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This paper is devoted to the study of second-order Duffing
equation

We deal with the second-order Duffing equation

where

As is well known, time map is the right tool to build an approach to the study of periodic solution of Equation (1) (see [

Consider the auxiliary autonomous system

and suppose that

Obviously, the orbits of system (2) are curves

Obviously, the orbits of system (2) are curves

where

In view of the assumptions (g_{0}), (g_{1}) and (G_{0}), there exists a

where

We recall an interesting result in [

and a kind of nonresonance condition for the time map

where

Now naturally, we consider the question whether Equation (1) has harmonic solution when we permit

cross resonance points and use a kind of nonresonance condition for time map. In the following we will give a positive answer. In order to state the main result of this paper, set

and assume that

Theorem 1.1 Assume that

Our main result is following.

Theorem 1.1 Assume that

In this case, we generalize the result in [

The remainer of the paper is organized as follows. In Section 2, we introduce some technical tools and present all the auxiliary results. In Section 3, we will give the proof of Theorem 1.1 by applying the phase-plane analysis methods and Poincaré-Bohl fixed point theorem.

we assume throughout the paper that

where

Dropping the hats for simplification of notations, we assume that

and

Thus,

and

We will prove Theorem 1.1 under conditions

Consider the equivalent system of (6):

Let

We now follow a method which was used by [

Lemma 2.1 Assume that conditions

By Lemma 2.1, we can define Poincaré map

It is obvious that the fixed points of the Poincaré map

Lemma 2.2 Assume that

where

This result has been proved in [

Using Lemma 2.2, we see that

Denote by

Thus, we can rewrite the Poincaré map in the form

where

For the convenience, two lemmas in [

Lemma 2.3 Assume that

Lemma 2.4 Assume that

Lemma 2.5 Assume that

Proof. Without loss of generality, we may assume that

sufficiently large

Throughout the lemma, we always assume that

(1) We shall first estimate

(2) We now estimate

Then,

Therefore, for

Note that

Since

where

to

By (10) and (11), we have

Let

that is,

Consequently,

Integrating both sides of the above inequality from

Recalling the conditions

for

From [

for

In the following, we deal with

By (13), we derive

On the other hand, from (11) we have

As a result,

Accordingly,

Meanwhile, following

Combining (16)-(19), we get

for

Using the same arguments as above, we can get

By the conditions (20), (21), we have

Recalling

The proof is complete.

In this section, we establish the existence of harmonic solutions for Equation (1) by appealing to Poincaré-Bohl theorem [

From Lemma 2.5 and condition

which implies

Thus, the image