In this section we state the main results and give its proof.
Theorem 3.1 Suppose the assumptions (H1)-(H4) hold. Then (1.3) has at least one T-periodic solution provided .
Proof Let , suppose , then
Integrating both sides of the second equation of (3.1) from 0 to T, we have
which implies that there exists a point such that
Now we claim that there must exist a point such that
In fact, let , , then from (3.2), . If holds, (3.3) is clearly true. Now assume .
Set , according to (H1), one has
namely , then there must exist a such that . That is, . According to (H2), (3.3) holds. If holds, by a similar method, we can show that (3.3) is also true.
Let , where and n is an integer. Then
Thus we get
On the other hand, multiplying both sides of the second equation of (3.1) by and integrating over , we have
where . Furthermore, in view of (H4), we have
Denote , then
Together with (3.6) and the fact yields
In view of Lemma 2.3, we obtain
where , are defined by Lemma 2.3, and
From (3.3), there exists a point such that . Let , we have , , and ; then, by Lemma 2.2,
By the Minkowski inequality, we get
Combining (3.7) and (3.8), we obtain
Since , there is a constant independent of λ such that , i.e., .
By the first equation of (3.1), we have
together with , which implies that there is a constant such that . Hence
By the two equations of (3.1) and Hölder’s inequality, we obtain
where , , . By (3.10), (3.11), .
Let . If , then . In view of , we have
So . Together with (H2) yields
Let . Then and Ω is a bounded open set of X. So (1) and (2) of Lemma 2.1 are satisfied.
In the next step we show that condition (3) of Lemma 2.1 holds. Define a linear isomorphism by , and let
The direct computation and (H2) show that for ,
Thus, for , which implies
Condition (3) of Lemma 2.1 holds. By Lemma 2.1, the equation has a solution. This completes the proof of Theorem 3.1. □
We can use a similar method to conclude the following result, the details are omitted.
Theorem 3.2 Suppose (H3), (H4), and the following assumptions hold:
() , , and , , , and .
() There is a constant such that if and only if .
Then (1.3) has at least one T-periodic solution if .