# Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

Authors: Yanning Wang; Jianwen Zhou; Yongkun Li
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scale T:Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))), Δ-a.e.t∈a,bTκ2, u(a)-u(b)=0, Tα(u)(a)-Tα(u)(b)=0, where Tα(u)(t) denotes the conformable fractional derivative of u of order α at t, σ is the forward jump operator, a,b∈T,0<a<b,p>1, and F:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.


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## Body

## 1. Introduction

A time scaleT is an arbitrary nonempty closed subset of the real numbers, which has the topology inherited from the real numbers with the standard topology. In order to unify and generalize continuous and discrete analysis, the calculus on time scales was initiated by Hilger in 1990 (see [1, 2]). In view of the fact that time scale calculus can be used to model dynamic processes whose time domains are more complicated than the set of integers or real numbers, it plays an important role in various equations and systems arising in economy, biology, ecology, astronomy, and so on (see [1, 3, 4]). During the last decade, there has been a great deal of interest in the study of dynamic equations on time scales and the research in this area is rapidly growing, see [5–10] and the references therein. Nevertheless, these studies are all about integer order dynamic equations on time scales. The existence and multiplicity of solutions for fractional dynamic equations on time scales has received considerably less attention (see [11, 12]).It is well known that the fractional calculus refers to differentiation and integration of an arbitrary (noninteger) order. The theory is owed to mathematicians such as Leibniz, Liouville, Riemann, Letnikov, and Grunwald. In this day and age, fractional calculus is one of the most intensively developing areas of mathematical analysis, including several definitions of fractional calculus like Riemann-Liouville fractional calculus, Caputo fractional calculus, Grunwald-Letnikov fractional calculus, Hadamard fractional calculus, Riesz fractional calculus, Weyl fractional calculus, Kolwankar-Gangal fractional calculus and so on. Especially, in [13], the authors introduce the conformable fractional calculus on R. In order to unify and generalize the Hilger calculus and the conformable fractional calculus on R, the authors introduce the conformable fractional calculus on time scales and study its properties in [14]. The generation of fractional differential equations is born with the birth of the fractional calculus. Fractional differential equations have gained importance due to their numerous applications in many fields of science and engineering including fluid flow, electrical networks, probability and statistics, viscoelasticity, chemical physics and signal processing, and so on, see [15–18] and references therein.On one hand, there have been many approaches to study solutions of boundary value problems for the fractional differential equations such as lower and upper solution method, monotone iterative method ([19]), fixed-point theorems ([20]), Leray-Schauder theory ([21]), critical point theory ([22]) and so on. But until now, as far as I am concerned, no researchers have applied the critical point theory to study conformable fractional differential equations on time scales. Since it is often very difficult to establish a suitable space and variational functional for conformable fractional differential equations on time scales.On the other hand, Sobolev spaces are regarded as one of most fundamental tools, especially in the use of variational methods to solve boundary value problems in ordinary and partial differential equations and difference equations, see [23–26]. Sobolev’s spaces on a closed interval [0,T] of R are well known in [24]. For the sake of the study for differential equations on time scales, the authors defined the Sobolev spaces on time scales and studied some of their important properties in [25]. In [26], Jiao and Zhou developed a Caputo fractional derivative space and some of their properties which can be used to study Caputo fractional differential equation boundary value problems via critical point theory.In view of the above reasons, our main purpose of this paper is to construct fractional Sobolev spaces on time scales via conformable fractional calculus and investigate some of their important properties. As an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equations boundary value problem on time scale T.This paper is organized as follows. In Section2, we introduce the definition of conformable fractional calculus and their important properties. In Section 3, we construct fractional Sobolev spaces on time scales via conformable fractional calculus and investigate some of their important properties. Section 4, as an application of the fractional Sobolev’s spaces on time scales, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for p-Laplacian conformable fractional differential equations boundary value problem on time scale T. By making a variational structure on the fractional Sobolev’s spaces on time scales, we can reduce the problem of finding solutions of a p-Laplacian conformable fractional differential equation boundary value problem on time scale T to the one of seeking the critical points of a corresponding functional.


## 2. Conformable Fractional Calculus on Time Scales and Their Properties

In this section, we introduce some definitions of conformable fractional derivative and integral and study some of their important properties.Throughout this paper, we assumea,b∈T,0<a<b. We start by the definitions of conformable fractional derivative.For convenience, we denote the intervals[a,b]T,[a,b)T and (a,b]T in T by (1)a,bT=a,b∩T,a,bT=a,b∩T,a,bT=a,b∩T,respectively. Note that [a,b]Tκ=[a,b]T if b is left-dense and [a,b]Tκ=[a,b)T=[a,ρ(b)]T if b is left-scattered. We denote [a,b]Tκ2=([a,b]Tκ)κ, therefore [a,b]Tκ2=[a,b]T if b is left-dense and [a,b]Tκ2=[a,ρ(b)]Tκ if b is left-scattered.Lett∈T and δ>0. We define the δ-neighborhood of t as Vt≔(t-δ,t+δ)∩T. We begin to introduce a new notation: the conformable fractional derivative of order α∈(0,1] for functions defined on arbitrary time scales.Definition 1 (Definition  1, [14]).
Letf:T→R, t∈Tκ, and α∈(0,1]. For t>0, we define Tα(f)(t) to be the number (provided it exists) with the property that given any ϵ>0, there is a δ-neighborhood Vt⊂T of t,δ>0, such that (2)fσt-fst1-α-Tαftσt-s≤ϵσt-s∀s∈Vt.We call Tα(f)(t) the conformable fractional derivative of f of order α at t, and we define the conformable fractional derivative at 0 as Tα(f)(0)=limt→0+⁡Tα(f)(t).Definition 2 (Definition  23, [14]).
Letα∈(n,n+1],n∈N, and f:T→R be n times delta differentiable at t∈Tκn. We define the conformable fractional derivative of f of order α as Tα(f)(t)≔Tα-n(fΔn)(t).Definition 3 (Definition  26, [14]).
Letf:T→R be a regulated function. Then the α-fractional integral of f,0<α≤1, is defined by ∫f(t)Δαt≔∫f(t)tα-1Δt.Definition 4 (Definition  28, [14]).
Supposef:T→R is a regulated function. Denote the indefinite α-fractional integral of f of order α,α∈(0,1], as follows: Fα(t)=∫f(t)Δαt. Then, for all a,b∈T, we define the Cauchy α-fractional integral by ∫abf(t)Δαt=Fα(b)-Fα(a).TheΔ-measure μΔ and Δ-integration are defined the same as those in [27].Definition 5 (Definition  2.3, [28]).
LetB⊂T. B is called Δ-null set if μΔ(B)=0. Say that a property P holds Δ-almost everywhere (Δ-a.e.) on B, or for Δ-almost all (Δ-a.a.) t∈B if there is a Δ-null set E0⊂B such that P holds for all t∈B∖E0.Definition 6.
Assumef:T→R is a function. Let A is a Δ-measurable subset of T. f is α-integrable on A if and only if tα-1f is integrable on A, and ∫Af(t)Δαt=∫Af(t)tα-1Δt.The conformable fractional calculus off:T→R has the following important properties. Before the statement of the properties, we denote(3)Crda,bT,R=f:a,bT⟶R,fis  rd-continuous  ona,bT,C0a,bT,R=f:a,bT⟶R,fis  continuous  ona,bTwith  compact  support  ina,bT,Crdαa,bT,R=fis  conformal  fractional  differentiable  of  orderαona,bTandTαf∈Crda,bT,R,C0,rdαa,bT,R=f∈Crdαa,bT,R:fa=fb=0,Ca,b;rdαa,bT,R=f∈Crdαa,bT,R:fa=fb.Lemma 7 (Theorem  4, [14]).
Letα∈(0,1]. Assume f:T→R and let t∈Tκ. The following properties hold:(i)
Iff is conformal fractional differentiable of order α at t>0, then f is continuous at t.(ii)
Iff is continuous at t and t is right-scattered, then f is conformable fractional differentiable of order α at t with (4)Tαft=fΔtt1-α.(iii)
Ift is right-dense, then f is conformable fractional differentiable of order α at t if and only if the limit (5)lims→t⁡ft-fst-st1-αexists as a finite number. In this case,(6)Tαft=f′tt1-α.(iv)
Iff is conformable fractional differentiable of order α at t, then (7)fσt=ft+μttα-1Tαft.Lemma 8 (Theorem  15, [14]).
Assumef,g:T→R are conformable fractional differentiable of order α. Then(i)
the sumf+g:T→R is conformable fractional differentiable with Tα(f+g)=Tα(f)+Tα(g);(ii)
for anyλ∈R, λf:T→R is conformable fractional differentiable with Tα(λf)=λTα(f);(iii)
iff and g are continuous, then the product fg:T→R is conformable fractional differentiable with Tα(fg)=Tα(f)g+(f∘σ)Tα(g)=Tα(f)(g∘σ)+(f)Tα(g);(iv)
iff is continuous, then 1/f is conformable fractional differentiable with (8)Tα1f=-Tαfff∘σ;(v)
iff and g are continuous, then f/g is conformable fractional differentiable with (9)Tαfg=Tαfg-fTαggg∘σ.Lemma 9 (Theorem  25, [14]).
Letα∈(n,n+1],n∈N. The following relation holds:(10)Tαft=t1+n-αfΔ1+nt.Remark 10.
In (10), when n=0, we have Tα(f)(t)=t1-αfΔ(t),α∈(0,1].Lemma 11 (Theorem  30, [14]).
Letα∈(0,1]. Then, for any rd-continuous function f:T→R, there exists a function Fα:T→R such that Tα(Fα)(t)=f(t) for all t∈Tκ. Function Fα is said to be an α-antiderivative of f.Lemma 12 (Theorem  31, [14]).
Letα∈(0,1],a,b,c∈T,λ∈R and f,g:T→R be two rd-continuous functions. Then,(i)
∫ab[f(t)+g(t)]Δαt=∫abf(t)Δαt+∫abg(t)Δαt;(ii)
∫abλf(t)Δαt=λ∫abf(t)Δαt;(iii)
∫abf(t)Δαt=-∫baf(t)Δαt;(iv)
∫abf(t)Δαt=∫acf(t)Δαt+∫cbf(t)Δαt;(v)
∫aaf(t)Δαt=0;(vi)
if there existg:T→R with |f(t)|≤g(t) for all t∈[a,b]T, then ∫abftΔαt≤∫abg(t)Δαt;(vii)
iff(t)>0 for all t∈[a,b]T, then ∫abf(t)Δαt≥0.Lemma 13 (Theorem  33, [14]).
LetT be a time scale. If Tα(f)(t)≥0 for all t∈[a,b]T, then f is an increasing function on [a,b]T.Theorem 14.
Letf:T→R be a continuous function on [a,b]T that is conformal fractional differentiable of order α on [a,b)T and satisfies f(a)=f(b). Then there exist ξ,τ∈[a,b]Tκ such that Tα(f)(ξ)≤0≤Tα(f)(τ).Proof.
Since the functionf is continuous on the compact set [a,b]T, f assumes its minimum m and its maximum M. Therefore there exist ξ,τ∈[a,b]T such that m=f(ξ) and M=f(τ). Since f(a)=f(b), we may assume that ξ,τ∈[a,b]Tκ. By Lemma 13, we have Tα(f)(ξ)≤0≤Tα(f)(τ). The proof is complete.Theorem 15 (mean value theorem).
Letf be a continuous function on [a,b]T which is conformal fractional differentiable of order α on [a,b]Tκ. Then there exist ξ,τ∈[a,b]Tκ such that (11)ξα-1Tαfξ≤fb-fab-a≤τα-1Tαfτ.Proof.
It follows from Lemma9 that (12)Tαt=t1-α,0<α<1;1,α=1.Let h(t)=f(t)-f(a)-fb-fa/b-a(t-a). Then, the function h is continuous function on [a,b]T which is conformal fractional differentiable of order α on (a,b]T and h(a)=h(b). Combining Lemma 8 and (12), we have (13)Tαht=Tαft-fb-fab-a,α=1;Tαft-fb-fab-at1-α,0<α<1.Applying Theorem 14 to h, there exist ξ,τ∈[a,b)T such that Tα(h)(ξ)≤0≤Tα(h)(τ). That is (14)ξα-1Tαfξ≤fb-fab-a≤τα-1Tαfτ.The proof is complete.Similar to the [Definition2.9, [26]], we give the following definition of absolutely continuous function.Definition 16.
A functionf:[a,b]T→R is said to be absolutely continuous on [a,b]T (i.e., f∈AC([a,b]T,R)), if for every ϵ>0, there exists δ>0 such that if {[ak,bk]T}k=1n is a finite pairwise disjoint family of subintervals of [a,b]T satisfying ∑k=1n(bk-ak)<δ, then ∑k=1nfρbk-fak<ϵ.Lemma 17 (Theorem2.10, [28]).
A functionf:[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(15)ft=fa+∫a,tTfΔsΔs,∀t∈a,bT.Theorem 18.
Assume functionf:[a,b]T→R is absolutely continuous on [a,b]T, then f is conformable fractional differentiable of order αΔ-a.e. on [a,b]T and the following equality is valid:(16)ft=fa+∫a,tTTαfsΔαs,∀t∈a,bT.Proof.
According to Lemma17, f is delta differentiable Δ-a.e. on [a,b]T. Then, by Remark 10, f is conformable fractional differentiable of order αΔ-a.e. on [a,b]T. Therefore, Definition 4 implies that(17)ft=fa+∫a,tTTαfsΔαs,∀t∈a,bT.The proof is complete.Lemma 19 (Theorem  2.11, [28]).
A functionf,g:[a,b]T→R is absolutely continuous on [a,b]T, then fg is absolutely continuous on [a,b]T and the following equality is valid:(18)∫a,bTfΔg+fσgΔtΔt=fbgb-faga=∫a,bTfgΔ+fΔgσtΔt.Theorem 20.
Assume functionf,g:[a,b]T→R is absolutely continuous on [a,b]T, then fg is absolutely continuous on [a,b]T and the following equality is valid:(19)∫abTαfg+fσTαgtΔαt=fbgb-faga=∫abfTαg+TαfgσtΔαt.Proof.
The result is obtained by applying Lemmas8 and 19 and Theorem 18 to the function f and g. The proof is complete.Definition 21 (Definition  2.4, [28]).
LetE⊂T be a Δ-measurable set and let p∈R¯=[-∞,+∞] be such that p≥1 and let f:E→R¯ be a Δ-measurable function. Say that f belongs to LΔp(E,R) provided that either(20)∫EftpΔt<+∞ifp∈R,or there exists a constant C∈R such that(21)f≤CΔ-a.e.onEifp=+∞.Definition 22.
LetE⊂T be a Δ-measurable set and let p∈R¯=[-∞,+∞] be such that p≥1 and let f:E→R¯ be a Δ-measurable function. Say that f belongs to Lα,Δp(E,R) provided that either(22)∫EftpΔαt<+∞ifp∈R,or there exists a constant C∈R such that(23)f≤CΔ-a.e.onEifp=+∞.Lemma 23 (Theorem  2.5, [28]).
Letp∈R¯ be such that p≥1. Then the set LΔp([a,b]T,R) is a Banach space together with the norm defined for f∈LΔp([a,b]T,R) as(24)fLΔpa,bT,R=∫a,bTftpΔt1/p,p∈R;inf⁡C∈R:f≤CΔ-a.e.ona,bT,p=+∞.Moreover, LΔ2([a,b]T,R) is a Hilbert space together with the inner product given for every (f,g)∈LΔp([a,b]T,R)×LΔp([a,b]T,R) by(25)f,gLΔ2a,bT,R=∫a,bTftgtΔt.Theorem 24.
Letp∈R¯ be such that p≥1. Then the set Lα,Δp([a,b]T,R)(a≥0) is a Banach space together with the norm defined for f∈Lα,Δp([a,b]T,R) as(26)fLα,Δpa,bT,R=∫a,bTftpΔαt1/p,p∈R;inf⁡C∈R:f≤CΔ-a.e.ona,bT,p=+∞.Moreover, Lα,Δ2([a,b]T,R) is a Hilbert space together with the inner product given for every (f,g)∈Lα,Δp([a,b]T,R)×Lα,Δp([a,b]T,R) by(27)f,gLα,Δ2a,bT,R=∫a,bTftgtΔαt.Proof.
Let{un}n∈N⊂Lα,Δp([a,b]T,R) be a Cauchy sequence, then we have (28)∫a,bTumt-untpΔαt=∫a,bTumt-untptα-1Δt=∫a,bTumttα-1/p-unttα-1/ppΔt⟶0m,n⟶∞.From Lemma 23 and (28), there exists u0∈Lα,Δp([a,b]T,R) such that (29)∫a,bTunttα-1/p-u0ttα-1/ppΔt⟶0n⟶∞.Therefore, we can get(30)un-u0Lα,Δpa,bT,R=∫a,bTunt-u0tpΔαt=∫a,bTunt-u0tptα-1Δt=∫a,bTunttα-1/p-u0ttα-1/ppΔt⟶0n⟶∞.Thus, the space Lα,Δp([a,b]T,R) is a Banach space together with the norm ·Lα,Δp([a,b]T,R).
Clearly,Lα,Δ2([a,b]T,R) is a Hilbert space together with the inner product given for every (f,g)∈Lα,Δp([a,b]T,R)×Lα,Δp([a,b]T,R) by(31)f,gLα,Δ2a,bT,R=∫a,bTftgtΔαt.Lemma 25 (Proposition  2.6, [28]).
Supposep∈R¯ and p≥1. Let p′∈R¯ be such that 1/p+1/p′=1. Then, if f∈LΔp([a,b]T,R) and g∈LΔp′([a,b]T,R), then f·g∈LΔ1([a,b]T,R) and(32)f·gLΔ1a,bT,R≤fLΔpa,bT,R·gLΔp′a,bT;R.Theorem 26.
Supposep∈R¯ and p≥1. Let p′∈R¯ be such that 1/p+1/p′=1. Then, if f∈Lα,Δp([a,b]T,R) and g∈Lα,Δp′([a,b]T,R), then f·g∈Lα,Δ1([a,b]T,R) and(33)f·gLα,Δ1a,bT,R≤fLα,Δpa,bT,R·gLα,Δp′a,bT;R.Proof.
By Lemma25, we assert(34)f·gLα,Δ1a,bT,R=∫a,bTfgΔαt=∫a,bTfgtα-1Δt=∫a,bTftα-1/pgtα-1/p′Δt≤∫a,bTftα-1/ppΔt1/p∫a,bTgtα-1/p′pΔt1/p′=∫a,bTfptα-1Δt1/p∫a,bTgp′tα-1Δt1/p′=fLα,Δpa,bT,R·gLα,Δp′a,bT;R.The proof is complete.Lemma 27 (Proposition  2.7, [28]).
Ifp∈R and p≥1, then, the set C0([a,b]T,R) is dense in LΔp([a,b]T,R).Theorem 28.
Letf∈Lα,Δ1([a,b]T,R) be such that the following equality is true:(35)∫a,bTftutΔαt=0,∀u∈C0a,bT,R,then(36)f≡0,Δ-a.e.ona,bT.Proof.
For everyϵ>0, the density of C0([a,b]T,R) in LΔp([a,b]T,R) guarantees the existence of f1∈C0([a,b]T,R) such that(37)∫a,bTf1t-ftΔαt<ϵand so, by Lemma 12, (35) and (37), we deduce that for every u∈C0([a,b]T,R), it is true that (38)∫a,bTftutΔαt≤uCa,bT,R∫a,bTf1t-fttα-1Δt=ϵaα-1uCa,bT,R.Because the sets (39)A1=t∈a,b∣f1t≥ϵ,A2=t∈a,b∣f1t≤-ϵare compact and disjoint subsets of [a,b]T, Urysohn’s lemma allows constructing a function u0:[a,b]T→R which belongs to C0([a,b]T,R) and it verifies (40)u0t=1,t∈A1,-1,t∈A2,ut≤1on⁡a,bTso that, by defining A=A1∪A2, Theorem 15, (38), (39), and (40), we have that (41)∫a,bTf1tΔαt=∫a,bTf1tu0tΔαt-∫a,bT∖Af1tu0tΔαt+∫a,bT∖Af1tΔαt≤ϵaα-1+2ϵaα-1b-a.As a consequence of the arbitrary choice of ϵ>0, by (41), we achieve (36). The proof is complete.Theorem 29.
Letf∈Lα,Δ1([a,b]T,R). Then, a necessary and sufficient condition for the validity of the equality(42)∫abftTαhtΔαt=0,∀h∈C0,rdαa,bT,Ris the existence of a constant C∈R such that(43)f≡CΔ-a.e.ona,bT.Proof.
Iff≡CΔ-a.e. on [a,b]T, for any h∈C0,rdα([a,b]T,R), from Lemma 12 and the definition of C0,rdα([a,b]T,R), one has(44)∫abftTαhtΔαt=C∫abTαhtΔαt=Chb-ha=0.Conversely, take u∈C0([a,b]T,R), by defining h,g:[a,b]T→R as (45)ht=ut-∫a,bTusΔαs∫a,bT1Δαs,t∈a,bT,-∫a,bTusΔαs∫a,bT1Δαs,t=b,gt=∫a,bTusΔαs,the fundamental theorem of conformable fractional calculus establishes that g∈C0,rdα([a,b]T,R) and so equality (42) yields (46)0=∫a,bTftut-∫a,bTusΔαs∫a,bT1ΔαsΔαt=∫a,bTft-∫a,bTfsΔαs∫a,bT1ΔαsutΔαt.Thereby, Theorem 28 and (46) allow to deduce (43) with C=∫[a,b]Tf(s)Δαs/∫[a,b]T1Δαs. The proof is complete.Next, we introduce the conformable fractional calculus on time scales for vector-valued functions and study some of their important properties.Definition 30.
Assumef:T→RN is a function, f(t)=(f1(t),f2(t),…,fN(t)) and let t∈Tκ. Then one defines Tα(f)(t)=(Tα(f1)(t),Tα(f2)(t),…,Tα(fN)(t)) (provided it exists). One calls Tα(f)(t) the conformable fractional derivative of f of order α at t. The function f is conformal fractional differentiable of order α provided Tα(f) exists for all t∈Tκ. The function Tα(f):Tκ→RN is then called the conformable fractional derivative of f of order α.Definition 31.
LetT be a time scale, α∈(n,n+1],n∈N, and let f:T→RN be n times delta differentiable at t∈Tκn. We define the conformable fractional derivative of f of order α as Tα(f)(t)≔Tα-n(fΔn)(t).Definition 32.
Assumef:T→RN is a function and f(t)=(f1(t),f2(t),…,fN(t)). Let A be a Δ-measurable subset of T. Then f is α-integrable on A if and only if fi(i=1,2,…,N) are α-integrable on A, and ∫Af(t)Δαt=(∫Af1(t)Δαt,∫Af2(t)Δαt,…,∫AfN(t)Δαt).From Definitions30 and 32, we have the following theorems.Theorem 33.
Letα∈(0,1]. Assume f:T→RN and let t∈Tκ. The following properties hold:(i)
Iff is conformal fractional differentiable of order α at t>0, then f is continuous at t.(ii)
Iff is continuous at t and t is right-scattered, then f is conformable fractional differentiable of order α at t with (47)Tαft=1μtfσt-ftt1-α.(iii)
Ift is right-dense, then f is conformable fractional differentiable of order α at t if and only if the limit (48)lims→t⁡1t-sft-fst1-αexists as a finite number. In this case,(49)Tαft=lims→t⁡1t-sft-fst1-α.(iv)
Iff is conformable fractional differentiable of order α at t, then (50)fσt=ft+μttα-1Tαft.Theorem 34.
Assumef,g:T→RN are conformable fractional differentiable of order α. Then,(i)
the sumf+g:T→RN is conformable fractional differentiable with Tα(f+g)=Tα(f)+Tα(g);(ii)
for anyλ∈R, λf:T→RN is conformable fractional differentiable with Tα(λf)=λTα(f);(iii)
iff and g are continuous, then the product fg:T→RN is conformable fractional differentiable with Tα(fg)=Tα(f)g+(f∘σ)Tα(g)=Tα(f)(g∘σ)+(f)Tα(g).Theorem 35.
Letα∈(0,1],a,b,c∈T,λ∈R and f,g:T→RN be two rd-continuous functions. Then,(i)
∫ab[f(t)+g(t)]Δαt=∫abf(t)Δαt+∫abg(t)Δαt;(ii)
∫abλf(t)Δαt=λ∫abf(t)Δαt;(iii)
∫abf(t)Δαt=-∫baf(t)Δαt;(iv)
∫abf(t)Δαt=∫acf(t)Δαt+∫cbf(t)Δαt;(v)
∫aaf(t)Δαt=0;(vi)
if there existg:T→R with ft≤gt for all t∈[a,b]T, then ∫abftΔαt≤∫abgtΔαt.Definition 36 (Definition  2.6, [25]).
A functionf: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.As we know from general theory of Sobolev’s spaces, another important class of functions is just the absolutely continuous functions on time scales. Similar to Definition2.11, [23], we give the following definition of absolutely continuous function.Definition 37.
A functionf:[a,b]T→RN,f(t)=(f1(t),f2(t),…,fN(t)). We say f is absolutely continuous on [a,b]T (i.e. f∈AC([a,b]T,RN)), if for every ϵ>0, there exist δ>0 such that if {[ak,bk]T}k=1n is a finite pairwise disjoint family of subintervals of [a,b]T satisfying ∑k=1n(bk-ak)<δ, then ∑k=1nfρbk-fak<ϵ.Remark 38 (Remark  2.1, [25]).
By Definitions22 and 30, we have that f∈AC([a,b]T,RN) if and only if fi∈AC([a,b]T,R)(i=1,2,…,N).Absolutely continuous functions have the following properties.Combining Definitions30, 36, Theorems 18 and 20, we have the following theorems.Theorem 39.
Assume functionf:[a,b]T→RN is absolutely continuous on [a,b]T, then f is conformable fractional differentiable of order αΔ-a.e. on [a,b]T and the following equality is valid:(51)ft=fa+∫a,bTTαfsΔαs,∀t∈a,bT.Theorem 40.
A functionf,g:[a,b]T→RN is absolutely continuous on [a,b]T, then fg is absolutely continuous on [a,b]T and the following equality is valid:(52)∫a,bTTαft,gt+fσt,TαgtΔαt=fb,gb-fa,ga=∫a,bTft,Tαgt+Tαft,gσtΔαt.


## 3. Conformable Fractional Sobolev’s Spaces on Time Scales and Their Properties

In this section, we develop the fractional Sobolev’s spaces on time scales via conformable fractional calculus and their important properties.Forp∈R,p≥1, we set the space(53)Lα,Δpa,bT,RN=u:a,bT⟶RN:∫a,bTftpΔαt<+∞with the norm(54)fLα,Δp=∫a,bTftpΔαt1/p.We can obtain the following theorem.Theorem 41.
Assumep≥1 is a real number. Then the space Lα,Δp([a,b]T,RN) is a Banach space together with the norm ·Lα,Δp. Especially, when p=2, Lα,Δ2([a,b]T,RN) is a Hilbert space together with the inner product given for every (f,g)∈Lα,Δp([a,b)TN,RN)×Lα,Δp([a,b]T,RN) by(55)f,gLα,Δ2=∫a,bTft,gtΔαt,where (·,·) denotes the inner product in RN.Proof.
Let{un}n∈N⊂Lα,Δp([a,b]T,RN) be a Cauchy sequence, un(t)=(un1(t),un2(t),…,unN(t)). Hence, one has (56)um-unLα,Δp=∫a,bTunt-umtpΔαt1/p=∫a,bT∑i=1Nunit-umit2p/2Δαt1/p⟶0m,n⟶∞.It follows from (56) that(57)umi-uniLα,Δpa,bT,R=∫a,bTunit-umitpΔαt1/p⟶0m,n⟶∞for any i∈{1,2,…,N}. Therefore, {uni}n∈N (i=1,2,…,N) are the Cauchy sequences of Lα,Δp([a,b]T,R). By Theorem 24, there exist u¯i∈Lα,Δp([a,b]T,R)(i=1,2,…,N) such that (58)uni-u¯iLΔpa,bT,R⟶0n⟶∞.We let u¯(t)=(u¯1(t),u¯2(t),…,u¯N(t)). On one hand, we can get(59)∫a,bTu¯pΔαt=∫a,bT∑i=1Nu¯i2p/2Δαt≤Np/2∫a,bT∑i=1Nu¯ipΔαt=Np/2∑i=1N∫a,bTu¯ipΔαt<+∞.Thus, u¯∈LΔp([a,b]T,RN). On the other hand, according to (58), we have(60)∫a,bTunt-u¯tpΔαt=∫a,bT∑i=1Nunit-u¯it2p/2≤Np/2∫a,bT∑i=1Nunit-u¯itpΔαt=Np/2∑i=1N∫a,bTunit-u¯itpΔαt=Np/2∑i=1Nuni-u¯iLα,Δpa,bT,R⟶0n⟶∞.From (60), un→u¯ in Lα,Δp([a,b]T,RN); therefore, the space Lα,Δp([a,b]T,Rn) is a Banach space together with the norm ·Lα,Δp.
Evidently,Lα,Δ2([a,b]T,RN) is a Hilbert space together with the inner product given for every (f,g)∈Lα,Δp([a,b)TN,RN)×Lα,Δp([a,b]T;RN) by(61)f,gLα,Δ2=∫a,bTft,gtΔαt.The proof is complete.Now, we introduce the conformable fractional Sobolev’s spaces on[a,b]T. For the sake of convenience, in the sequel, we will let uσ(t)=u(σ(t)) and(62)Crda,bT,RN=f:a,bT⟶RN,fis  rd-continuous  ona,bT,Crdαa,bT,RN=fis  conformal  fractional  differentiable  of  orderαona,bTandTαf∈Crda,bT,RN,Ca,b;rdαa,bT,RN=f∈Crdαa,bT,RN:fa=fb.Definition 42.
Letp∈R be such that p≥1 and u:[a,b]T→RN. One says that u∈WΔ;a,bα,p([a,b]T,RN) if and only if u∈Lα,Δp([a,b]T,RN),Tαu∈Lα,Δp([a,b]T,RN), and there exists g:[a,b]Tκ→RN such that g∈Lα,Δp([a,b]T,RN) and (63)∫a,bTut,TαϕtΔαt=-∫a,bTgt,ϕσtΔαt,∀ϕ∈Ca,b;rdαa,bT,RN.For p∈R,p≥1, we denote (64)VΔ;a,bα,pa,bT,RN=u∈ACa,bT,RN:Tαu∈Lα,Δpa,bT,RN,ua=ub.Remark 43.
Theorems39 and 40 imply that for every p∈R with p≥1, (65)VΔ;a,bα,pa,bT,RN⊂WΔ;a,bα,pa,bT,RN.Next, we will point out that both sets are, as class of functions, equivalent. For the sake of this purpose, we prove the following theorems firstly.Theorem 44.
Supposef∈Lα,Δ1([a,b]T,RN). Then, a necessary and sufficient condition for the validity of the equality (66)∫a,bTft,TαhtΔαt=0,∀h∈Ca,b;rdαa,bT,RNis the existence of a constant C∈RN such that (67)f≡CΔ-a.e.ona,bT.Proof.
By Definition30, f∈LΔ1([a,b]T,RN),f(t)=(f1(t),f2(t),…,fN(t)) implies fi∈LΔ1([a,b]T,R)(i=1,2,…,N).
On one hand, Iff≡C=(C1,C2,…,CN)Δ-a.e.on[a,b]T, for any h∈Ca,b;rdα([a,b]T,R),h(t)=(h1(t),h2(t),…,hN(t)), Combining Theorem 35 with the definition of Ca,b;rdα([a,b]T,R), Definitions 30 and 32, we have(68)∫abft,TαhtΔαt=∫ab∑i=1NCiTαhitΔαt=∑i=1N∫abCiTαhitΔαt=∑i=1NCi∫abTαhtΔαt=∑i=1NCihib-hia=0.On the other hand, ∀h1∈Ca,b;rdα([a,b]T,R), let h(t)=(h1(t),0,…,0), then h∈Ca,b;rdα([a,b]T,RN) and (69)∫a,bTft,TαhtΔαt=0.That is (70)∫a,bTf1tTαh1tΔαt=0.Because C0,rdα([a,b]T,R)⊂Ca,b;rdα([a,b]T,R), by Theorem 29, there exists C1∈R such that (71)f1≡C1Δ-a.e.  ona,bT.Analogically, there exists Ci∈R(i=2,3,…,N) such that (72)fi≡CiΔ-a.e.  ona,bT.Therefore, (73)f≡C1,C2,…,CNΔ-a.e.  ona,bT.The proof is finished.Now, we show the characterization of functions inWΔ;a,bα,p([a,b]T,RN) in terms of functions in VΔ;a,bα,p([a,b]T,RN).Theorem 45.
Assume thatu∈WΔ;a,bα,p([a,b]T,RN) for some p∈R with p≥1, and that (63) holds for g∈Lα,Δp([a,b]T,RN). Then, there exists a unique function x∈VΔ;a,bα,p([a,b]T,RN) such that(74)x=u,Tαx=gΔ-a.e.ona,bT.Proof.
Let{ej}j=1N denote the canonical basis of RN. We can take ϕ≡ej in (63), which guarantees (75)∫a,bTgt,ejΔαt=0j=1,2,…,N.So, it is easy to get that(76)∫a,bTgtΔαt=0.We define function v:[a,b]T→RN as(77)vt=∫a,bTgsΔαs.Equations (76) and (77) imply that v∈VΔ;a,bα,p([a,b]T,RN) and by Theorem 40, we assert that(78)∫a,bTvt-ut,hΔtΔαt=-∫a,bTTαvt-gt,hσtΔαtfor every h∈Ca,b;rdα([a,b]T,RN). It follows from Theorem 44 and (78) that there exists C0∈RN such that (79)v-u≡C0,on⁡a,bT.As a result of the fundamental theorem of conformable fractional calculus we can choose the function x:[a,b]T→RN defined as (80)xt=vt-C0,t∈a,bT.Then, the function x is the unique function in VΔ;a,bα,p([a,b]T,RN) for which (74) is valid. The proof is complete.If we identifyu∈WΔ;a,bα,p([a,b]T,RN) with its absolutely continuous representative x∈VΔ;a,bα,p([a,b]T,RN) for which (74) holds, then the set WΔ;a,bα,p([a,b]T,RN) can be endowed with the structure of Banach space.Theorem 46.
Supposep∈R and p≥1. Then the set WΔ;a,bα,p([a,b]T,RN) is a Banach space together with the norm defined as(81)uWΔ;a,bα,p=∫a,bTuσtpΔαt+∫a,bTTαutpΔαt1/p,∀u∈WΔ;a,bα,pa,bT,RN.Moreover, the set HΔ;a,bα=WΔ;a,bα,2([a,b]T,RN) is a Hilbert space together with the inner product (82)u,vHΔ;a,bα=∫a,bTuσt,vσtΔαt+∫a,bTTαut,TαvtΔαt,∀u,v∈HΔ;a,b1.Proof.
Let{un}n∈N be a Cauchy sequence in WΔ;a,bα,p([a,b]T,RN). According to the definition of WΔ;a,bα,p([a,b]T,RN), we have {un}n∈N⊂Lα,Δp([a,b]T,RN) and there exist gn:[a,b]κ→RN such that {gn}n∈N⊂Lα,Δp([a,b]T,RN) and (83)∫a,bTunt,TαϕtΔαt=-∫a,bTgnt,ϕσtΔαt,∀ϕ∈Ca,b;rdαa,bT,RN.Therefore, by Theorem 45, there exists {xn}n∈N⊂VΔ;a,bα,p([a,b]T,RN) such that(84)xn=un,Tαxn=gnΔ-a.e.  ona,bT.Equations (83) and (84) imply that (85)∫a,bTxnt,TαϕtΔαt=-∫a,bTTαxnt,ϕσtΔαtfor all ϕ∈Ca,b;rdα(a,bT,RN). On account of {un}n∈N is a Cauchy sequence in WΔ;a,bα,p([a,b]T,RN), by (81), we can get(86)∫a,bTunσt-umσtpΔαt⟶0m,n⟶∞,(87)∫a,bTTαunt-TαumtpΔαt⟶0m,n⟶∞.It follows from Theorems 33 and 35 and (86) and (87) that (88)∫a,bTunt-umtpΔαt=∫a,bTunσt-umσt-μttα-1Tαunt-TαumtpΔαt≤2p-1∫a,bTunσt-umσtpΔαt+2p-1apα-1σb-ap∫a,bTTαunt-TαumtpΔαt⟶0m,n⟶∞.For another, from Theorem 41, (87), and (88), there exist u,g∈Lα,Δp([a,b]T,RN) such that(89)un-uLα,Δp⟶0n⟶∞,Tαun-gLα,Δp⟶0n⟶∞.Combining (85) with (89), one has(90)∫a,bTut,TαϕtΔαt=-∫a,bTgt,ϕσtΔαtfor all ϕ∈Ca,b;rdα(a,bT,RN). By (90), we claim that u∈WΔ;a,bα,p([a,b]T,RN). Furthermore, it follows from Theorems 33 and 35 and (89) that (91)∫a,bTunσt-uσtpΔαt=∫a,bTunt-ut+μttα-1Tαunt-TαutpΔαt=∫a,bTunt-ut+μttα-1Tαunt-gtpΔαt≤2p-1∫a,bTunt-utpΔαt+2p-1apα-1σb-ap∫a,bTTαunt-gt2Δαt⟶0n⟶∞.Hence, by Remark 43, (89), (91), and Theorem 45, there exists x∈VΔ;a,bα,p([a,b]T,RN)⊂WΔ;a,bα,p([a,b]T,RN) such that (92)un-xWΔ;a,bα,p⟶0n⟶∞.So, WΔ;a,bα,p([a,b]T,RN) is a Banach space.
Clearly, the setHΔ;a,bα is a Hilbert space together with the inner product (93)u,vHΔ;a,bα=∫a,bTuσt,vσtΔαt+∫a,bTTαut,TαvtΔαt∀u,v∈HΔ;a,bα.The proof is finished.Next, we will show some important properties of the Banach spaceWΔ;a,bα,p([a,b]T,RN).Theorem 47.
The spaceWΔ;a,bα,p([a,b]T,RN) is a reflexive and uniformly convex Banach space when p∈(1,+∞).Proof.
By Clarkson inequality, we have(94)12uσ+vσLα,Δpp+12uσ-vσLα,Δpp≤12uσLα,Δpp+12vσLα,Δpp,p∈2,+∞,(95)12Tαu+TαvLα,Δpp+12Tαu-TαvLα,Δpp≤12TαuLα,Δpp+12TαvLα,Δpp,p∈2,+∞,(96)12uσ+vσLα,Δpq+12uσ-vσLα,Δpq≤12uσLα,Δp+12vσLα,Δpq-1,p∈1,2,(97)12Tαu+TαvLα,Δpq+12Tαu-TαvLα,Δpq≤12TαuLα,Δp+12TαvLα,Δpq-1,p∈1,2,where 1/p+1/q=1. Thus, ∀ϵ>0, for any u,v∈WΔ;a,bα,p([a,b]T,RN),uWΔ;a,bα,p=vWΔ;a,bα,p=1 and u-vWΔ;a,bα,p≥ϵ, if 2≤p<+∞, by (94) and (95), one has (98)12u+vWΔ;a,b1,pp=12uσ+vσLα,Δpp+12Tαu+TαvLα,Δpp≤12uσLα,Δpp+12vσLα,Δpp+12TαuLα,Δpp+12TαvLα,Δpp-12uσ-vσLα,Δpp-12Tαu-TαvLα,Δpp=12uWΔ;a,b1,pp+12vWΔ;a,b1,pp-12u-vWΔ;a,b1,pp≤1-ϵ2p;if p∈(1,2], by (96) and (97), we have (99)12u+vWΔ;a,b1,pq=12u+vWΔ;a,b1,ppq/p=12u+vWΔ;a,b1,ppq-1=12uσ+vσLα,Δpp+12Tαu+TαvLα,Δppq-1=12uσ+vσLα,Δpq+12Tαu+TαvLα,Δpq≤12uσLα,Δpp+12vσLα,Δppq-1+12TαuLα,Δpp+12TαvLα,Δppq-1-12uσ-vσLα,Δpq-12Tαu-TαvLα,Δpq≤2-ϵ2q.Combining (98) with (99), WΔ;a,bα,p([a,b]T,RN) is a uniformly convex Banach space. By functional analysis, we know that a uniformly convex Banach space is reflexive. The proof is complete.Theorem 48.
There existsK=K(p)>0 such that the inequality(100)u∞≤KuWΔ;a,bα,pholds for all u∈WΔ;a,bα,p([a,b]T,RN), where u∞=maxt∈[0,T]Tut.
Moreover, if∫[a,b]Tu(t)Δαt=0, then(101)u∞≤KTαuLα,Δp.Proof.
Going to the components ofu, we can suppose that N=1. If u∈WΔ;a,bα,p([a,b]T,RN), from Theorem 45, U(t)=∫[a,t]Tu(s)Δαs is absolutely continuous on [a,b]T. On the basis of Theorem 15, there exists ζ∈[a,b]T such that(102)uζ≤Ub-Uab-amax⁡b1-α,1=max⁡b1-α,1b-a∫a,bTusΔαs.Hence, for t∈[a,b]T, using Theorems 15 and 26 and (102), we obtain (103)ut=uζ+∫ζ,tTTαusΔαs≤uζ+∫a,bTTαusΔαs≤max⁡b1-α,1b-a∫a,bTusΔαs+max⁡aα-1,1b-a1/q∫a,bTTαuspΔαs1/p,where 1/p+1/q=1. If ∫[a,b]Tu(t)Δαt=0, by (103), (101) holds. In the general case, for t∈[a,b]T, from Theorem 33 and Hölder’s inequality, we have (104)
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,where K=max⁡b1-α,1/b-a+aα-1σb-amax⁡b1-α,1/b-a+1[max⁡{aα-1,1}(b-a)]1/q. By (104), (100) holds. The proof is complete.Remark 49.
It follows from Theorem48 that WΔ;a,bα,p([a,b]T,RN) is continuously immersed into C([a,b]T,RN) with the norm ·∞.Theorem 50.
If the sequence{uk}k∈N⊂WΔ;a,bα,p([a,b]T,RN) converges weakly to u in WΔ;a,bα,p([a,b]T,RN), then {uk}k∈N converges strongly in C([a,b]T,RN) to u.Proof.
Owing touk⇀u in WΔ;a,bα,p([a,b]T,RN), {uk}k∈N is bounded in WΔ;a,bα,p([a,b]T,RN) and, hence, in C([a,b]T,RN). It follows from Remark 49 that uk⇀u in C([a,b]T,RN). For t1,t2∈[a,b]T,t1≤t2, there exists C1>0 such that (105)ukt2-ukt1≤∫t1,t2TTαuksΔαs≤max⁡t1α-1,1t2-t11/q∫t1,t2TTαukspΔαs1/p≤t2-t11/qmax⁡t1α-1,11/qukWΔ;a,bα,p≤C1t2-t11/q.In view of this, the sequence {uk}k∈N is equicontinuous. According to Ascoli-Arzela theorem, {uk}k∈N is relatively in C([a,b]T,RN). By the uniqueness of the weak limit in C([a,b]T,RN), every uniformly convergent subsequence of {uk}k∈N converges to u. Consequently, {uk}k∈N converges strongly in C([a,b]T,RN) to u. The proof is complete.Remark 51.
It follows from Theorem50 that WΔ;a,bα,p([a,b]T,RN) is compact immersed into C([a,b]T,RN) with the norm ·∞.Theorem 52.
LetL:[a,b]T×RN×RN→R,(t,x,y)→L(t,x,y) be Δ-measurable in t for each (x,y)∈RN×RN and continuously conformal fractional differentiable of order α in (x,y) for Δ-almost every t∈[0,T]T. If there exist a^∈C(R+,R+),b^∈Lα,Δ1([a,b]T,R+) and c^∈Lα,Δq([a,b]T,R+)(1<q<+∞) such that for Δ-almost t∈[a,b]T and every (x,y)∈RN×RN, one has (106)Lt,x,y≤a^xb^t+yp,Lxt,x,y≤a^xb^t+yp,Lyt,x,y≤a^xc^t+yp-1,where 1/p+1/q=1, then the functional Φ:WΔ;a,bα,p([a,b]T,RN)→R defined by (107)Φu=∫a,bTLσt,uσt,TαutΔαtis continuously differentiable on WΔ;a,bα,p([a,b]T,RN) and (108)Φ′u,v=∫a,bTLxσt,uσt,Tαut,vσtΔαt+∫a,bTLyσt,uσt,Tαut,TαvtΔαt.Proof.
It suffices to prove thatΦ has at every point u a directional derivative Φ′(u)∈(WΔ;a,bα,p([a,b]T,RN))∗ given by (108) and that the mapping (109)Φ′:WΔ;a,bα,pa,bT,RN⟶WΔ;a,bα,pa,bT,RN∗is continuous.
As a matter of fact, on one hand, it follows from (106) that Φ is everywhere finite on WΔ;a,bα,p([a,b]T,RN). We denote, for u and v fixed in WΔ;a,bα,p([a,b]T,RN),t∈[a,b]T,λ∈[-1,1], (110)Gλ,t=Lσt,uσt+λvσt,Tαut+λTαvt,Ψλ=∫a,bTGλ,tΔαt=Φu+λv.In the light of (106), we obtain (111)DλGλ,t≤DxLσt,uσt+λvσt,Tαut+λTαvt,vσt+DyLσt,uσt+λvσt,Tαut+λTαvt,Tαvt≤a^uσt+λvσtb^σt+Tαut+λTαvtpvσt+a^uσt+λvσtc^σt+Tαut+λTαvtp-1Tαvt≤a¯b^σt+Tαut+Tαvtpvσt+a¯c^σt+Tαut+Tαvtp-1Tαvt≜dt,where (112)a¯=maxλ,t∈-1,1×a,bT⁡a^ut+λvt.By the definition of function d, d∈Lα,Δ1([a,b]T,R+). Since b^∈Lα,Δ1([a,b]T,R+),(|Tα(u)|+|Tα(v)|)p∈Lα,Δ1([a,b]T,R),c^∈Lα,Δq([a,b]T,R+), one has (113)DλGλ,t≤dt,(114)Ψ′0=∫a,bTDλG0,tΔαt=∫a,bTDxLσt,uσt,Tαut,vσtΔαt+∫a,bTDyLσt,uσt,Tαut,TαvtΔαt.On the other hand, (106) implies that(115)DxLσt,uσt,Tαut≤a^uσtb^σt+Tαutp≜ψ1t,DyLσt,uσt,Tαut≤a^uσtc^σt+Tαutp-1≜ψ2t,hence ψ1∈Lα,Δ1([a,b]T,R+),ψ2∈Lα,Δq([a,b]T,R+). Therefore, from Theorem 48, (114), and (115), there exist positive constants C2,C3,C4 such that(116)∫a,bTDxLσt,uσt,Tαut,vσt+DyLσt,uσt,Tαut,TαvtΔαt≤C2v∞+C3TαvLΔp≤C4vWΔ,T1,pand Φ has a directional derivative at u and Φ′(u)∈(WΔ;a,bα,p([a,b]T,RN))∗ given by (108).
Furthermore, (106) implies that the mapping from WΔ;a,bα,p([a,b]T,RN) into Lα,Δ1([a,b]T,RN)×Lα,Δq([a,b]T,RN) defined by (117)u⟶DxL·,uσ,Tαu,DyL·,uσ,Tαuis continuous, so that Φ′ is continuous from WΔ;a,bα,p([a,b]T,RN) into (WΔ;a,bα,p([a,b]T,RN))∗. The proof is finished.


## 4. An Application of the SpaceWΔ;a,bα,pa,bT,RN

In this section, as an application of the Sobolev’s spaceWΔ;a,bα,p([a,b]T,RN). We present a recent approach via variational methods and critical point theory to obtain the existence and multiplicity of solutions for the p-Laplacian conformable fractional differential equations boundary value problem on time scale T(118)TαTαup-2Tαut=∇Fσt,uσt,Δ-a.e.t∈a,bTκ2,ua-ub=0,Tαua-Tαub=0,where Tα(u)(t) denotes the conformable fractional derivative of u of order α at t, σ is the forward jump operator, a,b∈T,0<a<b,p>1 and F:[a,b]T×RN→R satisfies the following assumption:(A)
F(t,x) is Δ-measurable in t for every x∈RN and continuously differentiable in x for Δ-a.e. t∈[a,b]T and there exist a^∈C(R+,R+),b^∈Lα,Δ1([a,b]T,R+) such that (119)Ft,x≤a^xb^t,∇Ft,x≤a^xb^tfor allx∈RN and Δ-a.e. t∈[a,b]T, where ∇F(t,x) denotes the gradient of F(t,x) in x.By making a variational structure on WΔ;a,bα,p([a,b]T,RN), we can reduce the problem of finding solutions of (118) to the one of seeking the critical points of a corresponding functional.Problem (118) covers the p-Laplacian system (for when T=R,α=1) (120)u′tp-2u′t′=∇Ft,ut,a.e.t∈a,b,ua-ub=0,u′a-u′b=0,and the second-order Hamiltonian system on time scale T (for when p=2,α=1) (121)uΔ2t=∇Fσt,uσt,Δ-a.e.t∈a,bTκ2,ua-ub=0,uΔa-uΔb=0,as well as the second-order Hamiltonian system (for when p=2,α=1,T=R) (122)u′′t=∇Ft,ut,a.e.t∈a,b,ua-ub=0,u′a-u′b=0,and the second-order discrete Hamiltonian system (for when p=2,T=Z,α=1,b-a≥3) (123)Δ2ut=∇Ft,ut,t∈a,b-1∩Z,ua-ub=0,Δua-Δub=0.Consider the functionalφ:WΔ;a,bα,p([a,b]T,RN)→R defined by (124)φu=1p∫a,bTTαutpΔαt+∫a,bTFσt,uσtΔαt.We will show the following theorems.Theorem 53.
The functionalφ is continuously differentiable on WΔ;a,bα,p([a,b]T,RN) and (125)φ′u,v=∫a,bTTαup-2Tαut,TαvtΔαt+∫a,bT∇Fσt,uσt,vσtΔαtfor all v∈WΔ;a,bα,p(a,bT,RN).Proof.
We chooseL(t,x,y)=(1/p)yp+F(t,x) for all x,y∈RN and t∈a,bT in Theorem 52. It is easy to get that the functional φ is continuously differentiable on WΔ;a,bα,p(a,bT,RN) and(126)φ′u,v=∫a,bTTαup-2Tαut,TαvtΔαt+∫a,bT∇Fσt,uσt,vσtΔαtfor all v∈WΔ;a,bα,p(a,bT,RN) by condition (A) and Theorem 52. The proof is complete.Theorem 54.
Ifu∈WΔ;a,bα,p(a,bT,RN) is a critical of φ in WΔ;a,bα,p(a,bT,RN), that is, φ′(u)=0, then u is a weak solution of problem (118).Proof.
Sinceφ′(u)=0, it follows from Theorem 53 that (127)∫a,bTTαup-2Tαut,TαvtΔαt+∫a,bT∇Fσt,uσt,vσtΔαt=0for all v∈WΔ;a,bα,p(a,bT,RN). That is (128)∫a,bTTαup-2Tαut,TαvtΔαt=-∫a,bT∇Fσt,uσt,vσtΔαtfor all v∈WΔ;a,bα,p(a,bT,RN). Using condition (A) and Definition 42, we obtain that Tαup-2Tα(u)∈WΔ;a,bα,p(a,bT,RN). It follows from Theorem 45 and (74) that there exists a unique function x∈VΔ;a,bα,p(a,bT,RN) such that(129)x=u,TαTαxp-2Tαxt=∇Fσt,uσtΔ-a.e.  ona,bTκ.Equation (129) imply that (130)xa-xb=0,Tαxa-Tαxb=0.We identify u∈WΔ;a,bα,p(a,bT,RN) with its absolutely continuous representative
x ∈ V Δ ; a , b α , p ( a , b T , R N ) for which (129) holds. Then, u is solution of problem (118). The proof is complete.In order to prove the existence and multiplicity of solutions for (118), we need the following definitions and theorems.Definition 55 (see [24], P81).
LetX be a real Banach space, I∈C1(X,R) and c∈R. I is said to be satisfying (PS)c-condition on X if the existence of a sequence {xn}⊆X such that I(xn)→c and I′(xn)→0 as n→∞, implies that c is a critical value of I.Definition 56 (see [24], P81).
LetX be a real Banach space and I∈C1(X,R). I is said to be satisfying PS condition on X if any sequence {xn}⊆X for which I(xn) is bounded and I′(xn)→0 as n→∞, possesses a convergent subsequence in X.Remark 57.
It is clear that the PS condition implies the(PS)c-condition for each c∈R.Lemma 58 (Theorem4.7, [24]).
LetX be a Banach space and let Φ∈C1(X,R). Assume that X splits into a direct sum of closed subspace X=X-⊕X+ with (131)dim⁡X-<∞,supSR-⁡Φ<infX+⁡Φ,where SR-={u∈X-:u=R}. Let(132)BR-=u∈X-:u≤R,M=h∈CBR-,X:hs=sifs∈SR-,c=infh∈M⁡maxs∈BR-⁡Φhs.Then, if Φ satisfies the (PS)c-condition, c is a critical value of Φ.As shown in [29], a deformation lemma can be proved with the weaker condition (C) replacing the usual PS condition, and it turns out that the Saddle Point Theorem (Lemma 58) holds under (C) condition.For the sake of convenience, in the sequel, we denote·=·WΔ;a,bα,p.Foru∈WΔ;a,bα,p(a,bT,RN), let u¯=∫a,bT1Δαt-1∫a,bTu(t)Δt and u~(t)=u(t)-u¯. Set W~Δ;a,bα,p(a,bT,RN)={u∈WΔ;a,bα,p(a,bT,RN):∫[a,b]Tu(t)Δαt=0}, then WΔ;a,bα,p(a,bT,RN)=W~Δ;a,bα,p(a,bT,RN)⊕RN and we have the following theorem.Theorem 59.
In Banach spaceWΔ;a,bα,p(a,bT,RN), for u∈WΔ;a,bα,p(a,bT,RN), u→∞ if and only if (133)u¯p+∫a,bTTαutpΔαt1/p⟶∞.Proof.
Firstly, we prove the necessity. Indeed, from Theorems15 and 48, we get(134)∫a,bTuσtpΔαt=∫a,bTu¯+uσ~tpΔαt≤2p∫a,bTu¯p+uσ~tpΔαt≤2pmax⁡aα-1,1b-au¯p+u~∞p≤2pmax⁡aα-1,1b-au¯p+Kp∫a,bTTαutpΔαt.This inequality implies(135)u=∫a,bTTαutpΔαt+∫a,bTuσtpΔαt1/p≤2pmax⁡aα-1,1b-au¯p+Kp∫a,bTTαutpΔαt+∫a,bTTαutpΔαt1/p≤2pmax⁡aα-1,1b-a+Kp+11/pu¯p+∫a,bTTαutpΔαt1/p.That is u→∞⇒(|u¯|p+∫[a,b]TTαutpΔαt)1/p→∞.
Secondly, we prove the sufficiency. From Theorem15 and Hölder inequality we get(136)u¯=∫a,bT1Δαt-1∫a,bTuσtΔαt≤∫a,bT1Δαt-1/p∫a,bTuσtpΔαt1/p.Therefore, we have(137)u¯p+∫a,bTTαutpΔαt1/p≤∫a,bT1Δαt-1∫a,bTuσtpΔαt+∫a,bTTαutpΔαt1/p≤∫a,bT1Δαt-1+11/pu.That is (|u¯|p+∫[a,b]T|Tα(u)(t)|pΔαt)1/p→∞⇒u→∞. This completes the proof.Theorem 60.
Assume that (A) and the following conditions are satisfied:(F1)
there exist0<μ<p,M>0 such that (138)∇Ft,x,x≥μFt,xfor all|x|>M and Δ-a.e. t∈[a,b]T.(F2)
there existsgσ∈Lα,Δ1([a,b]T,R) such that (139)Ft,x≤gtfor allx∈RN and Δ-a.e. t∈[a,b]T.(F3)
there exists a subsetD of [a,b]T with μΔ(D)>0 such that (140)Ft,x⟶-∞asx⟶∞forΔ-a.e. t∈D.Then problem (118) has at least one solution.Proof.
We will use the Saddle Point Theorem (Lemma58) to prove Theorem 60. It suffices to prove that(i)
φ satisfies (C)-condition,(ii)
φ(u)→-∞ as u∈RN,u→∞,(iii)
φ(u)→+∞ as u∈W~Δ;a,bα,p([a,b]T,RN),u→∞.
Firstly, we prove (i).
Let{un}⊂WΔ;a,bα,p([a,b]T,RN) be (C)-sequence of φ, that is φ(un) is bounded and 1+unφ′un→0 as n→∞. Then there exists a positive constant C5 such that (141)φun≤C5,1+unφ′un≤C5for all n∈N. In accordance with (A), (F1), (124), and (125), we have(142)C5+pC5≥1+unφ′un-pφun≥φ′un,un-pφun=∫a,bT∇Fσt,unσt,unσt-pFσt,unσtΔαt=∫unσt>M∇Fσt,unσt,unσt-pFσt,unσtΔαt+∫unσt≤M∇Fσt,unσt,unσt-pFσt,unσtΔαt≥μ-p∫unσt>MFσt,unσtΔαt-p+Mmaxx≤M⁡a^x∫a,bTb^σtΔαtfor all n∈N. This inequality gives that there exists a constant C~6 such that (143)∫unσt>MFσt,unσtΔαt≥C~6for all n∈N. From condition (A), we have (144)∫unσt≤MFσt,unσtΔαt≥-∫unσt≤Ma^unσtb^σtΔαt≥-maxx≤Ma^x∫a,bTb^σtΔαt.Combining (143) with (144), one has (145)∫a,bTFσt,unσtΔαt=∫unσt>MFσt,unσtΔαt+∫unσt≤MFσt,unσtΔαt≥C6,where C6=C~6-max|x|≤M⁡a^(|x|)∫[a,b]Tb^σ(t)Δαt. Moreover, by (124), (141), and (145), one has(146)C5≥φun≥1p∫a,bTTαuntpΔαt+C6for all n∈N. This shows that there exists constant C7 such that (147)∫a,bTTαuntpΔαt≤C7for all n∈N. It follows from Theorem 48 and (147) that there exists constant C8 such that (148)u~n∞≤C8for all n∈N.
Now, we will assert the sequence{u¯n} is bounded. If not, we may assume, without loss of generality, that u¯n→∞ as n→∞. Set vn=un/un=u¯n/un+u~n/un=v¯n+v~n, then {vn} is bounded in WΔ;a,bα,p([a,b]T,RN). By Remark 51, there is a subsequence of {vn} (for simplicity denoted again by {vn}) such that(149)vn⇀vinWΔ;a,bα,pa,bT,RN,vn⟶vinCa,bT,RN.Combining with (148), {u~n} is bounded in C([a,b]T,RN), hence, v∈RN,v≠0. So |un(t)|→∞ as n→∞ for all t∈[a,b]T. By (F3), we have(150)limsupn→∞⁡∫a,bTFσt,unσtΔαt≤limsupn→∞⁡∫DFσt,unσtΔαt+∫a,bTgσtΔαt⟶-∞,which contradicts (145).
Consequently, by Theorem59, {un} is bounded in WΔ;a,bα,p([a,b]T,RN). Again, from Remark 51, there is a subsequence of {un} (for simplicity denoted again by {un}) such that (151)un⇀uinWΔ;a,bα,pa,bT,RN,(152)un⟶uinCa,bT,RN.According to (152), {un} is bounded in C([a,b]T,RN), then from condition (A), there exists positive constant C9 such that(153)∫a,bT∇Fσt,unσt,uσt-unσtΔαt≤∫a,bT∇Fσt,unσtuσt-unσtΔαt≤C9∫a,bTb^σtuσt-unσtΔαt≤C9u-un∞∫a,bTb^σtΔαt.Combining with (152), we get(154)∫a,bT∇Fσt,unσt,uσt-unσtΔαt⟶0asn⟶∞.Be aware of(155)φ′un,u-un=∫a,bTTαuntp-2Tαunt,Tαut-TαuntΔαt+∫a,bT∇Fσt,unσt,uσt-unσtΔαt,φ′un,u-un⟶0asn⟶∞;one has(156)∫a,bTTαuntp-2Tαunt,Tαut-TαuntΔαt⟶0asn⟶∞.Moreover, it is easy to derive from (152) that(157)∫a,bTunσtp-2unσt,uσt-unσtΔαt⟶0asn⟶∞.Define (158)ϕu=1pup=1p∫a,bTuσtpΔαt+∫a,bTTαutpΔαt.Then one has (159)ϕ′un,u-un=∫a,bTTαuntp-2Tαunt,Tαut-TαuntΔαt+∫a,bTunσtp-2unσt,uσt-unσtΔαt⟶0as n→∞. By Hölder inequality, we get(160)0≤un-uunp-1-up-1≤ϕ′un-ϕ′u,un-u.This combined with (159) yields un→u. It is easy to derive from (151) and Theorem 47 that un→u in WΔ;a,bα,p([a,b]T,RN).
Secondly, we prove (ii).
For anyu∈RN, combining (124) with (F2) and (F3), we have(161)φu=∫a,bTFσt,uσtΔαt≤∫DFσt,uσtΔαt+∫a,bT∖DgσtΔαt≤∫DFσt,uσtΔαt+∫a,bTgσtΔαt⟶-∞as n→∞. This shows (ii) holds.
Thirdly, we prove (iii).
Fors∈R,|x|≥M and Δ-a.e. t∈[a,b]T, we set (162)Gs=Ft,sx,Hs=G′s-μsGs.Make use of (F1), when s≥M/x, (163)Hs=1s∇Ft,sx,sx-μFt,sx≥0holds. In addition, using (162), G(s) satisfies(164)G′s=Hs+μsGs.
So, whens≥M/x,(165)Ft,sx=sμFt,x+∫1sτ-μHτdτ.Besides, depending on condition (A) and (163), for x≥M and Δ-a.e. t∈[a,b]T, we assert(166)MxμFt,x≥Ft,xMx≥-a0b^t,where a0=maxx≤Ma^(x). This implies that(167)Ft,x≥-a0b^txMμ+1for all x∈RN and Δ-a.e. t∈[a,b]T. Moreover, by virtue of Theorem 48 and (124), we obtain(168)φu≥1p∫a,bTTαutpΔαt-a0Mμu∞μ∫a,bTb^σtΔαt-a0∫a,bTb^σtΔαt=12p∫a,bTTαutpΔαt+12p∫a,bTTαutpΔαt-a0Mμu∞μ∫a,bTb^σtΔαt-a0∫a,bTb^σtΔαt≥12p∫a,bTTαutpΔαt+12pK-pu∞p-a0Mμu∞μ∫a,bTb^σtΔαt-a0∫a,bTb^σtΔαt≥12p∫a,bTTαutpΔαt+12pK-pmax⁡aα-1,1-1∫a,bTuσtpΔαt-a0KpMμuμ∫a,bTb^σtΔαt-a0∫a,bTb^σtΔαt≥min⁡12p,12pK-pmax⁡aα-1,1-1up-a0KpMμuμ∫a,bTb^σtΔαt-a0∫a,bTb^σtΔαtfor all u∈W~Δ,T1,p([a,b]T,RN). Since 0<μ<p, (iii) holds.
It follows from Lemma58 and Theorem 54 that Theorem 60 holds.Example 61.
LetT=⋃k=0∞[2k,2k+1],α=1/2,a=1,b=2015,N=5. Consider the 4-Laplacian conformable fractional differential equation boundary value problem on time scale T(169)T1/2T1/2u2T1/2ut=∇Fσt,uσt,Δ-a.e.t∈1,2015Tκ2,u1-u2015=0,Tαu1-Tαu2015=0,where F(t,x)=-|x|2-|((1,1,2,2,3),x)|.
In virtue ofF(t,x)=-|x|2-|((1,1,2,2,3),x)|, ∇F(t,x)=-2x-1,1,2,2,3,x/1,1,2,2,3,x(1,1,2,2,3), all conditions of Theorem 60 hold with p=4,μ=3,M=1,g(t)≡0. It is easy to derive from Theorem 60 that problem (169) has at least one solution. Moreover, 0 is not the solution of problem (169). Thus, problem (169) has at least one nontrivial solution.According to Theorem60, we have the following corollary.Corollary 62.
Assume that(A), (F1), and the following condition are satisfied.(F4)
F(t,x)→-∞asx→∞ uniformly for Δ-a.e. t∈[a,b]T.Then problem (118) has at least one solution.Theorem 63.
Suppose that(A),(F1) and the following condition are satisfied:(F5)
∫[a,b]TF(σ(t),x)Δαt→-∞ as x→∞.(F6)
Assume that-F(t,·) is (β,γ)-subconvex with γ>0Δ-a.e. t∈[a,b]T, that is, (170)Ft,βx+y≥γFt,x+Ft,yfor allx,y∈RN and Δ-a.e. t∈[a,b]T.Then problem (118) has at least one solution.Proof.
It follows directly from (F5) that (171)φu⟶-∞asu∈RN,u⟶∞.Analogous to the proof of Theorem 60, we can obtain (172)φu⟶+∞asu∈W~Δ;a,bα,pa,bT,RN,u⟶∞.
Next, we will verifyφ satisfies (C)-condition. Let {un}⊂WΔ;a,bα,p([a,b]T,RN) be a (C)-sequence; that is, φ(un) is bounded and 1+unφ′(un)→0 as n→∞. Using the same method as that of (145), (147), and (148) in the proof of (i) in Theorem 60, there exist constants C10,C11 and C12 such that(173)∫a,bTFσt,unσtΔαt≥C10,∫a,bTTαuntpΔαt≤C11,(174)u~n∞≤C12for all n∈N. By means of (F6), we obtain(175)C10≤∫a,bTFσt,unσtΔαt≤1γ∫a,bTFσt,βu¯nΔαt-∫a,bTFσt,-u~nσtΔαt≤1γ∫a,bTFσt,βu¯nΔαt-maxx≤C12⁡a^x∫a,bTb^σtΔαtfor all n∈N, this conclusion shows that {u¯n} is bounded. Combining with (174), we are sure {un} is bounded. In a way similar to the proof of (i) in Theorem 60 we can prove that {un} has a convergent subsequence, thus, φ satisfies condition (C).
It follows from Lemma58 and Theorem 54 that Theorem 63 holds.Example 64.
LetT=R,α=1,a=π,b=2π,N=3. Consider the 5-Laplacian differential equation boundary value problem(176)u′t3u′t′=∇Ft,ut,a.e.t∈π,2π,uπ-u2π=0,u′π-u′2π=0,where F(t,x)=-|x|3-|((2014,2015,2016),x)|.
In virtue ofF(t,x)=-x3-2014,2015,2016,x, and ∇Ft,x = -3xx-2014,2015,2016,x/2014,2015,2016,x (2014,2015,2016), all conditions of Theorem 63 hold p=5,μ=4,M=2. It follows directly from Theorem 63 that problem (176) has at least one solution. Furthermore, 0 is not the solution of problem (176). Therefore, problem (176) has at least one nontrivial solution.



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