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

Authors: You-Hui Su; Wan-Tong Li
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

This paper is concerned with the second-order Hamiltonian system on time scales𝕋 of the form uΔΔ(ρ(t))+μb(t)|u(t)|μ−2u(t)+∇¯H(t,u(t))=0,Δ-a.e. t∈[0,T]𝕋, u(0)−u(T)=uΔ(ρ(0))−uΔ(ρ(T))=0, where 0,T∈𝕋. 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.


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## Body

## 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 [2–6]. 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, 7–9]. 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 [10–21] 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 [22–24] 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 that0,T are points in 𝕋, for an interval (0,T)𝕋, we always mean (0,T)∩𝕋. Other types of intervals are defined similarly. We say that a property holds for Δ-a.e. t∈A⊂𝕋 or Δ-a.e. on A⊂𝕋, whenever there exists a set E⊂A with null Lebesgue Δ-measure such that this property holds for every t∈A∖E. We refer the reader to [3, 24, 25] for a broad introduction on Lebesgue Δ-measure.In this paper, motivated by references [26, 27], we consider the following second-order Hamiltonian system on time scales 𝕋 of the form
(1.1)uΔΔ(ρ(t))+μb(t)|u(t)|μ-2u(t)+∇¯H(t,u(t))=0,Δ-a.e.t∈[0,T]𝕋,u(0)-u(T)=uΔ(ρ(0))-uΔ(ρ(T))=0,
where T>0,μ>2, b∈C([0,T]𝕋,ℝ), H:[0,T]𝕋×ℝn→ℝ,(t,x)→H(t,x) is measurable in t for every x∈ℝn and continuously differentiable in x for Δ-a.e. t∈[0,T]𝕋 and ∇¯H(t,u)=DuH(t,u). 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 HT1(𝕋) 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 solutionu of problems (1.1); we mean u:𝕋κ→ℝn which is delta differential; uΔ and uΔΔ are both continuous Δ-a.e. on 𝕋κ∩𝕋κ, and u satisfies problems (1.1).Now, we present some basic definitions which can be found in [3–5, 28]. Another excellent source on dynamical systems on measure chains is the book [6].A time scale𝕋 is a nonempty closed subset of ℝ. For t∈𝕋, the forward and back jump operators σ,ρ:𝕋→𝕋 are well defined, respectively, by(1.2)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t}.
In this definition, one puts inf∅:=sup𝕋and sup∅:=inf𝕋, where ∅ denotes the empty set. A point t∈𝕋 is called left-dense if ρ(t)=t, left-scattered if ρ(t)<t, right-dense if σ(t)=t, and right-scattered if σ(t)>t.If𝕋 has a right-scattered minimum m, define 𝕋κ=𝕋-{m}; otherwise, set 𝕋κ=𝕋.If 𝕋 has a left-scattered maximum M, define 𝕋κ=𝕋-{M}; otherwise, set 𝕋κ=𝕋. The forward graininess is μ(t):=σ(t)-t. Similarly, the backward graininess is ν(t):=t-ρ(t).Iff:𝕋→ℝ is a function and t∈𝕋κ, then the delta derivative [4] of f at the point t is defined to be the number fΔ(t) (provided it exists) with the property that, for any ϵ>0, there is a neighborhood U⊂𝕋 of t such that
(1.3)|f(σ(t))-f(s)-fΔ(t)(σ(t)-s)|≤ϵ|σ(t)-s|,∀s∈U.Iff:𝕋→ℝ and t∈𝕋κ, then the nabla derivative of f at the point t is defined by the number f∇(t) (provided it exists) with the property that, for any ϵ>0, there is a neighborhood U⊂𝕋 of t such that
(1.4)|f(ρ(t))-f(s)-f∇(t)(ρ(t)-s)|≤ϵ|ρ(t)-s|,∀s∈U.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].Leta:=inf{s:s∈𝕋}andb:=sup{s:s∈𝕋}, defined a function E:[a,b]→ℝ by
(1.5)E(t):=sup{s∈𝕋:s≤t},t∈[a,b].Now, suppose thatf:𝕋κ→ℝ is arbitrary function, if f∘E is measurable on the real interval [a,b) in the usual Lebesgue senses, then we say f is measurable; if f∘E is integrable on the real interval [a,b) in the usual Lebesgue senses, then we say f is integrable. Let L1(𝕋) denote the set of such integrable functions on 𝕋. Furthermore, for any f∈L1(𝕋), we defined the integral of f by
(1.6)∫stfΔ:=∫stf∘Edτfors,t∈𝕋,with the norm defined by(1.7)∥f∥L1(𝕋)=∫ab|f|Δforf∈L1(𝕋).
We use the notation ∫stfΔ to denote the Lebesgue integral of a function f between s,t∈𝕋 (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
(1.8)L2(𝕋):={f∈L1(𝕋):|f|2∈L1(𝕋)},∥f∥L2(𝕋)=(∫ab|f|2Δ)1/2forf∈L2(𝕋).It is shown in [31] that L2(𝕋) is completed with respect to the norm ∥f∥L2(𝕋).Next, define the norm∥·∥ on Crd1(𝕋) by
(1.9)∥f∥=(∥f∥L2(𝕋)2+∥fΔ∥L2(𝕋κ)2)1/2forf∈Crd1(𝕋).The spaceH1(𝕋) is the completion of Crd1(𝕋) with respect to the norm ∥·∥ (see [31, Definition 4.1 and Remark 4.2]). The space H1(𝕋) is a time scale analog to the usual Sobolev space H1(I) on a real interval I.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 replaceu:𝕋→ℝ with u:𝕋→ℝn, then the above discussion still holds.
The rest of the paper is organized as follows. In Section2, 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 inHT1(𝕋) spaces for second-order Hamiltonian systems on time scales.LetHT1(𝕋) be the Hilbert space given by
(2.1)HT1(𝕋)={u:[0,T]𝕋→ℝn∣uis absolutely continuous,u(0)=u(T),uΔ(t)∈L2([0,T]𝕋κ,ℝn)},with the norm defined by(2.2)∥u∥=(∫0T|u(t)|2Δ+∫0T|uΔ(t)|2Δ)1/2foru∈HT1(𝕋).
Moreover, we define
(2.3)∥u∥L1(𝕋)=∫0T|u(t)|Δ,∥u∥L2(𝕋)=(∫0T|u(t)|2Δ)1/2,∥u∥∞=supt∈[0,T]𝕋|u|.
We also define inner product on HT1(𝕋) by
(2.4)(u,v)=∫0T[u(t)·v(t)+uΔ(t)·vΔ(t)]Δ.
For u∈HT1(𝕋), let
(2.5)u¯(t)=1T∫0Tu(t)Δ,ũ(t)=u(t)-u¯(t),
and let H̃T1(𝕋) be the subspace of HT1(𝕋) given by H̃T1(𝕋)={u∈HT1(𝕋)∣u¯(t)=0}.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.
Letp∈ℝ¯ be such that p≥1. Then, for every q∈[1,+∞), the immersion HT1(𝕋)↪Lq(𝕋) 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{um}m∈ℕ⊂HT1(𝕋) and u∈HT1(𝕋). If {um}m∈ℕ converges weakly in HT1(𝕋) to u, then {um}m∈ℕ converges uniformly to u on [0,T]𝕋.Lemma 2.3.
Ifu∈HT1(𝕋), then
(2.6)∥u∥∞≤c1∥u∥.
In particular, if ∫0Tu(t)Δ=0, then
(2.7)∥u∥∞≤T1/2∥uΔ(t)∥L2(𝕋).Lemma 2.4.
LetL:[0,T]𝕋×ℝn×ℝn→ℝ,(t,x,y)→L(t,x,y) be measurable in t for each [x,y]∈ℝn×ℝn and continuously differentiable in [x,y] for Δ-a.e. t∈[0,T]𝕋. If there exist a∈C(ℝ+,ℝ+),b1∈L1([0,T]𝕋,ℝ+), and c∈L2([0,T]𝕋,ℝ+), such that, for Δ-a.e. t∈[0,T]𝕋and each [x,y]∈ℝn×ℝn, one has
(2.8)|L(t,x,y)|≤a(|x|)(b1(t)+|y|2),|DxL(t,x,y)|≤a(|x|)(b1(t)+|y|2),|DyL(t,x,y)|≤a(|x|)(c(t)+|y|),
then the functional φ defined by φ(u)=∫0TL(t,u(t),uΔ(t))Δ is continuously differential on HT1(𝕋) and
(2.9)〈φ′(u),v〉=∫0T[DxL(t,u(t),uΔ(t))·v(t)+DyL(t,u(t),uΔ(t))·vΔ(t)]Δ.Proof.
In the following, we will prove thatφ has a directional derivative φ′(u)∈(HT1(𝕋))* given by (2.9) and that the mapping
(2.10)φ′:HT1(𝕋)→(HT1(𝕋))*,u→φ′(u)
is continuous.
(i) It follows easily from (2.8) that φ is everywhere finite on HT1(𝕋). Fixing u and v in HT1(𝕋), we define
(2.11)F(λ,t)=L(t,u(t)+λv(t),uΔ(t)+λvΔ(t))fort∈[0,T]𝕋,λ∈[-1,1],ψ(λ)=φ(u+λv)=∫0TF(λ,t)Δ=∫0TF(λ,t)∘Edt=∫0TF(λ,E(t))dt.
We will apply Leibniz formula of differentiation under integral sign toψ. According to assumption (2.8), one obtains
(2.12)|DλF(λ,E(t))|≤|(DxL(t,u+λv,u∆+λv∆)∘E)·(v∘E)+(DyL(t,u+λv,u∆+λv∆)∘E)·(v∆∘E)|≤a(|u(E(t))+λv(E(t))|)[(b1(t)+|u∆+λv∆|2)|v|+(c(t)+|u∆+λv∆|)|v∆|]∘E≤a0[(b1(t)+(|u∆|+|u∆|)2)|v|+(c(t)+|u∆|+|v∆|)|v∆|]∘E,
where a0=max(λ,t)∈[-1,1]×[0,T]a(|u(E(t))+λv(E(t))|).
It is obvious thatb1∘E∈L1([0,T],ℝ+), (|uΔ∘E|+|vΔ∘E|)2∈L1([0,T],ℝ+), c∘E∈L2([0,T],ℝ+). v∈HT1(𝕋) implies that vΔ∘E∈L2([0,T],ℝ+) and v∘E∈L1([0,T],ℝ+) hold, hence we have
(2.13)|DλF(λ,E(t))|≤d∘E∈L1([0,T],ℝ+).
In view of Leibniz formula and (1.6), we get
(2.14)ψ′(0)=〈φ′(u),v〉=∫0TDλF(0,E(t))dt=∫0TDxL(E(t),u(E(t)),uΔ(E(t)))·v(E(t))dt+∫0TDyL(E(t),u(E(t)),uΔ(E(t)))·vΔ(E(t))dt=∫0T[DxL(t,u(t),uΔ(t))·v(t)+DyL(t,u(t),uΔ(t))·vΔ(t)]Δ.
Moreover
(2.15)|DxL(t,u,uΔ)|≤a(|u|)(b1(t)+|uΔ|2)∈L1([0,T]𝕋,ℝ+),|DyL(t,u,uΔ)|≤a(|u|)(c(t)+|uΔ|)∈L2([0,T]𝕋,ℝ+).
Thus, from Lemma 2.3,
(2.16)∫0TDλF(0,t)Δ=∫0T[DxL(t,u,uΔ)·v+DyL(t,u,uΔ)·vΔ]Δ≤c1∥v∥∞+c2∥vΔ∥L2(𝕋)≤c3∥v∥,
and φ has a directional derivative φ′(u)∈(HT1(𝕋))*, given by (2.9).
(ii) According to the theorem of Krasnosel'skii [33], assumption (2.8) implies that the mapping from HT1(𝕋) into L1×L2 defined by u→(DxL(t,u,uΔ),DyL(t,u,uΔ)) is continuous, thus, φ′ is continuous from HT1(𝕋) into (HT1(𝕋))*, and the proof is completed.We also need the following theorem, which was the generalized mountain pass theorem.Lemma 2.5 (see [34]).
LetE be a real Hilbert space with E=E1⊕E2 and E2=E1⊥. Suppose I∈C1(E,R), satisfies (PS), and(I1)
I(u)=1/2(Au,u)+m(u), where Au=A1P1u+A2P2u and Ai:Ei→Ei is bounded and self-adjoint, i=1,2,(I2)
m′(u) is compact, and(I3)
there exist a subspaceẼ⊂E and sets S⊂E,Q⊂Ẽ and constants α>ω such that(i)
S⊂E1 and I|S≥α;(ii)
Q is bounded and I|∂Q≤ω;(iii)
S and ∂Q link.Then I possesses a critical value c≥α.


## 3. Existence Results

In this section, by using the minimax methods in critical theory, we establish the existence of at leastone nonzero periodic solution for second-order Hamiltonian system (1.1) on time scales.Throughout this section, the following is assumed.(H1)
Suppose that there existt1,t2∈[0,T]𝕋 and b∈C([0,T]𝕋,ℝ) satisfying b(t)≥(1/2)b(t0)>0 for all t∈[t1,t2]𝕋, where t0∈[t1,t2]𝕋. In addition, ∫0Tb(t)Δ=0.(H2)
∫0TH(t,x)Δ≥0 for all x∈ℝn and Δ-a.e. t∈[0,T]𝕋.(H3)
Assume that there existsg∈L1([0,T]𝕋,ℝ+) such that |∇¯H(t,x)|≤g(t) for all x∈ℝn and Δ-a.e. t∈[0,T]𝕋.(H4)
Assume that there existα0∈(0,1/2T2) and r0>0 such that |H(t,x)|≤α0|x|2 for all ∥x∥≤r0 and Δ-a.e. t∈[0,T]𝕋.(H5)
Assume thatσ(ρ(t))=t.If(3.1)L(t,x,y)=L(t,u(t),uΔ(t))=12|uΔ(t)|2-b(t)|u(t)|μ-H(t,u(t)),
then by Lemma 2.4, the functional φ is given by
(3.2)φ(u)=12∫0T|uΔ(t)|2Δ-∫0Tb(t)|u(t)|μΔ-∫0TH(t,u(t))Δ,
which is continuously differentiable on HT1(𝕋). Moreover
(3.3)〈φ′(u),v〉=∫0TuΔ(t)·vΔ(t)Δ-μ∫0Tb(t)|u(t)|μ-2u(t)·v(t)Δ-∫0T∇¯H(t,u(t))·v(t)Δ∀u,v∈HT1(𝕋).
That is,  for all  u,v∈HT1(𝕋), we get
(3.4)〈φ′(u),v〉=-∫0TuΔΔ(ρ(t))·v(t)Δ-μ∫0Tb(t)|u(t)|μ-2u(t)·v(t)Δ-∫0T∇¯H(t,u(t)).v(t)Δ=-∫0T(uΔΔ(ρ(t))+μb(t)|u(t)|μ-2u(t)+∇¯H(t,u(t)))·v(t)Δ.Hence,u∈HT1(𝕋) is a solution of problem (1.1) if and only if u is a critical point of φ.Lemma 3.1.
Let a sequence{un(t)}⊂HT1(𝕋) be such that φ′(un(t))→0 and let {un(t)} be bounded in HT1(𝕋), then {un(t)} has a convergent subsequence in HT1(𝕋).Proof.
Since{un(t)} is bounded in HT1(𝕋), it follows from [30, Theorem 4.12] that there exists a subsequence (still denoted by {un(t)}) which weakly converges to u0∈HT1(𝕋). By Lemma 2.2, we have
(3.5)un→u0in[0,T]𝕋.
Hence, for t∈[0,T]𝕋, there exists an M>0 such that |un(t)|≤M,n=1,2,….
Lemma2.1 leads to
(3.6)HT1(𝕋)↪L2([0,T]𝕋,ℝn)is compact.
Hence
(3.7)un→u0inL2([0,T]𝕋,ℝn).
Since
(3.8)〈φ′(un(t))-φ′(um(t)),un(t)-um(t)〉=∫0T(unΔ(t)-umΔ(t))·(unΔ(t)-umΔ(t))Δ-μ∫0Tb(t)(|un(t)|μ-2un(t)-|um(t)|μ-2um(t))·(un(t)-um(t))Δ-∫0T(∇¯H(t,un(t))-∇¯H(t,um(t)))·(un(t)-um(t))Δ,
in view of (H1) and (H3), one has
(3.9)∫0T|unΔ(t)-umΔ(t)|2Δ≤∥φ′(un(t))-φ′(um(t))∥∥un(t)-um(t)∥+2μMμ-1∥un(t)-um(t)∥∞∫0T|b(t)|Δ+2∥un(t)-um(t)∥∞∫0Tg(t)Δ→0,asn,m→∞.
Consequently
(3.10)∥un(t)-um(t)∥2=∫0T|unΔ(t)Δ-umΔ(t)|2Δ+∫0T|un(t)-um(t)|2Δ→0asn,m→∞,
which implies that {un} is a Cauchy sequence in HT1(𝕋). By the completeness of HT1(𝕋), we obtain that {un} is a convergent sequence in HT1(𝕋); the proof is completed.Now, we list our main result.Theorem 3.2.
Suppose thatμ>2, (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 Lemma2.5 hold with respect to φ.
First, we show thatφ satisfies the (PS) condition.
From Lemma3.1, we only need to prove that {un} is bounded. Otherwise, there exists a subsequence (still denoted by un) such that ∥un∥→∞. Let vn=un/∥un∥, then {vn} is bounded; it has a subsequence (we still denote {vn}) which weakly converges to v0. In view of Lemma 2.2, {vn} uniformly converges to v0 in [0,T]𝕋. Since ∥vn∥=1for alln∈ℕ, one has ∥vn∥≠0.
According to∥un∥→∞ as n→∞, μ>2 and (H3), for all vn,v∈HT1(𝕋), we get
(3.11)μ|∫0Tb(t)|vn(t)|μ-2vn(t)·v(t)Δ|≤∥un∥1-μ|-〈φ′(un),v〉|+∥un∥2-μ|∫0TvnΔ(t)·vΔ(t)Δ|+∥un∥1-μ|-∫0T∇¯H(t,un(t))·v(t)Δ|≤∥un∥1-μ∥φ′(un)∥∥v∥+C1∥un∥2-μ∥vn∥∥v∥+∥un∥1-μ∥v∥∞∫0Tg(t)Δ→0asn→∞.
Thus, it follows from Lebesgue dominated convergence theorem on time scales [28] that
(3.12)|∫0Tb(t)|v0(t)|μ-2v0(t)·v(t)Δ|=0∀v∈HT1(𝕋).
By the arbitrariness of v, one has
(3.13)b(t)=0forΔ-a.e.t∈[0,T]𝕋,
which contradicts the condition (H1). Hence φ satisfies the (PS) condition.
Second, if
(3.14)Au=u(t),m(u)=-∫0Tb(t)|u(t)|μΔ-∫0TH(t,u(t))Δ-12∫0Tu2(t)Δ,
then it is easy to verify that (I1) and (I2) hold.
Third, we will prove thatφ satisfies the condition (I3) in Lemma 2.5.
For arbitraryu∈H̃T1(𝕋) with ∥u∥≤r0, (H4) and (2.7) imply
(3.15)φ(u)=12∫0T|uΔ(t)|2Δ-∫0Tb(t)|u(t)|μΔ-∫0TH(t,u(t))Δ≥12∫0T|uΔ(t)|2Δ-∥u(t)∥∞μ∫0T|b(t)|Δ-α0∫0T|u(t)|2Δ≥12T∥u∥∞2-∥u(t)∥∞μ∫0T|b(t)|Δ-α0T∥u(t)∥∞2≥(12T-α0T)∥u∥∞2-∥u(t)∥∞μ∫0T|b(t)|Δ.
Choose ρ>0 small enough such that
(3.16)α≜(12T-α0T)ρ2-ρμ∫0T|b(t)|Δ>0,
thus
(3.17)φ(u)≥α>∀u∈H̃T1(𝕋)with∥u∥=ρ.
If
(3.18)S={u∈H̃T1(𝕋),∥u∥=ρ},
then φ|s≥α>0; this implies that the condition (i) of Lemma 2.5 holds.
It is known that (H1) and (H2) lead to(3.19)φ(x)=-∫0Tb(t)|x|μΔ-∫0TH(t,x)Δ≤0∀x∈ℝn.
Choose 0≠e∈H̃T1(𝕋) such that e(t)=0 for all t∈[0,T]𝕋∖[t1,t2]𝕋 and
(3.20)∫0Tb(t)e(t)Δ=∫t1t2b(t)e(t)Δ=0.
For arbitrary u∈HT1(𝕋), let
(3.21)φ1(u)=12∫0T|uΔ(t)|2Δ,φ2(u)=-∫0Tb(t)|u(t)|μΔ,φ3(u)=-∫0TH(t,u(t))Δ, then
(3.22)φ(u)=φ1(u)+φ2(u)+φ3(u).
For all x∈ℝn and r≥0, in terms of (H3), one has
(3.23)φ3(x+re)-φ3(x)=∫0T∫10∇¯H(t,x+sre(t))·re(t)dsΔ≤r∫0Tg(t)|e(t)|Δ≤r∥e∥∞∥g∥L1(𝕋).
Since -φ3(x)≥0for allx∈ℝn, we have
(3.24)φ3(x+re)≤r∥e∥∞∥g∥L1(𝕋)∀x∈ℝn,r≥0.
In terms of (3.20) and Hölder's inequality on time scales, one obtains
(3.25)∫t1t2-b(t)(|x|2+r2|e(t)|2)Δ=∫t1t2-b(t)(|x|+r|e(t)|)2Δ≤(∫t1t2-b(t)(|x|+r|e(t)|)μΔ)2/μ×(∫t1t2-b(t)Δ)(μ-2)/μ∀x∈ℝn,r≥0.
Thus, by using ∫0Tb(t)|x|μΔ=0, (3.25), and Hölder's inequality on time scales again,    for all x∈ℝn and r≥0, we obtain
(3.26)φ2(x+re)=-∫0Tb(t)(|x+re(t)|μ-|x|μ)Δ=-∫0t1b(t)|x+re(t)|μΔ-∫t1t2b(t)|x+re(t)|μΔ-∫t2Tb(t)|x+re(t)|μΔ+∫0Tb(t)|x|μΔ=-∫t1t2b(t)|x+re(t)|μΔ+∫t1t2b(t)|x|μΔ≤(∫t1t2-b(t)(|x|2+r2|e(t)|2)Δ)μ/2(∫t1t2-b(t)Δ)(2-μ)/2+∫t1t2b(t)|x|μΔ≤-∫t1t2b(t)(|x|+r|e(t)|)μΔ+∫t1t2b(t)|x|μΔ≤0.
(3.24) and (3.26) lead to
(3.27)φ(x+re)=φ1(x+re)+φ2(x+re)+φ3(x+re)≤12r2∫0T|eΔ|2Δ+r∥e∥∞∥g∥L1(𝕋)∀x∈ℝn,r≥0,
which means that there exists r3>0 such that
(3.28)φ(x+re)≤α2∀x∈ℝn,r∈[0,r3].
For all R≥0 and r≥0, let
(3.29)h(R,r)≜(∫t1t2-b(t)(|R|2+r2|e(t)|2)Δ)μ/2(∫t1t2-b(t)Δ)(2-μ)/2+∫t1t2b(t)|R|μΔ=-(∫t1t2b(t)(|R|2+r2|e(t)|2)Δ)μ/2(∫t1t2b(t)Δ)(2-μ)/2+∫t1t2b(t)|R|μΔ.
In view of ∫t1t2b(t)>0,  for all R≥0 and r≥0, we get
(3.30)∂h∂r(R,r)=-rμ∫t1t2b(t)|e(t)|2Δ(∫t1t2b(t)(R2+r2|e(t)|2)Δ)μ/2-1(∫t1t2b(t)Δ)(2-μ)/2≤-rμ∫t1t2b(t)|e(t)|2Δ(∫t1t2R2b(t)Δ)μ/2-1(∫t1t2b(t)Δ)(2-μ)/2≤-rμRμ-2∫t1t2b(t)|e(t)|2Δ.
(3.30) and h(R,0)=0 imply that
(3.31)h(R,r)≤-12r2μRμ-2∫t1t2b(t)|e(t)|2Δ∀R≥0,r≥0.
Note that there exists R1>ρ>0 such that
(3.32)∫0T|eΔ(t)|2Δ-μRμ-2∫t1t2b(t)|e(t)|2Δ≤-2r3∥e∥∞∥g∥L1(𝕋)∀R≥R1.
Therefore, by using (3.24), (3.26), (3.31), and (3.32),  for x=R,r∈[r3,R] and R≥R1, one has
(3.33)φ(x+re)=φ1(x+re)+φ2(x+re)+φ3(x+re)≤12r2∫0T|eΔ(t)|2Δ+h(R,r)+r∥e∥∞∥g∥L1(𝕋)≤12r2∫0T|eΔ(t)|2Δ-12r2μRμ-2∫t1t2b(t)|e(t)|2Δ+r∥e∥∞∥g∥L(1)≤-rr3(r-r3)∥e∥∞∥g∥L1(𝕋)≤0.
Thus, (3.28) and (3.33) can lead to
(3.34)φ(x+re)≤α2,∀|x|=R,r∈[0,R],R≥R1.
For all x∈ℝn and R≥0, denote
(3.35)f(x,R)≜-(∫t1t2b(t)(|x|2+R2|e(t)|2)Δ)μ/2(∫t1t2b(t)Δ)(2-μ)/2+∫t1t2b(t)|x|μΔ.
By using the similar way to inequality (3.26), one has
(3.36)φ2(x+Re)≤f(x,R)∀x∈ℝn,R≥0,
for all x∈ℝn and R≥0, since
(3.37)∂f∂R(x,R)=-Rμ∫t1t2b(t)|e(t)|2Δ(∫t1t2b(t)(|x|2+R2|e(t)|2)Δ)μ/(2-1)(∫t1t2b(t)Δ)(2-μ)/2≤-μRμ-1(∫t1t2b(t)|e(t)|2Δ)μ/2(∫t1t2b(t)Δ)(2-μ)/2.
In view of (3.37) and f(x,0)=0, for all x∈ℝn and R≥0, we get
(3.38)f(x,R)≤-Rμ(∫t1t2b(t)|e(t)|2Δ)μ/2(∫t1t2b(t)Δ)(2-μ)/2.
From (3.24), (3.36), and (3.38), for all x∈ℝn and R≥0, we obtain
(3.39)φ(x+Re)=φ1(x+Re)+φ2(x+Re)+φ3(x+Re)≤12R2∫0T|eΔ(t)|2Δ-Rμ(∫t1t2b(t)|e(t)|2Δ)μ/2(∫t1t2b(t)Δ)(2-μ)/2+R∥e∥∞∥g∥L1(𝕋),
which implies that there exists R2>R1, such that
(3.40)φ(x+Re)≤0∀x∈ℝn,R≥R2.
Now let
(3.41)Q={x+re∣x∈ℝn,∥x∥=R,r∈[0,R],R≥R2}.
Thus, (3.19), (3.34), and (3.40) lead to φ|∂Q≤(1/2)α, which means that the condition (ii) of Lemma 2.5 is satisfied.
It is easy to see thatS and ∂Q 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(4.1)𝕋=[0,0.3]∪{0.4,0.45,0.5,0.55,0.6}∪[0.7,1],T=1.
Consider the following second-order Hamiltonian system on time scales 𝕋 of the form
(4.2)uΔΔ(ρ(t))+μb(t)|u(t)|μ-2u(t)+∇¯H(t,u(t))=0,Δ-a.e.t∈[0,1]𝕋,u(0)-u(1)=uΔ(0)-uΔ(1)=0,
where μ>2 is a constant; let ε>0 is arbitrary small, H(t,u)=εu2(t)forΔ-a.e.t∈[0,1]𝕋 and
(4.3)b(t)={t,t∈[0,0.3]𝕋,-1.5t+0.75,t∈[0.3,0.7]𝕋,t-1,t∈[0.7,1]𝕋.
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.



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*Source: 328479-2009-07-02.xml*