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

Abstract

This paper is concerned with the second-order Hamiltonian system on time scales 𝕋 of the form 𝑢^ΔΔ(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)=𝑢^Δ(𝜌(0))−𝑢^Δ(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.

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 that 0,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 𝑢^ΔΔ(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)=𝑢^Δ(𝜌(0))−𝑢^Δ(𝜌(𝑇))=0, (1.1) where 𝑇>0,𝜇>2, 𝑏∈𝐶([0,𝑇]_𝕋,ℝ), 𝐻∶[0,𝑇]_𝕋×ℝ^𝑛→ℝ,(𝑡,𝑥)→𝐻(𝑡,𝑥) is measurable in 𝑡 for every 𝑥∈ℝ^𝑛 and continuously differentiable in 𝑥 for Δ-a.e. 𝑡∈[0,𝑇]_𝕋 and ∇𝐻(𝑡,𝑢)=𝐷_𝑢𝐻(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 𝐻¹_𝑇(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 solution 𝑢 of problems (1.1); we mean 𝑢∶𝕋_𝜅→ℝ^𝑛 which is delta differential; 𝑢^Δ and 𝑢^ΔΔ are both continuous Δ-a.e. on 𝕋^𝜅∩𝕋_𝜅, 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 scale 𝕋 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

𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡}. (1.2) 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.

If 𝕋 has a right-scattered minimum 𝑚, define 𝕋_𝜅=𝕋−{𝑚}; otherwise, set 𝕋_𝜅=𝕋.If 𝕋 has a left-scattered maximum 𝑀, define 𝕋^𝜅=𝕋−{𝑀}; otherwise, set 𝕋^𝜅=𝕋. The forward graininess is 𝜇(𝑡)∶=𝜎(𝑡)−𝑡. Similarly, the backward graininess is 𝜈(𝑡)∶=𝑡−𝜌(𝑡).

If 𝑓∶𝕋→ℝ is a function and 𝑡∈𝕋^𝜅, then the delta derivative [4] of 𝑓 at the point 𝑡 is defined to be the number 𝑓^Δ(𝑡) (provided it exists) with the property that, for any 𝜖>0, there is a neighborhood 𝑈⊂𝕋 of 𝑡 such that |𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)|≤𝜖|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (1.3)

If 𝑓∶𝕋→ℝ and 𝑡∈𝕋_𝜅, 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 |𝑓(𝜌(𝑡))−𝑓(𝑠)−𝑓^∇(𝑡)(𝜌(𝑡)−𝑠)|≤𝜖|𝜌(𝑡)−𝑠|,∀𝑠∈𝑈. (1.4)

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 𝑎∶=inf{𝑠∶𝑠∈𝕋}and𝑏∶=sup{𝑠∶𝑠∈𝕋}, defined a function 𝐸∶[𝑎,𝑏]→ℝ by 𝐸(𝑡)∶=sup{𝑠∈𝕋∶𝑠≤𝑡},𝑡∈[𝑎,𝑏]. (1.5)

Now, suppose that 𝑓∶𝕋^𝜅→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 ^𝑡_𝑠𝑓Δ∶=^𝑡_𝑠𝑓∘𝐸𝑑𝜏for𝑠,𝑡∈𝕋, (1.6)

with the norm defined by ‖𝑓‖_𝐿1(𝕋)=^𝑏_𝑎|𝑓|Δfor𝑓∈𝐿¹(𝕋). (1.7) We use the notation ∫^𝑡_𝑠𝑓Δ 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 𝐿²(𝕋)∶={𝑓∈𝐿¹(𝕋)∶|𝑓|²∈𝐿¹(𝕋)},‖𝑓‖_𝐿2(𝕋)=(^𝑏_𝑎|𝑓|²Δ)^1/2for𝑓∈𝐿²(𝕋). (1.8)

It is shown in [31] that 𝐿²(𝕋) is completed with respect to the norm ‖𝑓‖_𝐿2(𝕋).

Next, define the norm ‖⋅‖ on 𝐶¹_𝑟𝑑(𝕋) by ‖𝑓‖=(‖𝑓‖²_𝐿2(𝕋)+‖𝑓^Δ‖²_{𝐿2(𝕋𝜅)})^1/2for𝑓∈𝐶¹_𝑟𝑑(𝕋). (1.9)

The space 𝐻¹(𝕋) is the completion of 𝐶¹_𝑟𝑑(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 replace 𝑢∶𝕋→ℝ with 𝑢∶𝕋→ℝ^𝑛, 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 𝐻¹_𝑇(𝕋) spaces for second-order Hamiltonian systems on time scales.

Let 𝐻¹_𝑇(𝕋) be the Hilbert space given by 𝐻¹_𝑇(𝕋)={𝑢∶[0,𝑇]_𝕋→ℝ^𝑛𝑢∣𝑢isabsolutelycontinuous,𝑢(0)=𝑢(𝑇),_Δ(𝑡)∈𝐿²([0,𝑇]_𝕋𝜅,ℝ^𝑛)}, (2.1)

with the norm defined by ‖𝑢‖=(^𝑇₀|𝑢(𝑡)|²Δ+^𝑇₀|𝑢^Δ(𝑡)|²Δ)^1/2for𝑢∈𝐻¹_𝑇(𝕋). (2.2) Moreover, we define ‖𝑢‖_𝐿¹_(𝕋)=^𝑇₀|𝑢(𝑡)|Δ,‖𝑢‖_𝐿²_(𝕋)=(^𝑇₀|𝑢(𝑡)|²Δ)^1/2,‖𝑢‖_∞=sup_{𝑡∈[0,𝑇]𝕋}|𝑢|. (2.3) We also define inner product on 𝐻¹_𝑇(𝕋) by (𝑢,𝑣)=^𝑇₀[𝑢(𝑡)⋅𝑣(𝑡)+𝑢^Δ(𝑡)⋅𝑣^Δ(𝑡)]Δ. (2.4) For 𝑢∈𝐻¹_𝑇(𝕋), let 1𝑢(𝑡)=𝑇^𝑇₀𝑢(𝑡)Δ,𝑢(𝑡)=𝑢(𝑡)−𝑢(𝑡), (2.5) and let 𝐻¹_𝑇(𝕋) be the subspace of 𝐻¹_𝑇(𝕋) given by 𝐻¹_𝑇(𝕋)={𝑢∈𝐻¹_𝑇(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. Let 𝑝∈ℝ be such that 𝑝≥1. Then, for every 𝑞∈[1,+∞), the immersion 𝐻¹_𝑇(𝕋)↪𝐿^𝑞(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 {𝑢_𝑚}_𝑚∈ℕ⊂𝐻¹_𝑇(𝕋) and 𝑢∈𝐻¹_𝑇(𝕋). If {𝑢_𝑚}_𝑚∈ℕ converges weakly in 𝐻¹_𝑇(𝕋) to 𝑢, then {𝑢_𝑚}_𝑚∈ℕ converges uniformly to 𝑢 on [0,𝑇]_𝕋.

Lemma 2.3. If 𝑢∈𝐻¹_𝑇(𝕋), then ‖𝑢‖_∞≤𝑐₁‖𝑢‖. (2.6) In particular, if ∫^𝑇₀𝑢(𝑡)Δ=0, then ‖𝑢‖_∞≤𝑇^1/2‖𝑢^Δ(𝑡)‖_𝐿²_(𝕋). (2.7)

Lemma 2.4. Let 𝐿∶[0,𝑇]_𝕋×ℝ^𝑛×ℝ^𝑛→ℝ,(𝑡,𝑥,𝑦)→𝐿(𝑡,𝑥,𝑦) be measurable in 𝑡 for each [𝑥,𝑦]∈ℝ^𝑛×ℝ^𝑛 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]∈ℝ^𝑛×ℝ^𝑛, one has |𝐿(𝑡,𝑥,𝑦)|≤𝑎(|𝑥|)(𝑏₁(𝑡)+|𝑦|²),|𝐷_𝑥𝐿(𝑡,𝑥,𝑦)|≤𝑎(|𝑥|)(𝑏₁(𝑡)+|𝑦|²),|𝐷_𝑦𝐿(𝑡,𝑥,𝑦)|≤𝑎(|𝑥|)(𝑐(𝑡)+|𝑦|), (2.8) then the functional 𝜑 defined by 𝜑(𝑢)=∫^𝑇₀𝐿(𝑡,𝑢(𝑡),𝑢^Δ(𝑡))Δ is continuously differential on 𝐻¹_𝑇(𝕋) and ⟨𝜑^′(𝑢),𝑣⟩=^𝑇₀[𝐷_𝑥𝐿(𝑡,𝑢(𝑡),𝑢^Δ(𝑡))⋅𝑣(𝑡)+𝐷_𝑦𝐿(𝑡,𝑢(𝑡),𝑢^Δ(𝑡))⋅𝑣^Δ(𝑡)]Δ. (2.9)

Proof. In the following, we will prove that 𝜑 has a directional derivative 𝜑^′(𝑢)∈(𝐻¹_𝑇(𝕋))^∗ given by (2.9) and that the mapping 𝜑^′∶𝐻¹_𝑇(𝕋)→(𝐻¹_𝑇(𝕋))^∗,𝑢→𝜑^′(𝑢) (2.10) is continuous.
(i) It follows easily from (2.8) that 𝜑 is everywhere finite on 𝐻¹_𝑇(𝕋). Fixing 𝑢 and 𝑣 in 𝐻¹_𝑇(𝕋), we define
𝐹(𝜆,𝑡)=𝐿(𝑡,𝑢(𝑡)+𝜆𝑣(𝑡),𝑢^Δ(𝑡)+𝜆𝑣^Δ(𝑡))for𝑡∈[0,𝑇]_𝕋,𝜆∈[−1,1],𝜓(𝜆)=𝜑(𝑢+𝜆𝑣)=^𝑇₀𝐹(𝜆,𝑡)Δ=^𝑇₀𝐹(𝜆,𝑡)∘𝐸𝑑𝑡=^𝑇₀𝐹(𝜆,𝐸(𝑡))𝑑𝑡. (2.11)
We will apply Leibniz formula of differentiation under integral sign to 𝜓. According to assumption (2.8), one obtains
|𝐷_𝜆𝐹(𝜆,𝐸(𝑡))|≤|(𝐷_𝑥𝐿(𝑡,𝑢+𝜆𝑣,𝑢^▵+𝜆𝑣^▵)∘𝐸)⋅(𝑣∘𝐸)+(𝐷_𝑦𝐿(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, (2.12) where 𝑎₀=max_{(𝜆,𝑡)∈[−1,1]×[0,𝑇]}𝑎(|𝑢(𝐸(𝑡))+𝜆𝑣(𝐸(𝑡))|).
It is obvious that 𝑏₁∘𝐸∈𝐿¹([0,𝑇],ℝ⁺), (|𝑢^Δ∘𝐸|+|𝑣^Δ∘𝐸|)²∈𝐿¹([0,𝑇],ℝ⁺), 𝑐∘𝐸∈𝐿²([0,𝑇],ℝ⁺). 𝑣∈𝐻¹_𝑇(𝕋) implies that 𝑣^Δ∘𝐸∈𝐿²([0,𝑇],ℝ⁺) and 𝑣∘𝐸∈𝐿¹([0,𝑇],ℝ⁺) hold, hence we have
|𝐷_𝜆𝐹(𝜆,𝐸(𝑡))|≤𝑑∘𝐸∈𝐿¹([0,𝑇],ℝ⁺). (2.13) In view of Leibniz formula and (1.6), we get 𝜓^′(0)=⟨𝜑^′(𝑢),𝑣⟩=^𝑇₀𝐷_𝜆=𝐹(0,𝐸(𝑡))𝑑𝑡_𝑇0𝐷_𝑥𝐿(𝐸(𝑡),𝑢(𝐸(𝑡)),𝑢^Δ+(𝐸(𝑡)))⋅𝑣(𝐸(𝑡))𝑑𝑡_𝑇0𝐷_𝑦𝐿(𝐸(𝑡),𝑢(𝐸(𝑡)),𝑢^Δ(𝐸(𝑡)))⋅𝑣^Δ=(𝐸(𝑡))𝑑𝑡_𝑇0[𝐷_𝑥𝐿(𝑡,𝑢(𝑡),𝑢^Δ(𝑡))⋅𝑣(𝑡)+𝐷_𝑦𝐿(𝑡,𝑢(𝑡),𝑢^Δ(𝑡))⋅𝑣^Δ(𝑡)]Δ. (2.14)
Moreover
|𝐷_𝑥𝐿(𝑡,𝑢,𝑢^Δ)|≤𝑎(|𝑢|)(𝑏₁(𝑡)+|𝑢^Δ|²)∈𝐿¹([0,𝑇]_𝕋,ℝ⁺),|𝐷_𝑦𝐿(𝑡,𝑢,𝑢^Δ)|≤𝑎(|𝑢|)(𝑐(𝑡)+|𝑢^Δ|)∈𝐿²([0,𝑇]_𝕋,ℝ⁺). (2.15) Thus, from Lemma 2.3, ^𝑇₀𝐷_𝜆𝐹(0,𝑡)Δ=^𝑇₀[𝐷_𝑥𝐿(𝑡,𝑢,𝑢^Δ)⋅𝑣+𝐷_𝑦𝐿(𝑡,𝑢,𝑢^Δ)⋅𝑣^Δ]Δ≤𝑐₁‖𝑣‖_∞+𝑐₂‖𝑣^Δ‖_𝐿²_(𝕋)≤𝑐₃‖𝑣‖, (2.16) and 𝜑 has a directional derivative 𝜑^′(𝑢)∈(𝐻¹_𝑇(𝕋))^∗, given by (2.9).
(ii) According to the theorem of Krasnosel'skii [33], assumption (2.8) implies that the mapping from 𝐻¹_𝑇(𝕋) into 𝐿¹×𝐿² defined by 𝑢→(𝐷_𝑥𝐿(𝑡,𝑢,𝑢^Δ),𝐷_𝑦𝐿(𝑡,𝑢,𝑢^Δ)) is continuous, thus, 𝜑^′ is continuous from 𝐻¹_𝑇(𝕋) into (𝐻¹_𝑇(𝕋))^∗, and the proof is completed.

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

Lemma 2.5 (see [34]). Let 𝐸 be a real Hilbert space with 𝐸=𝐸₁⊕𝐸₂ and 𝐸₂=𝐸^⟂₁. Suppose 𝐼∈𝐶¹(𝐸,𝑅), satisfies (PS), and(I1)𝐼(𝑢)=1/2(𝐴𝑢,𝑢)+𝑚(𝑢), where 𝐴𝑢=𝐴₁𝑃₁𝑢+𝐴₂𝑃₂𝑢 and 𝐴_𝑖∶𝐸_𝑖→𝐸_𝑖 is bounded and self-adjoint, 𝑖=1,2,(I2)𝑚^′(𝑢) is compact, and(I3)there exist a subspace 𝐸⊂𝐸 and sets 𝑆⊂𝐸,𝑄⊂𝐸 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 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 𝑡₁,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, ∫^𝑇₀𝑏(𝑡)Δ=0.(H2)∫^𝑇₀𝐻(𝑡,𝑥)Δ≥0 for all 𝑥∈ℝ^𝑛 and Δ-a.e. 𝑡∈[0,𝑇]_𝕋.(H3)Assume that there exists 𝑔∈𝐿¹([0,𝑇]_𝕋,ℝ⁺) such that |∇𝐻(𝑡,𝑥)|≤𝑔(𝑡) for all 𝑥∈ℝ^𝑛 and Δ-a.e. 𝑡∈[0,𝑇]_𝕋.(H4)Assume that there exist 𝛼₀∈(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 that 𝜎(𝜌(𝑡))=𝑡.

If 𝐿(𝑡,𝑥,𝑦)=𝐿(𝑡,𝑢(𝑡),𝑢^Δ1(𝑡))=2|𝑢^Δ(𝑡)|²−𝑏(𝑡)|𝑢(𝑡)|^𝜇−𝐻(𝑡,𝑢(𝑡)), (3.1) then by Lemma 2.4, the functional 𝜑 is given by 1𝜑(𝑢)=2^𝑇₀|𝑢^Δ(𝑡)|²Δ−^𝑇₀𝑏(𝑡)|𝑢(𝑡)|^𝜇Δ−^𝑇₀𝐻(𝑡,𝑢(𝑡))Δ, (3.2) which is continuously differentiable on 𝐻¹_𝑇(𝕋). Moreover ⟨𝜑^′(𝑢),𝑣⟩=^𝑇₀𝑢^Δ(𝑡)⋅𝑣^Δ(𝑡)Δ−𝜇^𝑇₀𝑏(𝑡)|𝑢(𝑡)|^𝜇−2−𝑢(𝑡)⋅𝑣(𝑡)Δ_𝑇0∇𝐻(𝑡,𝑢(𝑡))⋅𝑣(𝑡)Δ∀𝑢,𝑣∈𝐻¹_𝑇(𝕋). (3.3) That is, for all 𝑢,𝑣∈𝐻¹_𝑇(𝕋), we get ⟨𝜑^′(𝑢),𝑣⟩=−^𝑇₀𝑢^ΔΔ(𝜌(𝑡))⋅𝑣(𝑡)Δ−𝜇^𝑇₀𝑏(𝑡)|𝑢(𝑡)|^𝜇−2𝑢(𝑡)⋅𝑣(𝑡)Δ−^𝑇₀∇𝐻(𝑡,𝑢(𝑡)).𝑣(𝑡)Δ=−^𝑇₀(𝑢^ΔΔ(𝜌(𝑡))+𝜇𝑏(𝑡)|𝑢(𝑡)|^𝜇−2𝑢(𝑡)+∇𝐻(𝑡,𝑢(𝑡)))⋅𝑣(𝑡)Δ. (3.4)

Hence, 𝑢∈𝐻¹_𝑇(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 {𝑢_𝑛(𝑡)}⊂𝐻¹_𝑇(𝕋) be such that 𝜑^′(𝑢_𝑛(𝑡))→0 and let {𝑢_𝑛(𝑡)} be bounded in 𝐻¹_𝑇(𝕋), then {𝑢_𝑛(𝑡)} has a convergent subsequence in 𝐻¹_𝑇(𝕋).

Proof. Since {𝑢_𝑛(𝑡)} is bounded in 𝐻¹_𝑇(𝕋), it follows from [30, Theorem 4.12] that there exists a subsequence (still denoted by {𝑢_𝑛(𝑡)}) which weakly converges to 𝑢₀∈𝐻¹_𝑇(𝕋). By Lemma 2.2, we have 𝑢_𝑛→𝑢₀in[0,𝑇]_𝕋. (3.5) Hence, for 𝑡∈[0,𝑇]_𝕋, there exists an 𝑀>0 such that |𝑢_𝑛(𝑡)|≤𝑀,𝑛=1,2,….
Lemma 2.1 leads to
𝐻¹_𝑇(𝕋)↪𝐿²([0,𝑇]_𝕋,ℝ^𝑛)iscompact. (3.6) Hence 𝑢_𝑛→𝑢₀in𝐿²([0,𝑇]_𝕋,ℝ^𝑛). (3.7)
Since
⟨𝜑^′(𝑢_𝑛(𝑡))−𝜑^′(𝑢_𝑚(𝑡)),𝑢_𝑛(𝑡)−𝑢_𝑚=(𝑡)⟩_𝑇0(𝑢^Δ_𝑛(𝑡)−𝑢^Δ_𝑚(𝑡))⋅(𝑢^Δ_𝑛(𝑡)−𝑢^Δ_𝑚(𝑡))Δ−𝜇^𝑇₀𝑏(𝑡)(|𝑢_𝑛(𝑡)|^𝜇−2𝑢_𝑛(𝑡)−|𝑢_𝑚(𝑡)|^𝜇−2𝑢_𝑚(𝑡))⋅(𝑢_𝑛(𝑡)−𝑢_𝑚−(𝑡))Δ_𝑇0(∇𝐻(𝑡,𝑢_𝑛(𝑡))−∇𝐻(𝑡,𝑢_𝑚(𝑡)))⋅(𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡))Δ, (3.8) in view of (H1) and (H3), one has ^𝑇₀|𝑢^Δ_𝑛(𝑡)−𝑢^Δ_𝑚(𝑡)|²Δ≤‖𝜑^′(𝑢_𝑛(𝑡))−𝜑^′(𝑢_𝑚(𝑡))‖‖𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)‖+2𝜇𝑀^𝜇−1‖𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)‖_∞^𝑇₀|𝑏(𝑡)|Δ+2‖𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)‖_∞^𝑇₀𝑔(𝑡)Δ→0,as𝑛,𝑚→∞. (3.9) Consequently ‖𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)‖²=^𝑇₀|𝑢^Δ_𝑛(𝑡)Δ−𝑢^Δ_𝑚(𝑡)|²Δ+^𝑇₀|𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)|²Δ→0as𝑛,𝑚→∞, (3.10) which implies that {𝑢_𝑛} is a Cauchy sequence in 𝐻¹_𝑇(𝕋). By the completeness of 𝐻¹_𝑇(𝕋), we obtain that {𝑢_𝑛} is a convergent sequence in 𝐻¹_𝑇(𝕋); the proof is completed.

Now, we list our main result.

Theorem 3.2. Suppose that 𝜇>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 Lemma 2.5 hold with respect to 𝜑.
First, we show that 𝜑 satisfies the (PS) condition.
From Lemma 3.1, we only need to prove that {𝑢_𝑛} is bounded. Otherwise, there exists a subsequence (still denoted by 𝑢_𝑛) such that ‖𝑢_𝑛‖→∞. Let 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, then {𝑣_𝑛} is bounded; it has a subsequence (we still denote {𝑣_𝑛}) which weakly converges to 𝑣₀. In view of Lemma 2.2, {𝑣_𝑛} uniformly converges to 𝑣₀ in [0,𝑇]_𝕋. Since ‖𝑣_𝑛‖=1for all𝑛∈ℕ, one has ‖𝑣_𝑛‖≠0.
According to ‖𝑢_𝑛‖→∞ as 𝑛→∞, 𝜇>2 and (H3), for all 𝑣_𝑛,𝑣∈𝐻¹_𝑇(𝕋), we get 𝜇|^𝑇₀𝑏(𝑡)|𝑣_𝑛(𝑡)|^𝜇−2𝑣_𝑛(𝑡)⋅𝑣(𝑡)Δ|≤‖𝑢_𝑛‖^1−𝜇|−⟨𝜑^′(𝑢_𝑛),𝑣⟩|+‖𝑢_𝑛‖^2−𝜇|^𝑇₀𝑣^Δ_𝑛(𝑡)⋅𝑣^Δ(𝑡)Δ|+‖𝑢_𝑛‖^1−𝜇|−^𝑇₀∇𝐻(𝑡,𝑢_𝑛(𝑡))⋅𝑣(𝑡)Δ|≤‖𝑢_𝑛‖^1−𝜇‖𝜑^′(𝑢_𝑛)‖‖𝑣‖+𝐶₁‖𝑢_𝑛‖^2−𝜇‖𝑣_𝑛‖‖𝑣‖+‖𝑢_𝑛‖^1−𝜇‖𝑣‖_∞^𝑇₀𝑔(𝑡)Δ→0as𝑛→∞. (3.11) Thus, it follows from Lebesgue dominated convergence theorem on time scales [28] that |^𝑇₀𝑏(𝑡)|𝑣₀(𝑡)|^𝜇−2𝑣₀(𝑡)⋅𝑣(𝑡)Δ|=0∀𝑣∈𝐻¹_𝑇(𝕋). (3.12) By the arbitrariness of 𝑣, one has 𝑏(𝑡)=0forΔ-a.e.𝑡∈[0,𝑇]_𝕋, (3.13) which contradicts the condition (H1). Hence 𝜑 satisfies the (PS) condition.
Second, if
𝐴𝑢=𝑢(𝑡),𝑚(𝑢)=−^𝑇₀𝑏(𝑡)|𝑢(𝑡)|^𝜇Δ−^𝑇₀1𝐻(𝑡,𝑢(𝑡))Δ−2^𝑇₀𝑢²(𝑡)Δ, (3.14) then it is easy to verify that (I1) and (I2) hold.
Third, we will prove that 𝜑 satisfies the condition (I3) in Lemma 2.5.
For arbitrary 𝑢∈𝐻¹_𝑇(𝕋) with ‖𝑢‖≤𝑟₀, (H4) and (2.7) imply 1𝜑(𝑢)=2^𝑇₀|𝑢^Δ(𝑡)|²Δ−^𝑇₀𝑏(𝑡)|𝑢(𝑡)|^𝜇Δ−^𝑇₀≥1𝐻(𝑡,𝑢(𝑡))Δ2^𝑇₀|𝑢^Δ(𝑡)|²Δ−‖𝑢(𝑡)‖^𝜇_∞^𝑇₀|𝑏(𝑡)|Δ−𝛼₀^𝑇₀|𝑢(𝑡)|²Δ≥12𝑇‖𝑢‖²_∞−‖𝑢(𝑡)‖^𝜇_∞^𝑇₀|𝑏(𝑡)|Δ−𝛼₀𝑇‖𝑢(𝑡)‖²_∞1≥(2𝑇−𝛼₀𝑇)‖𝑢‖²_∞−‖𝑢(𝑡)‖^𝜇_∞^𝑇₀|𝑏(𝑡)|Δ. (3.15) Choose 𝜌>0 small enough such that 1𝛼≜(2𝑇−𝛼₀𝑇)𝜌²−𝜌^𝜇𝑇₀|𝑏(𝑡)|Δ>0, (3.16) thus 𝜑(𝑢)≥𝛼>∀𝑢∈𝐻¹_𝑇(𝕋)with‖𝑢‖=𝜌. (3.17) If 𝑆={𝑢∈𝐻¹_𝑇(𝕋),‖𝑢‖=𝜌}, (3.18) 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 𝜑(𝑥)=−^𝑇₀𝑏(𝑡)|𝑥|^𝜇Δ−^𝑇₀𝐻(𝑡,𝑥)Δ≤0∀𝑥∈ℝ^𝑛. (3.19) Choose 0≠𝑒∈𝐻¹_𝑇(𝕋) such that 𝑒(𝑡)=0 for all 𝑡∈[0,𝑇]_𝕋⧵[𝑡₁,𝑡₂]_𝕋 and ^𝑇₀𝑏(𝑡)𝑒(𝑡)Δ=^𝑡2_𝑡1𝑏(𝑡)𝑒(𝑡)Δ=0. (3.20) For arbitrary 𝑢∈𝐻¹_𝑇(𝕋), let 𝜑₁1(𝑢)=2^𝑇₀|𝑢^Δ(𝑡)|²Δ,𝜑₂(𝑢)=−^𝑇₀𝑏(𝑡)|𝑢(𝑡)|^𝜇Δ,𝜑₃(𝑢)=−^𝑇₀𝐻(𝑡,𝑢(𝑡))Δ, (3.21) then 𝜑(𝑢)=𝜑₁(𝑢)+𝜑₂(𝑢)+𝜑₃(𝑢). (3.22) For all 𝑥∈ℝ^𝑛 and 𝑟≥0, in terms of (H3), one has 𝜑₃(𝑥+𝑟𝑒)−𝜑₃(𝑥)=^𝑇₀⁰₁∇𝐻(𝑡,𝑥+𝑠𝑟𝑒(𝑡))⋅𝑟𝑒(𝑡)𝑑𝑠Δ≤𝑟^𝑇₀𝑔(𝑡)|𝑒(𝑡)|Δ≤𝑟‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋). (3.23) Since −𝜑₃(𝑥)≥0for all𝑥∈ℝ^𝑛, we have 𝜑₃(𝑥+𝑟𝑒)≤𝑟‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋)∀𝑥∈ℝ^𝑛,𝑟≥0. (3.24) In terms of (3.20) and Hölder's inequality on time scales, one obtains ^𝑡2_𝑡1−𝑏(𝑡)(|𝑥|²+𝑟²|𝑒(𝑡)|²)Δ=^𝑡2_𝑡1−𝑏(𝑡)(|𝑥|+𝑟|𝑒(𝑡)|)²Δ≤(^𝑡2_𝑡1−𝑏(𝑡)(|𝑥|+𝑟|𝑒(𝑡)|)^𝜇Δ)^2/𝜇×(^𝑡2_𝑡1−𝑏(𝑡)Δ)^{(𝜇−2)/𝜇}∀𝑥∈ℝ^𝑛,𝑟≥0. (3.25) Thus, by using ∫^𝑇₀𝑏(𝑡)|𝑥|^𝜇Δ=0, (3.25), and Hölder's inequality on time scales again, for all 𝑥∈ℝ^𝑛 and 𝑟≥0, we obtain 𝜑₂(𝑥+𝑟𝑒)=−^𝑇₀𝑏(𝑡)(|𝑥+𝑟𝑒(𝑡)|^𝜇−|𝑥|^𝜇)Δ=−^𝑡1₀𝑏(𝑡)|𝑥+𝑟𝑒(𝑡)|^𝜇Δ−^𝑡2_𝑡1𝑏(𝑡)|𝑥+𝑟𝑒(𝑡)|^𝜇Δ−^𝑇_𝑡2𝑏(𝑡)|𝑥+𝑟𝑒(𝑡)|^𝜇Δ+^𝑇₀𝑏(𝑡)|𝑥|^𝜇Δ=−^𝑡2_𝑡1𝑏(𝑡)|𝑥+𝑟𝑒(𝑡)|^𝜇Δ+^𝑡2_𝑡1𝑏(𝑡)|𝑥|^𝜇Δ≤(^𝑡2_𝑡1−𝑏(𝑡)(|𝑥|²+𝑟²|𝑒(𝑡)|²)Δ)^𝜇/2(^𝑡2_𝑡1−𝑏(𝑡)Δ)^(2−𝜇)/2+^𝑡2_𝑡1𝑏(𝑡)|𝑥|^𝜇Δ≤−^𝑡2_𝑡1𝑏(𝑡)(|𝑥|+𝑟|𝑒(𝑡)|)^𝜇Δ+^𝑡2_𝑡1𝑏(𝑡)|𝑥|^𝜇Δ≤0. (3.26) (3.24) and (3.26) lead to 𝜑(𝑥+𝑟𝑒)=𝜑₁(𝑥+𝑟𝑒)+𝜑₂(𝑥+𝑟𝑒)+𝜑₃≤1(𝑥+𝑟𝑒)2𝑟^2𝑇₀|𝑒^Δ|²Δ+𝑟‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋)∀𝑥∈ℝ^𝑛,𝑟≥0, (3.27) which means that there exists 𝑟₃>0 such that 𝛼𝜑(𝑥+𝑟𝑒)≤2∀𝑥∈ℝ^𝑛,𝑟∈[0,𝑟₃]. (3.28) For all 𝑅≥0 and 𝑟≥0, let ℎ(𝑅,𝑟)≜(^𝑡2_𝑡1−𝑏(𝑡)(|𝑅|²+𝑟²|𝑒(𝑡)|²)Δ)^𝜇/2(^𝑡2_𝑡1−𝑏(𝑡)Δ)^(2−𝜇)/2+^𝑡2_𝑡1𝑏(𝑡)|𝑅|^𝜇Δ=−(^𝑡2_𝑡1𝑏(𝑡)(|𝑅|²+𝑟²|𝑒(𝑡)|²)Δ)^𝜇/2(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2+^𝑡2_𝑡1𝑏(𝑡)|𝑅|^𝜇Δ. (3.29) In view of ∫^𝑡2_𝑡1𝑏(𝑡)>0, for all 𝑅≥0 and 𝑟≥0, we get 𝜕ℎ𝜕𝑟(𝑅,𝑟)=−𝑟𝜇^𝑡2_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ(^𝑡2_𝑡1𝑏(𝑡)(𝑅²+𝑟²|𝑒(𝑡)|²)Δ)^𝜇/2−1(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2≤−𝑟𝜇^𝑡2_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ(^𝑡2_𝑡1𝑅²𝑏(𝑡)Δ)^𝜇/2−1(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2≤−𝑟𝜇𝑅^{𝜇−2𝑡2}_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ. (3.30) (3.30) and ℎ(𝑅,0)=0 imply that 1ℎ(𝑅,𝑟)≤−2𝑟²𝜇𝑅^{𝜇−2𝑡2}_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ∀𝑅≥0,𝑟≥0. (3.31) Note that there exists 𝑅₁>𝜌>0 such that ^𝑇₀|𝑒^Δ(𝑡)|²Δ−𝜇𝑅^{𝜇−2𝑡2}_𝑡1𝑏(𝑡)|𝑒(𝑡)|²2Δ≤−𝑟₃‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋)∀𝑅≥𝑅₁. (3.32) Therefore, by using (3.24), (3.26), (3.31), and (3.32), for 𝑥=𝑅,𝑟∈[𝑟₃,𝑅] and 𝑅≥𝑅₁, one has 𝜑(𝑥+𝑟𝑒)=𝜑₁(𝑥+𝑟𝑒)+𝜑₂(𝑥+𝑟𝑒)+𝜑₃≤1(𝑥+𝑟𝑒)2𝑟^2𝑇₀|𝑒^Δ(𝑡)|²Δ+ℎ(𝑅,𝑟)+𝑟‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋)≤12𝑟^2𝑇₀|𝑒^Δ(𝑡)|²1Δ−2𝑟²𝜇𝑅^{𝜇−2𝑡2}_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ+𝑟‖𝑒‖_∞‖𝑔‖_𝐿(1)𝑟≤−𝑟₃(𝑟−𝑟₃)‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋)≤0. (3.33) Thus, (3.28) and (3.33) can lead to 𝛼𝜑(𝑥+𝑟𝑒)≤2,∀|𝑥|=𝑅,𝑟∈[0,𝑅],𝑅≥𝑅₁. (3.34) For all 𝑥∈ℝ^𝑛 and 𝑅≥0, denote 𝑓(𝑥,𝑅)≜−(^𝑡2_𝑡1𝑏(𝑡)(|𝑥|²+𝑅²|𝑒(𝑡)|²)Δ)^𝜇/2(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2+^𝑡2_𝑡1𝑏(𝑡)|𝑥|^𝜇Δ. (3.35) By using the similar way to inequality (3.26), one has 𝜑₂(𝑥+𝑅𝑒)≤𝑓(𝑥,𝑅)∀𝑥∈ℝ^𝑛,𝑅≥0, (3.36) for all 𝑥∈ℝ^𝑛 and 𝑅≥0, since 𝜕𝑓𝜕𝑅(𝑥,𝑅)=−𝑅𝜇^𝑡2_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ(^𝑡2_𝑡1𝑏(𝑡)(|𝑥|²+𝑅²|𝑒(𝑡)|²)Δ)^𝜇/(2−1)(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2≤−𝜇𝑅^𝜇−1(^𝑡2_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ)^𝜇/2(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2. (3.37) In view of (3.37) and 𝑓(𝑥,0)=0, for all 𝑥∈ℝ^𝑛 and 𝑅≥0, we get 𝑓(𝑥,𝑅)≤−𝑅^𝜇(^𝑡2_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ)^𝜇/2(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2. (3.38) From (3.24), (3.36), and (3.38), for all 𝑥∈ℝ^𝑛 and 𝑅≥0, we obtain 𝜑(𝑥+𝑅𝑒)=𝜑₁(𝑥+𝑅𝑒)+𝜑₂(𝑥+𝑅𝑒)+𝜑₃≤1(𝑥+𝑅𝑒)2𝑅^2𝑇₀|𝑒^Δ(𝑡)|²Δ−𝑅^𝜇(^𝑡2_𝑡1𝑏(𝑡)|𝑒(𝑡)|²Δ)^𝜇/2(^𝑡2_𝑡1𝑏(𝑡)Δ)^(2−𝜇)/2+𝑅‖𝑒‖_∞‖𝑔‖_𝐿1(𝕋), (3.39) which implies that there exists 𝑅₂>𝑅₁, such that 𝜑(𝑥+𝑅𝑒)≤0∀𝑥∈ℝ^𝑛,𝑅≥𝑅₂. (3.40)
Now let
𝑄={𝑥+𝑟𝑒∣𝑥∈ℝ^𝑛,‖𝑥‖=𝑅,𝑟∈[0,𝑅],𝑅≥𝑅₂}. (3.41) 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 that 𝑆 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 𝕋=[0,0.3]∪{0.4,0.45,0.5,0.55,0.6}∪[0.7,1],𝑇=1. (4.1) Consider the following second-order Hamiltonian system on time scales 𝕋 of the form 𝑢^ΔΔ(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)=𝑢^Δ(0)−𝑢^Δ(1)=0, (4.2) 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 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. (4.3) 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).