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

Authors: Jiang Zhu; Ying Zhu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

TheΔ-power function and fractional Δ-integrals and fractional Δ-differential are defined, and then the
definitions and properties of Δ-Mittag-Leffler function are given.
The properties of fractional Δ-integrals and fractional Δ-differential on time scales are discussed in detail. After that, the
existence of the solution and the dependency of the solution upon the
initial value for Cauchy type problem with fractional Δ-derivative
are studied. Also the explicit solutions to homogeneous fractional Δ-differential equations and nonhomogeneous fractional Δ-differential
equations are derived by using Laplace transform method.


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## Body

## 1. 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 [5–11]. To unify differential equations and difference equations, Higer proposed firstly the time scale and built the relevant basic theories (see [12–15]). 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 fractionalΔ-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.


## 2. Preliminaries

First, we present some preliminaries about time scales in [12].Definition 1 (see [12]).
A time scale𝕋 is a nonempty closed subset of the real numbers. Throughout this paper, 𝕋 or 𝕋i(i=1,2,…,n) denotes a time scale.Definition 2 (see [12]).
Let𝕋 be a time scale. For t∈𝕋 one defines 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}. If σ(t)>t, one says that t is right-scattered, while if ρ(t)<t, one says that t is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if t<sup𝕋 and σ(t)=t, then t is called right-dense, and if t>inf𝕋 and ρ(t)=t, then t is called left-dense.Definition 3 (see [12]).
A functionf: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.Definition 4 (see [12]).
Lethk:𝕋2→ℝ, k∈ℕ0 be defined by
(1)h0(t,s)=1∀t,s∈𝕋
and then recursively by
(2)hk+1(t,s)=∫st‍hk(τ,s)Δτ∀t,s∈𝕋.Definition 5 (see [12]).
One defines the Cauchy functiony:𝕋×𝕋kn→ℝ for the linear dynamic equation
(3)Ly=0,whereLy=yΔn+∑i=1n‍piyΔn-i,
to be for each fixed s∈𝕋kn the solution of the initial value problem
(4)Ly=0,yΔi(σ(s),s)=0,0≤i≤n-2,yΔn-1(σ(s),s)=1.Remark 6 (see [12]).
Note that(5)y(t,s):=hn-1(t,σ(s))
is the Cauchy function of yΔn=0.Theorem 7 (variation of constants [12]).
Letf∈Crd; then the solution of the initial value problem
(6)Ly=f(t),yΔi(t0)=0,0≤i≤n-1,
is given by
(7)y(t)=∫t0t‍y(t,s)f(s)Δs,
where y(t,s) is the Cauchy function for
(8)Ly=0,whereLy=yΔn+∑i=1n‍piyΔn-i.Definition 8 (see [2]).
The factorial polynomial is defined as(9)(t)(n)=∏j=0n-1‍(t-j)=Γ(t+1)Γ(t+1-n).
For arbitrary ν, define
(10)t(ν)=Γ(t+1)Γ(t+1-υ),
where Γ denotes gamma function (see [3]).Definition 9 (see [12]).
One says that a functionp:𝕋→ℝ is regressive provided that
(11)1+μ(t)p(t)≠0
for all t∈𝕋k holds. The set of all regressive and rd-continuous functions f:𝕋→ℝ  will be denoted by
(12)ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ).Theorem 10 (see [12]).
Ifp∈ℛ, then the function ⊖p defined by
(13)⊖p(t):=-p(t)1+μ(t)p(t)∀t∈𝕋k
is also an element of ℛ.Definition 11 (see [12]).
Ifp∈ℛ, then one defines the Δ-exponential function by
(14)ep(t,s)=exp(∫st‍1μ(τ)Log(1+μ(τ)p(τ))Δτ)fors,t∈𝕋.Definition 12 (see [12]).
Ifp∈ℛ, then the first order linear dynamic equation
(15)yΔ=p(t)y
is called regressive.Theorem 13 (see [12]).
Suppose that (15) is regressive and fix t0∈𝕋. Then ep(·,t0) is a solution of the initial value problem
(16)yΔ=p(t)y,y(t0)=1
on 𝕋.Theorem 14 (see [12]).
Ifp∈ℛ, then
(17)ep(σ(t),s):=epσ(t,s)=(1+μ(t)p(t))ep(t,s).Definition 15 (see [12]).
Assume thatx:𝕋→ℝ is regulated. Then the Δ-Laplace transform of x is defined by
(18)ℒ{x}(z,t0)=∫t0∞‍x(t)e⊖zσ(t,t0)Δt
for z∈𝒟{x}, where 𝒟{x} consists of all complex numbers z∈ℂ for which the improper integral exists.Definition 16 (uniqueness of the inverse [12]).
If the functionsf:𝕋→ℝ and g:𝕋→ℝ have the same Laplace transform, then f=g.In order to give fractional integral and derivative on a time scale, we need to define fractional power functionhα(t,s) 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]).
Let𝕋 be a time scale that sup𝕋=∞ and fix t0∈𝕋. For a given f:[t0,∞)𝕋→ℂ, the solution of the shifting problem
(19)uΔt(t,σ(s))=-uΔs(t,s),t,s∈𝕋,t≥s≥t0,u(t,t0)=f(t),t∈𝕋,t≥t0
is denoted by f^ and is called the shift (or delay) of f.Example 18 (see [18]).
Considerhk(·,r)^(t,s)=hk(t,s) for all t,s∈𝕋, independent of r.Definition 19 (see [18]).
For given functionsf,g:𝕋→ℝ, their convolution f*g is defined by
(20)(f*g)(t)=∫t0t‍f^(t,σ(s))g(s)Δs,
where f^ is the shift of f introduced in Definition 17.Theorem 20 (associativity of the convolution [18]).
The convolution is associative; that is,(21)(f*g)*h=f*(g*h).Theorem 21 (see [18]).
Iff is delta differentiable, then
(22)(f*g)Δ=fΔ*g+f(t0)g,
and if g is delta differentiable, then
(23)(f*g)Δ=f*gΔ+fg(t0).Theorem 22 (see [18]).
Iff and g are infinitely often Δ-differentiable, then for all k∈N0(24)(f*g)Δk=fΔk*g+∑ν=0k-1‍fΔν(t0)gΔk-1-ν=f*gΔk+∑ν=0k-1‍fΔνgΔk-1-ν(t0),(f*g)Δk(t0)=∑ν=0k-1‍fΔν(t0)gΔk-1-ν(t0).Theorem 23 (convolution theorem [18]).
Suppose thatf,g:𝕋→ℝ are locally Δ-integrable functions on 𝕋 and their convolution f*g is defined by (20). Then,
(25)ℒ{f*g}(z)=ℒ{f}(z)·ℒ{g}(z),z∈𝒟{f}∩𝒟{g}.Theorem 24 (see [12]).
Assume thatx:𝕋→ℂ is a mapping, such that xΔk is regulated. Then
(26)ℒ{xΔk}(z,t0)=zkℒ{x}(z,t0)-∑i=0k-1‍zk-i-1xΔi(t0)for those regressivez∈ℂ satisfying
(27)limt→∞{xΔi(t)e⊖z(t,t0)}=0,i=0,1,…,k-1.Theorem 25 (see [12]).
Assume thathk(t,t0), k∈ℕ0 are defined as in Definition 4. Then
(28)ℒ{hk(·,t0)}(z,t0)=1zk+1
for those regressive z∈ℂ satisfying
(29)limt→∞{hk(t,t0)e⊖z(t,t0)}=0.


## 3.Δ-Power Function and Fractional Integral and Derivative on Time Scales

In this section, inspired by property ofhk(·,t0) in Theorem 25 for k∈ℕ0, we define fractional Δ-power functions hα(t,s) for α∈ℝ by using inversion of Δ-Laplace transform and give definitions of fractional integral and derivative on time scales.Definition 26.
One defines fractional generalizedΔ-power function hα(t,t0) on time scales
(30)hα(t,t0)=ℒ-1{1zα+1}(t)
to those suitable regressive z∈ℂ∖{0} such that ℒ-1 exist for α∈ℝ, t≥t0. Fractional generalized Δ-power function hα(t,s) on time scales is defined as the shift of hα(t,t0); that is,
(31)hα(t,s)=hα(·,t0)^(t,s)(t≥s≥t0).Applying the initial value theorem of Laplace transform (see, e.g., [15, Theorem 1.3]), for α>0, we have
(32)hα(t0,t0)=limz→∞z·1zα+1=0.Theorem 27.
Forα,β∈ℝ, one has
(33)(hα(·,t0)*hβ(·,t0))(t)=hα+β+1(t,t0).Proof.
According to convolution theorem,(34)ℒ{(hα(·,t0)*hβ(·,t0))(t)}(z,t0)=ℒ{hα(t,t0)}(z,t0)ℒ{hβ(t,t0)}(z,t0)=1zα+1·1zβ+1=1zα+β+2=ℒ{hα+β+1(t,t0)}(z,t0).
By the uniqueness of inverse transform for Laplace transform, we obtain
(35)(hα(·,t0)*hβ(·,t0))(t)=∫t0t‍hα(·,t0)^(t,σ(τ))hβ(τ,s)Δτ=hα+β+1(t,t0).Moreover, if we takeα=0, then
(36)(1*hβ)(t,t0)=∫t0t‍hβ(τ,s)Δτ=hβ+1(t,t0).
That is,
(37)hβ+1Δ(t,t0)=hβ(t,t0).Now, we will give the definitions of fractionalΔ-integral and Δ-derivative which are the main context in this section.Definition 28.
LetΩ be a finite interval on a time scale 𝕋, t0,t∈Ω. For α≥0 and for a function f:𝕋→ℝ, the Riemann-Liouville fractional Δ-integral of order α is defined by IΔ,t00f(t)=f(t) and
(38)(IΔ,t0αf)(t)=(hα-1(·,t0)*f)(t)=∫t0t‍hα-1(·,t0)^(t,σ(τ))f(τ)Δτ=∫t0t‍hα-1(t,σ(τ))f(τ)Δτ,
for α>0, t>t0.
When𝕋=ℝ, hα-1(x,a)=(x-a)α-1/Γ(α), according to Definition 17,
(39)hα-1(·,a)^(x,t)=(x-t)α-1Γ(α)
satisfy
(40)∂hα-1(·,a)^(x,t)∂x=-∂hα-1(·,a)^(x,t)∂thα-1(·,a)^(x,a)=hα-1(x,a).
When𝕋=ℤ, hα-1(x,a)=(x-a)(α-1)/Γ(α), according to Definition 17,
(41)hα-1(·,a)^(x,t)=(x-t)(α-1)Γ(α)
satisfy
(42)hα-1(·,a)^(x+1,t+1)-hα-1(·,a)^(x,t+1)=-hα-1(·,a)^(x,t+1)+hα-1(·,a)^(x,t),hα-1(·,a)^(x,a)=hα-1(x,a).
As an especial case of Definition28, we have the following examples.Example 29 (see [3]).
When𝕋=ℝ, the fractional Δ-integral of order α is defined by
(43)(Ia+αf)(x):=1Γ(α)∫ax‍1(x-t)1-αf(t)dt(x>a).Example 30.
When𝕋=ℤ, Consider the following.(1)
Thenth integral of f is defined by
(44)aΔ-nx(t)=1Γ(n)∑s=at-n‍(t-σ(s))(n-1)x(s).Heret-n≥a, σ(s)=s+1.Note that power function(t-σ(s))(n-1)/Γ(n) vanishes at s≥t-n+1. So
(45)aΔ-nx(t)=1Γ(n)∑s=at-1‍(t-σ(s))(n-1)x(s)=1Γ(n)∑s=at-n‍(t-σ(s))(n-1)x(s),fort-n≥a.(2)
Theγth fractional sum of f is defined by
(46)aΔ-γx(t)=1Γ(γ)∑s=at-1‍(t-σ(s))(γ-1)x(s),hhhhhhhhhhhhhhhhhfort≥a-[-γ].Definition 31.
Letα≥0, m=[α]+1, and f:𝕋→ℝ. For s,t∈𝕋km with s<t, the Riemann-Liouville fractional Δ-derivative of order α is defined by the expression
(47)DΔ,sαf(t):=DΔmIΔ,sm-αf(t),
if it exists.Throughout this paper, we denotefΔn=DΔnf=DΔ,snf, n∈ℕ, and, for α<0,   DΔ,sαf(t) means IΔ,s-αf(t) and IΔ,sαf(t) means DΔ,s-αf(t).Finally, we present the definition ofΔ-Mittag-Leffler function which is an important tool for solving fractional difference equation.Definition 32.
Δ-Mittag-Leffler function is defined by
(48)ΔFα,β(λ,t,t0)=∑j=0∞‍λjhjα+β-1(t,t0),
provided that the right series is convergent, where α,β>0, λ∈ℝ.Example 33.
When0<λ<1, for any  j,  |hjα+β-1(t,t0)|<M, we can obtain that the series ∑j=0∞λjhjα+β-1(t,t0) is convergent.Example 34.
When𝕋=ℝ, ∑j=0∞λjhjα+β-1(t,t0)=∑j=0∞λj((t-t0)jα+β-1/Γ(jα+β)). Since  ∑j=0∞λj((t-t0)jα+β-1/Γ(jα+β)) is convergent for any t≥t0, ∑j=0∞λjhjα+β-1(t,t0) is convergent. That is, ΔFα,β(λ,t,t0) is defined as t≥t0.As to the Laplace transform ofΔ-Mittag-Leffler function, we have the following theorem.Theorem 35.
The Laplace transform ofΔ-Mittag-Leffler function is
(49)ℒ{ΔFα,β(λ,t,t0)}(z,t0)=zα-βzα-λ(|λ|<|z|α).Proof.
According to the definition of Laplace transform, it is obtained that(50)ℒ{ΔFα,β(λ,t,t0)}(z,t0)=∫t0Δ∞‍Fα,β(λ,t,t0)·e⊖zσ(t,t0)Δt=∫t0∞‍∑j=0∞‍λjhjα+β-1(t,t0)·e⊖zσ(t,t0)Δt=∑j=0∞‍λj∫t0∞‍hjα+β-1(t,t0)·e⊖zσ(t,t0)Δt=∑j=0∞‍λjℒ{hjα+β-1(t,t0)}(z,t0)=∑j=0∞‍λj1zjα+β=z-β∑j=0∞‍λj(z-α)j=zα-βzα-λ(|λ|<|z|α).By differentiatingk times with respect to λ on both sides of the formula in Theorem 35, we get the following result:
(51)ℒ{∂k∂λkFΔα,β(λ,t,t0)}(z,t0)=k!zα-β(zα-λ)k+1.


## 4. Properties of FractionalΔ-Integral and Δ-Derivative on Time Scales

In this section, we mainly give the properties of fractionalΔ-integral and Δ-derivative on time scales which are needed in the following sections.Theorem 36.
Letα>0, m=[a]+1, and β∈ℝ. Then
(52)(1)IΔ,t0αhβ-1(t,t0)=hβ+α-1(t,t0),(2)DΔ,t0αhβ-1(t,t0)=hβ-α-1(t,t0).Proof.
(1) According to Definition28 and Theorem 27, we have
(53)IΔ,t0αhβ-1(t,t0)=(hα-1(·,t0)*hβ-1(·,t0))(t)=hα+β-1(t,t0).
(2) By Definition31, it is obtained that
(54)DΔ,t0αhβ-1(t,t0)=DΔmIΔ,t0m-αhβ-1(t,t0)=Dmhm+β-α-1(t,t0).
Then
(55)Dmhm+β-α-1(t,t0)=hβ-α-1(t,t0).In particular, ifβ=1, α>0, then the Riemann-Liouville fractional Δ-derivatives of a constant are, in general, not equal to zero:
(56)DΔ,t0α1=h-α(t,t0)(0<α<1).On the other hand, forj=1,2,…,m,
(57)DΔ,t0αhα-j(t,s)=0.
In fact,
(58)DΔ,t0αhα-j(t,s)=DΔmIΔ,t0m-αhα-j(t,s)=DΔmhm-j(t,s)=0.
From Theorem 36, we derive the following result in [3] when 𝕋=ℝ.Corollary 37 (see [3]).
Ifα≥0 and β>0, then
(59)(Ia+α(t-a)β-1)(x)=Γ(β)Γ(β+α)(x-a)β+α-1(α>0),(Da+α(t-a)β-1)(x)=Γ(β)Γ(β-α)(x-a)β-α-1(α≥0).
In particular, if β=1 and α≥0, then the Riemann-Liouville fractional derivatives of a constant are, in general, not equal to zero:
(60)(Da+α1)(x)=(x-a)-αΓ(1-α)(0<α<1).
On the other hand, for j=1,2,…,[α]+1,
(61)(Da+α(t-a)α-j)(x)=0.As to the fractional sum and difference, we have the following result, which is an improvement of Lemma 3.1 in [1].Corollary 38.
Letμ∈ℝ∖{…,-2,-1}. Then
(62)(1)Δa-ν(t-a)(μ)=μ(-ν)(t-a)(u+υ),fort≥a-[-ν],(2)Δaν(t-a)(μ)=μ(ν)(t-a)(u-υ),fort≥a+1.Lemma 39 (Taylor’s formula).
Letn∈ℕ. Suppose that the function f is n times differentiable on 𝕋kn. Let α∈𝕋kn-1, t∈𝕋, and t>α. Then one has
(63)f(t)=∑k=0n‍hk(t,α)fΔk(α)+∫αt‍hn(t,σ(τ))fΔn+1(τ)Δτ.Proof.
Letg(t):=fΔn+1(t). Then f solves the initial value problem
(64)xΔn+1=g(t),xΔk(α)=fΔk(α),0≤k≤n.
Note that the Cauchy function foryΔn+1=0 is y(t,s)=hn(t,σ(s)). By the variation of constants formula in Theorem 7,
(65)f(t)=u(t)+∫αt‍y(t,σ(τ))g(τ)Δτ,
where u solves the initial value problem
(66)uΔn+1=0,uΔm(α)=fΔm(α),0≤m≤n.
To validate the claim that u(t)=∑k=0n‍hk(t,α)fΔk(α), set
(67)ω(t):=∑k=0n‍hk(t,α)fΔk(α).
By the properties of hk(t,α), ωΔn+1(t)=0. We have moreover that
(68)ωΔm(t)=∑k=mn‍hk-m(t,α)fΔk(α),
so that
(69)ωΔm(α)=∑k=mn‍hk-m(α,α)fΔk(α)=fΔm(α),
for 0≤m≤n. We consequently have that ω also solves (66), whence u≡ω by uniqueness.Lemma 40.
(1) Forα>0, n=[α]+1, let f be a function which is n times Δ-differentiable on 𝕋kn with fΔn rd-continuous over 𝕋, and it is valid that
(70)IΔ,t0αf(t)=∑k=0n-1‍hk+α(t,t0)fΔk(t0)+(hn+α-1(·,t0)*fΔn)(t).
(2) Forα≥0, n=[α]+1, let f be a function which is n times Δ-differentiable on 𝕋kn with fΔn rd-continuous over 𝕋 and DΔ,t0αf exists almost on 𝕋, and it is valid that
(71)DΔ,t0αf(t)=∑k=0n-1‍hk-α(t,t0)fΔk(t0)+(hn-α-1(·,t0)*fΔn)(t).Proof.
By Taylor’s formula(72)f(t)=∑k=0n-1‍hk(t,t0)fΔk(t0)+∫t0t‍hn-1(t,σ(τ))fΔn(τ)Δτ,
we have
(73)IΔ,t0αf(t)=∑k=0n-1hk+α(t,t0)fΔk(t0)‍+(hα-1(·,t0)*(hn-1(·,t0)*fΔn))(t)=∑k=0n-1hk+α(t,t0)fΔk(t0)‍+((hα-1(·,t0)*hn-1(·,t0))*fΔn)(t)=∑k=0n-1hk+α(t,t0)fΔk(t0)‍+(hα+n-1(·,t0)*fΔn)(t).
Besides,
(74)DΔ,t0αf(t)=DΔmIΔ,t0m-αf(t)=DΔm[∑k=0n-1‍hk+m-α(t,t0)fΔk(t0)+(hm+n-α-1(·,t0)*fΔn)(t)∑k=0n-1‍]=∑k=0n-1‍hk-α(t,t0)fΔk(t0)+(hn-α-1(·,t0)*fΔn)(t),
where
(75)DΔm(hm+n-α-1(·,t0)*fΔn)(t)=(hn-α-1(·,t0)*fΔn)(t)+∑k=0m‍hm+n-α-1-k(t0,t0)gΔm-k=(hn-α-1(·,t0)*fΔn)(t).When𝕋=ℝ, there is the following corollary.Corollary 41 (see [3]).
Letα≥0 and n=[α]+1. If y(x)∈ACn[a,b], then the fractional derivative Da+αy exists almost everywhere on [a,b] and can be represented in the form
(76)(Da+αy)(x)=∑k=0n-1‍y(k)(a)Γ(1+k-α)(x-a)k-α+1Γ(n-α)∫ax‍y(n)(t)dt(x-t)α-n+1.Theorem 42.
Forα>0 and β>0, then (IΔ,t0αIΔ,t0βf)(t)=(IΔ,t0α+βf)(t).Proof.
According to Definition28, Theorems 20 and 27,
(77)(IΔ,t0αIΔ,t0βf)(t)=(hα-1(·,t0)*(hβ-1(·,t0)*f))(t)=((hα-1(·,t0)*hβ-1(·,t0))*f)(t)=(hα+β-1(·,t0)*f)(t)=(IΔ,t0α+βf)(t).Theorem 43.
Forα>0, n is a positive integer; if f is Δ-differentiable and the highest order derivative is rd-continuous over 𝕋, then it is valid that
(78)(1)DΔnDΔ,t0αf(t)=DΔ,t0n+αf(t),(2)DΔnIΔ,t0αf(t)=IΔ,t0α-nf(t).Proof.
(1) Suppose thatf is a function which is s times Δ-differentiable on 𝕋ks with fΔs rd-continuous over 𝕋. By Lemma 40(2),
(79)DΔnDΔ,t0αf(t)=DΔn[∑k=0s-1‍hk-α(t,t0)fΔk(t0)+(hs-1-α(·,t0)*fΔs)(t)]=∑k=0s-1hk-α-n(t,t0)fΔk(t0)+(hs-1-α-n(·,t0)*fΔs)(t)‍=DΔ,t0n+αf(t).
By a similar way, we can get (2).When𝕋=ℤ, we have the following corollary.Corollary 44.
Letf:ℕa→ℝ be given. For any k∈ℕ0 and μ>0with M-1<μ≤M, one has
(80)ΔkΔa-μf(t)=Δak-μf(t),fort≥a-[-μ],ΔkΔaμf(t)=Δak+μf(t),fort≥a+1.Theorem 45.
Forα>0, n is a positive integer; if f is Δ-differentiable and the highest order derivative is rd-continuous over 𝕋, then it is valid that
(81)(1)DΔ,t0n+αf(t)=DΔ,t0αDΔ,t0nf(t)+∑k=0n-1‍hk-α-n(t,t0)fΔk(t0),(2)DΔ,t0n-αf(t)=IΔ,t0αDΔ,t0nf(t)+∑k=0n-1‍hk+α-n(t,t0)fΔk(t0).Proof.
(1) In the proof of Theorem43(1), if we take s=n+m, then we have
(82)DΔnDΔ,t0αf(t)=DΔ,t0n+αf(t)=∑k=0n+m-1hk-α-n(t,t0)fΔk(t0)‍+(hs-1-α-n(·,t0)*fΔs)(t).
As DΔnf(t) is m times Δ-differentiable on 𝕋km+n, we have
(83)DΔ,t0α(DΔnf(t))=∑k=0m-1‍hk-α(t,t0)fΔn+k(t0)+(h-α+m-1(·,t0)*fΔs)(t).
By (82) and (83), if f is at least n times Δ-differentiable with the highest order derivative rd-continuous over 𝕋, then we have
(84)DΔnDΔ,t0αf(t)=DΔ,t0α(DΔnf(t))+∑k=0n-1‍h-α-n+k(t,t0)fΔk(t0).
Thus
(85)DΔnDΔ,t0αf(t)=DΔ,t0α(DΔnf(t))
is valid if and only if
(86)fΔk(t0)=0,k=0,…,n-1.
(2) Similarly, we have(87)IΔ,t0αDΔnf(t)=∑k=0m-1‍hk+α(t,t0)fΔk+n(t0)+(hα+m-1(·,t0)*fΔm+n)(t),DΔnIΔ,t0αf(t)=IΔ,t0α-nf(t)=∑k=0n+m-1‍hk+α-n(t,t0)fΔk(t0)+(hα+m-1(·,t0)*fΔm+n)(t).
Therefore
(88)DΔnIΔ,t0αf(t)=IΔ,t0αDΔnf(t)+∑k=0n-1‍hk+α-n(t,t0)fΔk(t0).
Provided that f is at least n times Δ-differentiable with the highest order derivative rd-continuous over 𝕋.
Thus(89)DΔnIΔ,t0αf(t)=IΔ,t0αDΔnf(t)
is valid if and only if
(90)fΔk(t0)=0,k=0,…,n-1.In particular, there are corollaries for𝕋=ℝ and for 𝕋=ℤ.Corollary 46 (see [3]).
Letα>0 and β>0 be such that n-1<α≤n, m-1<β≤m(n,m∈ℕ) and α+β<n, and let f∈L1(a,b) and fm-α∈ACm([a,b]). Then one has the following index rule:
(91)(Da+αDa+βf)(x)=(Da+α+βf)(x)-∑j=1m‍(Da+β-jf)(a+)(x-a)-j-αΓ(1-j-α).Corollary 47.
Letf:ℕa→ℝ be given. For any k∈ℕ0 and ν>0,
(92)Δa-νΔkf(t)=Δak-υf(t)-∑j=0k-1‍Δjf(a)Γ(υ-k+j+1)(t-a)(υ-k+j),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhfort≥a-[-ν].Theorem 48.
Letf be Δ-differentiable and let its highest order derivative be rd-continuous over 𝕋. When α,β>0, M=[β]+1, one has the following:
(93)(1)(DΔ,t0βIΔ,t0αf)(t)=(IΔ,t0α-β)f(t),(2)(IΔ,t0αDΔ,t0βf)(t)=(DΔ,t0β-α)f(t)hhhhhhhhhhhhhhhhhh-∑k=1M‍hα-k(t,t0)DΔ,t0β-kf(t0).Proof.
According to Theorem42 and (24), we have
(94)(DΔ,t0βIΔ,t0αf)(t)=(DΔMIΔ,t0M-βIΔ,t0αf)(t)=DΔM(hM+α-β-1(·,t0)*g)(t)=(hα-β-1(·,t0)*g)(t)=(IΔ,t0α-β)f(t).
In addition,
(95)IΔ,t0αDΔ,t0βf(t)=IΔ,t0αDΔMIΔ,t0M-βf(t)=DΔMIΔ,t0αIΔ,t0M-βf(t)-∑k=0M-1‍hα-M+k(t,t0)DΔ,t0k+β-Mf(t0)=IΔ,t0α-βf(t)-∑k=1M‍hα-k(t,t0)DΔ,t0β-kf(t0).As a direct corollary of Theorem48, we get Lemma 2.5 in [3].Corollary 49 (see [3]).
LetR(α)>0, n=[R(α)]+1, and let fn-α(x)=(Ia+n-αf)(x). If f(x)∈L1(a,b) and fn-α(x)∈ACn[a,b], then
(96)(Ia+αDa+αf)(x)=f(x)-∑j=1n‍fn-α(n-j)(a)Γ(α-j+1)(x-a)α-j
holds almost everywhere on [a,b].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.Corollary 50.
Letf:ℕa→ℝ be given. For any k∈ℕ0 and ν,μ>0 with M-1<μ≤M,
(97)Δa-νΔaμf(t)=Δaμ-υf(t)-∑j=0M-1‍Δaj-(M-μ)f(a)Γ(υ-k+j+1)(t-a)(υ-M+j).Next, we will give the Laplace transform of fractional integral and derivative on time scales.Theorem 51.
(1) Letα>0 and f:𝕋→ℝ be locally Δ-integrable. For s,t∈𝕋 with t>t0, one has
(98)ℒ{IΔ,t0αf(t)}(z,t0)=1zαℒ{f(t)}(z,t0).
(2) Letα>0 and f:𝕋→ℝ be locally Δ-integrable. For t0,t∈𝕋km with t>s, one has
(99)ℒ{DΔ,t0αf(t)}(z,t0)=zαℒ{f(t)}(z,t0)-∑j=1m‍zj-1DΔ,t0α-jf(t0).Proof.
(1) According to Definition28 and convolution theorem, we have
(100)ℒ{IΔ,t0αf(t)}(z,t0)=ℒ{(hα-1(·,t0)*f)(t)}(z,t0)=ℒ{hα-1(t,t0)}(z,t0)ℒ{f(t)}(z,t0)=1zαℒ{f(t)}(z,t0).
(2) By Definition31 and (26) and taking the Laplace transform of fractional integral into account, we get
(101)ℒ{DΔ,t0αf(t)}(z,t0)=ℒ{DΔ,t0mIΔ,t0m-αf(t)}(z,t0)=zmℒ{IΔ,t0m-αf(t)}(z,t0)-∑j=0m-1‍zm-j-1DjIΔ,t0m-αf(t0)=zm1zm-αℒ{f(t)}(z,t0)-∑j=0m-1‍zm-j-1DΔ,t0j-m+αf(t0)=zαℒ{f(t)}(z,t0)-∑j=1m‍zj-1DΔ,t0α-jf(t0).


## 5. Cauchy Type Problem with Riemann-Liouville Fractional Derivative

In this section, we consider Cauchy type problem with Riemann-Liouville fractional derivative(102)DΔ,t0αy(t)=f(t,y(t))(α>0),(103)DΔ,t0α-ky(t0)=bk(k=1,…,m=-[-α]).
In the space LΔα[a,b) defined for α>0 by
(104)LΔα[a,b):={y∈LΔ[a,b):DΔ,t0α∈LΔ[a,b)}.
Here LΔ[a,b):=LΔ,1[a,b) is the space of Δ-Lebesgue summable functions in a finite interval 𝕋~:=[a,b)𝕋.In the following, we prove that Cauchy type problem and nonlinear Volterra integral equation are equivalent in the sense that ify(t)∈LΔ[a,b) satisfies one of these relations, then it also satisfies the other.Theorem 52.
Letα>0, m=-[-α], 𝕋~:=[a,b]𝕋, t0,t∈𝕋~km. Let G be an open set in ℝ and let f:𝕋~×G→ℝ be a function such that f(t,y)∈LΔ[a,b) for any y∈G. If y(t)∈LΔ[a,b), then Cauchy type problem (102) and (103) is equivalent to
(105)y(t)=∑k=1m‍hα-k(t,t0)bk+IΔ,t0αf(t,y(t)).Proof.
First we prove the necessity. We applyIΔ,t0α to both sides of (102) and get by Theorem 48(106)IΔ,t0αDΔ,t0αy(t)=y(t)-∑k=1m‍hα-k(t,t0)DΔ,t0α-ky(t0)=(hα-1(·,t0)*f(·,·))(t).=IΔ,t0αf(t,y(t)).
Thus
(107)y(t)=∑k=1m‍hα-k(t,t0)bk+IΔ,t0αf(t,y(t)).
Now we prove the sufficiency. Applying the operator DΔ,t0α to both sides of (105) and by (57) and Theorem 48(1), we have
(108)DΔ,t0αy(t)=∑k=1m‍DΔ,t0αhα-k(t,t0)bk+DΔ,t0αIΔ,t0αf(t,y(t))=f(t,y(t)).
Now we show that the relations in (103) also hold. For this, we apply the operators DΔ,t0α-k  (k=1,…,m) to both sides of (105):
(109)DΔ,t0α-ky(t)=∑j=1m‍DΔ,t0α-khα-j(t,t0)bj+DΔ,t0α-kIΔ,t0αf(t,y(t))=∑j=1m‍DΔ,t0m-kIΔ,t0m-αhα-j(t,t0)bj+IΔ,t0kf(t,y(t))=∑j=1m‍DΔ,t0m-khm-j(t,t0)bj+IΔ,t0kf(t,y(t)).
Since
(110)DΔ,t0m-khm-j(t,t0)={0,k<j,1,k=j,hk-j(t,t0),k>j,DΔ,t0α-ky(t0)=bk(k=1,…,m).In the following, we bring in Lipschitzian-type condition:(111)|f(t,y1(t))-f(t,y2(t))|≤A|y1(t)-y2(t)|,
where A>0does not depend on t∈[a,b)𝕋. We will derive a unique solution to the Cauchy problem (102)-(103).Theorem 53.
Let the condition of Theorem52 be valid, let f(t,y) satisfy the Lipschitzian condition (111), and |f(t,y)|≤M, ΔFα,1(A,t,t0) is defined on 𝕋~:=[a,b]𝕋, where A is the Lipschitzian constant in (111). Then there exists a unique solution y(t) to initial value problem (102)-(103) in the space LΔα[a,b).Proof.
Since the Cauchy type problem (102)-(103) and the nonlinear Volterra integral equation (105) are equivalent, we only need to prove that there exists a unique solution to (105).
We define function sequences:(112)yl(t)=y0(t)+IΔ,t0αf(t,yl-1(t))(l=1,2,…,),
where
(113)y0(t)=∑k=1m‍hα-k(t,t0)bk.
We obtain, by induction that,
(114)|yl(t)-yl-1(t)|≤MAl-1hlα(t,t0).
In fact, for l=1, as |f(t,y(t))|≤M, we have
(115)|y1(t)-y0(t)|≤MIΔ,t0αh0(t,t0)=Mhα(t,t0).
If
(116)|yl-1(t)-yl-2(t)|≤MAl-2h(l-1)α(t,t0),
then
(117)|yl(t)-yl-1(t)|≤A(IΔ,t0α|yl-1-yl-2|)(t)≤A(IΔ,t0αMAl-2h(l-1)α(·,t0))(t)=MAl-1(IΔ,t0αh(l-1)α(·,t0))(t)=MAl-1hlα(t,t0).
Let
(118)y(t)=liml→∞(yl(t)-y0(t))=∑l=1∞‍(yl(t)-yl-1(t)),
and we have
(119)∑l=1∞‍|yl(t)-yl-1(t)|≤∑l=1∞‍MAl-1hlα(t,t0)=MA∑l=1∞‍Alhlα(t,t0)<MA∑l=1∞‍Alhlα(b,t0)=MA[FΔα,1(A,b,t0)-1].
By Weierstrass discriminance, we obtainyl(t) convergent uniformly. Next we will show the uniqueness. Assume that z(t) is another solution to (105); that is,
(120)z(t)=y0(t)+IΔ,t0αf(t,z(t)).
As
(121)|f(t,y(t))|≤M,|y0(t)-z(t)|≤IΔ,t0α|f(t,z(t))|≤Mhα(t,t0).
If
(122)|yl-1(t)-z(t)|≤MAl-1hlα(t,t0),
then
(123)|yl(t)-z(t)|≤(IΔ,t0α|f(·,yl-1(·))-f(·,z(·))|)(t)≤(IΔ,t0αA|yl-1-z|)(t)≤(IΔ,t0αAMAl-1hlα(·,t0))(t)≤MAl(IΔ,t0αhlα(·,t0))(t)=MAlh(l+1)α(t,t0).
By mathematical induction, we have
(124)|yl(t)-z(t)|≤MAlh(l+1)α(t,t0)
and then get z(t)=y(t)  owing to the uniqueness of the limit. This completes the proof of the theorem.Next we consider the generalized Cauchy type problem:(125)DΔ,t0αy(t)=f(t,y(t),DΔ,t0α1y(t),…,DΔ,t0αly(t))hhhhhhhhhhhhhh(0=α0≤α1≤⋯≤αl<α),(126)DΔ,t0α-ky(t0)=bk(k=1,…,n,n=[α]+1).Theorem 54.
Letf:[a,b)𝕋×G→ℝ be a function such that f(t,y,y1,…,yl)∈LΔ[a,b) for any (y,y1,…,yl)∈G. If y(t)∈LΔ[a,b), then y(t) satisfies a.e. the relations (126) and (125) if and only if it satisfies a.e. the integral equation
(127)y(t)=∑k=1n‍hα-k(t,t0)bk+(IΔ,t0αf(τ,y(τ),DΔ,t0α1y(τ),…,DΔ,t0αly(τ)))(t).Assume thatf satisfies generalized Lipschitzian condition
(128)|f(t,y,y1,…,yl)-f(t,z,z1,…,zl)|≤A[∑j=1l‍|yj-zj|].According to Theorem54 and by a similar proof to that of Theorem 53, we have the following theorem.Theorem 55.
Let the condition of Theorem54 be valid and let f(t,y,y1,…,yl) satisfy the Lipschitzian condition (128). Then there exists a unique solution y(t) to the generalized Cauchy type problem.


## 6. The Dependency of the Solution upon the Initial Value

We consider fractional differential initial value problem again:(129)DΔ,t0αy(t)=f(t,y(t)),DΔ,t0αy(t0)=η,
where 0<α<1.Using Theorem52, we have
(130)y(t)=ηhα-1(t,t0)+(IΔ,t0αf(τ,y(τ)))(t).
Suppose that z(t) is the solution to the initial value problem:
(131)DΔ,t0αy(t)=f(t,y(t)),DΔ,t0αy(t0)=η-.
We can derive the dependency of the solution upon the initial value.Theorem 56.
Let𝕋~:=[a,b]𝕋, t0,t∈𝕋~k and suppose that f satisfy the Lipschitz condition; that is,
(132)|f(t,z(t))-f(t,y(t))|≤A|z(t)-y(t)|.
Then one has
(133)|z(t)-y(t)|≤|η--η|Fα,αΔ(A,b,t0).Proof.
By the proof of Theorem53, we know that y(t)=limm→∞ym(t), z(t)=limm→∞zm(t), where(134)y0(t)=ηhα-1(t,t0),ym(t)=y0(t)+(IΔ,t0αf(τ,ym-1(τ)))(t),z0(t)=η-hα-1(t,t0),zm(t)=z0(t)+(IΔ,t0αf(τ,zm-1(τ)))(t).
Using the Lipschitz condition, we have
(135)|z1(t)-y1(t)|≤|η--η|hα-1(t,t0)+(IΔ,t0α|f(τ,z0(τ))-f(τ,y0(τ))|)(t)≤|η--η|hα-1(t,t0)+(IΔ,t0αA|z0(τ)-y0(τ)|)(t)=|η--η|hα-1(t,t0)+|η--η|A(IΔ,t0αhα-1(τ,t0))(t)=|η--η|hα-1(t,t0)+|η--η|Ah2α-1(t,t0)=|η--η|[hα-1(t,t0)+Ah2α-1(t,t0)].
Suppose that
(136)|zm-1(t)-ym-1(t)|≤|η--η|∑j=0m-1‍Ajhjα+α-1(t,t0).
Then
(137)|zm(t)-ym(t)|≤|η--η|hα-1(t,t0)+(IΔ,t0α|f(τ,zm-1(τ))-f(τ,ym-1(τ))|)(t)≤|η--η|hα-1(t,t0)+(IΔ,t0αA|zm-1(τ)-ym-1(τ)|)(t)≤|η--η|hα-1(t,t0)+(IΔ,t0αA|η--η|∑j=0m-1‍Ajhjα+α-1(τ,t0))(t)=|η--η|hα-1(t,t0)+A|η--η|∑j=0m-1‍Aj(IΔ,t0αhjα+α-1(τ,t0))(t)=|η--η|hα-1(t,t0)+|η--η|∑j=0m-1‍Aj+1h(j+1)α+α-1(t,t0)=|η--η|∑j=0m‍Ajhjα+α-1(t,t0).
According to mathematical induction, we have
(138)|zm(t)-ym(t)|≤|η--η|∑j=0m‍Ajhjα+α-1(t,t0).
Taking the limit m→∞, we obtain that
(139)|z(t)-y(t)|≤|η--η|∑j=0∞‍Ajhjα+α-1(t,t0)≤|η--η|Fα,αΔ(A,b,t0).As a special case, when fractional equation is linear, we can obtain its explicit solutions and we will explain it in next section.


## 7. Homogeneous Equations with Constant Coefficients

In this section, we apply the Laplace transform method to derive explicit solutions to homogeneous equations of the form(140)hh∑k=1m‍Ak[DΔ,t0αky(t)]+A0y(t)=0(t>t0;m∈ℕ;0<α1<⋯<αm)
with the Liouville fractional derivatives DΔ,t0αky(k=1,…,m). Here Ak∈ℝ(k=0,…,m) are real constants, and, generally speaking, we can take Am=1.In order to solve the equation, we need the following Laplace transform formula:(141)ℒ{DΔ,t0αy(t)}(z,t0)=zαℒ{y(t)}(z,t0)-∑j=1l‍djzj-1(l-1<α≤l∈ℕ),(142)dj=DΔ,t0α-jy(t0)(j=1,…,l).
First, we derive explicit solutions to (140) with m=1:
(143)DΔ,t0αy(t)-λy(t)=0(t>t0;l-1<α≤l;l∈ℕ;λ∈ℝ).
In order to prove our result, we need the following definition and lemma.Definition 57.
The functionWα(t) is defined by
(144)Wα(t)=det((DΔ,t0α-kyj)(t))k,j=1n(n=[α]+1,a≤t≤b).Lemma 58.
The solutionsy1(t),y2(t),…,yn(t) are linearly independent if and only if  Wα(t*)≠0 at some point t*∈[a,b].Proof.
We first prove sufficiency. If, to the contrary,yj(t)(j=1,2,…,n) are linearly dependent on Ω, then there exist n constants {cj}j=1n, not all zero, such that
(145)((DΔ,t0α-kyj)(t))k,j=1n(c1c2⋮cn)≡0
holds, and thus Wα(t)≡0 which leads to a contradiction. Therefore, if Wα(t*)≠0 at some point t*∈Ω, then y1(t),y2(t),…,yn(t) are linearly independent. Now we prove the necessity. Suppose, to the contrary, for t∈Ω,  Wα(t)=0. Consider
(146)((DΔ,t0α-kyj)(t*))k,j=1nC=0,where t*∈Ω, C=(c1c2⋮cn). As Wα(t*)=0, the equations have nontrivial solution cj(j=1,2,…,n). Now we construct a function using these constants:
(147)y(t)=∑j=1n‍cjyj(t),
and we get that y(t) is a solution. From (146), we obtain that y(t) satisfies initial value condition
(148)(DΔ,t0α-ky(t*))=0,k=1,…,n.
However, y(t)=0 is also a solution to equation satisfying the initial value condition. By the uniqueness of solution, we have
(149)∑j=1n‍cjyj(t)=0,
and, thus yj(t)(j=1,2,…,n) are linearly dependant, which leads to a contradiction. Thus, if the solutions y1(t),y2(t),…,yn(t) are linearly independent, then Wα(t*)≠0 at some point t*∈Ω.There hold the following statements.Theorem 59.
Letl-1<α≤l(l∈ℕ) and λ∈ℝ. Then the functions
(150)yj(t)=Fα,α+1-jΔ(λ,t,t0)(=∑k=0∞‍λkh^kα+α-j(t,t0))hhhhhhhhhhhhhhhhhhhhhhhhhhhL(j=1,…,l)
yield the fundamental system of solutions to (143). Moreover, yj(t), j=1,2,…,l, satisfy
(151)DΔ,t0α-kyj(t0)=0(k,j=1,…,l;k≠j),DΔ,t0α-kyk(t0)=1(k=1,…,l).Proof.
Applying the Laplace transform to (143) and taking (141) into account, we have
(152)ℒ{y(t)}(z,t0)=∑j=1l‍djzj-1zα-λ,
where dj(j=1,…,l) are given by (142).
Formula (49) with β=α+1-j yields
(153)ℒ{Fα,α+1-jΔ(λ,t,t0)}(z,t0)=zj-1zα-λ(|λ|<|z|α).
Thus, from (152), we derive the following solution to (143):
(154)y(t)=∑j=1l‍djyj(t),yj(t)=Fα,α+1-jΔ(λ,t,t0).
It is easily verified that the functions yj(t) are solutions to (143):
(155)DΔ,t0α[Fα,α+1-jΔ(λ,t,t0)]=λFα,α+1-jΔ(λ,t,t0)(j=1,…,l).
In fact,
(156)DΔ,t0α[Fα,α+1-jΔ(λ,t,t0)]=DΔ,t0α[∑k=0∞‍λkhkα+α-j(t,t0)]=∑k=0∞‍λkhkα-j(t,t0)=∑k=-1∞‍λk+1h(k+1)α-j(t,t0)=λ0h-j(t,t0)+∑k=0∞‍λk+1h(k+1)α-j(t,t0)=λ∑k=0∞‍λkhkα+α-j(t,t0)=λFα,α+1-jΔ(λ,t,t0).
Moreover,
(157)DΔ,t0α-kyj(t)=DΔ,t0α-k[∑s=0∞‍λsh^sα+α-j(t,t0)]=∑s=0∞‍λshsα+k-j(t,t0).
It follows from (157) that
(158)DΔ,t0α-kyj(t0)=0(k,j=1,…,l;k>j),DΔ,t0α-kyk(t0)=1(k=1,…,l).
If k<j, then
(159)DΔ,t0α-kyj(t)=∑s=1∞‍λshsα+k-j(t,t0)=∑s=0∞‍λs+1hsα+α+k-j(t,t0),
and since α+k-j≥α+1-l>0 for any k,j=1,…,l, the following relations hold:
(160)DΔ,t0α-kyj(t0)=0(k,j=1,…,l;k<j).
By (158) and (160), Wα(t0)=1. Then yj(t), j=1,…,l, yield the fundamental system of solutions to (143).Corollary 60.
Consider that(161)DΔ,t0αy(t)-λy(t)=0(t>t0;0<α≤1;λ∈ℝ)
has its solution given by
(162)y(t)=Fα,αΔ(λ,t,t0),
while
(163)DΔ,t0αy(t)-λy(t)=0(t>t0;1<α≤2;λ∈ℝ)
has the fundamental system of solutions given by
(164)y1(t)=Fα,αΔ(λ,t,t0),y2(t)=Fα,α-1Δ(λ,t,t0).Next we derive the explicit solutions to (140) with m=2 of the form
(165)hlhDΔ,t0αy(t)-λDΔ,t0βy(t)-μy(t)=0,(t>t0;l-1<α≤l;l∈ℕ;0<β<α)
with λ,μ∈ℝ.Theorem 61.
Letl-1<α≤l(l∈ℕ), 0<β<α, and λ,μ∈ℝ. Then the functions
(166)yj(t)=∑k=0∞‍μkk!∂k∂λkFα-β,α+kβ+1-jΔ(λ,t,t0)
yield the fundamental system of solutions to (165) provided that the series in (166) are convergent. Moreover, if α+1-l>β>l-1, then yj(t), j=1,2,…,l, in (166) satisfy (151).Proof.
Letm-1<β≤m(m≤l;m∈ℕ). Applying the Laplace transform to (165) and using (141) as in (152), we obtain
(167)ℒ{y(t)}(z,t0)=∑j=1l‍djzj-1zα-λzβ-μ,
where dj=DΔ,t0α-jy(t0)-λDΔ,t0β-jy(t0)(j=1,…,m), dj=DΔ,t0α-jy(t0)(j=m+1,…,l).
Forz∈ℂ and |μz-β/(zα-β-λ)|<1, we have
(168)1zα-λzβ-μ=z-βzα-β-λ·11-μz-β/(zα-β-λ)=∑k=0∞‍μkz-β-kβ(zα-β-λ)k+1,
and hence (167) has the following representation:
(169)ℒ{y(t)}(z,t0)=∑j=1l‍dj∑k=0∞‍μkzj-1-β-kβ(zα-β-λ)k+1.
By (51), for z∈ℂ and |λzβ-α|<1, we have
(170)zj-1-β-kβ(zα-β-λ)k+1=z(α-β)-(α+kβ+1-j)(zα-β-λ)k+1=1k!ℒ{∂k∂λkFα-β,α+kβ+1-jΔ(λ,t,t0)}(z,t0).
From (169) and (170), we derive the solution to (165) as
(171)y(t)=∑j=1l‍djyj(t),
where yj(t)(j=1,…,l) are given by (166). For q,j=1,…,l, the direct evaluation yields
(172)DΔ,t0α-qyj(t)=DΔ,t0α-q∑k=0∞‍μkk!∂k∂λkFα-β,α+kβ+1-jΔ(λ,t,t0)=∑k=0∞‍μkk!∂k∂λkDΔ,t0α-q[∑s=0∞‍λsh^s(α-β)+α+kβ-j(t,t0)]=∑k=0∞‍μkk!∂k∂λk∑s=0∞‍λsh^s(α-β)+kβ+q-j(t,t0).
For q>j, DΔ,t0α-qyj(t0)=0, and for q=j, DΔ,t0α-qyj(t0)=1. Thus we have Wα(t0)=1. It follows from Lemma 58 that the functions yj(t), j=1,2,…,l in (166) are linearly independent solutions, and then they yield the fundamental system of solutions to (165). Furthermore, if q<j, then we rewrite (172) as follows:
(173)DΔ,t0α-qyj(t)=DΔ,t0α-qh^α-j(t,t0)+∑s=1∞‍λsh^s(α-β)+q-j(t,t0)+∑k=1∞‍μkk!∂k∂λkh^kβ+q-j(t,t0)+∑k=1∞‍μkk!∂k∂λk∑s=1∞‍λsh^s(α-β)+kβ+q-j(t,t0)=∑s=1∞‍λsh^s(α-β)+q-j(t,t0)+∑k=1∞‍μkk!∂k∂λkh^kβ+q-j(t,t0)+∑k=1∞‍μkk!∂k∂λk∑s=1∞‍λsh^s(α-β)+kβ+q-j(t,t0).
If α+1-l>β>l-1, then s(α-β)+q-j≥(α-β)+1-l>0 for k=0, q, j=1,…,l,s∈ℕ+, and kβ+q-j≥β+1-l>0 fors=0, q, j=1,…,l,k∈ℕ+. Besides, we also have s(α-β)+kβ+q-j≥(α-β)+β+1-l=α+1-l>0 for q,j=1,…,l,s,k∈ℕ+. These imply that DΔ,t0α-qyj(t0)=0. Thus the relations in (151) are valid. The proof is finished.Corollary 62.
Consider that(174)hhhhhhhhhDΔ,t0αy(t)-λDΔ,t0βy(t)=0(t>t0;l-1<α≤l;l∈ℕ;0<β<α)
has its fundamental system of solution given by
(175)yj(t)=Fα-β,α+1-jΔ(λ,t,t0)(j=1,…,l).Finally, we find the explicit solutions to (140) with any m∈ℕ∖{1,2}. It is convenient to rewrite (140) in the form
(176)DΔ,t0αy(t)-λDΔ,t0βy(t)-∑k=0m-2‍AkDΔ,t0αky(t)=0(t>t0;m∈ℕ∖{1,2};0=α0<α1<⋯<αm-2<β<α;λ,A0,…,Am-2∈ℝ).Theorem 63.
Letm∈ℕ∖{1,2}, l-1<α≤l(l∈ℕ), let β and α1,…,αm-2 be such that α>β>αm-2>⋯>α1>α0=0, and let λ,A0,…,Am-2∈ℝ. Then the functions
(177)yj(t)=∑n=0∞‍(∑k0+⋯+km-2=n‍)1k0!⋯km-2!×[∏ν=0m-2‍(Aν)kν]∂n∂λnFα-β,α+1-j+∑ν=0m-2(β-αν)kνΔ(λ,t,t0)
yield the fundamental system of solutions to (176) provided that the series in (177) are convergent. The inner sum is taken over all k0,…,km-2∈ℕ0 such that k0+⋯+km-2=n. Moreover, if α+1-l>β>αm-2+l-1, then yj(t), j=1,2,…,l, in (177) satisfy (151).Proof.
Letlm-1-1<β≤lm-1, lk-1<αk<lk(k=1,…,m-2;0≤l1≤⋯≤lm-1≤l). Applying the Laplace transform to (176) and using (141) as in (167), we obtain
(178)ℒ{y(t)}(z,t0)=∑j=1l‍djzj-1zα-λzβ-∑k=0m-2Akzαk,
where
(179)dj=DΔ,t0α-jy(t0)-λDΔ,t0β-jy(t0)-∑k=1m-2‍AkDΔ,t0αk-jy(t0)(j=1,…,l1),dj=DΔ,t0α-jy(t0)-λDΔ,t0β-jy(t0)-∑k=2m-2‍AkDΔ,t0αk-jy(t0)(j=l1+1…,l2),⋮dj=DΔ,t0α-jy(t0)-λDΔ,t0β-jy(t0)(j=lm-2+1,…,lm-1),dj=DΔ,t0α-jy(t0)(j=lm-1+1,…,l).
Here ∑k=mnAk:=0(m>n). For z∈ℂ and |∑k=0m-2Akzαk-β/(zα-β-λ)|<1, we have
(180)1zα-λzβ-∑k=0m-2Akzαk=z-βzα-β-λ·1(1-∑k=0m-2Akzαk-β/(zα-β-λ))=∑n=0∞‍z-β(zα-β-λ)n+1(∑k=0m-2‍Akzαk-β)n=∑n=0∞‍(∑k0+⋯+km-2=n‍)n!k0!···km-2!×[∏ν=0m-2‍(Aν)kν]z-β-∑ν=0m-2(β-αν)kν(zα-β-λ)n+1,
if we also take into account the following relation:
(181)(x0+⋯+xm-2)n=(∑k0+⋯+km-2=n‍)n!k0!···km-2!∏ν=0m-2‍xυkν,
where the summation is taken over all k0,…,km-2∈ℕ0 such that k0+⋯+km-2=n.
In addition, forz∈ℂ and |λzβ-α|<1, we have
(182)zj-1-β-∑ν=0m-2(β-αν)kν(zα-β-λ)n+1=z(α-β)-(α+1-j+∑ν=0m-2(β-αν)kν)(zα-β-λ)n+1=1n!ℒΔ,t0{∂n∂λnFα-β,α+1-j+∑ν=0m-2(β-αν)kν∇(λ;t,t0)}(z).
From (178), (180), and (182), we derive the solution to (176), as
(183)y(t)=∑j=1l‍djyj(t),
which shows that arbitrary solution y(t) can be expressed by yj(t), j=1,…,l, where yj(t)(j=1,…,l) are given by (177). For q, j=1,…,l, the direct evaluation yields
(184)DΔ,t0α-qyj(t)=DΔ,t0α-q{∑n=0∞‍(∑k0+···+km-2=n‍)1k0!···km-2![∏ν=0m-2‍(Aν)kν]×∂n∂λnFα-β,α+1-j+∑ν=0m-2(β-αν)kνΔ(λ;t,t0)(∑k0+···+km-2=n‍)}=∑n=0∞‍(∑k0+···+km-2=n‍)1k0!···km-2![∏ν=0m-2‍(Aν)kν]×∂n∂λnDΔ,t0α-q[∑s=0∞‍λsh^s(α-β)+α-j+∑ν=0m-2(β-αν)kν(t,t0)]=∑n=0∞‍(∑k0+···+km-2=n‍)1k0!···km-2![∏ν=0m-2‍(Aν)kν]×∂n∂λn∑s=0∞‍λsh^s(α-β)+∑ν=0m-2(β-αν)kν+q-j(t,t0).
For q>j, DΔ,t0α-qyj(t0)=0, and for q=j, DΔ,t0α-qyj(t0)=1. Thus we have Wα(t0)=1. It follows from Lemma 58 that the functions yj(t), j=1,2,…,l in (177) are linearly independent solutions and then they yield the fundamental system of solutions to (176). Furthermore, if q<j, then we rewrite (184) as follows:
(185)DΔ,t0α-qyj(t)=DΔ,t0α-qh^α-j(t,t0)+∑s=1∞‍λsh^s(α-β)+∑ν=0m-2(β-αν)kν+q-j(t,t0)+∑n=1∞‍(∑k0+⋯+km-2=n‍)1k0!···km-2!×[∏ν=0m-2‍(Aν)kν]∂n∂λnh^∑ν=0m-2(β-αν)kν+q-j(t,t0)+∑n=1∞‍(∑k0+⋯+km-2=n‍)1k0!···km-2!×[∏ν=0m-2‍(Aν)kν]∂n∂λn∑s=1∞‍λsh^s(α-β)+∑ν=0m-2(β-αν)kν+q-j(t,t0)=∑s=1∞‍λsh^s(α-β)+q-j(t,t0)+∑n=1∞‍(∑k0+⋯+km-2=n‍)1k0!···km-2!×[∏ν=0m-2‍(Aν)kν]∂n∂λnh^∑ν=0m-2(β-αν)kν+q-j(t,t0)+∑n=1∞‍(∑k0+⋯+km-2=n‍)1k0!···km-2!×[∏ν=0m-2‍(Aν)kν]∂n∂λn∑s=1∞‍λsh^s(α-β)+∑ν=0m-2(β-αν)kν+q-j(t,t0).
If α+1-l>β>αm-2+l-1, then s(α-β)+q-j≥(α-β)+1-l>0 for n=0,q, j=1,…,l,s∈ℕ+, and ∑ν=0m-2(β-αν)kν+q-j≥β-αi+1-l≥β-αm-2+1-l>0 for s=0, q, j=1,…,l,n∈ℕ+. Besides, we also have s(α-β)+∑ν=0m-2(β-αν)kν+q-j ≥(α-β) +β -αi+1 -l ≥(α-β)+β-αm-2+1-l>0 for q,j=1,…,l,s,n∈ℕ+. These imply that DΔ,t0α-qyj(t0)=0. Thus the relations in (151) are valid. The result follows.


## 8. Nonhomogeneous Equations with Constant Coefficients

In Section7, 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
(186)∑k=1m‍AkDΔ,t0αky(t)+A0y(t)=f(t)(t>t0;0<α1<⋯<αm;m∈ℕ)
with real Ak∈ℝ(k=0,…,m).By (141)-(142), for suitable functions y, the Laplace transform of DΔ,t0αy is given by
(187)ℒ{DΔ,t0αy(t)}(z,t0)=zαℒ{y(t)}(z,t0).
Applying the Laplace transform to (186) and taking (187) into account, we have
(188)[A0+∑k=1m‍Akzαk]ℒ{y(t)}(z,t0)=ℒ{f(t)}(z,t0).
Using the inverse Laplace transform ℒΔ-1 from here, we obtain a particular solution to (186) in the form
(189)y(t)=ℒ-1[ℒ{f(t)}(z,t0)A0+∑k=1mAkzαk](t).Using the Laplace convolution formula(190)ℒ{f*g}(z,t0)=ℒ{f}(z,t0)ℒ{g}(z,t0),
we can introduce the Laplace fractional analog of the Green function as follows:
(191)Gα1,…,αm(t)=ℒ-1{1Pα1,…,αm(z)}(t),Pα1,…,αm(z)=A0+∑k=1m‍Akzαk,
and we can express a particular solution of (152) in the form of the Laplace convolution Gα1,…,αm(t) and hr(t,t0):
(192)y(t)=(f*Gα1,…,αm)(t).
Generally speaking, we can consider (186) with Am=1. First we derive a particular solution to (186) with m=1 in the form
(193)DΔ,t0αy(t)-λy(t)=f(t)(t>t0;α>0).Theorem 64.
Letα>0, λ∈ℝ. Then (193) is solvable, and its particular solution has the form
(194)y(t)=(f*Fα,αΔ(λ,s,t0))(t)
provided that the integral in the right-hand side of (194) is convergent.Proof.
Equation (193) is the same as (186) with m=1, α1=α, and A1=1, A0=-λ and (191) takes the form
(195)Gα(t)=ℒ-1{1zα-λ}(t)=Fα,αΔ(λ,t,t0).
Thus (192), with Gα1,…,αm(t)=Gα(t), yields (194). Theorem is proved.Next we derive a particular solution to (186) with m=2 of the form
(196)DΔ,t0αy(t)-λDΔ,t0βy(t)-μy(t)=f(t),hhhhhhhhhhhhhh(t>t0;α>β>0).Theorem 65.
Letα>β>0, λ,μ∈ℝ. Then (196) is solvable, and its particular solution has the form
(197)y(t)=(f*Gα,β;λ,μ(s))(t),(198)Gα,β;λ,μ(t)=∑k=0∞‍μkk!∂k∂λkFα-β,α+kβΔ(λ,t,t0)
provided that the series in (198) and the integral in (197) are convergent.Proof.
Equation (196) is the same as (186) with m=2, α2=α, α1=β, A2=1, A1=-λ, and A0=-μ, and (191) is given by
(199)Gα,β(t)=ℒ-1{1zα-λzβ-μ}(t).
According to (168) for z∈ℂ and |μz-β/(zα-β-λ)|<1, we have
(200)Gα,β;λ,μ(t)=ℒ-1{∑n=0∞‍μnz-β-nβ(zα-β-λ)n+1}(t).
In addition, for z∈ℂ and |λzβ-α|<1, we have
(201)z-β-nβ(zα-β-λ)n+1=1n!ℒ{∂n∂λnFα-β,α+nβΔ(λ,t,t0)}(z,t0),
and hence (200) takes the following form:
(202)Gα,β;λ,μ(t)=∑n=0∞‍μnn!∂n∂λnFα-β,α+nβ(λ,t,t0)Δ.
Thus the result in (197) follows from (192) with Gα1,…,αm(t)=Gα,β;λ,μ(t).Finally, we find a particular solution to (186) with any m∈ℕ∖{1,2}. It is convenient to rewrite (186) just as (176) in the form
(203)DΔ,t0αy(t)-λDΔ,t0βy(t)-∑k=1m-2‍AkDΔ,t0αky(t)-A0y(t)hhhhhhhhhhhhhhhhhhhhhhl=h^r(t,t0)(t>t0)
with m∈ℕ∖{1,2}, 0<α1<⋯<αm-2<β<α, and λ,A0,…,Am-2∈ℝ.Theorem 66.
Letm∈ℕ∖{1,2}, 0=α0<α1<⋯<αm-2<β<α, and let λ,A0,…,Am-2∈ℝ. Then (203) is solvable, and its particular solution has the form
(204)y(t)=(f*Gα1,…,αm-2,β,α;λ(s))(t),(205)Gα1,…,αm-2,β,α;λ(t)=∑n=0∞(∑k0+⋯+km-2=n‍)1k0!···km-2!‍×[∏υ=0m-2‍(Aυ)kυ]∂n∂λnFα-β,α+∑υ=0m-2(β-αυ)kυΔ(λ,t,t0)
provided that the series (205) and integral in (204) are convergent. The inner sum is taken over all k0,…,km-2 such that k0+⋯+km-2=n.Proof.
Equation (203) is the same equation as (186) with αm=α, αm-1=β, Am=1,…,Am-1=-λ, and with -Ak instead of Ak for k=0,…,m-2. Since α0=0, (191) takes the form
(206)Gα1,…,αm-2,β,α;λ(t)=ℒ-1{1zα-λαβ-∑k=0m-2Akzαk}(t).
For z∈ℂ and |∑k=0m-2Akzαk-β/(zα-β-λ)|<1, in accordance with (180), we have
(207)Gα1,…,αm-2,β,α;λ(t)=ℒ-1{∑n=0∞‍(∑k0+⋯+km-2=n‍)n!k0!···km-2!×[∏υ=0m-2‍(Aυ)kυ]z-β-∑υ=0m-2(β-αυ)kυ(zα-β-λ)n+1(∑k0+⋯+km-2=n‍)}(t).
For z∈ℂ and |λzβ-α|<1, we have
(208)z-β-∑υ=0m-2(β-αυ)kυ(zα-β-λ)n+1=1n!ℒ{∂n∂λnFα-β,α+∑υ=0m-2(β-αυ)kυΔ(λ,t,t0)}(z,t0).
The proof is finished.As in the case of ordinary differential equations, a general solution to the nonhomogeneous equation (186) is a sum of a particular solution to this equation and of the general solution to the corresponding homogeneous equation (140). Therefore, the results established in Section 7 and in Section 8 can be used to derive general solutions to the nonhomogeneous equations (193), (196), and (203). The following statements can thus be derived from Theorems 59, 64, 61, 65 and Theorems 63 and 66, respectively.Theorem 67.
Letl-1<α≤l(l∈ℕ), λ∈ℝ. Then (193) is solvable, and its general solution is given by
(209)y(t)=∑j=1l‍cjΔFα,α+1-j(λ,t,t0)+f(t),
where cj(j=1,…,l) are arbitrary real constants.Theorem 68.
Letl-1<α≤l(l∈ℕ), 0<β<α, λ,μ∈ℝ. Then (196) is solvable, and its general solution has the form
(210)y(t)=∑j=1l‍cj∑n=0∞‍μnn!∂n∂λnFα-β,α+nβ+1-jΔ(λ,t,t0)+(f*Gα,β,λ,μ(s))(t),
where Gα,β,λ,μ(t) is given by (198) and cj(j=1,…,l) are arbitrary real constants.Theorem 69.
Letm∈ℕ∖{1,2}, l-1<α≤l(l∈ℕ), let β and α1,α2,…,αm-2 be such that α>β>αm-2>⋯>α1>α0=0 and α-l+1≥β, and let λ,A0,…,Am-2∈ℝ. Then (203) is solvable, and its general solution is given by
(211)y(t)=∑j=1l‍cj∑n=0∞‍(∑k0+⋯+km-2=r‍)1k0!⋯km-2!×[∏υ=0m-2‍(Aυ)kυ]·∂n∂λnFα-β,α+1-j+∑υ=0m-2(β-αυ)kυΔ(λ,t,t0)+(f*Gα1,…,αm-2,β,α;λ(s))(t),
where Gα1,…,αm-2,β,α;λ(t) is given by (205) and cj(j=1,…,l) are arbitrary real constant.


## 9. Conclusions

In this paper, we first give a generalized definition of fractionalΔ-power function on general time scale by inversion of Laplace transform and shift transform. The fractional Δ-power function on a time scale is an important basis of fractional Δ-integral and fractional Δ-differential on time scales. Then, based on the fractional Δ-power function, we give a new definition of Riemann-Liouville fractional Δ-integral and Riemann-Liouville fractional Δ-derivative on time scales. Some of properties of Riemann-Liouville fractional Δ-integral and Riemann-Liouville fractional Δ-derivative on time scales are studied in detail. On this basis, equivalencies of Cauchy type problem with Riemann-Liouville fractional Δ-derivative and nonlinear Volterra integral equation are obtained. By employing Laplace transform, we derive explicit solutions to homogeneous and nonhomogeneous equations of Riemann-Liouville fractional Δ-derivative with constant coefficient. We give the conditions when the solutions of linear fractional differential equation will be linearly independent and when these linearly independent solutions form the fundamental system of solutions. On the other hand, we know that continuous fractional differential equation theory and discrete fractional difference equation theory have been studied by many authors, the existence and uniqueness of the solution to the boundary value problems for fractional differential equations have been studied a lot by many methods involving partial order method, fixed point method, lower and upper solutions method, transform method, and so on. For example, for the recent developments about continuous fractional differential equations and discrete fractional difference equations, one can refer to [3, 5–11] and the references therein. However, the fractional differential equation theory on time scales is still an open problem. Therefore, the authors believe that the present work will potentiate further research in the study of fractional differential equation theory on time scales in the future.



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