Fractional Cauchy Problem with Riemann-Liouville Fractional Delta Derivative on Time Scales

and Applied Analysis 3 Theorem 14 (see [12]). If p ∈ R, then e p (σ (t) , s) := e σ p (t, s) = (1 + μ (t) p (t)) ep (t, s) . (17) Definition 15 (see [12]). Assume that x : T → R is regulated. Then the Δ-Laplace transform of x is defined by


Introduction
The fractional differential equation theory is an important subject of mathematics, which includes continuous fractional differential equations and discrete fractional difference equations.The theory of fractional differential equations has gained considerable popularity and importance during the past three decades or so.Many applications in numerous seemingly diverse and widespread fields of science and engineering have been gained.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.About these advances, one can refer to [1,2], the books [3,4], and the references of them.For the recent developments about continuous fractional differential equations and discrete fractional difference equations, one can refer to [5][6][7][8][9][10][11].To unify differential equations and difference equations, Higer proposed firstly the time scale and built the relevant basic theories (see [12][13][14][15]).Recently, some authors studied fractional calculus on time scales (see [16,17]), where Williams [16] gives a definition of fractional ∇-integral and ∇derivative on time scales to unify three cases of specific time scales.Bastos gives definitions of fractional Δ-integral and Δ-derivative on time scales by the inverse of Laplace transform in [17].
Inspired by these works, the aim of this paper is to give a new definition of fractional Δ-integral and Δ-derivative on general time scales and then study some fractional differential equations on time scales.To define the fractional Δ-integral and fractional Δ-derivative, we would need to obtain a definition of fractional order power functions on time scales to generalize the monomials.Different from definition of ∇power functions by axiomatization method in [16], we define fractional Δ-power functions on general time scales by using inversion of time scale Laplace transform and shift transform in Section 3, and Riemann-Liouville fractional Δ-integral and Riemann-Liouville fractional Δ-derivative on general time scales are also given.In Section 4, we present the properties of fractional Δ-integrals and fractional Δ-differential on time scales.Then in Section 5, Cauchy type problem with Riemann-Liouville fractional Δ-derivative is discussed.In Section 6, for the Riemann-Liouville fractional Δ-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 fractional Δ-differential equations with constant coefficients.In Section 8, we also use the Laplace transform method to find particular solutions and general solutions of the corresponding fractional Δ-differential nonhomogeneous equations.

Preliminaries
First, we present some preliminaries about time scales in [12].
Definition 1 (see [12]).A time scale T is a nonempty closed subset of the real numbers.Throughout this paper, T or T  ( = 1, 2, . . ., ) denotes a time scale.
Definition 2 (see [12]).Let T be a time scale.For  ∈ T one defines the forward jump operator  : T → T by () := inf{ ∈ T :  > }, while the backward jump operator  : T → T is defined by () := sup{ ∈ T :  < }.If () > , one says that  is right-scattered, while if () < , one says that  is leftscattered.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.
Definition 3 (see [12]).A function  : T → R is called regulated provided that its right-sided limits exist (finite) at all right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T.
Theorem 7 (variation of constants [12]).Let  ∈  rd ; then the solution of the initial value problem is given by where (, ) is the Cauchy function for Definition 8 (see [2]).The factorial polynomial is defined as For arbitrary ], define where Γ denotes gamma function (see [3]).
Definition 9 (see [12]).One says that a function  : for all  ∈ T  holds.The set of all regressive and rd-continuous functions  : T → R will be denoted by Theorem 10 (see [12]).If  ∈ R, then the function ⊖ defined by is also an element of R.
Then the Δ-Laplace transform of  is defined by for  ∈ D{}, where D{} consists of all complex numbers  ∈ C for which the improper integral exists.
Definition 16 (uniqueness of the inverse [12]).If the functions  : T → R and  : T → R have the same Laplace transform, then  = .
In order to give fractional integral and derivative on a time scale, we need to define fractional power function ℎ  (, ) which is derived by the inverse of Laplace transform and is introduced in the following section.Before this, we need definitions of shift and convolution and some properties about convolution, such as convolution theorem and associativity, which are introduced in [18].
Definition 17 (see [18]).Let T be a time scale that sup T = ∞ and fix  0 ∈ T. For a given  : [ 0 , ∞) T → C, the solution of the shifting problem is denoted by f and is called the shift (or delay) of .
Definition 19 (see [18]).For given functions ,  : T → R, their convolution  *  is defined by where f is the shift of  introduced in Definition 17.

Δ-Power Function and Fractional Integral and Derivative on Time Scales
In this section, inspired by property of ℎ  (⋅,  0 ) in Theorem 25 for  ∈ N 0 , we define fractional Δ-power functions ℎ  (, ) for  ∈ R by using inversion of Δ-Laplace transform and give definitions of fractional integral and derivative on time scales.
Definition 26.One defines fractional generalized Δ-power function ℎ  (,  0 ) on time scales to those suitable regressive  ∈ C \ {0} such that L −1 exist for  ∈ R,  ≥  0 .Fractional generalized Δ-power function ℎ  (, ) on time scales is defined as the shift of ℎ  (,  0 ); that is, Applying the initial value theorem of Laplace transform (see, e.g., [15,Theorem 1.3], for  > 0, we have Theorem 27.For ,  ∈ R, one has Proof.According to convolution theorem, By the uniqueness of inverse transform for Laplace transform, we obtain Moreover, if we take  = 0, then That is, Now, we will give the definitions of fractional Δ-integral and Δ-derivative which are the main context in this section. Definition 28.Let Ω be a finite interval on a time scale T,  0 ,  ∈ Ω.For  ≥ 0 and for a function  : T → R, the Riemann-Liouville fractional Δ-integral of order  is defined by  0 Δ, 0 () = () and for  > 0,  >  0 .When T = R, ℎ −1 (, ) = ( − ) −1 /Γ(), according to Definition 17, ( As an especial case of Definition 28, we have the following examples. Example 29 (see [3]).When T = R, the fractional Δ-integral of order  is defined by Example 30.When T = Z, Consider the following.
(2) The th fractional sum of  is defined by provided that the right series is convergent, where ,  > 0,  ∈ R.

Theorem 35. The Laplace transform of Δ-Mittag-Leffler function is
Proof.According to the definition of Laplace transform, it is obtained that By differentiating  times with respect to  on both sides of the formula in Theorem 35, we get the following result: (51)

Properties of Fractional Δ-Integral and Δ-Derivative on Time Scales
In this section, we mainly give the properties of fractional Δintegral and Δ-derivative on time scales which are needed in the following sections.
(1) According to Definition 28 and Theorem 27, we have (2) By Definition 31, it is obtained that Then In particular, if  = 1,  > 0, then the Riemann-Liouville fractional Δ-derivatives of a constant are, in general, not equal to zero: On the other hand, for  = 1, 2, . . ., , In fact, From Theorem 36, we derive the following result in [3] when Corollary 37 (see [3]).If  ≥ 0 and  > 0, then In particular, if  = 1 and  ≥ 0, then the Riemann-Liouville fractional derivatives of a constant are, in general, not equal to zero: On the other hand, for  = 1, 2, . . ., [] + 1, As to the fractional sum and difference, we have the following result, which is an improvement of Lemma 3.1 in [1].
Lemma 40.(1) For  > 0,  = []+1, let  be a function which is  times Δ-differentiable on T   with  Δ  rd-continuous over T, and it is valid that (2) For  ≥ 0,  = [] + 1, let  be a function which is  times Δ-differentiable on T   with  Δ  rd-continuous over T and   Δ, 0  exists almost on T, and it is valid that Proof.By Taylor's formula we have Besides, Abstract and Applied Analysis 7 where When T = R, there is the following corollary.
Corollary 41 (see [3]).Let  ≥ 0 and  = then the fractional derivative   +  exists almost everywhere on [, ] and can be represented in the form Theorem 42.For  > 0 and  > 0, then Proof.According to Definition 28, Theorem 20 and 27, Theorem 43.For  > 0,  is a positive integer; if  is Δdifferentiable and the highest order derivative is rd-continuous over T, then it is valid that Proof.
(1) Suppose that  is a function which is  times Δ-differentiable on T   with  Δ  rd-continuous over T. By Lemma 40 (2), By a similar way, we can get (2).
When T = Z, we have the following corollary.
Corollary 44.Let  : N  → R be given.For any  ∈ N 0 and  > 0 with  − 1 <  ≤ , one has Theorem 45.For  > 0,  is a positive integer; if  is Δdifferentiable and the highest order derivative is rd-continuous over T, then it is valid that Proof.
(1) In the proof of Theorem 43(1), if we take  =  + , then we have As   Δ () is  times Δ-differentiable on T  + , we have By ( 82) and (83), if  is at least  times Δ-differentiable with the highest order derivative rd-continuous over T, then we have Thus is valid if and only if (2) Similarly, we have Therefore Provided that  is at least  times Δ-differentiable with the highest order derivative rd-continuous over T. Thus is valid if and only if In particular, there are corollaries for T = R and for T = Z.
For fractional sum and difference, there is also the following theorem in [1].
It is different from Theorem 3.3 in [1], and from Theorem 48, we can get the following corollary.
(1) According to Definition 28 and convolution theorem, we have (2) By Definition 31 and (26) and taking the Laplace transform of fractional integral into account, we get

Cauchy Type Problem with Riemann-Liouville Fractional Derivative
In this section, we consider Cauchy type problem with Riemann-Liouville fractional derivative In the space   Δ [, ) defined for  > 0 by Here  Δ [, ) :=  Δ,1 [, ) is the space of Δ-Lebesgue summable functions in a finite interval T := [, ) T .
In the following, we prove that Cauchy type problem and nonlinear Volterra integral equation are equivalent in the sense that if () ∈  Δ [, ) satisfies one of these relations, then it also satisfies the other.
Proof.First we prove the necessity.We apply   Δ, 0 to both sides of (102) and get by Theorem 48 Thus Now we prove the sufficiency.Applying the operator   Δ, 0 to both sides of (105) and by (57) and Theorem 48(1), we have =  (,  ()) . (108) Now we show that the relations in (103) also hold.For this, we apply the operators  − Δ, 0 ( = 1, . . ., ) to both sides of (105): Since In the following, we bring in Lipschitzian-type condition:      (, We obtain, by induction that, Let and we have By Weierstrass discriminance, we obtain   () convergent uniformly.Next we will show the uniqueness.Assume that () is another solution to (105); that is, By mathematical induction, we have and then get () = () owing to the uniqueness of the limit.This completes the proof of the theorem.
Next we consider the generalized Cauchy type problem: According to Theorem 54 and by a similar proof to that of Theorem 53, we have the following theorem.
Theorem 55.Let the condition of Theorem 54 be valid and let (, ,  1 , . . .,   ) satisfy the Lipschitzian condition (128).Then there exists a unique solution () to the generalized Cauchy type problem.

The Dependency of the Solution upon the Initial Value
We consider fractional differential initial value problem again: where 0 <  < 1.
Using Theorem 52, we have Suppose that () is the solution to the initial value problem: We can derive the dependency of the solution upon the initial value.
Proof.By the proof of Theorem 53, we know that () = lim  → ∞   (), () = lim  → ∞   (), where Using the Lipschitz condition, we have Suppose that Then According to mathematical induction, we have Taking the limit  → ∞, we obtain that As a special case, when fractional equation is linear, we can obtain its explicit solutions and we will explain it in next section.

Homogeneous Equations with Constant Coefficients
In this section, we apply the Laplace transform method to derive explicit solutions to homogeneous equations of the form with the Liouville fractional derivatives    Δ, 0  ( = 1, . . ., ).Here   ∈ R ( = 0, . . ., ) are real constants, and, generally speaking, we can take   = 1.
In order to solve the equation, we need the following Laplace transform formula: First, we derive explicit solutions to (140) with  = 1: In order to prove our result, we need the following definition and lemma.

Nonhomogeneous Equations with Constant Coefficients
In Section 7, we have applied the Laplace transform method to derive explicit solutions to the homogeneous equations (140) with the Liouville fractional derivatives.Here we use this approach to find particular solutions to the corresponding nonhomogeneous equations
Finally, we present the definition of Δ-Mittag-Leffler function which is an important tool for solving fractional difference equation.