We adopt the Leray-Schauder degree theory and critical point theory to consider a second order Dirichlet boundary value problem on time scales and obtain some existence theorems of weak solutions for the previous problem.

1. Introduction

In recent years, differential equations on time scales have been studied extensively in the literature. There have been many approaches to study the existence and multiplicity of solutions for differential equations on time scales. The variational method is, to the best of our knowledge, novel, and it may open a new approach to deal with nonlinear problems on time scales. For more details about recent development in the direction, we refer the reader to [1–12] and the references therein. The two books [13, 14] by Bohner and Peterson summarize some excellent results on time scales.

In [1, 2], the authors utilized variational techniques and critical point theory to derive some sufficient conditions for the existence of positive solutions for the following second order dynamic equation with Dirichlet boundary conditions:
(1)-uΔΔ(t)=f(σ(t),uσ(t)),Δ-a.e.t∈J,u(a)=0=u(b).

In [8], Zhou and Li studied Sobolev's spaces on time scales, and given their properties, as applications, they presented variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian systems on time scales as follows:
(2)uΔ2(t)=∇F(σ(t),uσ(t)),Δ-a.e.t∈[0,T]𝕋κ2,u(0)=u(T),uΔ(0)=uΔ(T),
where F(t,x) is Δ-measurable in t for every x∈ℝN and continuously differentiable in x for Δ-a.e. t∈[0,T]𝕋.

Motivated by the works cited previously, in this paper, we study the second order Dirichlet boundary value problem on time scales. Consider
(3)-uΔΔ(t)=λuσ(t)+h(σ(t),uσ(t)),Δ-a.e.t∈J,u(0)=u(T)=0,
where we say that a property holds for Δ-almost every t∈A⊂𝕋 or Δ-almost everywhere on A⊂𝕋, Δ-a.e., whenever there exists a set E⊂A with null Lebesgue Δ-measure such that this property holds for every t∈A∖E, J≔[0,ρ(T))∩𝕋 and 𝕋⊂ℝ is an arbitrary bounded time scale such that min𝕋=0 and max𝕋=T.

We assume that λ is a parameter, h∈CAR([0,T)𝕋,ℝ), [0,T)𝕋≔[0,T)∩𝕋, where CAR([0,T)𝕋,ℝ) means that h fulfils the Carathéodory conditions (see [15, Definition 3.2.22] and [2, (i) and (ii) in (H2)]). By virtue of the Leray-Schauder degree theory, a result involving the existence of weak solutions is established. Next, we utilize two critical point theorems to investigate that problem (3) has at least one weak solution and infinitely many weak solutions with the parameter λ<0, respectively. The results obtained here improve some existing results in the literature.

2. Preliminaries and an Existence Theorem via Leray-Schauder Degree Theory

We first offer some related preliminaries concerning the basic definitions and results on time scales. For convenience, for f∈LΔ2([0,T)𝕋,ℝ), we denote ∫0T|f(s)|2Δs=∫[0,T)𝕋|f(s)|2Δs. Let Sobolev's space WΔ,T1,2 be defined by (see [1–3])
(4)WΔ,T1,2≔{x∈AC([0,T]𝕋,ℝ):xΔ∈LΔ2([0,T)𝕋,ℝ),x(0)=x(T)=0x∈AC([0,T]𝕋,ℝ):xΔ∈LΔ2([0,T)𝕋,ℝ)},
where AC([0,T]𝕋,ℝ) (see [3, Definition 2.9]) denotes the set of absolutely continuous functions on [0,T]𝕋. Then WΔ,T1,2 is a Hilbert space with the inner product
(5)(u,v)1=∫[0,T)𝕋uΔ(t)vΔ(t)Δt,
and let ∥·∥1 be the norm induced by the inner product (·,·)1 (see [1, page 1265] and [2, page 371]).

Lemma 1 (see [<xref ref-type="bibr" rid="B3">3</xref>, Proposition 3.7]).

The immersion WΔ,T1,2↪C([0,T]𝕋,ℝ) is compact.

We firstly need to consider the following eigenvalue problem:
(6)-uΔΔ(t)=λuσ(t),t∈[0,T]𝕋,u(0)=u(T)=0.
For each v∈WΔ,T1,2, multiply by vσ(t) on both sides of the previous equation in (6) and integrate over [0,T)𝕋 to obtain
(7)∫[0,T)𝕋-uΔΔ(t)vσ(t)Δt=λ∫[0,T)𝕋uσ(t)vσ(t)Δt.

By (7) of [8], we obtain
(8)∫[0,T)𝕋uΔ(t)vΔ(t)Δt=λ∫[0,T)𝕋uσ(t)vσ(t)Δt.

Lemma 2 (see [<xref ref-type="bibr" rid="B19">9</xref>, Lemma 3.4] and [<xref ref-type="bibr" rid="B5">13</xref>, Theorem 4.95]).

The eigenvalues of (6) may be arranged as 0<λ1<λ2<⋯, and λ1 can be expressed by
(9)λ1≔infu∈WΔ,T1,2,u≠0∫[0,T)𝕋|uΔ(t)|2Δt∫[0,T)𝕋|uσ(t)|2Δt.
By Lemma 2, if there exists u1∈WΔ,T1,2 and ∥u1∥1=1, then we have
(10)1λ1=∫[0,T)𝕋|u1σ(t)|2Δt.
Choosing φ1σ=λ1u1σ, we can easily obtain
(11)∫[0,T)𝕋|φ1σ(t)|2Δt=λ1∫[0,T)𝕋|u1σ(t)|2Δt=1,λ1=∫[0,T)𝕋|φ1Δ(t)|2Δt.
Moreover, we also have
(12)∫[0,T)𝕋φ1Δ(t)vΔ(t)Δt=λ1∫[0,T)𝕋φ1σ(t)vσ(t)Δt,∀v∈WΔ,T1,2.

Lemma 3 (see [<xref ref-type="bibr" rid="B4">16</xref>, Corollary 3.3]).

Let u∈WΔ,T1,2. Then the Wirtinger-type inequality is
(13)∫[0,T)𝕋|uσ(t)|2Δt≤1λ1∫[0,T)𝕋|uΔ(t)|2Δt,
where λ1 is defined by Lemma 2.

Denote an operator A:WΔ,T1,2→WΔ,T1,2 as follows
(14)(Au,v)=∫[0,T)𝕋uσ(t)vσ(t)Δt,∀u,v∈WΔ,T1,2.
By Hölders inequality and Lemma 3, we have
(15)|∫[0,T)𝕋uσ(t)vσ(t)Δt|≤∫[0,T)𝕋|uσ(t)vσ(t)|Δt≤(∫[0,T)𝕋|uσ(t)|2Δt)1/2(∫[0,T)𝕋|vσ(t)|2Δt)1/2≤1λ1∥u∥1∥v∥1,∀u,v∈WΔ,T1,2.
Then A is a bounded linear operator. By Lemma 1, A is compact. Clearly, A is also symmetric, (Au,u)>0 for uσ≢0. Consequently, the supremum 1/λ1 is achieved by Theorem 6.3.12 in [15]. Hence, (10)–(12) are true.

In what follows, we offer two lemmas involving Leray-Schauder degree.

Lemma 4 (see [<xref ref-type="bibr" rid="B7">15</xref>, Proposition 5.2.22]).

Let Ω be an open bounded set in a Banach space X and A∈𝒞(Ω¯,X). Let x0∈Ω be a unique solution in Ω¯ of the equation x=F(x). Assume that the Fréchet derivative A′(x0) exists and I-A′(x0) is continuously invertible. Then
(16)deg(I-A,Ω,0)=(-1)β,whereβ=∑λ∈σ(F′(x0))∩ℝ,λ>1m(λ),
and m(λ) is the multiplicity of the eigenvalue λ of the operator A′(x0).

Lemma 5 (see [<xref ref-type="bibr" rid="B7">15</xref>, Theorem 5.2.13]).

Let Ω be an open bounded set in a Banach space X. There exists a mapping deg(I-A,Ω,y0) defined for all A∈𝒞(Ω¯,X) and y0∈X such that x-A(x)≠y0, ∀x∈∂Ω. This mapping has the homotopy invariance property: if A,B∈𝒞(Ω¯,X) and H(t,x)=(1-t)Ax+tBx, t∈[0,1], and x∈Ω¯ are such that x-H(t,x)≠y0, for every x∈∂Ω and t∈[0,1], then deg(I-A,Ω,y0)=deg(I-B,Ω,y0).

Now, we list our hypotheses for (3).

λ is a parameter and λ≠λn, where λn are determined by Lemma 2.

h∈
CAR
([0,T)𝕋,ℝ); there exist f∈LΔ2([0,T)𝕋,ℝ), c>0 and α∈[0,1) such that
(17)h(t,x)≤f(t)+c|x|α,∀t∈[0,T)𝕋,x∈ℝ.

h is α-Lipschitz continuous with respect to the second variable; that is, there exists a constant c>0, such that
(18)|h(t,x1)-h(t,x2)|≤c|x1-x2|α,∀t∈[0,T)𝕋,x1,x2∈ℝ,αasin(H2).

Remark 6.

We can take h(t,x)=f(t)+c|x|α, where c,f,α are as in (H2). Clearly, it also satisfies (H3). In fact,
(19)|h(t,x1)-h(t,x2)|=c||x1|α-|x2|α|≤c|x1-x2|α,∀x1,x2∈ℝ.

It is clear that (3) is equivalent to the following integral equation:
(20)∫[0,T)𝕋uΔ(t)vΔ(t)Δt=λ∫[0,T)𝕋uσ(t)vσ(t)Δt+∫[0,T)𝕋h(σ(t),uσ(t))vσ(t)Δt,∀v∈WΔ,T1,2.
We define an operator S:WΔ,T1,2→WΔ,T1,2 as follows:
(21)(Su,v)=∫[0,T)𝕋h(σ(t),uσ(t))vσ(t)Δt,∀u,v∈WΔ,T1,2.

Lemma 7.

S is compact on WΔ,T1,2.

Proof.

We first prove that there exists a ball B(0,R)⊂WΔ,T1,2 (R>0) such that S maps B(0,R) into itself if R is large enough. Indeed, (H2) leads to
(22)∥Su∥1=sup∥v∥1=1|Su,v|=sup∥v∥1=1∫[0,T)𝕋h(σ(t),uσ(t))vσ(t)Δt≤sup∥v∥1=1(∫[0,T)𝕋|h(σ(t),uσ(t))|2Δt)1/2×(∫[0,T)𝕋|vσ(t)|2Δt)1/2≤1λ1(∫[0,T)𝕋|fσ(t)+c|uσ(t)|α|2Δt)1/2≤2λ1(∫[0,T)𝕋|fσ(t)|2Δt)1/2+2c2λ1(∫[0,T)𝕋|uσ(t)|2αΔt)1/2≤2λ1(∫[0,T)𝕋|fσ(t)|2Δt)1/2+2c2T1-αλ1(∫[0,T)𝕋|uσ(t)|2Δt)α/2≤2λ1(∫[0,T)𝕋|fσ(t)|2Δt)1/2+2c2T1-αλ11+α∥u∥1α.
Let η1≔2/λ1(∫[0,T)𝕋|fσ(t)|2Δt)1/2, and η2≔2c2T1-α/λ11+α. For any u∈B(0,R),
(23)∥Su∥1<R,providedη1+η2Rα<R.
Therefore, the above claim is true. Meanwhile, we also arrive at immediately S is uniformly bounded on B(0,R). On the other hand, we shall prove that S is equicontinuous on B(0,R). By (H3) and Hölders inequality, we have
(24)∥Su1-Su2∥1≤sup∥v∥1=1(∫[0,T)𝕋|h(σ(t),u1σ(t))-h(σ(t),u2σ(t))||vσ(t)|Δt)≤sup∥v∥1=1(∫[0,T)𝕋c|u1σ(t)-u2σ(t)|α|vσ(t)|Δt)≤csup∥v∥1=1(∫[0,T)𝕋|u1σ(t)-u2σ(t)|2Δt)α/2×(∫[0,T)𝕋|vσ(t)|2/(2-α)Δt)(2-α)/2≤T(1-α)/(2-α)csup∥v∥1=1(∫[0,T)𝕋|u1σ(t)-u2σ(t)|2Δt)α/2×(∫[0,T)𝕋|vσ(t)|2Δt)1/(2-α)≤T(1-α)/(2-α)cλ11/(2-α)+α/2∥u1-u2∥1α.
We have from Arzelà-Ascoli theorem S is compact on B(0,R)⊂WΔ,T1,2.

Theorem 8.

Assume that (H1)–(H3) hold; problem (3) has at least a weak solution.

Proof.

We will utilize Leray-Schauder degree to prove the result. One is invited to verify that the existence of a solution of (3) is equivalent to the existence of a solution of the operator equation
(25)u=λAu+Su,∀u∈WΔ,T1,2,
where A and S are defined by (14) and (21), respectively. For the reason that A is bounded, linear, and compact on WΔ,T1,2, we easily see that Fréchet derivative (λA)′ exists and I-(λA)′ is continuously invertible by (H1). Consequently, Lemma 4 implies
(26)deg(I-λA,B(0,R),0)≠0.
So, to complete the proof, we have to find an admissible homotopy connecting I-λA-S(·) and I-λA. Define
(27)H(τ,u)=u-λAu-τSu,τ∈[0,1],u∈WΔ,T1,2.
We shall prove that there exists R>0 such that for all u∈WΔ,T1,2, ∥u∥1=R, and τ∈[0,1], we obtain
(28)H(τ,u)≠0.
If the claim is false, we can find sequences {un}⊂WΔ,T1,2 and {tn}⊂[0,1] such that ∥un∥1→∞ and
(29)un-λAun-τnSun=0.
Set vn≔un/∥un∥1 and divide (29) by ∥un∥1 to get
(30)vn-λAvn-τnSun∥un∥1=0.
This is equivalent to
(31)∫[0,T)𝕋vnΔ(t)wΔ(t)Δt=λ∫[0,T)𝕋vnσ(t)wσ(t)Δt+τn∫[0,T)𝕋h(σ(t),unσ(t))∥un∥1wσ(t)Δt,
for each w∈WΔ,T1,2. Now, passing to suitable subsequences, without loss of generality, we may assume that τn→τ∈[0,1] and vn→v in WΔ,T1,2. Note that, similar with (22), we find
(32)|∫[0,T)𝕋h(σ(t),unσ(t))∥un∥1wσ(t)Δt|≤∫[0,T)𝕋|h(σ(t),unσ(t))|∥un∥1|wσ(t)|Δt≤(η1+η2∥un∥1α)∥w∥1∥un∥1⟶0,n⟶∞.
On the other hand, by the compactness of A, we have Avn→Av (see [15, Proposition 2.2.4(iii)]). In (30), let n→∞; we have v=λAv, and v∈WΔ,T1,2 satisfies ∥v∥1=1. However, this contradicts our assumption λ≠λn, n=1,2,…. This implies that (28) holds. By Lemma 5 and (26), we have
(33)deg(I-λA-S,B(0,R),0)=deg(I-λA,B(0,R),0)≠0.
Therefore, (3) has at least a weak solution.

3. Two Existence Theorems via Critical Point Theory

We still use the Sobolev’s space WΔ,T1,2 defined by (4), which is equipped with the inner product
(34)(u,v)2≔∫[0,T)𝕋uσ(t)vσ(t)Δt+∫[0,T)𝕋uΔ(t)vΔ(t)Δt,
and the corresponding norm
(35)∥u∥2≔(∫[0,T)𝕋|uσ(t)|2Δt+∫[0,T)𝕋|uΔ(t)|2Δt)1/2.
Now, we will establish the corresponding variational formulations for problem (3) as follows:
(36)φ(u)=12(∫[0,T)𝕋|uΔ(t)|2Δt-λ∫[0,T)𝕋|uσ(t)|2Δt)-∫[0,T)𝕋F(σ(t),uσ(t))Δt,
where F(t,ξ)=∫0ξh(t,s)ds.

We now list our hypotheses for (3).

F(t,x) is Δ-measurable in t for every x∈ℝ and continuously differentiable in x for t∈[0,T]𝕋, and there exist ϵ1∈C(ℝ+,ℝ+) and ϵ2∈LΔ1([0,T]𝕋,ℝ+)(37)|F(t,x)|≤ϵ1(|x|)ϵ2(t),|h(t,x)|≤ϵ1(|x|)ϵ2(t),

for all x∈ℝ and Δ-a.e. t∈[0,T]𝕋.

There exist c1>0 and p>0 such that
(38)|F(t,x)|≤c1|x|p,t∈[0,T]𝕋,u∈ℝ.

There exist ɛ1,ɛ2>0 such that
(39)h(t,x)x-2F(t,x)≥ɛ1|x|-ɛ2,t∈[0,T]𝕋,u∈ℝ.

lim|x|→∞(F(t,x)/|x|2)=+∞ uniformly for t∈[0,T]𝕋.

F(t,-x)=F(t,x), t∈[0,T]𝕋, u∈ℝ.

Remark 9.

Let
(40)F(t,x)=∑i=1n1(2i)2x2i,
where n>1 is a fixed positive integer. Then
(41)h(t,x)=∑i=1n12ix2i-1.
Direct computation shows
(42)lim|x|→∞F(t,x)|x|2=lim|x|→∞∑i=1n(1/(2i)2)x2i|x|2=+∞,lim|x|→∞h(t,x)x-2F(t,x)|x|=lim|x|→∞∑i=1n[1/2i-1/2i2]x2i|x|=+∞.
Clearly, (H4)–(H8) hold.

By (H4), we find φ∈C1(WΔ,T1,2,ℝ) (see [11, Theorem 2.27]), and for u,v∈WΔ,T1,2,
(43)φ′(u)v=∫[0,T)𝕋uΔ(t)vΔ(t)Δt-λ∫[0,T)𝕋uσ(t)vσ(t)Δt-∫[0,T)𝕋h(σ(t),uσ(t))vσ(t)Δt.
Clearly, the existence of weak solutions for (3) is equivalent to the existence of critical points for φ. In what follows, we take a¯≔min{1,-λ}>0, b¯≔max{1,-λ}>0.

Lemma 10 (see [<xref ref-type="bibr" rid="B12">17</xref>, Theorem 1.2]).

Suppose X is a reflexive Banach space with norm ∥·∥, and φ:X→ℝ∪{+∞} is coercive and weak (sequentially) lower semicontinuous; that is, the following conditions are fulfilled as follows:

φ(u)→∞ as ∥u∥→∞, u∈X.

For any u∈X, any sequence {um}⊂X such that um⇀u, there holds
(44)φ(u)≤liminfm→∞φ(um).

Then φ is bounded from below and attains its infimum on X.

Theorem 11.

If (H4) and (H5) with p∈(0,2) hold, (3) has at least a weak solution.

Proof.

Our working space WΔ,T1,2 is a Hilbert space, so it is reflexive. By Lemma 2.1 and Theorem 3.3 in [8], we see φ is weakly lower semicontinuous on WΔ,T1,2. On the other hand, by (H5), we have
(45)φ(u)≥a¯2∥u∥22-∫[0,T)𝕋c1|uσ(t)|pΔt≥a¯2∥u∥22-c1T(2-p)/2∥u∥2p,
and thus φ(u)→∞ as ∥u∥2→∞. Lemma 10 implies φ can attain its infimum in X; that is, (3) has at least a weak solution. This completes the proof.

If X is a Hilbert space, there exist (see [18]) {en}n=1∞⊂X and {fn}n=1∞⊂X* such that fn(em)=δn,m, X=span¯{en:n=1,2,…} and X*=span¯W*{fn:n=1,2,…}. For j,k∈ℕ, denote Xj≔span{ej}, Yk≔⊕j=1kXj, and Zk≔⊕j=k+1∞Xj¯. Clearly, X=⊕j∈ℕXj¯ with dimXj<∞ for all j∈ℕ. Since WΔ,T1,2 is a Hilbert space, we can choose an orthonormal basis {en} such that
(46)WΔ,T1,2=span¯{en:n=1,2…},Yk≔⨁j=1kXj,Zk≔⨁j=k+1∞Xj¯,whereXj≔span{ej}.

Definition 12 (see [<xref ref-type="bibr" rid="B11">19</xref>, Definition 1.1]).

Assume that X is a Banach space with norm ∥·∥; we say that φ∈C1(X,ℝ) satisfies Cerami condition (C) if for all d∈ℝ,

any bounded sequence {un}⊂X satisfying φ(un)→d, φ′(un)→0 possesses a convergent subsequence;

there exist δ,ξ,ρ>0 such that for any u∈φ-1([d-δ,d+δ]) with ∥u∥≥ξ, ∥φ′(u)∥·∥u∥≥ρ. Denote Sρ≔{u∈X:∥u∥=ρ}. We will introduce the following Fountain theorem under condition (C).

Lemma 13 (see [<xref ref-type="bibr" rid="B11">19</xref>, Proposition 1.2]).

Assume that φ∈C1(X,ℝ) satisfies condition (C), and φ(-u)=φ(u). For each k∈ℕ, there exists ρk>rk>0 such that

bk≔infu∈Zk∩Srkφ(u)→+∞, k→∞;

ak≔maxu∈Yk∩Sρkφ(u)≤0.

Then φ has a sequence of critical points un, such that φ(un)→+∞ as n→∞.

Lemma 14.

Suppose that (H4), (H5), and (H6) hold; then φ satisfies the condition (C).

Proof.

For all d∈ℝ, we assume that {un}∈WΔ,T1,2 is bounded and
(47)φ(un)⟶d,φ′(un)⟶0,n⟶∞.
Going, if necessary, to a subsequence, we can assume that un⇀u in WΔ,T1,2; then
(48)(φ′(un)-φ′(u))(un-u)=∫[0,T)𝕋|unΔ(t)-uΔ(t)|2Δt-λ∫[0,T)𝕋|unσ(t)-uσ(t)|2Δt-∫[0,T)𝕋[h(σ(t),unσ(t))-h(σ(t),uσ(t))]×[unσ(t)-uσ(t)]Δt.
Lemma 1 leads to
(49)∫[0,T)𝕋[h(σ(t),unσ(t))-h(σ(t),uσ(t))]×[unσ(t)-uσ(t)]Δt⟶0,n⟶∞.
It follows that un⇀u in WΔ,T1,2 and (φ′(un)-φ′(u))(un-u)→0, we see
(50)∫[0,T)𝕋|unΔ(t)-uΔ(t)|2Δt-λ∫[0,T)𝕋|unσ(t)-uσ(t)|2Δt⟶0,n⟶∞.

Clearly, it is equivalent to ∥·∥2, and then we have
(51)∥un-u∥2⟶0,n⟶∞.
Hence, condition (i) of Definition 12 holds. Next, we prove condition (ii) of Definition 12, suppose the contrary, there exists a sequence {un}⊂WΔ,T1,2 such that
(52)φ(un)⟶d,∥φ′(un)∥2·∥un∥2⟶0,n⟶∞,(53)∥un∥2⟶∞,n⟶∞.
By (52), there exists a constant ɛ3>0 such that
(54)φ(un)-12φ′(un)un≤c3.
On the other hand, (H6) implies
(55)φ(un)-12φ′(un)un=∫[0,T)𝕋[12h(σ(t),unσ(t))unσ(t)-F(σ(t),unσ(t))]Δt≥12ɛ1∫[0,T)𝕋|unσ(t)|Δt-12ɛ2T.
This, together with (54), leads to there is a constant ɛ4>0 such that maxt∈[0,T]𝕋|unσ(t)|≤ɛ4. By (H5), similar with (45), we have
(56)φ(un)≥a¯2∥un∥22-∫[0,T)𝕋c1|unσ(t)|pΔt≥a¯2∥un∥22-c1Tɛ4p,
and thus φ(un)→∞ if (53) holds, which contradicts φ(un)→d in (52). This proves that φ satisfies condition (C).

Theorem 15.

Under assumptions (H4)–(H8) with p>2 in (H5), problem (3) has infinitely many solutions.

Proof.

(H8) and Lemma 14 enable us to obtain that φ(u)=φ(-u) and φ satisfies the condition (C). For any u∈Yk, let
(57)∥u∥*≔(∫[0,T)𝕋|uσ(t)|2Δt)1/2,
and it is easy to verify that ∥·∥ defined by (57) is a norm of Yk. Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant ɛ5 such that ɛ5∥u∥2≤∥u∥*. In view of (H7), there exist ɛ6>b¯/2ɛ52 and ϱ>0 such that
(58)F(σ(t),uσ(t))≥ɛ6|uσ(t)|2-ϱ,t∈[0,T]𝕋,u∈ℝ.
This implies that
(59)φ(u)=12(∫[0,T)𝕋|uΔ(t)|2Δt-λ∫[0,T)𝕋|uσ(t)|2Δt)-∫[0,T)𝕋F(σ(t),uσ(t))Δt≤b¯2∥u∥22-ɛ6ɛ52∥u∥22+ϱT.
Since b¯/2-ɛ6ɛ52<0; then there exists positive constant dk such that
(60)φ(u)≤0,foreachu∈Yk,∥u∥≥dk.
Let βk≔supu∈Zk,∥u∥2=1|u(t)|, t∈[0,T]𝕋. Then by Lemma 3.8 of [20] and Lemma 1 we obtain βk→0, as k→∞. For any u∈Zk, note that p∈(1,2), in view of (H5), we find that
(61)φ(u)=12(∫[0,T)𝕋|uΔ(t)|2Δt-λ∫[0,T)𝕋|uσ(t)|2Δt)-∫[0,T)𝕋F(σ(t),uσ(t))Δt≥a¯2∥u∥22-c1∫[0,T)𝕋|uσ(t)|pΔt≥a¯2∥u∥22-c1T∥u∥∞p≥a¯2∥u∥22-c1Tβkp∥u∥2p.
Choosing ∥u∥2=rk≔1/βk, then rk→∞ as k→∞, then we have;
(62)φ(u)≥a¯2rk2-c1T⟶∞,ask⟶∞.
Hence, bk≔infu∈Zk,∥u∥=rkφ(u)→∞ as k→∞. Combining this and (60), we can take ρk≔max{dk,rk+1}, and thus ak≔maxu∈Yk,∥u∥=ρkφ(u)≤0. Up until now, we have proved that the functional φ satisfies all the conditions of Lemma 13; then φ has infinitely many solutions.

Acknowledgment

This research is supported by the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J13LI51).

AgarwalR. P.Otero-EspinarV.PereraK.ViveroD. R.Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methodsAgarwalR. P.Otero-EspinarV.PereraK.ViveroD. R.Multiple positive solutions of singular Dirichlet problems on time scales via variational methodsAgarwalR. P.Otero-EspinarV.PereraK.ViveroD. R.Basic properties of Sobolev's spaces on time scalesOtero-EspinarV.Pernas-CastañoT.Variational approach to second-order impulsive dynamic equations on time scalesSuY.-H.LiW.-T.Periodic solution for non-autonomous second order Hamiltonian systems on time scalesSuY.-H.LiW.-T.Periodic solution of second-order Hamiltonian systems with a change sign potential on time scalesSuY.-H.FengZ.A non-autonomous Hamiltonian system on time scalesZhouJ.LiY.Sobolev's spaces on time scales and its applications to a class of second order Hamiltonian systems on time scalesZhangQ.-G.SunH.-R.Variational approach for Sturm-Liouville boundary value problems on time scalesZhouJ.LiY.Variational approach to a class of second order Hamiltonian systems on time scalesLiY.ZhouJ.Existence of solutions for a class of damped vibration problems on time scalesZhangQ.HeX.SunH.Positive solutions for Sturm-Liouville BVPs on time scales via sub- supersolution and variational methodsBohnerM.PetersonA.BohnerM.PetersonA.DrábekP.MilotaJ.AgarwalR. P.Otero-EspinarV.PereraK.ViveroD. R.Wirtinger's inequalities on time scalesStruweM.FanX.HanX.Existence and multiplicity of solutions for ℝN-Laplacian equations in RNQianA.LiC.Infinitely many solutions for a Robin boundary value problemWillemM.