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
is called reversible with respect to
, if
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
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:
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:
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
We define the function
, implicitly by
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
with initial condition
. Define a transformation
by
It is easy to see that
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
where
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
for
,
.
Proof.
We only prove the second inequality since the first one can be proved similarly.
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
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
,
If
,
,
, then, we have
Moreover,
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
Since
, we have
. So
. Thus
is bounded and so
. Similarly, we have
For
, we have
Since
, it follows that
Let
. Since
, we know that for
sufficiently large
By the property of
, we have
for
sufficiently large.
If
, then by a direct computation, we have
where the sum is found for the indices satisfying
Without loss of generality, we assume that
Furthermore, we suppose that among
, there are
numbers which equal to 0, and among
, there are
numbers which equal to 0.
Since
we have
and then,
Obviously
Since
By the condition of (iii) we obtain
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:
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
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
It is easy to verify that
where
,
, and
and
are defined in the same way as
and
.
So, we have
where
So
Thus,
. In the same way, we have
.
Now we change system (2.35) to
where
and
. By the proof of Lemma 2.4, we know
and
. Thus,
and
where
with
,
.
2.4. Coordination Transformation
Lemma 2.6.
There exists a transformation of the form
such that the system (2.41) is changed into the form
where
satisfy:
Moreover, the system (2.44) is reversible with respect to the involution G:
.
Proof.
Let
then
It is easy to see that
Hence the map
with
is diffeomorphism for
. Thus, there is a function
such that
where
Under this transformation, the system (2.41) is changed to (2.44) with
Below we estimate
and
. We only consider
since
can be considered similarly or even simpler.
Obviously,
Note that
By the third conclusion of Lemma 2.3, we have that
In the same way as the above, we have
and so
By (2.54) and (2.56), noting that
, it follows that
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
which changes (2.44) to the form
where
with
and the new perturbations
satisfy:
Moreover, the system (2.59) is reversible with respect to the involution G:
.
Proof.
We choose
Then
Defined a transformation by
Then the system of (2.44) becomes
where
It is easy to very that
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
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
Since
then
Now, we define a transformation by
Then the system (2.59) has the form
where
Lemma 2.8.
The perturbations
and
satisfy the following estimates:
Proof.
By (2.73), (2.60) and noting that
, it follows that
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,
Consider a mapping
, which is reversible with respect to
. Moreover, a lift of
can be expressed in the form:
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
In addition, we assume that there is a function
satisfying
Moreover, suppose that there are two numbers
, and
such that
and
where
Then there exist
and
such that, if
and
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
then there exist
and
such that
has an invariant curve in
if
and
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
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
where
and
. Moreover,
and
satisfy
Case 1 (
is rational).
Let
, it is easy to see that
Since
only depends on
, and
, all conditions in Lemma 2.9 hold.
Case 2 (
is irrational).
Since
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.