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

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 bymeans 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 (PS)+ compact condition, Ambrosetti-Coti Zelati studied the periodic solutions of a fixed energy h ∈ R for Hamiltonian systems with singular potentialV ∈ C2(Rn \ 0,R):


Introduction
In recent years, dynamic equations on time scales have been studied intensively in the literature [1][2][3][4][5][6][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][5][6][7][8][9][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 ℎ ∈ R for Hamiltonian systems with singular potential  ∈  2 (R  \ 0, R): q     2 +  () = ℎ. ( After Ambrosetti-Coti Zelati, a lot of mathematicians studied singular Hamiltonian systems (see, e.g., [13][14][15]).Motivated by the above, in this paper, we consider the following second order Hamiltonian system with a fixed energy ℎ on time scale T: () + ∇ ( () ,   ()) = 0, where  Δ () denotes the delta (or Hilger) derivative of  at ,  Δ 2 () = ( Δ ) Δ (),  is the forward jump operator, ℎ ∈ R, and  : T × R  → R satisfies the following assumption: 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.

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

Journal of Mathematics
Definition 1 (see [3]).A time scale T is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The forward jump operator  : T → T is defined by () = inf{ ∈ T,  > } for all  ∈ T, while the backward jump operator  : T → T is defined by () = sup{ ∈ T,  < } for all  ∈ T. Finally, the graininess function Definition 2 (see [3]).Assume that  : T → R is a function and let  ∈ T  .Then we define  Δ () to be the number (provided it exists) with the property that given any  > 0, there is a neighborhood  of  (i.e.,  = ( − ,  + ) ∩ T for some  > 0) such that We call  Δ () the delta (or Hilger) derivative of  at .The function  is delta (or Hilger) differentiable on T  provided  Δ () exists for all  ∈ T  .The function  Δ : T  → R is then called the delta derivative of  on T  .Then we define the function   : T  → R by   () = (()) for all  ∈ T  .
Definition 3 (see [3]).For a function  : T → R we will talk about the second derivative Definition 4 (see [3]).A function  : T → R  is called rdcontinuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T.
Lemma 5 (see [3] Lemma 6 (see [3]).Assume the functions ,  : [, ] T → R are absolutely continuous on [, ] T ; then  is absolutely continuous on [, ] T and the following equality is valid: In the following, we adopt the notations used in [4].

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 T.
By making a variational structure on  1 Δ, , 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  1 Δ, is a Hilbert space with the norm defined by In addition, let  2 Δ (T, R  ) denote the Hilbert space of periodic functions on time scale T with values in R  , and the norm is defined by Let We define the equivalent norm in  = { ∈  1 Δ, =  1,2  Δ, (T, R  ), ( + (/2)) = −(), ∀ ∈ T} as follows: Let where 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 {0} and a  2 function  on  − {0} such that Then we have Let Then we have Lemma 12.The existence of a bounded minimizing sequence is insured when  is coercive in space H1 Δ, .

Lemma 13. ‖𝑢‖ → ∞ if and only if
Proof.On the one hand, by the definition of ‖‖, we have On the other hand, if then then so ‖‖ → ∞.The proof is complete.
Consider the functional  :  1 Δ, → R defined by For any V ∈  1 Δ, and 0 < || < 1, 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  1 Δ, .

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 that for all  ∈ R  and Δ-a.e.  ∈ T, where ∇(, ) denotes the gradient of (, ) in .
Theorem 16.Assume that conditions ( 0 ) and ( 1 ) hold, and the following two conditions are true: Then, systems ( 2)-( 3) have at least one periodic solution which minimizes the function .
Example 17.Let T = {1/ :  ∈ N} ∩ {0} and  = 1.Consider the following second order Hamiltonian system on time scale T of the form where ((), ) = 2|| 2 .It is easy to verify that (H 0 ) and all conditions of Theorem 16 are satisfied.By Theorem 16 we see that system (34) has at least one solution.

Proof of
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  0 .That is, systems (2)-( 3) have at least one nonconstant periodic solution with the given energy ℎ.
where ((), ) = || 2 (0 <  <  3 ).It is easy to verify that (H 0 ) 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 ℎ.