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 (PS)+ compact condition, Ambrosetti-Coti Zelati studied the periodic solutions of a fixed energy h∈R for Hamiltonian systems with singular potential V∈C2(Rn∖0,R):
(1)q¨+V′q=0,12q˙2+Vq=h.
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 h on time scale T:
(2)qΔ2t+∇Vσt,qσt=0,(3)12qΔt2+Vσt,qσt=h,
where qΔ(t) denotes the delta (or Hilger) derivative of q at t, qΔ2(t)=(qΔ)Δ(t), σ is the forward jump operator, h∈R, and V:T×Rn→R satisfies the following assumption:
V(t,x) is measurable in t for every x∈Rn and continuously differentiable in x for t∈T.
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 T is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The forward jump operator σ:T→T is defined by σ(t)=inf{s∈T,s>t} for all t∈T, while the backward jump operator ρ:T→T is defined by ρ(t)=sup{s∈T,s<t} for all t∈T. Finally, the graininess function μ:T→[0,∞) is defined by μ(t)=σ(t)-t.
Definition 2 (see [3]).
Assume that f:T→R is a function and let t∈Tκ. Then we define fΔ(t) to be the number (provided it exists) with the property that given any ϵ>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩T for some δ>0) such that
(4)fσt-fs-fΔtσt-sM≤ϵσt-sM∀s∈U.
We call fΔ(t) the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable on Tκ provided fΔ(t) exists for all t∈Tκ. The function fΔ:Tκ→R is then called the delta derivative of f on Tκ. Then we define the function fσ:Tκ→R by fσ(t)=f(σ(t)) for all t∈Tκ.
Definition 3 (see [3]).
For a function f:T→R we will talk about the second derivative fΔ2 provided fΔ is differentiable on Tκ2=(Tκ)κ with derivative fΔ2=(fΔ)Δ:Tκ2→R.
Definition 4 (see [3]).
A function f:T→RN is called rd-continuous 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]).
A function f:[a,b]T→R is absolutely continuous on [a,b]T if and only if f is delta differentiable Δ-a.e. on [a,b)T and
(5)ft=fa+∫a,bTfΔsΔs,∀t∈a,bT.
Lemma 6 (see [3]).
Assume the functions f, g:[a,b]T→R are absolutely continuous on [a,b]T; then fg is absolutely continuous on [a,b]T and the following equality is valid:
(6)∫a,bTfΔg+fσgΔtΔt=fbgb-faga=∫a,bTfgΔ+fΔgσtΔt.
In the following, we adopt the notations used in [4].
Lemma 7 (see [4]).
There exists K>0 such that the inequality
(7)u∞≤KuWΔ,T1,p
holds for all u∈WΔ,T1,p([0,T]T,RN), where u∞=maxt∈[0,T]T|u(t)|.
Moreover, if ∫[0,T)Tu(t)Δt=0, then
(8)u∞≤KuΔLΔp.
Lemma 8 (see [4]).
If the sequence {uk}k∈N⊂WΔ,T1,p([0,T]T,RN) converges weakly to u in WΔ,T1,p([0,T]T,RN), then {uk}k∈N converges strongly in C([0,T]T,RN) to u.
Lemma 9 (see [4]).
Assume that p≥1(p∈R¯). Then, for every q∈[1,+∞) the immersion WΔ1,p([a,b]T,Rn)↪LΔq([a,b]T,Rn) is compact.
Lemma 10 (see [11]).
If f is weakly lower semicontinuous on a reflexive Banach space X and has a bounded minimizing sequence, then f has a minimum on X.
3. Existence of Periodic Solutions3.1. 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 HΔ,T1, we can reduce the problem of finding solutions of (2) to the one of seeking the critical points of a corresponding functional.
Let
(9)HΔ,T1=WΔ,T1,2T,Rn=u:T⟶Rn∣u∈LΔ2T,Rn,MMuΔ∈LΔ2T,Rn,ut+T=ut,∀t∈T.
It is easy to see that HΔ,T1 is a Hilbert space with the norm defined by
(10)u=uHΔ,T1=∫0,TTut2Δt+∫0,TTuΔt2Δt1/2.
In addition, let LΔ2(T,Rn) denote the Hilbert space of T-periodic functions on time scale T with values in Rn, and the norm is defined by
(11)uLΔ,T2=∫0,TTut2Δt1/2.
Let
(12)W=u∈HΔ,T1∣ut≠0,∀t∈T,W0=u∈HΔ,T1=WΔ,T1,2T,Rn,t+T2MMut+T2=-ut,ut≠0,∀t∈T.
We define the equivalent norm in E={u∈HΔ,T1=WΔ,T1,2(T,Rn),u(t+T/2)=-u(t),∀t∈T} as follows:
(13)u=uE=∫0,TTuΔt2Δt1/2.
Let
(14)u~=u-u¯,H~Δ,T1=u∈HΔ,T1:u¯=0,
where u¯=1/T∫[0,T)Tu(t)Δt. It is easy to see that HΔ,T1=Rn⊕H~Δ,T1.
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 N of {0} and a C2 function U on N-{0} such that
U(x)→-∞ as x→0;
-V(x)≥|∇U(x)|2 for all x in N-{0}.
Let
(15)∂W=u∈HΔ,T1=WΔ,T1,2T,Rn,∃t0∈T,ut0=0.
Then we have
(16)∫0,TTVσt,unσtΔt⟶-∞,∀un⇀u∈∂W.
Let
(17)∂W0=u∈HΔ,T1=WΔ,T1,2T,Rn,ut+T2=-ut,MMt+T2∃t0∈T,ut0=0.
Then we have
(18)∫0,TTVσt,unσtΔt⟶-∞,000000000000000∀un⇀u∈∂W0.
Lemma 12.
The existence of a bounded minimizing sequence is insured when f is coercive in space H~Δ,T1.
Proof.
Since f is coercive, limu→∞f(u)=+∞. So we can choose a u0∈H~Δ,T1 and a positive number δ>0 such that f(u)≥2f(u0)>f(u0) as u>δ.
Let B={u∈H~Δ,T1∣u<δ}; it is easy to see that the set B is a weakly closed subset; there exists u1∈B such that
(19)fu1=infu∈Bfu≤fu0;
we obtain f(u1)=infu∈H~Δ,T1f(u). The proof is complete.
Lemma 13.
u→∞ if and only if (|u¯|2+∫[0,T)T|uΔ(t)|2Δt)1/2→∞.
Proof.
On the one hand, by the definition of u, we have
(20)u=uHΔ,T1=∫0,TTut2Δt+∫0,TTuΔt2Δt1/2=∫0,TTu¯+u~t2Δt+∫0,TTuΔt2Δt1/2≤∫0,TTu¯+u~∞2Δt+∫0,TTuΔt2Δt1/2≤∫0,TT4maxu¯,u~∞2Δt+∫0,TTuΔt2Δt1/2≤4Tmaxu¯,u~∞2+∫0,TTuΔt2Δt1/2≤4Tu¯2+1+TC∫0,TTuΔt2Δt1/2≤Mu¯2+∫0,TTuΔt2Δt1/2;
thus if u→∞, then (|u¯|2+∫[0,T)T|uΔ(t)|2Δt)1/2→∞.
On the other hand, if
(21)u¯2+∫0,TTuΔt2Δt1/2⟶∞,
then ∫[0,T)T|uΔ(t)|2Δt→∞ or |u¯|2→∞. If
(22)∫0,TTuΔt2Δt⟶∞,
then
(23)u=∫0,TTut2Δt+∫0,TTuΔt2Δt1/2⟶∞.
If |u¯|2→∞, then 1/T∫[0,T)T|u(t)|Δt→∞. Since
(24)∫0,TTutΔt≤T∫0,TTut2Δt1/2,
so u→∞. The proof is complete.
Lemma 14 (see [13]).
Let X be a Banach space, and let E⊂X be a weakly closed subset. Suppose that g(u) is defined on an open subset W⊂X and g(u)≠-∞ for any u∈W. Let g(u)=+∞ for u∈∂W. Assume that g(u)≠+∞ and is weakly lower semicontinuous on W¯∩E and that it is coercive on W∩E:
(25)gu⟶+∞,u⟶+∞,gun⟶+∞,un⇀u∈∂W.
Then g attains its infimum in W∩E.
Lemma 15 (see [13]).
The functional f(u) attains the infimum on W0; furthermore, the minimizer is nonconstant.
Consider the functional f:HΔ,T1→R defined by
(26)fu=12∫0,TTuΔt2h-Vσt,uσtΔt.
For any v∈HΔ,T1 and 0<|λ|<1, we have
(27)1λfu+λv-fuM=12∫0,TTuΔ+λvΔ2h-Vσt,uσ+λvσλΔtMM-12∫0,TTuΔ2h-Vσt,uσλΔtM=12∫0,TTuΔ2hλΔtMM-12∫0,TTuΔ2Vσt,uσ+λvσλΔtMM+∫0,TTuΔ,vΔhΔtMM-∫0,TTuΔ,vΔVσt,uσ+λvσΔtMM+12∫0,TTλvΔ2hΔtMM-12∫0,TTλvΔ2Vσt,uσ+λvσΔtMM-12∫0,TTuΔ2hλΔt+12∫0,TTuΔ2Vσt,uσλΔtM=-12∫0,TTuΔ2Vσt,uσ+λvσ-Vσt,uσλΔtMM+∫0,TTuΔ,vΔh-Vσt,uσ+λvσΔtMM+12∫0,TTλvΔ2h-Vσt,uσ+λvσΔt.
It follows from the dominated convergence theorem on time scales that
(28)f′u,v=limλ→01λfu+λv-fu=-12∫0,TTlimλ→0uΔ2Vσt,uσ+λvσ-Vσt,uσλΔtM+∫0,TTlimλ→0uΔ,vΔh-Vσt,uσ+λvσΔtM+12∫0,TTlimλ→0λvΔ2h-Vσt,uσ+λvσΔt=-12∫0,TTlimλ→0uΔ2Vσt,uσ+λvσ-Vσt,uσvσλvσΔtM+∫0,TTlimλ→0uΔ,vΔh-Vσt,uσ+λvσΔtM+12∫0,TTlimλ→0λvΔ2h-Vσt,uσ+λvσΔt=-12∫0,TTuΔ2∇Vσt,uσ,vσΔtM+∫0,TTuΔ,vΔh-Vσt,uσΔt.
From the preceding discussions, we know that the critical points of functional f are classical periodic solutions of systems (2) and (3). It is obvious that the functional f is continuously differentiable and weakly lower semicontinuous on HΔ,T1.
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 that
there exist a∈C(R+,R+),b∈LΔ1(T,R+) such that
(29)Vt,x≤axbt,∇Vt,x≤axbt
for all x∈Rn and Δ-a.e. t∈T, where ∇V(t,x) denotes the gradient of V(t,x) in x.
Theorem 16.
Assume that conditions (H0) and (H1) hold, and the following two conditions are true:
∫[0,T)TV(σ(t),x)Δt→+∞ as |x|→+∞;
there exists g∈L1([0,T],R+) such that gσ∈LΔ1([0,T)T,R+) and
(30)∇Vσt,x≤gσt.
Then, systems (2)-(3) have at least one periodic solution which minimizes the function f.
Proof.
By Lemma 7, there exists C1>0 such that
(31)u~∞2≤C1∫0,TTuΔt2Δt.
It follows from (H3), Lemma 7, and (31) that
(32)∫0,TTVσt,uσt-Vσt,u-ΔtM≤∫0,TT∫01∇Vσt,u-+su~σt,u~σtdsΔtM≤∫0,TT∫01gσtu~σtdsΔtM≤u~∞∫0,TTgσtΔtM≤C2∫0,TTuΔt2Δt1/2
for all u∈HΔ,T1, where C2=(C1)1/2∫[0,T)Tgσ(t)Δt. Therefore, we have
(33)fuM=12∫0,TTuΔt2h-Vσt,uσtΔtM=h2∫0,TTuΔt2ΔtMM-12∫0,TTuΔt2Vσt,uσtΔtM=h2∫0,TTuΔt2ΔtMM-12∫0,TTuΔt2Vσt,uσt-Vσt,u-MMMMMMMMMMMVσt,uσt-Vσt,u-+Vσt,u-ΔtM≥h2∫0,TTuΔt2Δt+C22∫0,TTuΔt2Δt3/2MM-12∫0,TTuΔt2×Vσt,u-Δt
for all u∈HΔ,T1. It follows from Lemmas 11 and 13 and (H2) that f(u)→+∞ as u→∞. Therefore, systems (2)-(3) have at least one periodic solution.
Example 17.
Let T={1/n:n∈N}∩{0} and T=1. Consider the following second order Hamiltonian system on time scale T of the form
(34)qΔ2t+∇Vt1-t,qt1-t=0,12qΔt2+Vt1-t,qt1-t=h,
where V(σ(t),x)=2t|x|2. It is easy to verify that (H0) 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 V∈C1(T×Rn∖{0},R) satisfies condition (H0) and the following two conditions:
there exist constants C3>0, α>2 such that for t∈[0,T]T the following inequality uniformly holds:
(35)limx→∞infVσt,xxα<-C3;
for t∈[0,T]T the following equality uniformly holds:
(36)Vσt,-x=Vσt,x,x≠0.
Then for any h>0, systems (2)-(3) have at least one nonconstant periodic solution with the given energy h.
3.3. Proof of Theorem 18Lemma 19.
Assume (H4) holds; then, for any weakly convergent sequence un⇀u∈∂W0, one has
(37)fun⟶+∞.
Proof.
Notice that (H4) implies Gordon’s strong force condition. It follows from (H4) and there exist constants 0<β<C3 and M>0 such that
(38)Vσt,u<-C3uα+βuα
for all u∈Rn with |u|>M and t∈[0,T]T. By un⇀u∈∂W and V satisfying Gordon’s strong force condition, we have
(39)∫0,TT-Vσt,unσtΔt⟶+∞,00000000000000000∀un⇀u∈∂W.
By un⇀u in the Hilbert space HΔ,T1, we know that un is bounded.
(1) If u≡0, then by Sobolev’s embedding theorem, we have the uniform convergence property:
(40)un∞⟶0,n⟶+∞.
By the symmetry of u(t+T/2)=-u(t), we have ∫[0,T)Tu(t)Δt=0. So, by Sobolev’s inequality
(41)∫0,TTuΔ2Δt≥12Tu∞2,
we have
(42)funM=12∫0,TTunΔt2h-Vσt,unσtΔtM≥6C3-βTun∞2+α⟶+∞,n⟶+∞.(2) If u≠0, then, by the weakly lower semicontinuity for the norm, we have
(43)limn→+∞infun≥u>0.
So, by Gordon’s lemma, we have
(44)limn→+∞inffun=limn→+∞inf12∫0,TTunΔt2h-Vσt,unσtΔt=+∞.
The proof is complete.
Lemma 20.
The functional f(u) is weakly lower semicontinuous on W0¯.
Proof.
Let
(45)f1u=h2∫0,TTuΔt2Δt,f2u=12∫0,TTuΔt2Vσt,uσtΔt.
Then, for given energy h, f1 is continuous and convex. Hence, f1 is weakly lower semicontinuous. On the other hand, let {un}n∈N⊂W0¯, un⇀u in W0¯. By Lemma 14, {uk}k∈N converges strongly in C([0,T]T,Rn) to u. By condition (H1), we have
(46)f2un-f2u=12∫0,TTunΔt2Vσt,unσtΔtMM-12∫0,TTuΔt2Vσt,uσtΔt≤12∫0,TTunΔt2Vσt,unσtMMMMMM-uΔt2Vσt,uσtΔt≤12∫0,TTunΔt2Vσt,unσtMMMMMM-uΔt2Vσt,unσtΔtM+12∫0,TTuΔt2Vσt,unσtMMMMMMM-uΔt2Vσt,uσtΔt≤12∫0,TTVσt,unσtunΔt2-uΔt2ΔtM+12∫0,TTuΔt2Vσt,unσt-Vσt,uσtΔt⟶0.
Thus, f2 is weakly continuous. Consequently, f=f1-f2 is weakly lower semicontinuous. The proof is complete.
Lemma 21.
W0¯ is a weakly closed subset of HΔ,T1.
Proof.
From Lemma 9, HΔ,T1=WΔ,T1,2(T,Rn)↪W0¯ is compact, so the proof is obvious.
Lemma 22.
The functional f(u) is coercive on W0.
Proof.
By the definition of f(u) and the assumption (H4), we have
(47)fu=12∫0,TTuΔt2h-Vσt,uσtΔt≥12∫0,TTuΔt2h+C3-βuαΔt≥h2uΔLΔ,T22,∀u∈W0;
that is, f(u)→+∞ as u→∞ for u∈W0; then f(u) is coercive in W0. The proof is complete.
Since f is lower semicontinuous and coercive, we obtain that f is bounded below and has a bounded minimizing sequence. By virtue of Lemmas 10 and 15, we know that f has at least one nonconstant critical point in W0. That is, systems (2)-(3) have at least one nonconstant periodic solution with the given energy h.
Example 23.
Let T={n/2:n∈N0} and T=1. Consider the second order Hamiltonian system on time scale T:
(48)qΔ2t+∇Vt+12,qt+12=0,12qΔt2+Vt+12,qt+12=h,
where V(σ(t),x)=λ|x|2(0<λ<C3). It is easy to verify that (H0) 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.
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.
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