Left fractional Sobolev space via Riemann$-$Liouville derivatives on time scales and its application to a fractional boundary value problem on time scales

We first prove the equivalence of two definitions of Riemann-Liouville fractional integral on time scales, then by the concept of fractional derivative of Riemann-Liouville on time scales, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Next, 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 the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary condition is studied, and three results of the existence of weak solutions for this problem is obtained.


Introduction
To unify the discrete analysis and continuous analysis, and allow a simultaneous treatment of differential and difference equations, Stefan Hilger [1] proposed the time scale theory and established its related basic theory [2,3]. So far, the study of dynamic equations on time scales has attracted worldwide attention.
It is well known that Sobolev space theory is established to study modern differential equation theory and many problems in the field of mathematical analysis. It has become an integral part of analytical mathematics. In order to study the solvability of boundary value problems of dynamic equations on time scales, Sobolev space theory on time scales is studied in [4][5][6][7].
On the one hand, 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) non-local phenomena [8][9][10][11][12][13]. 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.
On the other hand, recently, based on the concept of fractional derivative of Riemann−Liouville on time scales [14], the authors of [7] established the fractional Sobolev space on time scales. However, the authors in the recent work [15] pointed out that the definition of fractional integral on time scales proposed in [14] is not the natural one on time scales. And they developed a new notion of Riemann-Liouville fractional integral on time scales, which can well unify the discrete fractional calculus [16,17] and its continuous counterpart [18].
Motivated by the above discussion, in order to fix this defect of the fractional Sobolev space on time scales established in [7], in this paper, we want to contribute with the development of this new area on theories of fractional differential equations on time scales. More precisely, we first show that the concept of Riemann-Liouville fractional integral on time scales from [7] coincides with the ones from [19], which is significant for us to prove the semigroup properties of Riemann-Liouville fractional integral on time scales. Next, the left fractional Sobolev space in the sense of weak Riemann−Liouville derivatives on time scales was constructed via Riemann−Liouville derivatives on time scales. Then, as an application of our new theory, we we study the solvability of a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary condition by using variational methods and the critical point theory.

Preliminaries
In this section, we briefly collect some basic known notations, definitions, and results that will be used later.
A time scale T 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: [2] For t ∈ T we define the forward jump operator σ : T → T by σ(t) := inf{s ∈ T : s > t}, while the backward jump operator ρ : T → T is defined by ρ(t) := sup{s ∈ T : s < t}.
(2) If σ(t) > t, we say that t is right−scattered, while if ρ(t) < t, we say that t is left−scattered. Points that are right−scattered and left−scattered at the same time are called isolated.
(3) If t < sup T and σ(t) = t, we say that t is right−dense, while if t > inf T and ρ(t) = t, we say that t is left−dense. Points that are right−dense and left−dense at the same time are called dense.
(5) The derivative makes use of the set T k , which is derived from the time scale T as follows: If T has a left−scattered maximum M, then T k := T\{M}; otherwise, T k := T.
Definition 2.2. [2] Assume that f : T → R is a function and let t ∈ T k . Then we define f ∆ (t) to be the number (provided it exists) with the property that given any The set of functions f : T → R that are differentiable and whose derivative is rd−continuous is denoted by Theorem 2.3. [20] The convolution is commutative and associative, that is, for f, g, h ∈ F , Proposition 2.1. [21] f is an increasing continuous function on J. If F is the extension of f to the real interval J R given by [15] and Definition 2.1 in [22], we can present the right Riemann−Liouville fractional integral and derivative on time scales as follows:

Motivated by Definition 4 in
Definition 2.5. [15] (Fractional integral on time scales) Suppose h is an integrable function on J. Let 0 < α ≤ 1. Then the left fractional integral of order α of h is defined by The right fractional integral of order α of h is defined by where Γ is the gamma function. (2. 2) The right Riemann−Liouville fractional derivative of order α of h is defined by (21) in [15] and Theorem 2.1 in [23], we can present the right Caputo fractional derivative on time scales as follows:

Motivated by Definition 4 and Equation
The right Caputo fractional derivative of order α of h is defined by where g(t) = e ⊖z (t, 0). Theorem 2.4. [24] (Inversion formula of the Laplace transform) Suppose that F(z) is analytic in the region Re µ (z) > Re µ (c) and F(z) → 0 uniformly as |z| → ∞ in this region. Suppose F(z) has finitely many regressive poles of finite order {z 1 , z 2 , . . . , z n } and F R (z) is the transform of the function Res z=z i e z (t, 0)F(z), has transform F(z) for all z with Re(z) > c.
(ii) The equality holds for every t ∈ T.
[27] A function q : J R → R m is absolutely continuous iff there exist a constant c ∈ R m and a function ϕ ∈ L 1 such that In this case, we have q(a) = c and q ′ (t) = ϕ(t), t ∈ J R a.e..
Theorem 2.10. [5] (Integral representation) Let α ∈ (0, 1) and q ∈ L 1 . Then, q has a left−sided Riemann−Liouville derivative D α a + q of order α iff there exist a constant c ∈ R m and a function ϕ ∈ L 1 such that In this case, we have I 1−α a + q(a) = c and (D α a + q)(t) = ϕ(t), t ∈ J R a.e.. Definition 2.10. [4] Let p ∈R be such that p ≥ 1 and u : J →R. Say that u belongs to W 1,p ∆ (J) iff u ∈ L p ∆ (J 0 ) and there exists g : J k →R such that g ∈ L p ∆ (J 0 ) and where C 1 rd (J k ) is the set of all continuous functions on J such that they are ∆−differential on J k and their ∆−derivatives are rd−continuous on J k . Theorem 2.11. [4] Let p ∈R be such that p ≥ 1. Then, the set L p ∆ (J 0 ) is a Banach space together with the norm defined for every f ∈ L p ∆ (J 0 ) as Moreover, L 2 ∆ (J 0 ) is a Hilbert space together with the inner product given for every Theorem 2.12. [18] Fractional integration operators are bounded in L p (J R ), i.e., the following estimate holds.
This expression is called Hölder's inequality and Cauchy−Schwarz's inequality whenever p = 2.  (i) X is bounded; (ii) For any given Then, X is relatively compact.

Some fundamental properties of Left Riemann-Liouville fractional operators on time scales
Inspired by [29], we can obtain the consistency of Definition 2.5 and Definition 2.9 by using the above theory of the Laplace transform on time scales and the inverse Laplace transform on time scales.
Proof. Using the Laplace transform on time scale for (2.1), in view of Definition 2.5, Theorem 2.5, Theorem 2.6 and Definition 2.8, we have Taking the inverse Laplace transform on time scale for (2.1), with an eye to Definition 2.9, one arrives at The proof is complete.
Proof. Let h : T → R. In view of (2.1) and (2.2), we obtain The proof is complete. [19] and Theorem 3.1, one gets that

Proposition 3.2. For any function h that is integrable on J, the Riemann−Liouville
In a similarly way, one arrives at Consequently, we obtain that The proof is complete.

Proposition 3.3. For any function h that is integrable on J one has
The proof is complete.
The proof is complete.
where T a I α t (J) denotes the space of functions that can be represented by the left Riemann−Liouville ∆-integral of order α of a C(J)−function. ).

In view of Proposition 3.2, one gets
Inversely, suppose that f ∈ C(J) satisfies (3.2) and (3.3). Then, by applying Taylor's formula to Note that ϕ ∈ C(J) by (3.2). Now by Proposition 3.2, one sees that and hence From the uniqueness of solution to Abel's integral equation ( [30]), this implies that f − T a I α t ϕ ≡ 0. Hence, f = T a I α t ϕ and f ∈ T a I α t (J). The proof is complete.
Proof. Combining with Theorem 3.2 and Proposition 3.3, we can see that The proof is complete.
where p 1 and q 1 in the case when then the following integration by parts formulas hold.
Proof. (a) It follows from Definition 2.5 and Fubini's theorem on time scales that The proof is complete.
(b) It follows from Definition 2.6 and Fubini's theorem on time scales that The proof is complete.
(c) It follows from Definition 2.7, Fubini's theorem on time scales and Theorem 2.
The second relation is obtained in a similar way. The proof is complete.

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, let 0 < a < b. Inspired by Theorems 2.7−2.10, we give the following definition.
with c ∈ R N and ϕ ∈ L 1 ∆ . Then, we have the following result.
; that is, f has the representation (4.1). In such a case, f is (identified to) an absolutely continuous function. From the integral representation of Theorems 2.7 and 2.9, there exist a constant c ∈ R N and a function ϕ ∈ L 1 ∆ such that

By Proposition 3.2 and applying
3) The result follows from the ∆−differentiability of (4.3).
Conversely, let us assume that (4.1) holds true. From Proposition 3.2 and applying T a I 1−α t to (4.1) we obtain that is, f has the representation (4.1) with ϕ ∈ L p ∆ . Definition 4.2. Let 0 < α ≤ 1 and let 1 ≤ p < ∞. By left Sobolev space of order α we will mean the set W α,p ∆,a + = W α,p ∆,a + (J, R N ) given by Remark 4.2. A function g given in Definition 5.2 will be called the weak left fractional derivative of order 0 < α ≤ 1 of u; let us denote it by T u α a + . The uniqueness of this weak derivative follows from [4].
We have the following characterization of W α,p ∆,a + .
On the other hand, if u ∈ W α,p ∆,a + , then u ∈ L p ∆ and there exists a function g ∈ L p ∆ such that for any ϕ ∈ C ∞ c,rd . To show that u ∈ AC α,p ∆,a + ∩ L p ∆ it suffices to check (Theorem 4.1 and definition of AC α,p ∆,a + ) that u possesses the left Riemann−Liouville derivative of order α, which belongs to L p ∆ , that is, T a I 1−α t u is absolutely continuous on J and its delta derivative of α order (existing ∆ − a.e. on J) belongs to L p ∆ .
In view of (4.4) and (4.5), we get u is absolutely continuous and its delta derivative is equal ∆ − a.e. on [a, b] T to g ∈ L p ∆ . The proof is complete.
From the proof of Theorem 4.2 and the uniqueness of the weak fractional derivative the following theorem follows. Theorem 4.3. If 0 < α ≤ 1 and 1 ≤ p < ∞, then the weak left fractional derivative T u α a + of a function u ∈ W α,p ∆,a + coincides with its left Riemann−Liouville fractional derivative T a D α t u ∆ − a.e. on J.
By using the definition of W α,p ∆,a + with 0 < α ≤ 1 and Theorem 4.3, one can easily prove the following result.
In such a case, there exists the left Riemann−Liouville derivative T a D α t u of u and g = T a D α t u. Remark 4.4. Function g will be called the weak left fractional derivative of u ∈ W α,p ∆,a + of order α. Its uniqueness follows from [4]. From the above theorem it follows that it coincides with an appropriate Riemann−Liouville derivative.
Let us fix 0 < α ≤ 1 and consider in the space W α,p ∆,a + a norm · W α,p ∆,a + given by (Here · p L ∆ denotes the delta norm in L p ∆ (Theorem 2.11)).
That is to say, the fractional integration operator is bounded in L p ∆ .
For any fixed t ∈ [a, b] T , consider the functional H ξ * f : According to (4.8), it is obvious that . Therefore, by (4.8) and (4.9) and the Riesz representation theorem, and . Hence, we have by (4.10) and Definition 2.5 according to (4.11). Combining with (4.7) and (4.12), we obtain inequality (4.6). The proof is complete.
On the other hand, we will prove that there exists a constant M α,1 such that  1, 2, . . . , N) such that Hence, for a fixed t 0 ∈ J 0 , if ( T a I 1−α t u) i (t 0 ) 0 for all i = 1, 2, . . . , N, then we can take constants θ i such that Therefore, we have From the absolute continuity (Theorem 2.8) of ( T a I 1−α t u) i it follows that for any t ∈ J. Consequently, combining with Proposition 3.1 and Lemma 4.1, we see that In particular, So, where |θ| = max i∈{1,2,...,N} and, consequently, = 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 M α,1 such that (4.13) holds.
(2) When (1 − α)p ≥ 1, then (Remark 4.3) W α,p ∆,a + = AC α,p ∆,a + ∩ L p ∆ is the set of all functions belong to AC α,p ∆,a + that satisfy the condition ( T a I 1−α t u)(a) = 0. Hence, in the same way as in the case of (1 − α)p < 1 (putting c = 0), we obtain the inequality , with some M α,1 > 0 is obvious (it is sufficient to put M α,1 = 1 and use the fact that ( T a I 1−α t u)(a) = 0). The proof is complete. Now, we are in a position to prove some basic properties of the space W α,p ∆,a + . Theorem 4.6. The space W α,p ∆,a + is complete with respect to each of the norms · W α,p ∆,a + and · a,W α,p ∆,a + for any 0 < α ≤ 1, 1 ≤ p < ∞.
Proof. In view of Theorem 4.5, we only need to show that W α,p ∆,a + with the norm · a,W α,p ∆,a + is complete. Let {u k } ⊂ W α,p ∆,a + be a Cauchy sequence with respect to this norm. So, the sequences { T a I 1−α t u k (a)} and { T a D α t u k } are Cauchy sequences in R N and L p ∆ , respectively. Let c ∈ R N and ϕ ∈ L p ∆ be the limits of the above two sequences in R N and L p ∆ , respectively. Then the function belongs to W α,p ∆,a + and it is the limit of {u k } in W α,p ∆,a + with respect to · a,W α,p ∆,a + . 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), (c) in [32].
Proof. Let us consider W α,p ∆,a + with the norm · W α,p ∆,a + and define a mapping It is obvious that which means that the operator λ : u → u, T a D α t u is a isometric isomorphic mapping and the space W α,p ∆,a + is isometric isomorphic to the space Ω = u, T a D α t u : ∀u ∈ W α,p ∆,a + , which is a closed subset of L p ∆ × L p ∆ as W α,p ∆,a + is closed. Since L p ∆ is reflexive, the Cartesian product space L p ∆ × L p ∆ is also a reflexive space with respect to the norm v L p ∆,a + is reflexive with respect to the norm · W α,p ∆,a + . The proof is complete.
Proof. Let us consider W α,p ∆,a + with the norm · W α,p ∆,a + and the mapping λ defined in the proof of Theorem 4.7. Obviously, λ(W α,p ∆,a + ) is separable as a subset of separable space L p ∆ × L p ∆ . Since λ is the isometry, W α,p ∆,a + is also separable with respect to the norm · W α,p ∆,a + . The proof is complete. (4.14) if α > 1 p and 1 p + 1 q = 1, then Proof. In view of Remark 4.3 and Theorem 3.3, in order to prove inequalities (4.14) and (4.15), we only need to prove that for α > 1 p and 1 p + 1 q = 1.
Note that T a D α t u ∈ L p ∆ (J, R N ), the inequality (4.16) follows from Lemma 4.1 directly. We are now in a position to prove (4.17). For α > 1 p , choose q such that 1 p + 1 q = 1. For all u ∈ W α,p ∆,a + , since (t − s) (α−1)q is an increasing monotone function, by using Proposition 2.1, we find that t a (t − σ(s)) (α−1)q ∆s ≤ t a (t − s) (α−1)q ds. Taking into account of Proposition 2.2, we have The proof is complete.
Remark 4.5. (i) According to (4.14), we can consider W α,p ∆,a + with respect to the norm in the following analysis.
(ii) It follows from (4.14) and (4.15) that W α,p ∆,a + is continuously immersed into C(J, R N ) with the natural norm · ∞ . Proposition 4.2. Let 0 < α ≤ 1 and 1 < p < ∞. Assume that α > 1 p and the sequence {u k } ⊂ W α,p ∆,a + converges weakly to u in W α,p ∆,a + . Then, u k → u in C(J, R N ), i.e., u − u k ∞ = 0, as k → ∞. Proof. If α > 1 p , then by (4.15) and (5.5), the injection of W α,p ∆,a + into C(J, R N ), with its natural norm · ∞ , is continuous, i.e., u k → u in W α,p ∆,a + , then u k → u in C(J, R N ). Since u k ⇀ u in W α,p ∆,a + , it follows that u k ⇀ u in C(J, R N ). In fact, for any h ∈ C(J, , and thus h(u k ) → h(u), i.e., u k ⇀ u in C(J, R N ). By the Banach−Steinhaus theorem, {u k } is bounded in W α,p ∆,a + and, hence, in C(J, R N ). Now, we prove that the sequence {u k } is equi−continuous. Let 1 p + 1 q = 1 and t 1 , t 2 ∈ J with t 1 ≤ t 2 ,for all f ∈ L p ∆ (J, R N ), by using Proposition 2.2, Proposition 2.1, Theorem 2.14, and noting α > 1 p , we have Therefore, the sequence {u k } is equi−continuous since, for t 1 , t 2 ∈ J, t 1 ≤ t 2 , by applying (4.19) and (5.5), we have where 1 p + 1 q = 1 and C ∈ R + is a constant. By the Ascoli−Arzela theorem on time scales (Lemma 2.3), {u k } is relatively compact in C(J, R N ). By the uniqueness of the weak limit in C(J, R N ), every uniformly convergent subsequence of {u k } converges uniformly on J to u. The proof is complete.
Remark 4.6. It follows from Proposition 4.2 that W α,p ∆,a + is compactly immersed into C(J, R N ) with the natural norm · ∞ .
Then the functional χ defined by is continuously differentiable on W α,p ∆,a + , and ∀ u, v ∈ W α,p ∆,a + , one has Proof. It suffices to prove that χ has, at every point u, a directional derivative χ ′ (u) ∈ (W α,p ∆,a + ) * given by (4.20) and that the mapping ∆,a + ) * is continuous. The rest proof is similar to the proof of Theorem 1.4 in [33]. We will omit it here. The proof is complete. For the sake of the infinitely many critical points of ϕ, one introduces the genus properties as follows. First, we let (i) If Ξ n 0 and c n ∈ R, then c n is a critical value of ϕ; (ii) If there exists l ∈ N such that c n = c n+1 = · · · = c n+l = c ∈ R and c ϕ(0), then γ(K c ) ≥ l + 1.
Remark 5.1. [36] In view of Remark 7.3 in [36], one sees that if K c ∈ Ξ and γ(K c ) contains infinitely many distinct points. In other words, ϕ has infinitely many distinct critical points in E.
There have been many results using critical point theory to study boundary value problems of fractional differential equations ( [37][38][39][40][41][42][43]) and dynamic equations on time scales ( [44][45][46][47][48]), but the results of using critical point theory to study boundary value problems of fractional dynamic equations on time scales are still rare [6]. This section will explain that critical point theory is an effective way to deal with the existence of solutions of (5.1) on time scales.
In this section, we let N = 1. For purpose of the presence and proof of our main results, let's first define the functional ϕ : W α,p ∆,a + → R by It is easy to testify from (4.14), (5.5) and g ∈ C(J × R, R) that the functional ϕ is well defined on W α,p ∆,a + and ϕ ∈ C(W α,p ∆,a + , R). Moreover, for ∀u, v ∈ W α,p ∆,a + , one obtains Now, it is time for us to present and prove our main results as follows: Theorem 5.1. Let α ∈ 1 p , 1 , suppose that G satisfies the following conditions: (G 1 ) G(t, x) is ∆− measurable and continuously differentiable in x for t ∈ J and there exist a ∈ C(R + , for all x ∈ R and t ∈ J. (G 2 ) There are two constants µ > p 2 , M > 0 such that Proof. It is clear that ϕ(0) = 0, ϕ ∈ C 1 (W α,p ∆,a + , R), where W α,p ∆,a + is a real Banach space from Theorem 4.6, So now, we are in a position by using Mountain pass theorem (Lemma 5.1) to prove that step 1. ϕ satisfies the (PS ) condition in W α,p ∆,a + . The argument is as follows: Let {u k } ⊂ W α,p ∆,a + be a sequence such that where K > 0 is a constant. We first prove that {u k } is bounded in W α,p ∆,a + . From the continuity of µG(t, x) − xg(t, x), we obtain that there is a constant c > 0 such that Combining with (G 2 ), we obtain that Hence, taking account of (5.5), (5.2), (5.4), (5.6), (5.7) and Lemma 2.2, we have Then, combining with µ > p 2 and proof by contradiction, we know that {u k } is bounded in W α,p ∆,a + . Because W α,p ∆,a + is a reflexive Banach space (Theorem 4.7 and Theorem 4.6), going if necessary to a subsequence, we can assume u k ⇀ u in W α,p ∆,a + . As a result, in view of ϕ ′ (u k ) → 0 as k → ∞ and the definition of weak convergence, one sees (5.10) Furthermore, it follows from (4.15), (5.5) and Remark 4.6 that {u k } is bounded in C(J, R) and u k − u ∞ → 0, as k → ∞. Therefore, there is a constant c 1 > 0 such that |∇G(t, u k (t)) − ∇G(t, u(t))| ≤ c 1 , ∀t ∈ J, (5.11) which yields J 0 (∇G(t, u k (t)) − ∇G(t, u(t)))(u k (t) − u(t))∆t ≤ c 1 b u k − u ∞ step 2. ϕ satisfies the (A 1 ) condition in Lemma 5.1, which can be explained by the following reasons: Therefore, when s ≥ M |x| , we have ≤ 1 ̺p 2 (β + ̺ ςu 0 p ) p − λ 0 ης r 2 I |u 0 (t)| r 2 ∆t − β p ̺p 2 , 0 < s ≤ δ. (5.29) Because of 1 < r 2 < p 2 , (5.29) implies ϕ(ςu 0 ) < 0 for s > 0 small enough. Therefore, u * 0.
All in all, u * ∈ W α,p ∆,a + is a nontrivial critical point of ϕ, and consequently, u * is a nontrivial solution of KFBVP T (5.1). Hence, we complete the proof of Theorem 5.2.
Proof. Lemma 5.3 is a powerful tool for us to clarify our conclusion.

Conclusions
In this paper, a class of fractional Sobolev spaces on time scales is introduced through a new definition of the fractional derivative of Riemann −Liouville on time scales, and some basic properties of them are obtained. As an application, we study a class of Kirchhoff type fractional p-Laplace boundary value problems on time scales. The existence and multiplicity of nontrivial weak solutions are obtained by using mountain path theorem and genus properties. The results and methods of this paper can also be used to study the solvability of other boundary value problems on time scales. Nowadays, the concepts of fractional derivative on time scales in different senses are constantly put forward. Therefore, it is our future direction to study the theory and application of fractional Sobolev spaces on time scales introduced by fractional derivatives in other senses on time scales.