- Research Article
- Open access
- Published:
Lagrangian Stability of a Class of Second-Order Periodic Systems
Boundary Value Problems volume 2011, Article number: 845413 (2011)
Abstract
We study the following second-order differential equation: , where
(
),
and
are positive constants, and
satisfies
. Under some assumptions on the parities of
and
, by a small twist theorem of reversible mapping we obtain the existence of quasiperiodic solutions and boundedness of all the solutions.
1. Introduction and Main Result
In the early 1960s, Littlewood [1] asked whether or not the solutions of the Duffing-type equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ1_HTML.gif)
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 [2]. 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
.
When satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ2_HTML.gif)
that is, the differential equation (1.1) issemilinear, similar results also hold, but the proof is more difficult since there may be resonant case. We refer to [6–8] and the references therein.
In [8] Liu considered the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ3_HTML.gif)
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 [9] to the Pioncaré map, the author obtained the existence of quasi-periodic solutions and the boundedness of all solutions.
Then the result is generalized to a class of -Laplacian differential equation.Yang [10] considered the following nonlinear differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ4_HTML.gif)
where is bounded,
is periodic. The idea is also to change the original problem to Hamiltonian system and then use a twist theorem of area-preserving mapping to the Pioncaré map.
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.
Recently, Liu [6] studied the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ5_HTML.gif)
where is a positive constant and
is
-periodic in
. Under some assumption of
and
, the differential equation (1.5) has a reversible structure. Suppose that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ6_HTML.gif)
where and
. Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ7_HTML.gif)
where is a constant. Note that here and below we always use
to indicate some constants. Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ8_HTML.gif)
Then, the following conclusions hold true.
(i)There exist and a closed set
having positive measure such that for any
, there exists a quasi-periodic solution for (1.5) with the basic frequency
.
(ii)Every solution of (1.5) is bounded.
Motivated by the papers [6, 8, 10], we consider the following -Laplacian equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ9_HTML.gif)
where ,
, and
are constants. We want to generalize the result in [6] to a class of
-Laplacian-type differential equations of the form (1.9). The main idea is similar to that in [6]. 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.
Theorem 1.1.
Suppose that and
are of class
in their arguments and
-periodic with respect to t such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ10_HTML.gif)
Moreover, suppose that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ11_HTML.gif)
for all , for all
,
. Then every solution of (1.9) is bounded.
Remark 1.2.
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
We first recall the definitions of reversible system. Let be an open domain, and
be continuous. Suppose
is an involution (i.e.,
is a
-diffeomorphism such that
) satisfying
. The differential equations system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ12_HTML.gif)
is called reversible with respect to , if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ13_HTML.gif)
with denoting the Jacobian matrix of
.
We are interested in the special involution with
. Let
. Then
is reversible with respect to
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ14_HTML.gif)
Below we will see that the symmetric properties (1.10) imply a reversible structure of the system (1.9).
Let . Then
, where
satisfies
. Hence, the differential equation (1.9) is changed into the following planar system:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ15_HTML.gif)
By (1.10) it is easy to see that the system (2.4) is reversible with respect to the involution .
Below we will write the reversible system (2.4) as a form of small perturbation. For this purpose we first introduce action-angle variables .
Consider the homogeneous differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ16_HTML.gif)
This equation takes as an integrable part of (1.9). We will use its solutions to construct a pair of action-angle variables. One of solutions for (2.5) is the function as defined below. Let the number
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ17_HTML.gif)
We define the function , implicitly by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ18_HTML.gif)
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:
(i),
;
(ii);
(iii) is an odd periodic function with period
.
It is easy to verify that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ19_HTML.gif)
with initial condition . Define a transformation
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ20_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ21_HTML.gif)
Since the Jacobian matrix of is nonsingular for
, the transformation
is a local homeomorphism at each point
of the set
, while
is a global homeomorphism from
to
. Under the transformation
the system (2.4) is changed to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ22_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ23_HTML.gif)
with .
It is easily verified that and
and so the system (2.11) is reversible with respect to the involution
.
2.2. Some Lemmas
To estimate and
, we need some definitions and lemmas.
Lemma 2.1.
Let . If
and
satisfy (1.11), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ24_HTML.gif)
for ,
.
Proof.
We only prove the second inequality since the first one can be proved similarly.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ25_HTML.gif)
To describe the estimates in Lemma 2.1, we introduce function space , where
is a function of
.
Definition 2.2.
Let . We say
, if for
, there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ26_HTML.gif)
Lemma 2.3 (see [6]).
The following conclusions hold true:
(i)if , then
and
;
(ii)if and
, then
;
(iii)Suppose satisfy that, there exists
such that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ27_HTML.gif)
If ,
,
, then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ28_HTML.gif)
Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ29_HTML.gif)
Proof.
This lemma was proved in [6], but we give the proof here for reader's convenience. Since (i) and (ii) are easily verified by definition, so we only prove (iii). Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ30_HTML.gif)
Since , we have
. So
. Thus
is bounded and so
. Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ31_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ32_HTML.gif)
Since , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ33_HTML.gif)
Let . Since
, we know that for
sufficiently large
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ34_HTML.gif)
By the property of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ35_HTML.gif)
for sufficiently large.
If , then by a direct computation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ36_HTML.gif)
where the sum is found for the indices satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ37_HTML.gif)
Without loss of generality, we assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ38_HTML.gif)
Furthermore, we suppose that among , there are
numbers which equal to 0, and among
, there are
numbers which equal to 0.
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ39_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ40_HTML.gif)
and then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ41_HTML.gif)
Obviously
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ42_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ43_HTML.gif)
By the condition of (iii) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ44_HTML.gif)
In the same way we can consider and we omit the details.
2.3. Some Estimates
The following lemma gives the estimate for and
.
Lemma 2.4.
,
, where
.
Proof.
Since , we first consider
and
. By Lemma 2.1,
. Again
, using the conclusion (iii) of Lemma 2.3, we have
, where
. Note that
and
, we have
. In the same way we can prove
. Thus Lemma 2.4 is proved.
Since , we get
. So
for sufficiently large
. When
the system (2.11) is equivalent to the following system:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ45_HTML.gif)
It is easy to see that and
. Hence, system (2.34) is reversible with respect to the involution
.
We will prove that the Poincaré mapping can be a small perturbation of integrable reversible mapping. For this purpose, we write (2.34) as a small perturbation of an integrable reversible system. Write the system (2.34) in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ46_HTML.gif)
where ,
, with
and
defined in (2.11). It follows
, and so (2.35) is also reversible with respect to the involution
. Below we prove that
and
are smaller perturbations.
Lemma 2.5.
,
.
Proof.
If is sufficiently large, then
and so
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ47_HTML.gif)
It is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ48_HTML.gif)
where ,
, and
and
are defined in the same way as
and
.
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ49_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ50_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ51_HTML.gif)
Thus, . In the same way, we have
.
Now we change system (2.35) to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ52_HTML.gif)
where and
. By the proof of Lemma 2.4, we know
and
. Thus,
and
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ53_HTML.gif)
with ,
.
2.4. Coordination Transformation
Lemma 2.6.
There exists a transformation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ54_HTML.gif)
such that the system (2.41) is changed into the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ55_HTML.gif)
where satisfy:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ56_HTML.gif)
Moreover, the system (2.44) is reversible with respect to the involution G: .
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ57_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ58_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ59_HTML.gif)
Hence the map with
is diffeomorphism for
. Thus, there is a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ60_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ61_HTML.gif)
Under this transformation, the system (2.41) is changed to (2.44) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ62_HTML.gif)
Below we estimate and
. We only consider
since
can be considered similarly or even simpler.
Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ63_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ64_HTML.gif)
By the third conclusion of Lemma 2.3, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ65_HTML.gif)
In the same way as the above, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ66_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ67_HTML.gif)
By (2.54) and (2.56), noting that , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ68_HTML.gif)
Since , the system (2.44) is reversible in
with respect to the involution
. Thus Lemma 2.6 is proved.
Now we make average on the nonlinear term in the second equation of (2.44).
Lemma 2.7.
There exists a transformation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ69_HTML.gif)
which changes (2.44) to the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ70_HTML.gif)
where with
and the new perturbations
satisfy:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ71_HTML.gif)
Moreover, the system (2.59) is reversible with respect to the involution G: .
Proof.
We choose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ72_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ73_HTML.gif)
Defined a transformation by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ74_HTML.gif)
Then the system of (2.44) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ75_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ76_HTML.gif)
It is easy to very that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ77_HTML.gif)
which implies that the system (2.59) is reversible with respect to the involution G: . In the same way as the proof of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ78_HTML.gif)
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.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ79_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ80_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ81_HTML.gif)
Now, we define a transformation by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ82_HTML.gif)
Then the system (2.59) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ83_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ84_HTML.gif)
Lemma 2.8.
The perturbations and
satisfy the following estimates:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ85_HTML.gif)
Proof.
By (2.73), (2.60) and noting that , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ86_HTML.gif)
In the same way, . The estimates (2.74) for
follow easily from (2.60).
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 [13]. Later, Liu [14] improved the previous results. Let us first recall the theorem in [14].
Let be a finite part of cylinder
, where
, we denote by
the class of Jordan curves in
that are homotopic to the circle
. The subclass of
composed of those curves lying in
will be denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ87_HTML.gif)
Consider a mapping , which is reversible with respect to
. Moreover, a lift of
can be expressed in the form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ88_HTML.gif)
where is a real number,
is a small parameter, the functions
,
,
, and
are
periodic.
Lemma 2.9 (see [14, Theorem 2]).
Let with an integer n and the functions
,
,
, and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ89_HTML.gif)
In addition, we assume that there is a function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ90_HTML.gif)
Moreover, suppose that there are two numbers , and
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ91_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ92_HTML.gif)
Then there exist and
such that, if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ93_HTML.gif)
the mapping has an invariant curve in
, the constant
and
depend on
, and
. In particular,
is independent of
.
Remark 2.10.
If satisfy all the conditions of Lemma 2.9, then Lemma 2.9 still holds.
Lemma 2.11 (see [14, Theorem 1]).
Assume that and
,
and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ94_HTML.gif)
then there exist and
such that
has an invariant curve in
if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ95_HTML.gif)
The constants and
depend on
only.
We use Lemmas 2.9 and 2.11 to prove our Theorem 1.1. For the reversible mapping (2.86), ,
.
2.6. Invariant Curves
From (2.73) and (2.66), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ96_HTML.gif)
which yields that system (2.72) is reversible in with respect to the involution
. Denote by
the Poincare map of (2.72), then
is also reversible with the same involution
and has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ97_HTML.gif)
where and
. Moreover,
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ98_HTML.gif)
Case 1 ( is rational).
Let , it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ99_HTML.gif)
Since only depends on
, and
, all conditions in Lemma 2.9 hold.
Case 2 ( is irrational).
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F845413/MediaObjects/13661_2010_Article_61_Equ100_HTML.gif)
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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Jiang, S., Xu, J. & Zhang, F. Lagrangian Stability of a Class of Second-Order Periodic Systems. Bound Value Probl 2011, 845413 (2011). https://doi.org/10.1155/2011/845413
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DOI: https://doi.org/10.1155/2011/845413