^{1, 2}

^{2}

^{1}

^{2}

This paper is concerned with the second-order Hamiltonian system on time scales

The theory of calculus on time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 [

Recently, for the existence problems of positive solutions for dynamic equations on time scales, some authors have obtained many results; for details, see [

We make the blanket assumption that

In this paper, motivated by references [

There is a solution

Now, we present some basic definitions which can be found in [

A time scale

If

If

If

We refer the reader to [

Let

Now, suppose that

with the norm defined by

It is shown in [

Next, define the norm

The space

We refer the reader to [

If we replace

The rest of the paper is organized as follows. In Section

In this section, to interpret Hamiltonian systems on time scales in a functional-analytic setting, we introduce some lemmas, which will be used in the rest of the paper and be very important in proving the existence of periodic solutions in

Let

with the norm defined by

In the following, we will prove several lemmas which are very important in proving the existence of periodic solutions for problem (

Let

The proving is similar to the way as in proving of [

The following two lemmas are an immediate consequence of the [

let

If

Let

In the following, we will prove that

(i) It follows easily from (

We will apply Leibniz formula of differentiation under integral sign to

It is obvious that

Moreover

(ii) According to the theorem of Krasnosel'skii [

We also need the following theorem, which was the generalized mountain pass theorem.

Let

there exist a subspace

In this section, by using the minimax methods in critical theory, we establish the existence of at least

Throughout this section, the following is assumed.

Suppose that there exist

Assume that there exists

Assume that there exist

Assume that

If

Hence,

Let a sequence

Since

Lemma

Since

Now, we list our main result.

Suppose that

It suffices to show that all the conditions of Lemma

First, we show that

From Lemma

According to

Second, if

Third, we will prove that

For arbitrary

It is known that (H1) and (H2) lead to

Now let

It is easy to see that

In this section, we present a simple example to illustrate our result.

Let

This work was supported by the NSF of China (10571078) and the grant of XZIT (XKY2008311).