Right fractional Sobolev space via Riemann$-$Liouville derivatives on time scales and an application to fractional boundary value problem on time scales

Using the concept of fractional derivatives of Riemann$-$Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some imbeddings. Finally, as an application, we propose a recent method to study the existence of weak solutions of fractional boundary value problems on time scales by using variational method and critical point theory, and by constructing an appropriate variational setting, we obtain two existence results of the problem.


Introduction
In the past two decades, fractional calculus and fractional (order) differential equations have aroused widespread interest and 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 phenomenon ( [1][2][3][4][5]). It is the rise of these applications that revitalize the (1) In Definition 2.1, we put inf ∅ = sup T (i.e., σ(t) = t if T has a maximum t) and sup ∅ = inf T (i.e., ρ(t) = t if T has a minimum t), where ∅ denotes the empty set.
(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. ( [19]) 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 ε > 0, there is a neighborhood U of t (i.e, U = (t − δ, t + δ) ∩ T for some δ > 0) such that for all s ∈ U. We call f ∆ (t) the delta (or Hilger) derivative of f at t. Moreover, we say that f is delta (or Hilger) differentiable (or in short: differentiable) on T k provided f ∆ (t) exists for all t ∈ T k . The function f ∆ : T k → R is then called the (delta) derivative of f on T k . Definition 2.3. ( [18]) A function f : 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 C rd = C rd (T) = C rd (T, R).
The set of functions f : T → R that are differentiable and whose derivative is rd−continuous is denoted by C 1 rd = C 1 rd (T) = C 1 rd (T, R). Definition 2.4. ( [20]) Let J denote a closed bounded interval in T. A function F : J → R is called a delta antiderivative of function f : J 0 → R provided F is continuous on J, delta differentiable at J 0 , and F ∆ (t) = f (t) for all t ∈ J 0 . Then, we define the ∆−integral of f from a to b by b a f (t)∆t := F (b) − F (a). Theorem 2.1. ( [19]) If a, b ∈ T and f, g ∈ C rd (T), then  Motivated by Definition 4 in [14] and Definition 2.1 in [22], we can present the right Riemann−Liouville fractional integral and derivative on time scales as follows: The right fractional integral of order α of h is defined by where Γ is the gamma function. The right Riemann−Liouville fractional derivative of order α of h is defined by (2.2) Motivated by Definition 4 and Equation (21) in [14] and Theorem 2.1 in [4], we can present the right Caputo fractional derivative on time scales as follows: The right Caputo fractional derivative of order α of h is defined by where g(t) = e ⊖z (t, 0). [23]) (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 f (t) on R that corresponds to the transform Res z=z i e z (t, 0)F (z) has transform F (z) for all z with Re(z) > c.
Motivated by Definition 3.1 in [24], we can present the right Riemann−Liouville fractional integral on time scales as follows: Theorem 2.4. ( [25]) A function f : T → R is absolutely continuous on T if and only if the following conditions are satisfied: (i) f is ∆−differentiable ∆ − a.e. on J 0 and f ∆ ∈ L 1 (T).
(ii) The equality holds for every t ∈ T.
Theorem 2.5. ( [26]) A function q : J R → R m is absolutely continuous if and only if 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 2.6. ( [26]) (integral representation) Let α ∈ (0, 1] and q ∈ L 1 . Then, q has a right−sided Riemann−Liouville derivative D α b − q of order α if and only if there exist a constant d ∈ R m and a function ψ ∈ L 1 such that In this case, we have . Then, a necessary and sufficient condition for the validity of the equality is the existence of a constant c ∈ R such that f ≡ c ∆ − a.e. on J 0 . Definition 2.10. ( [15]) Let p ∈R be such that p ≥ 1 and u : J →R. Say that u belongs to W 1,p ∆ (J) if and only if u ∈ L p ∆ (J 0 ) and there exists g : J k →R such that g ∈ L p ∆ (J 0 ) and where and 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 .
Moreover, L 2 ∆ (J 0 ) is a Hilbert space together with the inner product given for every Theorem 2.8. ( [22]) 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.

Some fundamental properties of Right Riemann-Liouville fractional operators on time scales
Motivated by Theorem 2 in [14], we can present and prove the Cauchy result on time scales as follows: Proof. The proof is similar to the proof of Theorem 2 in [14], we will omit it here.
Proof. The proof is similar to the proof of Theorem 1.3 in [29], we will omit it here.
Inspired by [30], 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.
Taking the inverse Laplace transform on T for (2.1), with an eye to Definition 2.9, one arrives at The proof is complete.
Combining with [20], [24] and Theorem 3.3, we can prove that the analogues of Proposition 15, Proposition 16, Proposition 17, Corollary 18, Theorem 20 and Theorem 21 of the right Riemann−Liouville fractional operators on time scales remain intact under the new Definition 2.5.
Proof. Let h : T → R. In view of (2.1) and (2.2), we obtain The proof is complete.
Proof. Inspired by the proof of Proposition 3.4 in [24], with an view of Definition 2.9, we have Combining with (3.2) and Theorem 3.3, one gets that In a similarly way, one arrives at Consequently, we obtain that The proof is complete.
Proof. Taking account of Propositions 3.1 and 3.2, one can get The proof is complete.
where Id denotes the identity operator.
Proof. In view of Proposition 3.3, we have The proof is complete. ).

In view of Proposition 3.2, one gets
As a result, Inversely, suppose that f ∈ C(J) satisfies (3.3) and (3.4). Then, by applying Taylor's formula to . Now by Proposition 3.2, one sees that and hence From the uniqueness of solution to Abel's integral equation ( [34]), this implies that The proof is complete.
Theorem 3.5. Let α > 0 and f ∈ C(J) satisfy the condition in Theorem 3.4. Then, Proof. Combining with Theorem 3.4 and Proposition 3.3, we can see that The proof is complete.
Motivated by the proof of Equation (2.20) in [22], we can present and prove the following Theorem: Theorem 3.6. Let α > 0, p, q ≥ 1, and 1 p + 1 q ≤ 1 + α, where p = 1 and q = 1 in the case 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, inspired by the above discussion, we present and prove the following lemmas, propositions and theorems, which are of utmost significance for our main results. In the following, let 0 < a < b. Suppose a, b ∈ T.
Motivated by Theorems 2.3-2.6, we can propose the following definition.
; 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 Theorem 2.3, there exist a constant vector d ∈ R N and a function ψ ∈ L 1 ∆ such that

By Proposition 3.2 and applying
The result follows from the ∆−differentiability of (4.3).
Conversely, now, let us assume that (4.1) holds true. From Proposition 3.2 and applying and then, T f has an absolutely continuous representation and f has a right Riemann −Liouville derivative T t D α b f . This completes the proof.
(ii) It is easy to see that Theorem 4.1 implies the following one (for any 1 ≤ p < ∞): f has the right Riemann−Liouville Definition 4.2. Let 0 < α ≤ 1 and let 1 ≤ p < ∞. By right Sobolev space of order α we will mean the set W α,p Remark 4.2. A function g given above will be called the weak right fractional derivative of order 0 < α ≤ 1 of u; let us denote it by T u α b − . The uniqueness of this weak derivative follows from ( [15]).
We have the following characterization of W α,p ∆,b − .
For another thing, now, let us assume that u ∈ W α,p for any ϕ ∈ C ∞ c,rd . To show that u ∈ AC α,p ∆,b − ∩ L p ∆ it suffices to check (Theorem 4.1 and definition of AC α,p ∆,b − ) that u possesses the right Riemann−Liouville derivative of order α, belonging to L p ∆ , that is, u is absolutely continuous on [a, b] T 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 b a u is absolutely continuous and its delta derivative is equal to ∆ − a.e. on J to g ∈ L P ∆ .
From the proof of Theorem 4.2 and the uniqueness of the weak fractional derivative the following theorem follows.
By using the definition of W α,p ∆,b − with 0 < α ≤ 1 and Theorem 4.3, one can easily prove the following result.
In such a case, there exists the right Riemann−Liouville derivative T t D α b u of u and g = T t D α b u.
Remark 4.4. Function g will be called the weak right fractional derivative of order α of u ∈ W α,p ∆,b − . Its uniqueness follows from [15]. From the above theorem it follows that it coincides with the appropriate Riemann−Liouville derivative.
Let us fix 0 < α ≤ 1 and consider in the space (Here · p L ∆ denotes the delta norm in L p ∆ (Theorem 2.7)).
Proof. The conclusion follows from Theorem 2.8, Proposition 2.2 and Proposition 2.1. The proof is complete. Proof.
(1) Assume that (1 − α)p < 1. On the one hand, for u ∈ W α,p ∆,b − given by with d ∈ R N and ψ ∈ L p ∆ . Since (b − t) (α−1)p is an increasing monotone function, by using Proposition 2.1, we can write that Consequently, On the other hand, now, we will prove that there exists a constant M α,1 such that Indeed, let u ∈ W α,p ∆,b − and consider coordinate functions such that Hence, if, for all i = 1, 2, . . . , N, ( T t I 1−α b u) i (t 0 ) = 0, then we can take constants θ i such that for fixed t 0 ∈ J 0 . Therefore, we have From the absolute continuity (Theorem 2.4) of ( T t I 1−α b u) i it follows that for any t ∈ J. Consequently, combining with Proposition 3.1 and Lemma 4.1, we see that In particular, where |θ| = max i∈{1,2,...,N } |θ i | and M α,0 = |θ|(b−a) −α Γ(2−α) + 1. Thus, and, consequently, If some of or even all of ( T t I 1−α b u) i (t 0 ) = 0, from the above proof process, we can see that our conclusion is still valid. ( Consequently, in the same way as in the case of (1 − α)p < 1 (putting d = 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)(b) = 0). The proof is complete.
We are now in a position to state and prove some basic properties of the introduced space.
Theorem 4.6. The space W α,p ∆,b − is complete with respect to each of the norms · W α,p ∆,b − and · b,W α,p ∆,b − for any 0 < α ≤ 1 and 1 ≤ p < ∞. Proof. In view of Theorem 4.5, we only need to show that W α,p Cauchy sequence with respect to this norm. So, the sequences are Cauchy sequences in R N and L p ∆ , respectively. Let d ∈ R N and ψ ∈ L p ∆ be the limits of the above sequences in R N and L p ∆ , respectively. Then the function belongs to W α,p ∆,b − and is the limit of {u k } in W α,p ∆,b − with respect to · b,W α,p ∆,b − . (To assert that u ∈ L p ∆ it is sufficient to consider the cases (1 − α)p < 1 and (1 − α)p ≥ 1. In the second case In the proofs of the next two theorems we use the method presented in [32] P 121 Proposition 8.1 (b), (c).
Proof. Let us consider W α,p ∆,b − with the norm · W α,p ∆,b − and define a mapping λ : It is obvious that which means that the operator λ : u → u, T t D α b u is an isometric isomorphic mapping and Theorem 4.8. The space W α,p ∆,b − is separable with respect to the norm · W α,p ∆,b − for any 0 < α ≤ 1 and 1 ≤ p < ∞.
Proof. Let us consider W α,p ∆,b − with the norm · W α,p ∆,b − and the mapping λ defined in the proof of Theorem 4.7. Obviously, λ(W α,p ∆,b − ) is separable as a subset of separable space L p ∆ × L p ∆ . Since λ is the isometry, W α,p ∆,b − is also separable with respect to the norm · W α,p ∆,b − .
If α > 1 p and 1 p + 1 q = 1, then Proof. In view of Remark 4.3 and Theorem 3.5, in order to prove inequalities (4.6) and (4.7), we only need to prove that for α > 1 p and 1 , the inequality (4.8) follows from Lemma 4.1 directly.
We are now in a position to prove (4.9). For α > 1 p , choose q such that 1 p + 1 q = 1. For all u ∈ W α,p ∆,b − , since (s − σ(t)) (α−1)q is an increasing monotone function, by using Proposition 2.1, we find that This completes the proof.
Remark 4.5. (i) According to (4.6), we can consider W α,p ∆,b − with respect to the norm in the following analysis.
(ii) It follows from (4.6) and (4.7) that W α,p ∆,b − 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 } converges weakly to u in W α,p ∆,b − . Then, u k → u in C(J, R N ), i.e., u − u k ∞ = 0, as k → ∞. Proof. If α > 1 p , then by (4.7) and (4.10), the injection of W α,p ∆,b − into C(J, R N ), with its natural norm · ∞ , is continuous, i.e., By the Banach−Steinhaus theorem, {u k } is bounded in W α,p ∆,b − and, hence, in C(J, R N ). We are now in a position to prove that the sequence {u k } is equi−continuous. Let 1 Therefore, the sequence {u k } is equi−continuous since, for t 1 , t 2 ∈ [a, b] T , t 1 ≤ t 2 , by applying (4.11) and in view of (4.10), we have where 1 p + 1 q = 1 and C ∈ R + is a constant. By the Ascoli−Arzela theorem on time scales (Lemma 2.2), {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. Remark 4.6. It follows from Proposition 4.2 that W α,p ∆,b − is compactly immersed into C(J, R N ) with the natural norm · ∞ . Theorem 4.9. Let 1 < p < ∞, 1 p < α ≤ 1, 1 (ii) For ∆−almost every t ∈ J, L(t, x, y) is continuously differentiable in (x, y).
If there exists m 1 ∈ C(R + , R + ), m 2 ∈ L 1 ∆ (J, R + ) and m 3 ∈ L q ∆ (J, R + ), 1 < q < ∞, such that, for ∆−a.e. t ∈ J and every (x, y) ∈ R N × R N , one has Then the functional ϕ defined by Proof. It suffices to prove that ϕ has at every point u, a directional derivative ϕ ′ (u) ∈ (W α,p ∆,b − ) * given by (4.12) and that the mapping The rest of the proof is similar to the proof of [33] P 10 Theorem 1.4. We omit it here. The proof is complete.

An application
In this section, we present a recent approach via variational methods and critical point theory to get the existence of weak solutions for the following fractional boundary value problem (FBVP for short) on time scales where T t D α b and T a D α t are the right and the left Riemann−Liouville fractional derivative operators of order 0 < α ≤ 1 defined on T respectively, F : J × R N → R satisfies the following assumption: t ∈ J and there exist m 1 ∈ C(R + , R + ) and m 2 ∈ L 1 ∆ (J, R + ) such that for all x ∈ R N and ∆−a.e. t ∈ J, and ∇F (t, x) is the gradient of F at x.
By constructing a variational structure on W α,2 ∆,b − , we can reduce the problem of finding weak solutions of (5.1) to one of seeking the critical points of a corresponding functional.
In particular, when T = R, FBVP (5.1) reduces to the standard fractional boundary value problem of the following form When α = 1, FBVP (5.1) reduces to the second order Hamiltonian system on time scale and the induced norm From now on, ϕ which we defined in (5.2) will be considered as a functional on W α,2 ∆,b − with 1 2 < α ≤ 1. We have the following facts.
Theorem 5.1. The functional ϕ is continuously differentiable on W α,2 ∆,b − and x) for all x, y ∈ R N and t ∈ J. Then, by condition (H 1 ), L(t, x, y) meets all the requirements of Theorem 4.9. Therefore, by Theorem 4.9 it follows that the functional ϕ is continuously differentiable on W α,p ∆,b − and The proof is complete.
after applying (b) of Theorem 3.6 and Definition 5.6. Hence, we can give the definition of weak solution for FBVP (5.1) as follows.
By our above remarks, any solution u ∈ W α,2 ∆,b − of FBVP (5.1) is a weak solution provided that ∇F (·, u(·)) ∈ L 1 ∆ (J, R N ). Our task is now to establish a variational structure on W α,2 ∆,b − with α ∈ 1 2 , 1 , which enables us to reduce the existence of weak solutions of FBVP (5.1) to the one of finding critical points of the corresponding functional.
Proof. Because of ϕ ′ (u) = 0, it follows from Theorem 5 for all v ∈ W α,2 ∆,b − , and hence for all v ∈ C ∞ 0 (J, R N ). Therefore, according to Definition 5.2, u is a weak solution of FBVP (5.1) and the proof is complete.
According to Theorem 5.2, we see that in order to find weak solutions of FBVP (5.1), it suffices to obtain the critical points of the functional ϕ given by (5.2). We need to use some critical point theorems. For the reader's convenience, we present some necessary definitions and theorems and skip the proofs.
Let H be a real Banach space and C 1 (H, R N ) denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on H.  (ii) there exist ρ > 0 and σ > 0 such that ψ(z) ≥ σ for all z ∈ H with z = ρ, (iii) there exists z 1 in H with z 1 ≥ ρ such that ψ(z 1 ) < σ.
First, we use Theorem 5.3 to solve the existence of weak solutions for FBVP (5.1). Suppose that the assumption (H 1 ) is satisfied. Recall that, in our setting in (5.2), the corresponding functional ϕ on W α,2 ∆,b − given by is continuously differentiable according to Theorem 5.1 and is also weakly lower semi−continuous functional on W α,2 ∆,b − as the sum of a convex continuous function and a weakly continuous function.
In fact, in view of Proposition 4.2, if u k ⇀ u in W α,2 ∆,b − , then u k → u in C(J, R N ). As a result, F (t, u k (t)) → F (t, u(t)) ∆−a.e. t ∈ [a, b] T . Using the Lebesgue dominated convergence theorem, we obtain b a F (t, u k (t))∆t → b a F (t, u(t))∆t, which means that the functional u → b a F (t, u(t))∆t is weakly continuous on W α,2 ∆,b − . Furthermore, because fractional derivative operator on T is linear operator, the functional u → b a | T a D α t u(t)| 2 ∆t is convex and continuous on W α,2 ∆,b − . If ϕ is coercive, by Theorem 5.3, ϕ has a minimum so that FBVP (5.1) is solvable. It remains to find conditions under which ϕ is coercive on W α,2 ∆,b − , i.e., lim z →∞ ϕ(z) = +∞, for u ∈ W α,2 ∆,b − . We shall know that it suffices to require that F (t, x) is bounded by a function for ∆−a.e. t ∈ J and all x ∈ R N . Theorem 5.5. Let 1 2 < α ≤ 1, and suppose that F satisfies (H 1 ). If and c ∈ L 1 ∆ (J, R), then FBVP (5.1) has at least one weak solution which minimizes ϕ on W α,2 ∆,b − . Proof. Taking account of the arguments above, our task reduces to testify that ϕ is coercive on W α,2 ∆,b − . For u ∈ W α,2 ∆,b − , it follows from (5.4), (4.6) and the Hölder inequality on time scales that ϕ(u) Noting that a ∈ 0, Γ 2 (α+1) 2b 2α and γ ∈ (0, 2), we obtain ϕ(u) = +∞ as u → ∞, and so ϕ is coercive, which completes the proof.
. The following result follows immediately from Theorem 5.4.
Theorem 5.6. Let 1 2 < α ≤ 1, and suppose that F satisfies (H 1 ). If (H 2 ) F ∈ C(J × R N , R), and there are µ ∈ 0, 1 2 and M > 0 such that 0 < F (t, x) ≤ µ(∇F (t, x), x) for all x ∈ R N with |x| ≥ M and t ∈ J, uniformly for t ∈ J and x ∈ R N are satisfied, then FBVP (5.1) has at least one nonzero weak solution on W α,2 ∆,b − . Proof. We will prove that ϕ satisfies all the conditions of Theorem 5.4.