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.