist condition.
2. The Proof of Theorem
The proof of Theorem 1.1 is based on Moser's small twist theorem for reversible mapping. It mainly consists of two steps. The first one is to find an equivalent system, which has a nearly integrable form of a reversible system. The second one is to show that Pincaré map of the equivalent system satisfies some twist theorem for reversible mapping.
2.1. Action-Angle Variables
Below we will see that the symmetric properties (1.10) imply a reversible structure of the system (1.9).
The function will be extended to the whole real axis as explained below, and the extension will be denoted by . Define on by . Then, we define on such that is an odd function. Finally, we extend to by -periodicity. It is not difficult to verify that has the following properties:
2.2. Some Lemmas
Lemma 2.3 (see ).
The following conclusions hold true:
2.3. Some Estimates
2.4. Coordination Transformation
Thus Lemma 2.7 is proved.
Below we introduce a small parameter such that the system (2.4) is written as a form of small perturbation of an integrable.
2.5. Poincaré Map and Twist Theorems for Reversible Mapping
We can use a small twist theorem for reversible mapping to prove that the Pioncaré map has an invariant closed curve, if is sufficiently small. The earlier result was due to Moser [11, 12], and Sevryuk . Later, Liu  improved the previous results. Let us first recall the theorem in .
Lemma 2.9 (see [14, Theorem 2]).
Lemma 2.11 (see [14, Theorem 1]).
2.6. Invariant Curves
all the assumptions in Lemma 2.11 hold.
Thus, in the both cases, the Poincare mapping always have invariant curves for being sufficient small. Since , we know that for any , there is an invariant curve of the Poincare mapping, which guarantees the boundedness of solutions of the system (2.11). Hence, all the solutions of (1.9) are bounded.
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