Fractional Cauchy Problem with Riemann-Liouville Derivative on Time Scales LingWu and

and Applied Analysis 3 Proof. FromTheorem 4.13 in [9], we have that ∫ E f σ (s) Δs = ∫ E f σ (s) ds + ∑ i∈IE f σ (t i ) μ (t i ) = ∫ E f (s) ds + ∑ i∈IE f (σ (t i )) μ (t i ) = ∫ E f (s) ∇s, (15) where I E denotes the indices set of right-scattered points ofE. Definition 10 (see [8]). The increasing factorial function is defined as (x) (n) = x (x + 1) (x + 2) ⋅ ⋅ ⋅ (x + n − 1) , (16) where n is a positive integer and x is a real number, and the symbol [ x n ] is defined as [ x n ] = x (x + 1) (x + 2) ⋅ ⋅ ⋅ (x + n − 1) n! . (17) Definition 11 (see [8]). Let p > 0. The fractional sum whose lower limit is n 0 is defined as n0 ∇ −p n f (n) = n ∑ r=n0 [ p n − r ]f (r) . (18) Definition 12 (see [8]). Let p > 0 and m = [p] + 1. The fractional difference whose lower limit is n 0 is defined as n0 ∇ p n f (n) = ∇ m [ n0 ∇ −(m−p) n f (n)] . (19) In our discussion, we also need some information about ∇-exponential function. Definition 13 (see [2, Definition 3.4]). The function p is ]regressive if 1 − ] (t) p (t) ̸ = 0 (20) for all t ∈ T k . Define the ]-regressive class of functions on T k to be R] = {p : T 󳨀→ R | p is ld continuous and ]-regressive} . (21) For p ∈ R], define circle minus p by ⊖]p := − p 1 − p] . (22) Definition 14 (see [2, Definition 3.9]). For h > 0, let Z h := {z ∈ C : −π h < Im (z) < π h } , C h := {z ∈ C : z ̸ = 1 h } . (23) Define the ]-cylinder transformation ξ h : C h → Z h by ξ h (z) := − 1 h Log (1 − zh) , (24) where Log is the principal logarithm function. Definition 15 (see [2, Definition 3.10]). If p ∈ R], then one defines the nabla exponential function by e p (t, s) := exp(∫ t s ξ](τ) (p (τ)) ∇τ) , for s, t ∈ T , (25) where the ]-cylinder transformation ξ h is as in (24). Definition 16 (see [2, Definition 3.12]). If p ∈ R], then the first-order linear dynamic equation y ∇ = p (t) y (26) is called ]-regressive. Lemma 17 (see [2, Lemma 3.11]). If p ∈ R], then the semigroup property e p (t, u) e p (u, s) = e p (t, s) , ∀u, s, t ∈ T (27) is satisfied. Theorem 18 (see [2, Theorem 3.13]). Suppose that (26) is ]regressive and fix t 0 ∈ T .Then e p (⋅, t 0 ) is a solution of the initial value problem y ∇ = p (t) y, y (t 0 ) = 1 (28) on T . Theorem 19 (see [2, Theorem 3.15(ii)]). If p ∈ R], then e p (ρ (t) , s) := e ρ p (t, s) = (1 − ] (t) p (t)) e p (t, s) . (29) 3. ∇-Laplace Transform, Fractional Generalized ∇-Power Function, Fractional ∇-Integral and Derivative, and ∇-Mittag-Leffler Function In this section, we first define ∇-Laplace transform and discuss the properties of ∇-Laplace transform. By using the inverse∇-Laplace transform, we define fractional generalized ∇-power function, which is a basis of our definitions of fractional ∇-integral and fractional ∇-derivative. From now on, we always assume that t 0 ∈ T , sup T = ∞. Note that if we assume that z ∈ R] is a constant, then ⊖]z ∈ R] and ⊖]z t0) is well defined. With this in mind we make the following definition. Definition 20. Assume that x : T → R is regulated and t 0 ∈ T . Then, the Laplace transform of x is defined by L ∇,t0 {x} (z) = ∫ ∞ t0 x (t) e ρ ⊖]z (t, t 0 ) ∇t, (30) for z ∈ D{x}, where D{x} consists of all complex numbers z ∈ R] for which the improper integral exists. 4 Abstract and Applied Analysis The following result is needed frequently. Lemma 21. If z ∈ C is regressive, then e ρ ⊖Vz (t, t 0 ) = e ⊖Vz (t, t 0 ) 1 − ] (t) z = − ⊖Vz z e ⊖Vz (t, t 0 ) . (31) Proof. ByTheorem 19, we have e ρ ⊖Vz (t, t 0 ) = (1 − ] (t) (⊖Vz)) ⊖Vz (t, t0) = (1 − ] (t) (−z) 1 − V (t) z ) e ⊖Vz (t, t 0 ) = e ⊖Vz (t, t 0 ) 1 − V (t) z = − ⊖Vz z e ⊖Vz (t, t 0 ) . (32) This proves our claim. We now will use the Lemma 21 to find the Laplace transform of x(t) ≡ 1 as follows: L ∇,t0 {1} (z) = ∫ ∞ t0 1 ⋅ e ρ ⊖]z (t, t 0 ) ∇t


Introduction
The subject of fractional calculus (see [1]) has gained considerable popularity and importance during the past three decades or so due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.It does indeed provide several potentially useful tools for solving differential and integral equations and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables.
On the other hand, in real applications, it is not always a continuous case, but also a discrete case.So, an useful tool as that time scale is considered.In order to unify differential equations and difference equations, Higer proposed firstly the time scale and built the relevant basic theories (see [2][3][4]).Recently, some authors studied fractional calculus on time scales (see [5][6][7]).Williams [6] gives a definition of fractional integral and derivative on time scales to unify three cases of specific time scales, which improved the results in [5].Bastos gives definition of fractional Δ-integral and Δ-derivative on time scales in [7].In [8], the theory of fractional difference equations has been studied in detail.In the light of the above work, we will further study the theory of fractional integral and derivative on general time scales.From Theorem 3.1.3in [6], we know that the integer order ∇-integral on time scales is ĥ−1 (,  ())  () ∇ = ĥ−1 (,  0 ) *  () , where * is defined in Definition 31.For continuous case, the fractional ∇-integral (see, e.g., [3]) is defined by while for discrete case, the fractional sum (see, e.g., [8]) is defined by Thus, we expect that fractional ∇-integral general on time scales can be defined by   ∇, 0  () = ĥ−1 (,  0 ) *  () .
To do this, the key problem is that how to define generalized ∇-power function ĥ (,  0 ) on time scales.In [6], Williams, by using axiomatization method, gives a definition of fractional generalized ∇-power function.However, this definition has no specific form, but it only has an abstract expression.On the other hand, we find that some properties of ∇-power function ĥ (,  0 ) on time scales under the Laplace transform are important to define fractional generalized ∇-power function ĥ (,  0 ) on time scales.So, in Section 3, we will give a definition of ∇-Laplace transform, fractional generalized ∇power function ĥ (,  0 ) on time scales.Then by using these definitions, we define and study the Riemann-Liouville fractional ∇-integral, Riemann-Liouville fractional ∇-derivative, and ∇-Mittag-Leffler function on time scales.In Section 4, we present some properties of fractional ∇-integral and fractional ∇-differential on time scales.Then, in Section 5, Cauchy-type problem with the Riemann-Liouville fractional ∇-derivative is discussed.In Section 6, for the Riemann-Liouville fractional ∇-differential initial value problem, we discuss the dependency of the solution upon the initial value.In Section 7, by applying the Laplace transform method, we derive explicit solutions to homogeneous equations with constant coefficients.In Section 8, we also use the Laplace transform method to find particular solutions of the corresponding nonhomogeneous equations.

Preliminaries
First, we present some preliminaries about time scales in [2].
Definition 1 (see [2]).A time scale T is an nonempty closed subset of the real numbers.
If () > , we say that  is right-scattered, while if () < , we say that  is left-scattered.Points that are right-scattered and left-scattered at the same time are called isolated.Also, if  < sup T and () = , then  is called right-dense, and if  > inf T and () = , then  is called left-dense.Finally, the graininess function ] : T → [0, ∞) is defined by ] () :=  −  () .
We call  ∇ () the nabla derivative of  at .
Definition 4 (see [2]).A function  : T → R is called regulated provided its right-sided limits exist (finite) at all rightdense points in T and its left-sided limits exist (finite) at all left-dense points in T.
Definition 5 (see [2, page 100]).The generalized polynomials are the functions ĥ : and given ĥ for  ∈ N 0 , the function ĥ+1 is Definition 6 (see [3, page 38]).The generalized polynomials are the functions ℎ  : T 2 := T × T → R,  ∈ N 0 , defined recursively as follows: The function ℎ 0 is and given ℎ  for  ∈ N 0 , the function ℎ +1 is Theorem 7 (see [2], Taylor's Formula).Let  ∈ N. Suppose that the function  is such that Definition 8 (see [6]).A subset  ⊂ T is called a time scale interval, if it is of the form  =  ∩ T for some real interval  ⊂ R. For a time scale interval , a function  :  → R is said to be left dense absolutely continuous if for all  > 0 there exist  > 0 such that ∑  =1 |(  ) − (  )| <  whenever a disjoint finite collection of subtime scale intervals ∈ , then we denote  ∈   ∇ .According to Theorem 4.13 in [9], we have the following lemma.
Proof.From Theorem 4.13 in [9], we have that where   denotes the indices set of right-scattered points of .
Definition 10 (see [8]).The increasing factorial function is defined as where  is a positive integer and  is a real number, and the symbol [   ] is defined as Definition 11 (see [8]).Let  > 0. The fractional sum whose lower limit is  0 is defined as Definition 12 (see [8]).Let  > 0 and  = [] + 1.The fractional difference whose lower limit is  0 is defined as In our discussion, we also need some information about ∇-exponential function.
Definition 13 (see [2,Definition 3.4]).The function  is ]regressive if for all  ∈ T  .Define the ]-regressive class of functions on T  to be For  ∈ R ] , define circle minus  by Definition 14 (see [2, Definition 3.9]).For ℎ > 0, let Define the ]-cylinder transformation ξℎ : where Log is the principal logarithm function.where the ]-cylinder transformation ξℎ is as in (24).
Lemma 17 (see [2,Lemma 3.11]).In this section, we first define ∇-Laplace transform and discuss the properties of ∇-Laplace transform.By using the inverse ∇-Laplace transform, we define fractional generalized ∇-power function, which is a basis of our definitions of fractional ∇-integral and fractional ∇-derivative.
From now on, we always assume that  0 ∈ T, sup T = ∞.Note that if we assume that  ∈ R ] is a constant, then ⊖ ]  ∈ R ] and ê⊖ ]  (,  0 ) is well defined.With this in mind we make the following definition.
Definition 20.Assume that  : T → R is regulated and  0 ∈ T.Then, the Laplace transform of  is defined by for  ∈ D{}, where D{} consists of all complex numbers  ∈ R ] for which the improper integral exists.
The following result is needed frequently.
Proof.By Theorem 19, we have This proves our claim.
We now will use the Lemma 21 to find the Laplace transform of () ≡ 1 as follows: for all complex values of  ∈ R ] such that lim  → ∞ ê⊖ ]  (,  0 ) = 0 holds.The following two results are derived using integration by parts.
Theorem 22. Assume that  : Proof.Integration by parts and Lemma 21 directly yield provided that (35) holds.
Theorem 23.Assume that  : for ,  0 ∈ T; then Proof.By using integration by parts and Lemma 21, we obtain that provided that (41) holds.
Theorem 24.Assume that ĥ (,  0 ),  ∈ N 0 are defined as in Definition 5.Then, Proof.It follows from (33) that (43) holds for  = 0. Assume that (43) is valid for , and we will show that it is right for +1.
In fact, by using Theorem 23, we have that The claim follows by the principle of mathematical induction.
It is similar to the proof of Theorems 1.5 and 1.3 in [10], we get the following uniqueness result about the inverse of Laplace transform and initial value theorem.

Theorem 25 (uniqueness of the inverse). If the functions 𝑓 :
T → R and  : T → R have the same Laplace transform, then  = .
Applying the initial value theorem of Laplace transform, for  > 0, we have In particular, when  =  ∈ N, it follows from Theorems 24 and 25, we can know that ĥ (,  0 ) is usual power function on time scales for  ≥  0 defined in Definition 5.
Example 28.When T = R, the time scale power functions provided that ĥ (,  0 ) makes sense.In fact, it follows from Definition 27 that On the other hand, Thus, we have that (51) By using uniqueness of the inverse Laplace transform, we imply that Next, in order to define fractional generalized ∇-power function for general , we will present some preliminaries about convolution on time scales.In [11], the definitions of shift and convolution, and some properties about convolution, such as convolution theorem and associativity, are presented for delta case, and in the following, we give them similarly for nabla case.
Let T be a time scale such that sup T = ∞ and fix  0 ∈ T.
Definition 29.For a given  : [ 0 , ∞) T → C, the solution of the following shifting problem: is denoted by f and is called the shift of .
Example 30.Let  0 ∈ T.Then, for  ∈ N 0 , In fact, it is similar to the discussion for the delta case (refer to [3, page 38]), and we can prove that where ĝ is defined by Thus, according to Definition 29, we can derive the result.
Definition 31.For given functions ,  : T → R, their convolution  *  is defined by where f is the shift introduced in Definition 29.
Theorem 32 (associativity of the convolution).The convolution is associative; that is, Theorem 33.If  is nabla differentiable, then and if  is nabla differentiable, then Theorem 34 (convolution theorem).Suppose that ,  : T → R are locally ∇-integrable functions on T.Then, In the following, we will define fractional generalized ∇power function for general .
Definition 35.Fractional generalized ∇-power function ĥ (, ) on time scales is defined as the shift of ĥ (,  0 ), that is According to convolution theorem and Definition 27, we have By the uniqueness of inverse Laplace transform, we obtain that is, In particular, if  = 0, then that is, Thus, If  = 0, then that is, According to Theorem 33 and ( 47), (68), for  > 0, we have Now, we can give definitions of fractional ∇-integral and fractional ∇-derivative on time scales.
From now on, we will always denote Ω := [ 0 ,  1 ] T a finite interval on a time scale T (sup T = ∞).
Definition 39. ∇-Mittag-Leffler function is defined as provided that the right hand series is convergent, where  > 0, ,  ∈ R.
As to the Laplace transform of ∇-Mittag-Leffler function, we have the following theorem.
Theorem 40.The Laplace transform of ∇-Mittag-Leffler function is Proof.From the definition of Laplace transform, it is obtained that By differentiating  times with respect to  on both sides of the formula in the theorem above, we get the following result: (81)

Properties of Fractional ∇-Integral and Fractional ∇-Derivative
In this section, we mainly give the properties of fractional ∇integral and ∇-derivative on time scales which are often used in the following sections.
If () ∈   [, ], then the fractional derivative   +  exists almost everywhere on [, ] and can be represented in the following form: Similarly, for the fractional sum and difference, there is also the following corollary.
Proof.According to Definition 36 and (64), and using associativity of the convolution, we have The following assertion shows that the fractional differentiation is an operation inverse to the fractional integration from the left.
holds almost everywhere on Ω.
Proof.According to the definition of the fractional ∇-derivative and using (99), we get In the following, we will derive the composition relations between fractional ∇-differentiation and fractional ∇-integration operators.
holds almost everywhere on Ω.In particular, when  =  ∈ N and  > , then Proof.The proof is the same with the proof of Property 4, so we omit it. Thus, is valid if and only if Proof.Since () ∈  + ∇ (Ω), by (93) in Property 2 and (71), (68), we have On the other hand, from () ∈  + ∇ (Ω), we know that   ∇ () ∈   ∇ (Ω) and thus, we have Comparing with (108) and (109), we can get that which proves the result.
Proof.Applying Laplace transform to   ∇, 0   ∇ (), we have Using the uniqueness of the inverse Laplace transform, we can derive that The proof is finished.
(2) Applying Laplace transform to   ∇, 0   ∇, 0 (), we can get Proof.Let  = [] + 1.According to Property 3 and the definition of fractional ∇-derivative, we have In addition, Proof.According to Property 6 and Property 9, we have From Property 10, we derive the following result in [1] when T = R.
Next, we consider the generalized Cauchy-type problem as follows:
Using Theorem 47, we have where Suppose that () is the solution to the initial value problem as follows We denote ‖()‖ := max ∈Ω ().We can derive the dependency of the solution upon the initial value.
As a special case, when fractional equation is linear, we can obtain its explicit solutions and we will explain it in the next section.

Homogeneous Equations with Constant Coefficients
In this section, we apply the Laplace transform method to derive the fundamental system of solutions to homogeneous equations of the following form: with the Riemann-Liouville fractional derivatives    ∇, 0 ( = 1, . . ., ).Here,   ∈ R( = 0, . . ., ) are real constants, and, generally speaking, we can take   = 1.
The Laplace transform method is based on the relation (75) which is equivalent to the following one: First, we derive explicit solutions to (169) with  = 1 as follows: In order to prove our result, we also need the following definition and lemma.
There holds the following statements.