Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 328479 | 17 pages | https://doi.org/10.1155/2009/328479

Periodic Solution of Second-Order Hamiltonian Systems with a Change Sign Potential on Time Scales

Academic Editor: Yong Zhou
Received26 Nov 2008
Accepted04 Apr 2009
Published02 Jul 2009

Abstract

This paper is concerned with the second-order Hamiltonian system on time scales of the form a.e. , where . By using the minimax methods in critical theory, an existence theorem of periodic solution for the above system is established. As an application, an example is given to illustrate the result. This is probably the first time the existence of periodic solutions for second-order Hamiltonian system on time scales has been studied by critical theory.

1. Introduction

The theory of calculus on time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 [1]. It cannot only unify discrete and continuous calculus but also exhibit much more complicated dynamics on time scales [26]. In particular, dynamic equations on time scales have many important applications, such as, in the study of biological, heat transfer, stock market, and epidemic models [2, 5, 79]. Consequently, it has been attracted considerable amount of interest and is now a hot topic of still fairly theoretical exploration in mathematics.

Recently, for the existence problems of positive solutions for dynamic equations on time scales, some authors have obtained many results; for details, see [1021] and the references therein. To the best of our knowledge, there is no work on the existence of periodic solutions for second-order Hamiltonian systems on time scales. In particular, there is very little work [2224] on the existence of solutions of dynamic equations on time scales by using critical theory. Now, it is natural to use critical theory to consider the existence of periodic solutions for second-order Hamiltonian systems on time scales.

We make the blanket assumption that are points in for an interval we always mean Other types of intervals are defined similarly. We say that a property holds for -a.e. or -a.e. on whenever there exists a set with null Lebesgue -measure such that this property holds for every We refer the reader to [3, 24, 25] for a broad introduction on Lebesgue -measure.

In this paper, motivated by references [26, 27], we consider the following second-order Hamiltonian system on time scales of the form where , , is measurable in for every and continuously differentiable in for -a.e. and By using the minimax methods in critical theory, we establish the existence of at least one nonzero solution for the problem (1.1). Our results are even new for the special cases of difference equation and include the results of Tang and Wu [27] for differential equation. Moreover, we prove some lemmas, which will be very important in proving the existence of periodic solutions in spaces for many other second-order Hamiltonian systems on time scales. As an application, an example is given to illustrate the result.

There is a solution of problems (1.1); we mean which is delta differential; and are both continuous -a.e. on , and satisfies problems (1.1).

Now, we present some basic definitions which can be found in [35, 28]. Another excellent source on dynamical systems on measure chains is the book [6].

A time scale is a nonempty closed subset of For , the forward and back jump operators are well defined, respectively, by

In this definition, one puts and where denotes the empty set. A point is called left-dense if , left-scattered if , right-dense if , and right-scattered if

If has a right-scattered minimum define otherwise, set If has a left-scattered maximum define otherwise, set . The forward graininess is Similarly, the backward graininess is

If is a function and then the delta derivative [4] of at the point is defined to be the number (provided it exists) with the property that, for any , there is a neighborhood of such that

If and then the nabla derivative of at the point is defined by the number (provided it exists) with the property that, for any there is a neighborhood of such that

We refer the reader to [25] for measure on time scales; absolutely continuous on time scales can be found in [29]. We now provide the definition in [24, 30] and simply summarize the main points, which also be described in [31].

Let defined a function by

Now, suppose that is arbitrary function, if is measurable on the real interval in the usual Lebesgue senses, then we say is measurable; if is integrable on the real interval in the usual Lebesgue senses, then we say is integrable. Let denote the set of such integrable functions on . Furthermore, for any , we defined the integral of by

with the norm defined by We use the notation to denote the Lebesgue integral of a function between (when it is defined). That is, we use the same notation for the Lebesgue-type integral defined in [24, 30] as is commonly used in the time scale literature for a Riemann-type integral defined in terms of antiderivatives. A detailed discussion of the Lebesgue-type integral and it's relationship with the usual time scale integral is given in [24, 30]. With the Lebesgue integral defined, denote

It is shown in [31] that is completed with respect to the norm .

Next, define the norm on by

The space is the completion of with respect to the norm (see [31, Definition 4.1 and Remark 4.2]). The space is a time scale analog to the usual Sobolev space on a real interval .

We refer the reader to [32] for an introduction on basic properties of Sobolev's spaces on bounded time scales.

Remark 1.1. If we replace with , then the above discussion still holds.
The rest of the paper is organized as follows. In Section 2, we list some lemmas, which are important in proving the existence of periodic solutions. By applying these lemmas, we establish the existence of periodic solutions for problem (1.1). In the final section, an example is given to illustrate our main result.

2. Some Lemmas

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 spaces for second-order Hamiltonian systems on time scales.

Let be the Hilbert space given by

with the norm defined by Moreover, we define We also define inner product on by For let and let be the subspace of given by

In the following, we will prove several lemmas which are very important in proving the existence of periodic solutions for problem (1.1).

Lemma 2.1. Let be such that Then, for every the immersion is compact.

Proof. The proving is similar to the way as in proving of [32, Corollary 3.11], and we omit it here.

The following two lemmas are an immediate consequence of the [23, Proposition  3.6] (see also [23, Corollary  3.9]).

Lemma 2.2. let and If converges weakly in to , then converges uniformly to on

Lemma 2.3. If then In particular, if , then

Lemma 2.4. Let be measurable in for each and continuously differentiable in for -a.e. . If there exist , and , such that, for -a.e. and each , one has then the functional defined by is continuously differential on and

Proof. In the following, we will prove that has a directional derivative given by (2.9) and that the mapping is continuous.
(i) It follows easily from (2.8) that is everywhere finite on Fixing and in we define

We will apply Leibniz formula of differentiation under integral sign to . According to assumption (2.8), one obtains
where
It is obvious that , , . implies that and hold, hence we have
In view of Leibniz formula and (1.6), we get
Moreover
Thus, from Lemma 2.3, and has a directional derivative , given by (2.9).
(ii) According to the theorem of Krasnosel'skii [33], assumption (2.8) implies that the mapping from into defined by is continuous, thus, is continuous from into , and the proof is completed.

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

Lemma 2.5 (see [34]). Let be a real Hilbert space with and Suppose , satisfies (), and(I1) where and is bounded and self-adjoint, ,(I2) is compact, and(I3)there exist a subspace and sets and constants such that (i) and (ii) is bounded and (iii) and link.Then possesses a critical value

3. Existence Results

In this section, by using the minimax methods in critical theory, we establish the existence of at least one nonzero periodic solution for second-order Hamiltonian system (1.1) on time scales.

Throughout this section, the following is assumed.

(H1)Suppose that there exist and satisfying for all where In addition, (H2) for all and -a.e. (H3)Assume that there exists such that for all and -a.e. (H4)Assume that there exist and such that for all and -a.e. (H5)Assume that

If then by Lemma 2.4, the functional is given by which is continuously differentiable on . Moreover That is, for all , we get

Hence, is a solution of problem (1.1) if and only if is a critical point of .

Lemma 3.1. Let a sequence be such that and let be bounded in , then has a convergent subsequence in .

Proof. Since is bounded in , it follows from [30, Theorem 4.12] that there exists a subsequence (still denoted by ) which weakly converges to . By Lemma 2.2, we have Hence, for , there exists an such that
Lemma 2.1 leads to
Hence
Since
in view of (H1) and (H3), one has Consequently which implies that is a Cauchy sequence in . By the completeness of , we obtain that is a convergent sequence in ; the proof is completed.

Now, we list our main result.

Theorem 3.2. Suppose that (H1), (H2), (H3), (H4), and (H5) hold, then the problem (1.1) has at least one nonzero solution.

Proof. It suffices to show that all the conditions of Lemma 2.5 hold with respect to
First, we show that satisfies the (PS) condition.
From Lemma 3.1, we only need to prove that is bounded. Otherwise, there exists a subsequence (still denoted by ) such that . Let then is bounded; it has a subsequence (we still denote ) which weakly converges to . In view of Lemma 2.2, uniformly converges to in Since for all, one has
According to as , and (H3), for all we get Thus, it follows from Lebesgue dominated convergence theorem on time scales [28] that By the arbitrariness of , one has which contradicts the condition (H1). Hence satisfies the (PS) condition.
Second, if
then it is easy to verify that (I1) and (I2) hold.
Third, we will prove that satisfies the condition (I3) in Lemma 2.5.
For arbitrary with (H4) and (2.7) imply Choose small enough such that thus If then ; this implies that the condition (i) of Lemma 2.5 holds.
It is known that (H1) and (H2) lead to Choose such that for all and For arbitrary let then For all and , in terms of (H3), one has Since for all we have In terms of (3.20) and Hölder's inequality on time scales, one obtains Thus, by using (3.25), and Hölder's inequality on time scales again, for all and , we obtain (3.24) and (3.26) lead to which means that there exists such that For all and , let In view of for all and , we get (3.30) and imply that Note that there exists such that Therefore, by using (3.24), (3.26), (3.31), and (3.32), for and , one has Thus, (3.28) and (3.33) can lead to For all and , denote By using the similar way to inequality (3.26), one has for all and , since In view of (3.37) and for all and , we get From (3.24), (3.36), and (3.38), for all and , we obtain which implies that there exists , such that
Now let
Thus, (3.19), (3.34), and (3.40) lead to , which means that the condition (ii) of Lemma 2.5 is satisfied.
It is easy to see that and link. Hence, all the conditions of the generalized mountain pass theorem are satisfied. By Lemma 2.5, the problem (1.1) has at least one nonzero solution.

4. An Example

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

Let Consider the following second-order Hamiltonian system on time scales of the form where is a constant; let is arbitrary small, and It is easy to verify that all the conditions of Theorem 3.2 are satisfied. By Theorem 3.2, we see that the problem (4.2) has at least one nonzero solution.

Acknowledgment

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

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Copyright © 2009 You-Hui Su and Wan-Tong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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