# Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

Authors: Xing Hu; Yongkun Li
Journal: Journal of Function Spaces
 (2022)

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## Abstract

By the concept of fractional derivative of Riemann-Liouville on time scales, we first introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Then, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using fibering mapping and Nehari manifolds, the existence of weak solutions for a class of fractional boundary value problems on time scales is studied, and a result of the existence of weak solutions for this problem is obtained.


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## Body

## 1. Introduction

The Sobolev space theory was developed by the Soviet mathematician S.L. Sobolev in the 1930s. It was created for the needs of studying modern theories of differential equations and studying many problems in the fields related to mathematical analysis. It has become a basic content in mathematics. In order to study the existence of solutions of differential and difference equations under a unified framework, papers [1–3] study some Sobolev space theories on time scales.In the past few decades, fractional calculus and fractional differential equations have attracted widespread attention in the field of differential equations, as well as in applied mathematics and science. In addition to true mathematical interest and curiosity, this trend is also driven by interesting scientific and engineering applications that have produced fractional differential equation models to better describe (time) memory effects and (space) nonlocal phenomena [4–9]. It is the rise of these applications that give new vitality to the field of fractional calculus and fractional differential equations and call for further research in this field.In order to unify the discrete analysis and continuous analysis, Hilger [10] proposed the time scale theory and established its related basic theory [11, 12]. So far, the study of time scale theory has attracted worldwide attention. It has been widely used in engineering, physics, economics, population dynamics, cybernetics, and other fields [13–17].As far as we know, no one has studied the fractional Sobolev space and its properties on time scales through the Riemann-Liouville derivative. In order to fill this gap, the main purpose of this article is to establish the fractional Sobolev space on time scales via the Riemann-Liouville derivative and to study its basic properties. Then, as an application of our new theory, we study the solvability of a class of fractional boundary value problems on time scales.


## 2. Preliminaries

In this section, we briefly collect some basic known notations, definitions, and results that will be used later.A time scaleT is an arbitrary nonempty closed subset of the real set ℝ with the topology and ordering inherited from ℝ. Throughout this paper, we denote by T a time scale. We will use the following notations: Jℝ0=a,b, Jℝ=a,b, J0=Jℝ0∩T, J=Jℝ∩T, Jk=a,ρb∩T.Definition 1 (see [18]).
Fort∈T, we define the forward jump operator σ:T⟶T by σt≔infs∈T:s>t, while the backward jump operator ρ:T⟶T is defined by ρt≔sups∈T:s<t.Remark 2 (see [18]).
(1)
In Definition1, we put inf∅=supT (i.e., σt=t if T has a maximum t) and sup∅=infT (i.e., ρt=t if T has a minimum t), where ∅ denotes the empty set.(2)
Ifσt>t, we say that t is right-scattered, while if ρt<t, we say that t is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated.(3)
Ift<supT and σt=t, we say that t is right-dense, while if t>infT and ρt=t, we say that t is left-dense. Points that are right-dense and left-dense at the same time are called dense(4)
The graininess functionμ:T⟶0,∞ is defined by μt≔σt−t.(5)
The derivative makes use of the setTk, which is derived from the time scale T as follows: if T has a left-scattered maximum M, then Tk≔T\M; otherwise, Tk≔T.Definition 3 [18].
Assume thatf:T⟶ℝ is a function and let t∈Tk. Then, we define fΔt to be the number (provided it exists) with the property that given any ε>0, there is a neighborhood U of t (i.e., U=t−δ,t+δ∩T for some δ>0) such that
(1)fσt−fs−fΔtσt−s≤εσt−s,for all s∈U. We call fΔt the delta (or Hilger) derivative of f at t. Moreover, we say that f is delta (or Hilger) differentiable (or in short: differentiable) on Tk provided fΔt exists for all t∈Tk. The function fΔ:Tk⟶ℝ is then called the (delta) derivative of f on Tk.Definition 4 (see [18]).
A functionf:T⟶ℝ 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. The set of rd-continuous functions f:T⟶ℝ will be denoted by Crd=CrdT=CrdT,ℝ. The set of functions f:T⟶ℝ that are differentiable and whose derivative is rd-continuous is denoted by Crd1=Crd1T=Crd1T,ℝ.Definition 5 (see [19]).
LetJ denote a closed bounded interval in T. A function F:J⟶ℝ is called a delta antiderivative of function f:J⟶ℝ provided F is continuous on J, delta differentiable at J, and FΔt=ft for all t∈J. Then, we define the Δ-integral of f from a to b by ∫abftΔt≔Fb−Fa.Proposition 6 (see [20]).
f is an increasing continuous function on J. If F is the extension of f to the real interval Jℝ given by
(2)Fs≔fs,ifs∈T,ft,ifs∈t,σt∉T,then ∫abftΔt≤∫abFtdt.Definition 7 (see [19]) (fractional integral on time scales).
Supposeh is an integrable function on J. Let 0<α≤1. The left fractional integral of order α of h is defined by
(3)aTItαht≔∫att−sα−1ΓαhsΔs.
The right fractional integral of orderα of h is defined by
(4)tTIbαht≔∫tbs−tα−1ΓαhsΔs,where Γ is the gamma function.Definition 8 (see [19]) (Riemann-Liouville fractional derivative on time scales).
Lett∈T, 0<α≤1, and h:T⟶ℝ. The left Riemann-Liouville fractional derivative of order α of h is defined by
(5)aTDtαht≔IaTt1−αhtΔ=1Γ1−α∫att−s−αhsΔsΔ.
The right Riemann-Liouville fractional derivative of orderα of h is defined by
(6)aTDtαht≔ItTb1−αhtΔ=−1Γ1−α∫tbt−s−αhsΔsΔ.Proposition 9 (see [19]).
Let0<α≤1, we have aTDtα=Δ∘IaTt1−α.Proposition 10 (see [19]).
For any functionh that is integrable on J, the Riemann-Liouville Δ-fractional integral satisfies IvTtα∘IaTtβ=IaTtα+β for α>0 and β>0.Proposition 11 (see [19]).
For any functionh that is integrable on J, one has DaTtα∘IaTtαh=h.Corollary 12 (see [19]).
For0<α≤1, we have DaTtα∘DaTt−α=Id and IaTt−α∘IaTtα=Id, where Id denotes the identity operator.Theorem 13 (see [19]).
Letf∈CJ and α>0, then f∈IaTtαJ iff
(7)IaTt1−αf∈C1J,IaTt1−αftt=a=0.Theorem 14 (see [19]).
Letα>0 and f∈CJ satisfy the condition in Theorem 13. Then,
(8)IaTtα∘DaTtαf=f.Theorem 15 (see [2]).
A functionf:J⟶ℝN is absolutely continuous on J iff f is Δ-differentiable Δ-a.e. on J0 and
(9)ft=fa+∫a,tTfΔsΔs,∀t∈J.Theorem 16 (see [21]).
A functionf:T⟶ℝ is absolutely continuous on T iff the following conditions are satisfied:
(i)
f is Δ-differentiable Δ-a.e. on J0 and fΔ∈L1T(ii)
The equality(10)ft=fa+∫a,tTfΔsΔs,holds for every t∈T.Theorem 17 (see [22]).
A functionq:Jℝ⟶ℝm is absolutely continuous iff there exist a constant c∈ℝm and a function φ∈L1 such that
(11)qt=c+Ia+1φt,t∈Jℝ.
In this case, we haveqa=c and q′t=φt, t∈Jℝ a.e.Theorem 18 (see [2]) (integral representation).
Letα∈0,1 and q∈L1. Then, q has a left-sided Riemann-Liouville derivative Da+αq of order α iff there exist a constant c∈ℝm and a function φ∈L1 such that
(12)qt=1Γαct−a1−α+Ia+αφt,t∈Jℝa.e.
In this case, we haveIa+1−αqa=c and Da+αqt=φt, t∈Jℝ a.e.Theorem 19 (see [23]).
Letα>0, p,q≥1, and 1/p+1/q≤1+α, where p≠1 and q≠1 in the case when 1/p+1/q=1+α. Moreover, let
(13)IaTtαLp≔f:f=IaTtαg,g∈LpJ,ItTbαLp≔f:f=ItTbαg,g∈LpJ,then the following integration by part formulas hold.
(a)
Ifφ∈LpJ and ψ∈LqJ, then
(14)∫J0φtIaTtαψtΔt=∫J0ψtItTtαφtΔt.(b)
Ifg∈ItTbαLp and f∈ItTbαLp, then
(15)∫J0gtDaTtαftΔt=∫J0ftDtTbαgtΔt.Lemma 20 (see [1]).
Letf∈LΔ1J0. Then, the following
(16)∫J0f·φΔsΔs=0,foreveryφ∈C0,rd1Jk,holds iff there exists a constant c∈ℝ such that
(17)f≡cΔ−a.e.onJ0.Definition 21 (see [1]).
Letp∈ℝ¯ be such that p≥1 and u:J⟶ℝ¯. Say that u belongs to WΔ1,pJ iff u∈LΔpJ0 and there exists g:Jk⟶ℝ¯ such that g∈LΔpJ0 and
(18)∫J0u·φΔsΔs=−∫J0g·φσsΔs,∀φ∈C0,rd1Jk,with
(19)C0,rd1Jk≔f:J⟶ℝ:f∈Crd1Jk,fa=fb,where Crd1Jk is the set of all continuous functions on J such that they are Δ-differential on Jk and their Δ-derivatives are rd-continuous on Jk.Theorem 22 (see [1]).
Letp∈ℝ¯ be such that p≥1. Then, the set LΔpJ0 is a Banach space together with the norm defined for every f∈LΔpJ0 as
(20)fLΔp≔∫J0fpsΔs1/p,ifp∈ℝ,infC∈ℝ:f≤CΔ−a.e.onJ0,ifp=+∞.
Moreover,LΔ2J0 is a Hilbert space together with the inner product given for every f,g∈LΔ2J0×LΔ2J0 by
(21)f,gLΔ2≔∫J0fs·gsΔs.Theorem 23 (see [24]).
Fractional integration operators are bounded inLpJℝ, i.e., the following estimate
(22)Ia+αφLpa,b≤b−aReαReαΓαφLpJℝ,Reα>0,holds.Proposition 24 (see [1]).
Supposep∈ℝ¯ and p≥1. Let p′∈ℝ¯ be such that 1/p′+1/p′=1. Then, if f∈LΔpJ0 and g∈LΔp′J0, then f·g∈LΔ1J0 and
(23)f·gLΔ1≤fLΔp·gLΔp′.
This expression is called Hölder’s inequality and Cauchy-Schwarz’s inequality wheneverp=2.Theorem 25 (see [25]) (the first mean value theorem).
Letf and g be bounded and integrable functions on J, and let g be nonnegative (or nonpositive) on J. Let us set
(24)m=infft:t∈J0,M=supft:t∈J0.
Then, there exists a real numberΛ satisfying the inequalities m≤Λ≤M such that
(25)∫abftgtΔt=Λ∫abgtΔt.Corollary 26 (see [25]).
Letf be an integrable function on J and let m and M be the infimum and supremum, respectively, of f on J0. Then, there exists a number Λ between m and M such that ∫abftΔt=Λb−a.Theorem 27 (see [25]).
Letf be a function defined on J and let c∈T with a<c<b. If f is Δ-integrable from a to c and from c to b, then f is Δ-integrable from a to b and
(26)∫abftΔt=∫acftΔt+∫cbftΔt.Lemma 28 (see [26]) (a time scale version of the Arzelà-Ascoli theorem).
LetX be a subset of CJ,ℝ satisfying the following conditions:
(i)
X is bounded(ii)
For any givenε>0, there exists δ>0 such that t1,t2∈J, t1−t2<δ implies ft1−ft2<ε for all f∈X
Then,X is relatively compact.


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

In this section, we present and prove some lemmas, propositions, and theorems, which are of utmost significance for our main results.In the following, let0<a<b. Inspired by Theorems 15–18, we give the following definition.Definition 29.
Let0<α≤1. By ACΔ,a+α,1J,ℝN, we denote the set of all functions f:J⟶ℝN that have the representation
(27)ft=1Γαct−a1−α+aTItαφt,t∈JΔ−a.e.,with c∈ℝN and φ∈LΔ1.
Then, we have the following result.Theorem 30.
Let0<α≤1 and f∈LΔ1. Then, function f has the left Riemann-Liouville derivative DaTtaf of order α on the interval J iff f∈ACΔ,a+α,1J,ℝN; that is, f has the representation (27). In such a case,
(28)IaTt1−αfa=c,DaTtαft=φt,t∈JΔ‐a.e.Proof.
Let us assume thatf∈LΔ1 has a left-sided Riemann-Liouville derivative DaTtαf. This means that IaTt1−αf is (identified to) an absolutely continuous function. From the integral representation of Theorems 15 and 17, there exist a constant c∈ℝN and a function φ∈LΔ1 such that
(29)IaTt1−αft=c+IaTt1φt,t∈J,with IaTt1−αfa=c and aTIt1−αftΔ=DaTtαft=φt, t∈JΔ−a.e.
By Proposition10 and applying IaTtα to (29), we obtain
(30)IaTt1ft=IaTtαct+IaTt1IaTtαφt,t∈JΔ‐a.e.
The result follows from theΔ-differentiability of (30).
Conversely, let us assume that (27) holds true. From Proposition 10 and applying IaTt1−a to (27), we obtain
(31)IaTt1−αft=c+IaTt1φt,t∈JΔ‐a.e.,and then, IaTt1−αf has an absolutely continuous representation. Further, f has a left-sided Riemann-Liouville derivative DaTtαf. This completes the proof.Remark 31.
(i)
ByACΔ,a+α,p1≤p<∞, we denote the set of all functions f:J⟶ℝN possessing representation (27) with c∈ℝN and φ∈LΔp(ii)
It is easy to see that Theorem30 implies that for any 1≤p<∞, f has the left Riemann-Liouville derivative DaTtαf∈LΔp iff f∈ACΔ,a+α,p; that is, f has the representation (27) with φ∈LΔpDefinition 32.
Let0<α≤1 and let 1≤p<∞. By the left Sobolev space of order α, we will mean the set WΔ,a+α,p=WΔ,a+α,pJ,ℝN given by
(32)WΔ,a+α,p≔u∈LΔp;∃g∈LΔp,∀φ∈Cc,rd∞suchthat∫J0ut⋅DtTbαφtΔt=∫J0gt⋅φtΔt.Remark 33.
A functiong given in Definition 32 will be called the weak left fractional derivative of order 0<α≤1 of u; let us denote it by uTa+α. The uniqueness of this weak derivative follows from 1.We have the following characterization ofWΔ,a+α,p.Theorem 34.
If0<α≤1 and 1≤p<∞, then WΔ,a+α,p=ACΔ,a+α,p∩LΔp.Proof.
On the one hand, ifu∈ACΔ,a+α,p∩LΔp, then from Theorem 30, it follows that u has derivative DaTtαu∈LΔp. Theorem 19 implies that
(33)∫J0utDtTbαφtΔt=∫J0DaTtαutφtΔt,for any φ∈Cc,rd∞. So, u∈WΔ,a+α,p with uTa+α=g=DaTtαu∈LΔp.
On the other hand, ifu∈WΔ,a+α,p, then u∈LΔp, and there exists a function g∈LΔp such that
(34)∫J0utDtTbαφtΔt=∫J0gtφtΔt,for any φ∈Cc,rd∞. To show that u∈ACΔ,a+α,p∩LΔp, it suffices to check (Theorem 30 and definition of ACΔ,a+α,p) that u possesses the left Riemann-Liouville derivative of order α, which belongs to LΔp; that is, aTIt1−αu is absolutely continuous on J and its delta derivative of α order (existing Δ-a.e. on J) belongs to LΔp.
In fact, letφ∈Cc,rd∞, then φ∈DtTbαCrd and DtTbαφ=−tTIb1−αΔ. From Theorem 19, it follows that
(35)∫J0utDtTbαφtΔt=∫J0ut−ItTb1−αφΔtΔt=∫J0DaTt1−αIaTt1−αut−ItTb1−αφΔtΔt=∫J0IaTt1−αut−φΔtΔt=−∫J0IaTt1−αutφΔtΔt.
In view of (34) and (35), we get
(36)∫J0IaTt1−αutφΔtΔt=−∫J0gtφtΔt,for any φ∈Cc,rd∞. So, IaTt1−αu∈WΔ,a+1,p. Consequently, IaTt1−αu is absolutely continuous and its delta derivative is equal Δ-a.e. on a,bT to g∈LΔp. The proof is complete.From the proof of Theorem34 and the uniqueness of the weak fractional derivative, the following theorem follows.Theorem 35.
If0<α≤1 and 1≤p<∞, then the weak left fractional derivative uTa+α of a function u∈WΔ,a+α,p coincides with its left Riemann-Liouville fractional derivative DaTtαuΔ-a.e. on J.Remark 36.
(1)
If0<α≤1 and 1−αp<1, then ACΔ,a+α,p⊂LΔp and, consequently,
(37)WΔ,a+α,p=ACΔ,a+α,p∩LΔp=ACΔ,a+α,p.(2)
If0<α≤1 and 1−αp≥1, then WΔ,a+α,p=ACΔ,a+α,p∩LΔp is the set of all functions belonging to ACΔ,a+α,p that satisfy the condition IaTt1−αfa=0By using the definition ofWΔ,a+α,p with 0<α≤1 and Theorem 35, one can easily prove the following result.Theorem 37.
Let0<α≤1,1≤p<∞, and u∈LΔp. Then, u∈WΔ,a+α,p iff there exists a function g∈LΔp such that
(38)∫J0utDtTbαφtΔt=∫J0gtφtΔt,φ∈Cc,rd∞.In such a case, there exists the left Riemann-Liouville derivativeDaTtαu of u and g=DaTtαu.Remark 38.
Functiong will be called the weak left fractional derivative of u∈WΔ,a+α,p of order α. Its uniqueness follows from [1]. From the above theorem, it follows that it coincides with an appropriate Riemann-Liouville derivative.Let us fix0<α≤1 and consider in the space WΔ,a+α,p a norm ·WΔ,a+α,p given by
(39)uWΔ,a+α,pp=uLΔpp+DaTtαuLΔpp,u∈WΔ,a+α,p.(Here·LΔp denotes the delta norm in LΔp (Theorem 22)).Lemma 39.
Let0<α≤1 and 1≤p<∞, then
(40)IaTtαφLΔpp≤KpφLΔpp,where K=b−aα/Γα+1. That is to say, the fractional integration operator is bounded in LΔp.Proof.
The conclusion follows from Theorem23, Proposition 24, and Proposition 6. The proof is complete.Theorem 40.
If0<α≤1, then the norm ·WΔ,a+α,p is equivalent to the norm ·a,WΔ,a+α,p given by
(41)ua,WΔ,a+α,pp=IaTt1−αuap+DaTtαuLΔpp,u∈WΔ,a+α,p.Proof.
(1)
Assume that1−αp<1. On the one hand, in view of Remarks 31 and 36, for u∈WΔ,a+α,p, we can write it as(42)ut=1Γαct−a1−α+IaTtαφt,with c∈ℝN and φ∈LΔp. Since t−aα−1p is an increasing monotone function, by using Proposition 6, we can write that ∫J0t−aα−1pΔt≤∫Jℝ0t−aα−1pdt. And taking into account Lemma 39, we have
(43)uLΔpp=∫J01Γαct−a1−α+IaTtαφtpΔt≤2p−1cpΓpα∫J0t−aα−1pΔtIaTtαφLΔpp≤2p−1cpΓpα∫J0t−aα−1pdtIaTtαφLΔpp≤2p−1cpΓpα1α−1p+1b−aα−1p+1+KpφLΔpp,where K comes from Lemma 39. Noting that c=IaTt1−αua, φ=DaTtαu, one can obtain
(44)uLΔpp≤Lα,0cp+φLΔpp≤Lα,0IaTt1−αuap+DaTtαuLΔpp=Lα,0ua,WΔ,a+α,pp,where
(45)Lα,0=2p−1b−a1−1−αpΓpα1−1−αp+Kp.
Consequently,(46)uWΔ,a+α,pp=uLΔPP+DaTtαuLΔpp≤Lα,1ua,WΔ,a+α,pp,where Lα,1=Lα,0+1.
On the other hand, we will prove that there exists a constantMα,1 such that
(47)ua,WΔ,a+α,pp≤Mα,1uWΔ,a+α,pp,u∈WΔ,a+α,p.
Indeed, letu∈WΔ,a+α,p and consider coordinate functions IaTt1−αui of IaTt1−αu with i∈1,⋯,N. Lemma 39, Theorem 25, and Corollary 26 imply that there exist constants
(48)Λi∈inft∈a,bTIaTt1−αuit,supt∈a,bTIaTt1−αuit,i=1,2,⋯,N,such that
(49)Λi=1b−a∫abIaTt1−αuisΔs.
Hence, for a fixedt0∈J0, if IaTt1−αuit0≠0 for all i=1,2,⋯,N, then we can take constants θi such that
(50)θiIaTt1−αuit0=Λi=1b−a∫abIaTt1−αuisΔs.
Therefore, we have25IaTt1−αuit0=θib−a∫abIaTt1−αuisΔs.
From the absolute continuity (Theorem16) of aTIt1−αui, it follows that
(52)IaTt1−αuit=It1−αaTuit0+∫t0,tTIt1−αaTuisΔΔs,for any t∈J. Consequently, combining with Proposition 9 and Lemma 39, we see that
(53)It1−αaTuit=It1−αaTuit0+∫t0,tTIt1−αaTuisΔΔs≤θib−aIt1−αaTuLΔ1+∫t0,tTDtαaTusΔs≤θib−aIt1−αaTuLΔ1+DtαaTuLΔ1≤θib−ab−a1−αΓ2−αuLΔ1+DtαaTuLΔ1,for t∈J. In particular,
(54)It1−αaTuit≤θib−ab−a1−αΓ2−αuLΔ1+DtαaTuLΔ1.
So,(55)It1−αaTua≤Nθb−a−αΓ2−α+1uLΔ1+DtαaTuLΔ1≤NMα,0b−ap−1/puLΔp+DtαaTuLΔp,where θ=maxi∈1,2,⋯,Nθi and Mα,0=θb−a−α/Γ2−α+1. Thus,
(56)It1−αaTuap≤NpMα,0pb−ap−12p−1uLΔpp+DtαaTuLΔpp,and, consequently,
(57)ua,WΔ,a+α,pp=It1−αaTuap+DtαaTuLΔpp≤NpMα,0pb−ap−12p−1+1uLΔpp+DtαaTuLΔpp=Mα,1ua,WΔ,a+α,pp,where Mα,1=NpMα,0pb−ap−12p−1+1.
IfIt1−αaTuit0=0 for i belongs to some subset of 1,2,⋯,N, from the above argument process, one can easily see that there exists a constant Mα,1 such that (32) holds.
(2)
When1−αp≥1, then (Remark 36) WΔ,a+α,p=ACΔ,a+α,p∩LΔp is the set of all functions that belong to ACΔ,a+α,p that satisfy the condition aTIt1−αua=0. Hence, in the same way as in the case of 1−αp<1 (putting c=0), we obtain the inequality(58)uWΔ,a+α,pp≤Lα,1ua,WΔ,a+α,pp,withsomeLα,1>0.
The inequality,(59)ua,WΔ,a+α,pp≤Mα,1uWΔ,a+α,pp,withsomeMα,1>0,is obvious (it is sufficient to put Mα,1=1 and use the fact that It1−αaTua=0).
The proof is complete.Now, we are in a position to prove some basic properties of the spaceWΔ,a+α,p.Theorem 41.
The spaceWΔ,a+α,p is complete with respect to each of the norms ·WΔ,a+α,p and ·a,WΔ,a+α,p for any 0<α≤1, 1≤p<∞.Proof.
In view of Theorem40, we only need to show that WΔ,a+α,p with the norm ·a,WΔ,a+α,p is complete. Let uk⊂WΔ,a+α,p be a Cauchy sequence with respect to this norm. So, the sequences It1−αaTuka and DtαaTuk are Cauchy sequences in ℝN and LΔp, respectively.
Letc∈ℝN and φ∈LΔp be the limits of the above two sequences in ℝN and LΔp, respectively. Then, the function
(60)ut=cΓαt−aα−1+ItαaTφt,t∈JΔ−a.e.,belongs to WΔ,a+α,p and it is the limit of uk in WΔ,a+α,p with respect to ·a,WΔ,a+α,p. The proof is complete.The proof method of the following two theorems is inspired by the method used in the proof of Proposition 8.1 (b) and (c) in [27].Theorem 42.
The spaceWΔ,a+α,p is reflexive with respect to the norm ·WΔ,a+α,p for any 0<α≤1 and 1<p<∞.Proof.
Let us considerWΔ,a+α,p with the norm ·WΔ,a+α,p and define a mapping
(61)λ:WΔ,a+α,p∋u⟼u,DtαaTu∈LΔp×LΔp.
It is obvious that(62)uWΔ,a+α,p=λuLΔp×LΔp,where
(63)λuLΔp×LΔp=∑i=12λuiLΔpp1/p,λu=u,DtαaTu∈LΔp×LΔp,which means that the operator λ:u↦u,aTDtαu is an isometric isomorphic mapping and the space WΔ,a+α,p is isometric isomorphic to the space Ω=u,aTDtαu:∀u∈WΔ,a+α,p, which is a closed subset of LΔp×LΔp as WΔ,a+α,p is closed.
SinceLΔp is reflexive, the Cartesian product space LΔp×LΔp is also a reflexive space with respect to the norm ∥v∥LΔp×LΔp=∑i=12∥vi∥LΔpp1/p, where v=v1,v2∈LΔp×LΔp.
Thus,WΔ,a+α,p is reflexive with respect to the norm ∥·∥WΔ,a+α,p. The proof is complete.Theorem 43.
The spaceWΔ,a+α,p is separable with respect to the norm ∥·∥WΔ,a+α,p for any 0<α≤1 and 1≤p<∞.Proof.
Let us considerWΔ,a+α,p with the norm ∥·∥WΔ,a+α,p and the mapping λ defined in the proof of Theorem 42. Obviously, λWΔ,a+α,p is separable as a subset of separable space LΔp×LΔp. Since λ is the isometry, WΔ,a+α,p is also separable with respect to the norm ∥·∥WΔ,a+α,p. The proof is complete.Theorem 44.
Let1≤s≤r≤t<∞, u∈LΔsJ0∩LΔtJ0, then u∈LΔrJ0 and
(64)∥u∥LΔr≤∥u∥LΔsθ∥u∥LΔt1−θ,where θ∈0,1 with 1/r=θ/s+1−θ/t.Proof.
We will divide the proof into the following three major cases.(i)
Whenr=s, we can take θ=1, the conclusion is evident(ii)
Whenr=t, we can take θ=0, the conclusion is obvious(iii)
Let1≤s<r<t<∞
In this case, if there existm,n>0 such that r=s/m+t/n, then
(65)ur=us/m·ut/n.
In view ofu∈LΔsJ0∩LΔtJ0, we have
(66)∫J0us/mmΔx=∫J0usΔx<+∞,∫J0ut/nnΔx=∫J0utΔx<+∞.
Hence, we obtain that(67)us/m∈LΔmJ0,ut/n∈LΔnJ0.
Therefore, whenm,n satisfy the following conditions
(68)m,n>0,sm+tn=r,1m+1n=1,that is to say,
(69)m=t−st−r,n=t−sr−s,by Proposition 2.6 in P4 from [1], one obtains
(70)∫J0urΔx=∫J0us/m·ut/nΔx≤∫J0us/mmΔx1/m·∫J0ut/nnΔx1/n=∫J0usΔx1/m·∫J0utΔx1/n=uLΔss/m·uLΔtt/n<∞,so u∈LΔrJ0 and
(71)∥u∥LΔrr=∫J0urΔx≤∥u∥LΔss/m·∥u∥LΔtt/n.
Letθ=s/rm, then θ∈0,1, t/m=1−θ, θ/s+1−θ/t=1/rm+1/rn=1/r, and hence,
(72)∥u∥LΔr≤∥u∥LΔsθ∥u∥LΔt1−θ.
The proof is complete.Proposition 45.
Let0<α≤1 and 1<p<∞. For all u∈WΔ,a+α,p, if 1−α≥1/p or α>1/p, then
(73)∥u∥LΔp≤bαΓα+1aTDtαuLΔp,if α>1/p and 1/p+1/q=1, then
(74)∥u∥∞≤bα−1/pΓαα−1q+11/qaTDtαuLΔp.Proof.
In view of Remark36 and Theorem 14, in order to prove inequalities (73) and (74), we only need to prove that
(75)aTItαaTDtαuLΔp≤bαΓα+1aTDtαuLΔp,for 1−α≥1/p or α>1/p, and that
(76)aTItαaTDtαu∞≤bα−1/pΓαα−1q+11/qaTDtαuLΔp,for α>1/p and 1/p+1/q=1.
Note thataTDtαu∈LΔpJ,ℝN, the inequality (75) follows from Lemma 39 directly.
We are now in a position to prove (76). For α>1/p, choose q such that 1/p+1/q=1. For all u∈WΔ,a+α,p, since t−sα−1q is an increasing monotone function, by using Proposition 6, we find that ∫att−sα−1qΔs≤∫att−sα−1qds. Taking into account Proposition 24, we have
(77)aTItαaTDtαut=1Γα∣∫att−sα−1aTDtαusΔs∣≤1Γα∫att−sα−1qΔs1/q∥aTDtαu∥LΔp≤1Γα∫att−sα−1qds1/q∥aTDtαu∥LΔp≤b1/q+α−1Γαα−1q+11/qaTDtαuLΔp=bα−1/pΓαα−1q+11/qaTDtαuLΔp.
The proof is complete.Remark 46.
(i)
According to (73), we can consider WΔ,a+α,p with respect to the norm
(78)∥u∥WΔ,a+α,pp=∥aTDtαu∥LΔpp=∫J0aTDtαutpΔt1/p,in the following analysis.(ii)
It follows from (73) and (74) that WΔ,a+α,p is continuously immersed into CJ,ℝN with the natural norm ∥·∥∞Proposition 47.
Let0<α≤1 and 1<p<∞. Assume that α>1/p and the sequence uk⊂WΔ,a+α,p converges weakly to u in WΔ,a+α,p. Then, uk⟶u in CJ,ℝN, i.e., ∥u−uk∥∞=0, as k⟶∞.Proof.
Ifα>1/p, then by (74) and (78), the injection of WΔ,a+α,p into CJ,ℝN, with its natural norm ∥·∥∞, is continuous, i.e., uk⟶u in WΔ,a+α,p, then uk⟶u in CJ,ℝN.
Sinceuk⇀u in WΔ,a+α,p, it follows that uk⇀u in CJ,ℝN. In fact, for any h∈CJ,ℝN∗, if uk⟶u in WΔ,a+α,p, then uk⟶u in CJ,ℝN, and thus, huk⟶hu. Therefore, h∈WΔ,a+α,p∗, which means that CJ,ℝN∗⊂WΔ,a+α,p∗. Hence, if uk⇀u in WΔ,a+α,p, then for any h∈CJ,ℝN∗, we have h∈WΔ,a+α,p∗, and thus, huk⟶hu, i.e., uk⇀u in CJ,ℝN.
By the Banach-Steinhaus theorem,uk is bounded in WΔ,a+α,p and, hence, in CJ,ℝN. Now, we prove that the sequence uk is equicontinuous. Let 1/p+1/q=1 and t1,t2∈J with t1≤t2, for all f∈LΔpJ,ℝN, by using Proposition 24, Proposition 6, and Theorem 27, and noting α>1/p, we have
(79)aTIt1αft1−aTIt2αft2=1Γα∫at1t1−sα−1fsΔs−∫at2t2−sα−1fsΔs≤1Γα∫at1t1−sα−1fsΔs−∫at1t2−sα−1fsΔs+1Γα∫t1t2t2−sα−1fsΔs≤1Γα∫at1t1−sα−1−t2−sα−1∣fs∣Δs+1Γα∫t1t2t2−sα−1∣fs∣Δs≤1Γα∫at1t1−sα−1−t2−sα−1qΔs1/q∥f∥LΔp+1Γα∫t1t2t2−sα−1qΔs1/q∥f∥LΔp≤1Γα∫at1t1−sα−1q−t2−sα−1qΔs1/q∥f∥LΔp+1Γα∫t1t2t2−sα−1qΔs1/q∥f∥LΔp≤1Γα∫at1t1−sα−1q−t2−sα−1qds1/q∥f∥LΔp+1Γα∫t1t2t2−sα−1qds1/q∥f∥LΔp=∥f∥LΔpΓα1+α−1q1/qt1α−1q+1−t2α−1q+1+t2−t1α−1q+11/q+∥f∥LΔpΓα1+α−1q1/qt2−t1α−1q+11/q≤2∥f∥LΔpΓα1+α−1q1/qt2−t1α−1+1/q=2∥f∥LΔpΓα1+α−1q1/qt2−t1α−1/p.
Therefore, the sequenceuk is equicontinuous since, for t1,t2∈J, t1≤t2, by applying (79) and (78), we have
(80)ukt1−ukt2=aTIt1αaTDt1αukt1−aTIt2αaTDt2αukt2≤2t2−t1α−1/pΓα1+α−1q1/q∥aTDtαuk∥LΔp=2t2−t1α−1/pΓα1+α−1q1/q∥aTDtαuk∥LΔp≤2t2−t1α−1/pΓαα−1q+11/qaTDtαuLΔp=2t2−t1α−1/pΓαα−1q+11/qukWΔ,a+α,p≤Ct2−t1α−1/p,where 1/p+1/q=1 and C∈ℝ+ is a constant. By the Arzelà-Ascoli theorem on time scales (Lemma 28), uk is relatively compact in CJ,ℝN. By the uniqueness of the weak limit in CJ,ℝN, every uniformly convergent subsequence of uk converges uniformly on J to u. The proof is complete.Remark 48.
It follows from Proposition47 that WΔ,a+α,p is compactly immersed into CJ,ℝN with the natural norm ∥·∥∞.Theorem 49.
Let1<p<∞, 1/p<α≤1, 1/p+1/q=1, L:J×ℝN×ℝN⟶ℝ, t,x,y↦Lt,x,y satisfies the following:
(i)
For eachx,y∈ℝN×ℝN, Lt,x,y is Δ-measurable in t(ii)
ForΔ-almost every t∈J, Lt,x,y is continuously differentiable in x,y
If there existm1∈Cℝ+,ℝ+, m2∈LΔ1J,ℝ+, and m3∈LΔqJ,ℝ+, 1<q<∞, such that, for Δ-a.e. t∈J and every x,y∈ℝN×ℝN, one has
(81)∣Lt,x,y∣≤m1∣x∣m2t+yp,∣DxLt,x,y∣≤m1∣x∣m2t+yp,∣DyLt,x,y∣≤m1∣x∣m3t+yp−1.
Then, the functionalχ defined by
(82)χu=∫J0Lt,ut,aTDtαutΔt,is continuously differentiable on WΔ,a+α,p, and ∀u,v∈WΔ,a+α,p, one has
(83)χ′u,v=∫J0DxLt,ut,aTDtαut,vt+DyLt,ut,aTDtαut,aTDtαvtΔt.Proof.
It suffices to prove thatχ has, at every point u, a directional derivative χ′u∈WΔ,a+α,p∗ given by (84) and that the mapping
(84)χ′:WΔ,a+α,p∋u↦χ′u∈WΔ,a+α,p∗,is continuous. The rest proof is similar to the proof of Theorem 1.4 in [28]. We will omit it here. The proof is complete.


## 4. An Application

As an application of the concepts we introduced and the results obtained in Section3, in this section, we will use critical point theory to study the solvability of a class of boundary value problems on time scales. More precisely, our goal is to study the following fractional nonlinear Dirichlet problem on time scale T:
(85)DtTbαDαTtαutp−2DαTtαut=∇Ft,ut+σωtutq−2ut,Δ‐a.e.t∈J,uα=ub=0,where tTDbα and aTDtα are the right and the left Riemann-Liouville fractional derivative operators of order α defined on T, respectively, ∇Ft,u is the gradient of Ft,u at u and F∈CJ×ℝN,ℝ is homogeneous of degree r, σ is a positive parameter, ω∈CJ, 1<r<p<q and 1/p<α<1.We make the following assumptions:H1F:J×ℝN⟶ℝ is homogeneous of degree r, that is,
(86)Ft,su=srFt,us>0,for all t∈J, u∈ℝN;H2F±t,u=max±Ft,u,0≠0 for all u≠0.ByH1, Ft,u, we have
(87)u∇Fx,u=srFx,u,(88)∣Fx,u∣≤Kur,for some constant K>0.Our main results are as follows.Theorem 50.
Let1/p<α<1, 1<r<p<q and suppose that Ft,u satisfies the conditions H1 and H2. Then, there exists σ0>0 such that for all σ∈0,σ0, (85) has at least two nontrivial solutions.There have been many results using critical point theory to study boundary value problems of fractional differential equations [29–35] and dynamic equations on time scales [36–40], but the results of using critical point theory to study boundary value problems of fractional dynamic equations on time scales are still rare [3]. This section will explain that critical point theory is an effective way to deal with the existence of solutions of (85) on time scales.We will use the famous Nehari manifold and fibering map theory to prove our main results.We say thatu∈WΔ,a+α,p is a solution to the problem (85), if u satisfies the following equality:
(89)∫J0aTDtαutp−2aTDtαut,aTDtαvtΔt−∫J0∇Ft,ut,vtΔt−σ∫J0ωtutq−2ut,vtΔt=0,∀v∈WΔ,a+α,p.As a result, associated to the problem (85), we define the functional
(90)Jσu=1p∥u∥p−∫J0Ft,utΔt−σq∫J0ωtutqΔt=Ju−Hu−Mσu,where
(91)Ju=1p∫J0aTDtαutpΔt,Hu=∫J0Ft,utΔt,Mσu=σq∫J0ωtutqΔt.We need to show that the following lemma holds.Lemma 51.
(i)
The functionalJσ is well defined on WΔ,a+α,p(ii)
The functionalJσ is of class C1WΔ,a+α,p,ℝ, and for all u,v∈WΔ,a+α,p, we have(92)J′σu,v=∫J0aTDtαutp−2aTDtαut,aTDtαvtΔt−∫J0∇Ft,ut,vtΔt−σ∫J0ωtutq−2ut,vtΔt=J′u,v−H′u,v−M′σu,v,where
(93)J′u,v=∫J0aTDtαutp−2aTDtαut,aTDtαvtΔt,H′u,v=∫J0∇Ft,ut,vtΔt,M′σu,v=σ∫J0ωtutq−2ut,vtΔt.Proof.
(i)
From (33) in Proposition 45, (87), (88), and the equivalent norm, we obtain(94)Jσu=1p∥u∥p−∫J0Ft,utΔt−σq∫J0ωtutqΔt≤1p∥u∥p+∫J0∣Ft,ut∣Δt+σqmaxt∈Jωt∫J0utqΔt≤1pup+K∫J0utrΔt+σq∥ω∥∞∥u∥LΔqq≤1p∥u∥p+K∥u∥LΔrr+σq∥ω∥∞bαqΓqα+1aTDtαuLΔqq≤1p∥u∥p+KbαrΓrα+1aTDtαuLΔrr+σq∥ω∥∞bαqΓqα+1aTDtαuLΔqq≤1p∥u∥p+c1∥u∥r+c2∥u∥q,which implies that Jσ is well defined on WΔ,a+α,p.
(ii)
Let(95)Πu=1paTDtαup−Ft,ut−σqωtuq.
Then, we can easily show that for allu,v∈WΔ,a+α,p and for Δ-a.e. t∈J,
(96)lims⟶0Πut+svt−Πuts=lims⟶01s1paTDtαu+svp−Ft,u+sv−σqωtu+svq−1paTDtαup+Ft,ut+σqωtuq=aTDtαutp−2aTDtαut,aTDtαvt−∇Ft,ut,vt−σωtutq−2ut,vt.
Hence, in view of the Lagrange mean value theorem, (87) and (42), there exists a real number κ such that ∣κ∣≤∣s∣ and
(97)Πut+svt−Πuts=aTDtαu+κvp−2aTDtαu+κv,aTDtαvt−∇Ft,u+κv,v−σωtu+κvq−2u+κv,v≤aTDtαu+κvp−1aTDtαv−∇Ft,u+κv,v−σωtu+κvq−1v≤aTDtαu+κvp−1aTDtáv−ru+κvFt,u+κvv+σωtu+κvq−1v≤DaTtαu+κvp−1DaTtαv−ru+κvKu+κvrv+σωtu+κvq−1v≤DaTtαu+κvp−1DaTtαv−rKu+κvr−1v+σωtu+κvq−1v≤DaTtαup−1DaTtαv+DaTtαvp+rKur−1v+rKvr+σωtuq−1v+σωtvq.
On the other hand, in view of Hölder inequality on time scales, we see that(98)∫J0aTDtαutp−1aTDtαvtΔt≤∫J0aTDtαutp−1p/p−1Δtp−1/p∫J0aTDtαvtpΔt1/p=aTDtαup−1LΔp/p−1·aTDtαvLΔp,∫J0utς−1vt∣Δt≤∫J0utς−1ς/ς−1Δtp−1/p∫J0vtpΔt1/p=∥uς−1∣∥LΔς/ς−1·∥v∥LΔςfor ς=r or ς=q. Because ω is bounded, then, from the above inequalities, we conclude that the expression (97) is in LΔ1J.
As a result, in view of the dominated convergence theorem on time scales, one gets(99)lims⟶0Jσut+svt−Jσuts=∫J0aTDtαutp−2aTDtαut,aTDtαvtΔt−∫J0∇Ft,ut,vtΔt−σ∫J0ωtutq−2ut,vtΔt.
That is to say,Jσ is Ga^teaux differentiable.
In what follows, we prove that the Ga^teaux derivative of Jσ is continuous.
First, we verify thatH′:WΔ,a+α,p⟶WΔ,a+α,p∗ is continuous.
Taking into account (88), we have
(100)∫J0∇Ft,unt−∇Ft,ut2Δt≤∫J0∣∇Ft,unt∣+∣∇Ft,ut∣2Δt≤2∫J0∇Ft,unt2+∇Ft,ut2Δt≤2r2K2∫J0unt2r−1+ut2r−1Δt≤2r2K2∥un∥∞2r−1+∥u∥∞2r−1b−a,which combining with un⟶u in LΔ2 and Lebesgue’s dominated convergence theorem on time scales leads to
(101)∫J0∇Ft,unt−∇Ft,ut2Δt1/2⟶0,n⟶∞.
Namely,(102)∇Ft,un⟶∇Ft,u,inLΔ2J,ℝN.
Letun,u∈WΔ,a+α,p such that un⇀u in WΔ,a+α,p (n⟶∞). Using the Hölder inequality on time scales and (73) in Proposition 45, we can obtain
(103)H′un−H′uWΔ,a+α,p∗=supv∈WΔ,a+α,p,∥v∥=1H′un−H′uv=supv∈WΔ,a+α,p,∥v∥=1∫J0∇Ft,unt−∇Ft,utvtΔt≤supv∈WΔ,a+α,p,∥v∥=1∥∇F·,un·−∇F·,u·∥LΔ2∥v·∥LΔ2≤supv∈WΔ,a+α,p,∥v∥=1∥∇F·,un·−∇F·,u·∥LΔ2bαΓα+1aTDtαvLΔ2=Csupv∈WΔ,a+α,p,∥v∥=1∥∇F·,un·−∇F·,u·∥LΔ2⟶0.
So,H′u is continuous.
Next, we will proveJ∈C1WΔ,a+α,p,ℝ. For any given u,v∈WΔ,a+α,p, by the Hölder inequality on time scales, we have
(104)J′u,v=∫J0aTDtαutp−2aTDtαut,aTDtαvtΔt≤∫J0aTDtαutp−1aTDtαvtΔt≤∫J0aTDtαutp−1p/p−1Δtp−1/p∫J0aTDtαvtpΔt1/p=∫J0aTDtαutpΔtp−1/pv=up−1v.
That is,J′u is bounded. It is obvious that J′u is linear. Hence, for any u∈WΔ,a+α,p, J′u∈WΔ,a+α,p∗.
Defineg:WΔ,a+α,p⟶LΔp′J, gu=aTDtαu∣p−2aTDtαu, ∀u∈WΔ,a+α,p, where 1/p+1/p′=1. Now, it is time for us to demonstrate g is continuous in the following two cases:
(1)
Ifp∈2,∞, then, for u,v∈WΔ,a+α,p, using Hölder inequality on time scales, we can deduce that(105)∫J0gu−gvp′Δt=∫J0∣aTDtαu∣p−2aTDtαu−∣aTDtαv∣p−2aTDtαvp′Δt≤β∫J0aTDtαu−aTDtαvp′∣aTDtαu∣+∣aTDtαv∣p′p−2Δt≤β∫J0aTDtαu−aTDtαvp′p/p′Δtp′/p∫J0∣aTDtαu∣+∣aTDtαv∣p′p−2p/p′p−2Δtp′p−2/p=β∥aTDtαu−aTDtαv∥LΔpp′∥∣aTDtαu+aTDtαv∣∥LΔpp′p−2≤C¯∥u−v∥p′∥u∥+∥v∥p′p−2.(2)
Ifp∈1,2, then, for u,v∈WΔ,a+α,p, we have(106)∫J0gu−gvp′Δt=∫J0aTDtαu∣p−2aTDtαu−aTDtαv∣p−2aTDtαvp′Δt≤β∫J0aTDtαu−aTDtαvp′p−1Δt≤C¯1∥u−v∥.
Consequently, whenp>1, g is continuous.
Now, foru,v∈WΔ,a+α,p, we will show that
(107)∥J′u−J′v∥WΔ,a+α,p∗≤k∥gu−gv∥LΔp′.
In fact, foru,v∈WΔ,a+α,p, by Hölder inequality on time scales, we have
(108)∥J′u−J′v∥WΔ,a+α,p∗=supφ∈WΔ,a+α,p,∥φ∥=1J′u−J′v,v=supφ∈WΔ,a+α,p,∥φ∥=1∣∫J0aTDtαutp−2aTDtαut,aTDtαφtΔt−∫J0aTDtαvtp−2aTDtαvt,aTDtαφtΔt∣=supφ∈WΔ,a+α,p,∥φ∥=1∫J0gu−gv,aTDtαφtΔt≤supφ∈WΔ,a+α,p,∥φ∥=1∫J0gu−gvDtαφtaTΔt≤supφ∈WΔ,a+α,p,∥φ∥=1∫J0gu−gvp′Δt1/p′∫J0aTDtαφpΔt1/p≤∥gu−gv∥LΔp′∥φ∥=∥gu−gv∥LΔp′.
Combining with the continuity ofg, we see that J∈C1WΔ,a+α,p,ℝ.
In conclusion, (ii) is proven. The proof is complete.We deduce from Lemma51 and (87) that
(109)J′σu,u=∫J0aTDtαutpΔt−r∫J0Ft,utΔt−σ∫J0ωtutqΔt.It is easy to see that the energy functionalJσ is not bounded below on the space WΔ,a+α,p, but it is bounded below on a suitable subset of WΔ,a+α,p. In order to study problem (85), we define the constraint set
(110)Nσ≔u∈WΔ,a+α,p\0:J′σu,u=0.Note thatNσ contains every nonzero solution of (85), and u∈Nσ if and only if
(111)∥u∥p−r∫J0Ft,utΔt−σ∫J0ωtutqΔt=0.In order to get the existence of solutions, we decomposeNσ into three parts: corresponding to local minima, local maxima, and points of inflection are Δ-measurable sets defined as follows:
(112)Nσ+≔u∈Nσ:p−1∥u∥p−rr−1∫J0Ft,utΔt−σq−1∫J0ωtutqΔt>0,Nσ−≔u∈Nσ:p−1∥u∥p−rr−1∫J0Ft,utΔt−σq−1∫J0ωtutqΔt<0,Nσ0≔u∈Nσ:p−1∥u∥p−rr−1∫J0Ft,utΔt−σq−1∫J0ωtutqΔt=0.Next, we give some important attributes ofNσ+, Nσ− and Nσ0. Let p¯ be such that 1/p+1/p¯=1 and put
(113)η0=p−rΓqαα−1p¯+1q/pq−r∥ω∥∞b1+qα−1/pq−pΓrαα−1p¯+1r/pKrq−rb1+rα−1/pq−p/p−r.Then, we have the following crucial result.Lemma 52.
Ifσ∈0,η0, then Nσ0=∅.Proof.
We proceed by contradiction to show thatNσ0=∅ for all σ∈0,η0. If there exists u0∈Nσ0, then, in view of (111), we get
(114)p−r∥u0∥p−σq−r∫J0ωtu0tqΔt=0,(115)q−p∥u0∥p−rq−r∫J0Ft,u0tΔt=0.
By Proposition45 and (114), we have
(116)p−r∥u0∥p=σq−r∫J0ωtu0tqΔt≤σq−r∥ω∥∞b∥u0∥∞q≤bσq−r∥ω∥∞bα−1/pqΓqαα−1p¯+1q/paTDtαu0LΔpq=σq−r∥ω∥∞b1+α−1/pqΓqαα−1p¯+1q/pu0q,which implies that
(117)∥u0∥≥p−rΓqαα−1p¯+1q/p¯σq−r∥ω∥∞b1+qα−1/p1/q−p.
Moreover, combining with Proposition45, (88), and (115), one has
(118)q−p∥u0∥p=rq−r∫J0Ft,u0tΔt≤rq−r∫J0Ku0rΔt≤Krq−r∥u0∥∞rb≤Krq−rbbα−1/prΓrαα−1p¯+1r/p¯aTDtαu0LΔpr=Krq−rb1+α−1/prΓrαα−1p¯+1r/p¯u0r.
Hence,(119)∥u0∥≤Krq−rb1+α−1/prq−pΓrαα−1p¯+1r/p¯1/p−r.
It follows from (47) that
(120)σ≥p−rΓqαα−1p¯+1q/p∥u0∥q−pq−r∥ω∥∞b1+qα−1/p,and that
(121)∥u0∥1/q−p≥q−pΓrαα−1p¯+1r/p¯Krq−rb1+α−1/prq−p/p−r.
Combining with (120) and (120), we gain that
(122)σ≥p−rΓqαα−1p¯+1q/pq−r∥ω∥∞b1+qα−1/pq−pΓrαα−1p¯+1r/p¯Krq−rb1+α−1/prq−p/p−r=η0.
Namely,σ≥η0, which leads to a contradiction. This completes the proof of Lemma 52.Lemma 53.
Ifσ∈0,η0, then Jσ is coercive and bounded below on Nσ.Proof.
Letu∈Nσ, then, by (88) and Proposition 45, we gain
(123)∫J0Ft,utΔt≤K∫J0utrΔt≤K∥u∥∞rb≤Kb1+rα−1/pΓrαα−1p¯+1r/p¯aTDtαuLΔpr=Kb1+rα−1/pΓrαα−1p¯+1r/p¯∥u∥r.
Hence, in view of (111), one gets
(124)Jσu=1p∥u∥p−∫J0Ft,utΔt−σq∫J0ωtutqΔt=q−pqp∥u∥p−∫J0Ft,utΔt+rq∫J0Ft,utΔt=q−pqp∥u∥p−q−rq∫J0Ft,utΔt≥q−pqp∥u∥p−Kq−rb1+rα−1/pqΓrαα−1p¯+1r/p¯∥u∥r.
Sincer<p<q, Jσ is coercive and bounded below on Nσ. The proof of Lemma 53 is now completed.Now as we know, the Nehari manifold is closely related to the behavior of the functionsΘu:0,∞⟶ℝ defined as
(125)Θus=Jσsu.Such maps are called fibering maps. Foru∈WΔ,a+α,p, we define
(126)Θus=Jσsu=1p∥su∥p−∫J0Ft,utΔt−σq∫J0ωtsutqΔt=spp∥u∥p−sr∫J0Ft,utΔt−σsqq∫J0ωtutqΔt.Then, one obtains(127)Θu′s=sp−1∥u∥p−rsr−1∫J0Ft,utΔt−σsq−1∫J0ωtutqΔt,(128)Θ″us=p−1sp−2∥u∥p−rr−1sr−2∫J0Ft,utΔt−σq−1sq−2∫J0ωtutqΔt.Then, it is obvious to see thatsu∈Nσ iff Θ′us=0, and in particular, u∈Nσ iff Θ′u1=0.Before using fiber mapping to study the behavior of Nehari manifolds, let us introduce some symbols.(129)F±=u∈WΔ,a+α,p\0:∫J0Ft,utΔt≶0,F0=u∈WΔ,a+α,p\0:∫J0Ft,utΔt=0,Π±=u∈WΔ,a+α,p\0:∫J0ωtutqΔt≶0,Π0=u∈WΔ,a+α,p\0:∫J0ωtutqΔt=0.We will study the fibering mapΘu according to the signs of ∫J0ωtutqΔt and ∫J0Ft,utΔt. To this end, let us define ρu:0,∞⟶ℝ by setting
(130)ρus=sp−r∥u∥p−σsq−r∫J0ωtutqΔt.Hence, fors>0, one gets
(131)Θ′us=sp−1∥u∥p−rsr−1∫J0Ft,utΔt−σsq−1∫J0ωtutqΔt=sr−1ρus−r∫J0Ft,utΔt,which implies that su∈Nσ iff s is a solution of the following equation:
(132)ρus=r∫J0Ft,utΔt.Furthermore, obviously,ρu0=0 and
(133)ρ′us=p−rsp−r−1∥u∥p−σq−rsq−r−1∫J0ωtutqΔt.Lemma 54.
Ifu∈F0¯∩Π0¯, then Θu has no critical point.Proof.
In this case,Θu0=0 and Θ′us>0 for all s>0, which yields that Θu is strictly increasing and hence has no critical point. The proof is complete.Lemma 55.
Ifu∈F0¯∩Π+, then Θu possesses a unique critical point that corresponds to a global maximum point. Moreover, there exists s0>0 such that s0∈Nσ− and Jσs0u>0.Proof.
In this case, there exists a uniques¯∈0,∞ such that ρ′us¯=0. In addition, ρ′us>0 for s∈0,s¯ and ρ′us<0 for s∈s¯,∞. Note that ρu0=0 and ρus⟶−∞ as s⟶∞. So, for u∈F−, there exists a unique s0 such that ρus0=∫J0Ft,utΔt. Consequently, according to (131), we get Θ′us>0 for 0<s<s0, and Θ′us<0 for s>s0. That is, Θu is increasing on 0,s0 and decreasing on s0,∞. Therefore, Θu has exactly one critical point at s0, which is a global maximum point. Thus, by (128), s0u∈Nσ−. The proof is complete.Lemma 56.
Ifu∈F+∩Π0¯, then Θu possesses a unique critical point that corresponds to a global minimum point. Moreover, there exists s1>0 such that s1∈Nσ+ and Jσs1u<0.Proof.
In this case, it is easy to see thatρu0=0 and ρ′us>0 for all s>0, which implies that ρu is strictly increasing. Since u∈F+, there exists a unique s1>0 such that ρus1=∫J0Ft,utΔt. This implies that Θu is decreasing on 0,s1, increasing on s1,∞ and Θ′us1=0. Thus, Θu has exactly one critical point corresponding to global minimum point. Hence, s1u∈Nσ+. Moreover, since Jσ0=0, then we have Jσs1u<0. The proof is complete.Lemma 57.
Ifu∈F+∩Π+, then there exists η1>0 such that for σ∈0,η1, Θu has a positive value and Θu has exactly two critical points that correspond to the local minimum and local maximum. Moreover, there exists s2>0 such that s2∈Nσ+ and Jσs2u<0.Proof.
Letu∈WΔ,a+α,p. As in above, we define
(134)Ξus=spp∥u∥p−σsqq∫J0ωtutqΔt.
Then,(135)Ξ′us=sp−1∥u∥p−σsq−1∫J0ωtutqΔt.
LetΞ′us=0, we have
(136)s~=∥u∥pσ∫J0ωtutqΔt1/q−p,which is the maximum value point of Ξu. Moreover, one has
(137)Ξus~=s~pp∥u∥p−σs~qq∫J0ωtutqΔt=s~p1p∥u∥p−σs~q−pq∫J0ωtutqΔt=s~p1p∥u∥p−σ∥u∥p/σ∫J0ωtutqΔtq−p/q−pq∫J0ωtutqΔt=s~p1p∥u∥p−1q∥u∥p=s~p∥u∥p1p−1q=∥u∥pσ∫J0ωtutqΔt1/q−p∥u∥q−p/q−pp1p−1q=1p−1q∥u∥p+q−pσ∫J0ωtutqΔtp/q−p=1p−1q∥u∥qσ∫J0ωtutqΔtp/q−p,Ξ″us~=p−1s~p−2∥u∥p−σq−1s~q−2∫J0ωtutqΔt=s~p−2p−1∥u∥p−σq−1s~q−p∫J0ωtutqΔt=s~p−2p−1∥u∥p−σq−1∥u∥pσ∫J0ωtutqΔtq−p/q−p∫J0ωtutqΔt=s~p−2p−1∥u∥p−q−1∥u∥p=s~p−2∥u∥pp−q=p−q∥u∥pσ∫J0ωtutqΔtp−2/q−p∥u∥p=p−q∥u∥pq−2/q−pσ∫J0ωtutqΔtp−2/q−p<0.
In consideration of Proposition45, we deduce that
(138)Ξus~=1p−1q∥u∥qσ∫J0ωtutqΔtp/q−p≥q−pqp∥u∥qσ∥ω∥∞∥u∥∞qbp/q−p≥q−pqp∥u∥qbσ∥ω∥∞bα−1/pq/Γqαα−1p¯+1q/p∥u∥qp/q−p=q−pqpΓqαα−1p¯+1q/p¯σ∥ω∥∞b1+qα−1/pp/q−p≔δ,which is independent of u. We now prove that there exists η1>0 such that Θ′us~>0. Taking (16) and Proposition 45 into consideration, one obtains
(139)s~r∫J0Ft,utΔt≤s~r∫J0KutrΔt≤s~rK∥u∥∞rb≤∥u∥pσ∫J0gtutqΔtr/q−pKb1+α−1/prΓrαα−1p¯+1r/p¯∥u∥r=∥u∥pσ∫J0gtutqΔtr/q−pKb1+α−1/prΓrαα−1p¯+1r/p¯∥u∥r·q−p/rr/q−p=Kb1+α−1/prΓrαα−1p¯+1r/p¯∥u∥qσ∫J0gtutqΔtr/q−p=Kb1+α−1/prΓrαα−1p¯+1r/p¯pqq−pr/pMus~r/p.
Hence, we have(140)Θus~=Ξus~−s~r∫J0Ft,utΔt≥Ξus~−Kb1+α−1/prΓrαα−1p¯+1r/p¯pqq−pr/pΞus~r/p≥δ−δr/pKb1+α−1/prΓrαα−1p¯+1r/p¯pqq−pr/p=δr/p+p−r/p−δr/pKb1+α−1/prΓrαα−1p¯+1r/p¯pqq−pr/p=δr/pδp−r/p−Kb1+α−1/prΓrαα−1p¯+1r/p¯pqq−pr/p>0,for 0<σ<η1, where δ is the constant given in (138) and
(141)η1=q−pqpΓqαα−1p¯+1q/p∥ω∥∞b1+qα−1/p·Γrαα−1p¯+1r/p¯Kb1+rα−1/p·q−ppqr/pq−p/p−r.
The same arguments used in the proof of Lemma55 show that Θu has exactly two critical points which correspond to the local minimum and local maximum. Furthermore, there exists s2>0 such that s2u∈Nσ+ and Jσs2u<0. The proof of Lemma 57 is now completed.From now on, we defineσ0 by
(142)σ0=minη0,η1.Note that if0<σ<σ0, then all the above related lemmas are true.Lemma 58.
Letu be a local minimizer for Jσ on subsets Nσ+ or Nσ− of Nσ such that u∉Nσ0. Then, u is a critical of Jσ.Proof.
Sinceu is a minimizer for Jσ under the constraint
(143)Iσu≔J′σu,u.
Then, applying the theory of Lagrange multipliers, we get the existence ofη∈ℝ such that
(144)J′σu=ηI′σu.
Therefore, one has(145)J′σu,u=ηI′σu,u=ηΘ′′u1=0,but u∉Nσ0 and so Θ′′u1≠0. Hence, η=0, which gives the proof of Lemma 58. The proof is complete.In the following, we assume that1/2<α<1 and 1<r<p<q. Let σ0 be the constant given by (56). Then, the proof of Theorem 50 is based on the following two propositions.Proposition 59.
Suppose that assumptions of Theorem50 are satisfied. Then, for all 0<σ<σ0, Jσ achieves its minimum on Nσ+.Proof.
In view ofσ∈0,σ0 and Lemma 53, we have Jσ which is bounded below on Nσ and also on Nσ+. Therefore, there exists a minimizing sequence uk⊂Nσ+ such that
(146)limk⟶∞Jσuk=infu∈Nσ+Jσu.
AsJσ is coercive on Nσ, uk is a bounded sequence in WΔ,a+α,p up to a subsequence; there is uk⊂WΔ,a+α,p such that ukuσ weakly in WΔ,a+α,p.
Letu∈WΔ,a+α,p such that ∫J0Ft,utΔt>0. So, using Lemmas 56 and 57, there is s1>0 such that s1u∈Nσ+ and Jσu<0. Therefore, infu∈Nσ+Jσu<0.
Because ofuk⊂Nσ, we have
(147)Jσuk=1p+1q∥uk∥p−1+rq∫J0Ft,uktΔt,which yields that
(148)1+rq∫J0Ft,uktΔt=1p+1q∥uk∥p−Jσuk.
Lettingk go to infinity in the above equation, we obtain
(149)∫J0Ft,uσtΔt>0.
Now, we declare thatuk⟶uσ strongly in WΔ,a+α,p. Otherwise, we have
(150)∥uσ∥p<liminfk⟶∞∥uk∥p.
SinceΘ′uσs1=0, it follows from (150) that Θ′uks1>0 for sufficiently large k. Hence, we must have s1>1.
However,s1uσ∈Nσ+, and so,
(151)Jσs1uσ<Jσuσ≤limk⟶∞Jσuk=infu∈Nσ+Jσu,which gives a contradiction. Thus, uk⟶uσ strongly in WΔ,a+α,p; as a consequence, uσ∈Nσ=Nσ+∪Nσ0. In addition, it is easy to check by contradiction that uσ∈Nσ+. Therefore, from (149), uσ is a nontrivial solution of (85). The proof is complete.Proposition 60.
Suppose that assumptions of Theorem50 are satisfied. Then, for all 0<σ<σ0, Jσ achieves its minimum on Nσ−.Proof.
Letu∈Nσ−. Hence, by the result of Lemma 55, we obtain the existence of η1>0 such that Jσu≥η1. Therefore, there is a minimizing sequence vk⊂Nσ− such that
(152)limk⟶∞Jσvk=infu∈Nσ−Jσu>0.
Furthermore, in view of Lemma53, we know that Jσ is coercive, so vk is a bounded sequence in WΔ,a+α,p up to a subsequence, there is vk⊂WΔ,a+α,p such that vk⇀vσ weakly in WΔ,a+α,p.
Because ofvk⊂Nσ, then we have
(153)σ1r−1q∫J0ωtvktqΔt=Jσvk−1p−1r∥vk∥p.
Lettingk go to infinity in (153), it follows from (152) that
(154)∫J0ωtvktqΔt>0.
Therefore,vσ∈Π+, and so, Θvσ has a global maximum at some point s~. Consequently, s~vσ∈Nσ−.
On the other hand,vk∈Nσ− implies that 1 is a global maximum point for Θvk, i.e.,
(155)Jσs~uk=Θvks~≤Θvk1=Jσvk.
Now, as in the proof of Proposition59, we assert that vk⟶vσ in WΔ,a+α,p. Assuming it is not true, then
(156)∥vσ∥p<liminfk⟶∞∥vk∥p.It follows from 4.23 that
(157)Jσs~vσ=s~pp∥vσ∥p−s~r∫J0Ft,vσtΔt−σs~qq∫J0ωtvσtqΔt<liminfk⟶∞s~pp∥vk∥p−s~r∫J0Ft,vktΔt−σs~qq∫J0ωtvktqΔt≤limk⟶∞Jσs~vk=Θvks~≤infu∈Nσ−Jσvk=Θvk1=infu∈Nσ−Jσu,which gives a contradiction. Therefore, vk⟶vσ, and so, vσ∈Nσ−∪Nσ0.
Using Lemma52, we have Nσ0=∅, so vσ is a minimizer for Jσ on Nσ−.
On the other hand, by (22),vσ is a nontrivial solution of (1). Finally, since Nσ−∩Nσ+=∅, uσ and vσ are distinct. That is, the result of Theorem 50 holds true. The proof is complete.


## 5. Conclusions

In this paper, we introduced a class of fractional Sobolev spaces via the fractional derivative of Riemann-Liouville on time scales and obtain some of their basic properties. As an application, we use critical point theory to study the solvability of a class of fractional boundary value problems on time scales. The results and methods in this paper can also be used to study the solvability of other boundary value problems on time scales. At present, the concept of fractional derivatives in different meanings is constantly being proposed. Therefore, studying the theory and application of fractional Sobolev spaces on time scales in other meanings is our future direction.



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