Periodic Solutions for a Class of Singular Hamiltonian Systems on Time Scales

Abstract

We are concerned with a class of singular Hamiltonian systems on time scales. Some results on the existence of periodic solutions are obtained for the system under consideration by means of the variational methods and the critical point theory.

1. Introduction

In recent years, dynamic equations on time scales have been studied intensively in the literature [1–7]. Some ideas and methods have been developed to study the existence and multiplicity of solutions for dynamic equations on time scales, for example, the fixed point theory, the method of the upper and lower solutions, the coincidence degree theory, and so on.

However, not much work has been seen on the existence of solutions to dynamic equations on time scales through the variational method and the critical point theory; for details see [4–10] and the references therein. For example, authors of [11] give some results on the existence and multiplicity of periodic solutions which are obtained for the Hamiltonian system by means of the saddle point theorem, the least action principle, and the three-critical-point theorem. To the best of our knowledge, it is still worth making an attempt to extend variational methods to study the existence of periodic solutions for various Hamiltonian systems. Naturally, it is interesting and necessary to study the existence of periodic solutions for Hamiltonian systems on time scales.

Besides, in [12], using Lyusternik-Schnirelmann theory with classical compact condition, Ambrosetti-Coti Zelati studied the periodic solutions of a fixed energy for Hamiltonian systems with singular potential : After Ambrosetti-Coti Zelati, a lot of mathematicians studied singular Hamiltonian systems (see, e.g., [13–15]).

Motivated by the above, in this paper, we consider the following second order Hamiltonian system with a fixed energy on time scale : where denotes the delta (or Hilger) derivative of at , , is the forward jump operator, , and satisfies the following assumption: is measurable in for every and continuously differentiable in for .

The paper is organized as follows. In Section 2, we introduce some definitions and make some preparations for later sections. We summarize our main results on the existence of periodic solutions of the second order Hamiltonian system on time scales in Section 3.

2. Preliminaries

In this section, we will first recall some fundamental definitions and lemmas which are used in what follows.

Definition 1 (see [3]). A time scale is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from . The forward jump operator is defined by for all , while the backward jump operator is defined by for all . Finally, the graininess function is defined by .

Definition 2 (see [3]). Assume that is a function and let . Then we define to be the number provided it exists with the property that given any , there is a neighborhood of (i.e., for some such that We call the delta (or Hilger) derivative of at . The function is delta (or Hilger) differentiable on provided exists for all . The function is then called the delta derivative of on . Then we define the function by for all .

Definition 3 (see [3]). For a function we will talk about the second derivative provided is differentiable on with derivative .

Definition 4 (see [3]). A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in .

Lemma 5 (see [3]). A function is absolutely continuous on if and only if is delta differentiable -a.e. on and

Lemma 6 (see [3]). Assume the functions , are absolutely continuous on ; then is absolutely continuous on and the following equality is valid:

In the following, we adopt the notations used in [4].

Lemma 7 (see [4]). There exists such that the inequality holds for all , where .
Moreover, if , then

Lemma 8 (see [4]). If the sequence converges weakly to in , then converges strongly in to .

Lemma 9 (see [4]). Assume that . Then, for every the immersion is compact.

Lemma 10 (see [11]). If is weakly lower semicontinuous on a reflexive Banach space and has a bounded minimizing sequence, then has a minimum on .

3. Existence of Periodic Solutions

3.1. Variational Structure

In this section, we present a recent approach via variational methods and critical point theory to obtain the existence of periodic solutions for the second order Hamiltonian systems on time scale .

By making a variational structure on , we can reduce the problem of finding solutions of (2) to the one of seeking the critical points of a corresponding functional.

Let It is easy to see that is a Hilbert space with the norm defined by In addition, let denote the Hilbert space of -periodic functions on time scale with values in , and the norm is defined by Let We define the equivalent norm in as follows: Let where . It is easy to see that .

Lemma 11 (see [16]). The system (2) will be said to satisfy the strong force (SF) condition if and only if there exists a neighborhood of and a function on such that (i) as ;(ii) for all in .

Let Then we have Let Then we have

Lemma 12. The existence of a bounded minimizing sequence is insured when is coercive in space .

Proof. Since is coercive, . So we can choose a and a positive number such that as .
Let ; it is easy to see that the set is a weakly closed subset; there exists such that we obtain . The proof is complete.

Lemma 13. if and only if .

Proof. On the one hand, by the definition of , we have thus if , then .
On the other hand, if then or . If then If , then . Since so . The proof is complete.

Lemma 14 (see [13]). Let be a Banach space, and let be a weakly closed subset. Suppose that is defined on an open subset and for any . Let for . Assume that and is weakly lower semicontinuous on and that it is coercive on : Then attains its infimum in .

Lemma 15 (see [13]). The functional attains the infimum on ; furthermore, the minimizer is nonconstant.

Consider the functional defined by For any and , we have It follows from the dominated convergence theorem on time scales that From the preceding discussions, we know that the critical points of functional are classical periodic solutions of systems (2) and (3). It is obvious that the functional is continuously differentiable and weakly lower semicontinuous on .

3.2. Results on the Existence

In this subsection, we present two results on the existence of periodic solutions for the Hamiltonian system on time scales.

Throughout this subsection, we assume thatthere exist such that  for all and -a.e. , where denotes the gradient of in .

Theorem 16. Assume that conditions and hold, and the following two conditions are true: as ;there exists such that and Then, systems (2)-(3) have at least one periodic solution which minimizes the function .

Proof. By Lemma 7, there exists such that It follows from , Lemma 7, and (31) that for all , where . Therefore, we have for all . It follows from Lemmas 11 and 13 and that as . Therefore, systems (2)-(3) have at least one periodic solution.

Example 17. Let and . Consider the following second order Hamiltonian system on time scale of the form where . It is easy to verify that and all conditions of Theorem 16 are satisfied. By Theorem 16 we see that system (34) has at least one solution.

Theorem 18. Assume that satisfies condition and the following two conditions:there exist constants , such that for the following inequality uniformly holds: for the following equality uniformly holds: Then for any , systems (2)-(3) have at least one nonconstant periodic solution with the given energy .

3.3. Proof of Theorem 18

Lemma 19. Assume holds; then, for any weakly convergent sequence , one has

Proof. Notice that implies Gordon’s strong force condition. It follows from and there exist constants and such that for all with and . By and satisfying Gordon’s strong force condition, we have By in the Hilbert space , we know that is bounded.
If , then by Sobolev’s embedding theorem, we have the uniform convergence property: By the symmetry of , we have . So, by Sobolev’s inequality we have If , then, by the weakly lower semicontinuity for the norm, we have So, by Gordon’s lemma, we have The proof is complete.

Lemma 20. The functional is weakly lower semicontinuous on .

Proof. Let Then, for given energy , is continuous and convex. Hence, is weakly lower semicontinuous. On the other hand, let , in . By Lemma 14, converges strongly in to . By condition , we have Thus, is weakly continuous. Consequently, is weakly lower semicontinuous. The proof is complete.

Lemma 21. is a weakly closed subset of .

Proof. From Lemma 9, is compact, so the proof is obvious.

Lemma 22. The functional is coercive on .

Proof. By the definition of and the assumption , we have that is, as for ; then is coercive in . The proof is complete.

Since is lower semicontinuous and coercive, we obtain that is bounded below and has a bounded minimizing sequence. By virtue of Lemmas 10 and 15, we know that has at least one nonconstant critical point in . That is, systems (2)-(3) have at least one nonconstant periodic solution with the given energy .

Example 23. Let and . Consider the second order Hamiltonian system on time scale : where . It is easy to verify that and all conditions of Theorem 18 are satisfied. By Theorem 18 we see that system (34) has at least one nonconstant periodic solution with the given energy .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant 11361072.