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

Abstract

The Δ-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.

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 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.

2. 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_𝑖(𝑖=1,2,…,𝑛) denotes a time scale.

Definition 2 (see [12]). Let 𝕋 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 function 𝑓∶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]). Let ℎ_𝑘∶𝕋²→ℝ, 𝑘∈ℕ₀ be defined by ℎ₀(𝑡,𝑠)=1∀𝑡,𝑠∈𝕋 (1) and then recursively by ℎ_𝑘+1(𝑡,𝑠)=^𝑡_𝑠ℎ_𝑘(𝜏,𝑠)Δ𝜏∀𝑡,𝑠∈𝕋. (2)

Definition 5 (see [12]). One defines the Cauchy function 𝑦∶𝕋×𝕋^𝑘𝑛→ℝ for the linear dynamic equation 𝐿𝑦=0,where𝐿𝑦=𝑦^Δ𝑛+^𝑛_𝑖=1𝑝_𝑖𝑦^{Δ𝑛−𝑖}, (3) to be for each fixed 𝑠∈𝕋^𝑘𝑛 the solution of the initial value problem 𝐿𝑦=0,𝑦^Δ𝑖(𝜎(𝑠))𝑦,𝑠=0,0≤𝑖≤𝑛−2,_Δ𝑛−1(𝜎(𝑠),𝑠)=1. (4)

Remark 6 (see [12]). Note that 𝑦(𝑡,𝑠)∶=ℎ_𝑛−1(𝑡,𝜎(𝑠)) (5) is the Cauchy function of 𝑦^Δ𝑛=0.

Theorem 7 (variation of constants [12]). Let 𝑓∈𝐶_rd; then the solution of the initial value problem 𝐿𝑦=𝑓(𝑡),𝑦^Δ𝑖𝑡₀=0,0≤𝑖≤𝑛−1, (6) is given by 𝑦(𝑡)=^𝑡_𝑡0𝑦(𝑡,𝑠)𝑓(𝑠)Δ𝑠, (7) where 𝑦(𝑡,𝑠) is the Cauchy function for 𝐿𝑦=0,𝑤ℎ𝑒𝑟𝑒𝐿𝑦=𝑦^Δ𝑛+^𝑛_𝑖=1𝑝_𝑖𝑦^{Δ𝑛−𝑖}. (8)

Definition 8 (see [2]). The factorial polynomial is defined as (𝑡)^(𝑛)=^𝑛−1_𝑗=0=Γ()𝑡−𝑗𝑡+1Γ().𝑡+1−𝑛 (9) For arbitrary 𝜈, define 𝑡^(𝜈)=Γ(𝑡+1)Γ(),𝑡+1−𝜐 (10) where Γ denotes gamma function (see [3]).

Definition 9 (see [12]). One says that a function 𝑝∶𝕋→ℝ is regressive provided that 1+𝜇(𝑡)𝑝(𝑡)≠0 (11) for all 𝑡∈𝕋^𝑘 holds. The set of all regressive and rd-continuous functions 𝑓∶𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ). (12)

Theorem 10 (see [12]). If 𝑝∈ℛ, then the function ⊖𝑝 defined by (𝑡)𝑝(𝑡)⊖𝑝∶=−1+𝜇(𝑡)𝑝(𝑡)∀𝑡∈𝕋^𝑘 (13) is also an element of ℛ.

Definition 11 (see [12]). If 𝑝∈ℛ, then one defines the Δ-exponential function by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠1𝜇(𝜏)Log1+𝜇(𝜏)𝑝(𝜏)Δ𝜏for𝑠,𝑡∈𝕋. (14)

Definition 12 (see [12]). If 𝑝∈ℛ, then the first order linear dynamic equation 𝑦^Δ=𝑝(𝑡)𝑦 (15) is called regressive.

Theorem 13 (see [12]). Suppose that (15) is regressive and fix 𝑡₀∈𝕋. Then 𝑒_𝑝(⋅,𝑡₀) is a solution of the initial value problem 𝑦^Δ=𝑝(𝑡)𝑦,𝑦𝑡₀=1 (16) on 𝕋.

Theorem 14 (see [12]). If 𝑝∈ℛ, then 𝑒_𝑝(𝜎(𝑡),𝑠)∶=𝑒^𝜎_𝑝(𝑡,𝑠)=1+𝜇(𝑡)𝑝(𝑡)𝑒_𝑝(𝑡,𝑠). (17)

Definition 15 (see [12]). Assume that 𝑥∶𝕋→ℝ is regulated. Then the Δ-Laplace transform of 𝑥 is defined by ℒ{𝑥}𝑧,𝑡₀=^∞_𝑡0𝑥(𝑡)𝑒^𝜎_⊖𝑧𝑡,𝑡₀Δ𝑡 (18) 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 functions 𝑓∶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 function ℎ_𝛼(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]). Let 𝕋 be a time scale that sup𝕋=∞ and fix 𝑡₀∈𝕋. For a given 𝑓∶[𝑡₀,∞)_𝕋→ℂ, the solution of the shifting problem 𝑢^Δ𝑡(𝑡,𝜎(𝑠))=−𝑢^Δ𝑠(𝑡,𝑠),𝑡,𝑠∈𝕋,𝑡≥𝑠≥𝑡₀,𝑢𝑡,𝑡₀=𝑓(𝑡),𝑡∈𝕋,𝑡≥𝑡₀ (19) is denoted by 𝑓 and is called the shift (or delay) of 𝑓.

Example 18 (see [18]). Consider ℎ_𝑘(⋅,𝑟)(𝑡,𝑠)=ℎ_𝑘(𝑡,𝑠) for all 𝑡,𝑠∈𝕋, independent of 𝑟.

Definition 19 (see [18]). For given functions 𝑓,𝑔∶𝕋→ℝ, their convolution 𝑓∗𝑔 is defined by 𝑓∗𝑔(𝑡)=^𝑡_𝑡0𝑓(𝑡,𝜎(𝑠))𝑔(𝑠)Δ𝑠, (20) where 𝑓 is the shift of 𝑓 introduced in Definition 17.

Theorem 20 (associativity of the convolution [18]). The convolution is associative; that is, 𝑓∗𝑔∗ℎ=𝑓∗𝑔∗ℎ. (21)

Theorem 21 (see [18]). If 𝑓 is delta differentiable, then 𝑓∗𝑔^Δ=𝑓^Δ∗𝑔+𝑓𝑡₀𝑔, (22) and if 𝑔 is delta differentiable, then 𝑓∗𝑔^Δ=𝑓∗𝑔^Δ+𝑓𝑔𝑡₀. (23)

Theorem 22 (see [18]). If 𝑓 and 𝑔 are infinitely often Δ-differentiable, then for all 𝑘∈𝑁₀𝑓∗𝑔^Δ𝑘=𝑓^Δ𝑘∗𝑔+^𝑘−1_𝜈=0𝑓^Δ𝜈𝑡₀𝑔^{Δ𝑘−1−𝜈}=𝑓∗𝑔^Δ𝑘+^𝑘−1_𝜈=0𝑓^Δ𝜈𝑔^{Δ𝑘−1−𝜈}𝑡₀,𝑓∗𝑔^Δ𝑘𝑡₀=^𝑘−1_𝜈=0𝑓^Δ𝜈𝑡₀𝑔^{Δ𝑘−1−𝜈}𝑡₀. (24)

Theorem 23 (convolution theorem [18]). Suppose that 𝑓,𝑔∶𝕋→ℝ are locally Δ-integrable functions on 𝕋 and their convolution 𝑓∗𝑔 is defined by (20). Then, ℒ𝑧)𝑓𝑧)𝑔𝑧)𝑓𝑔.𝑓∗𝑔(=ℒ(⋅ℒ(,𝑧∈𝒟∩𝒟 (25)

Theorem 24 (see [12]). Assume that 𝑥∶𝕋→ℂ is a mapping, such that 𝑥^Δ𝑘 is regulated. Then ℒ𝑥^Δ𝑘𝑧,𝑡₀=𝑧^𝑘ℒ{𝑥}𝑧,𝑡₀−^𝑘−1_𝑖=0𝑧^{𝑘−𝑖−1}𝑥^Δ𝑖𝑡₀ (26)

for those regressive 𝑧∈ℂ satisfying lim_𝑡→∞𝑥^Δ𝑖(𝑡)𝑒_⊖𝑧𝑡,𝑡₀=0,𝑖=0,1,…,𝑘−1. (27)

Theorem 25 (see [12]). Assume that ℎ_𝑘(𝑡,𝑡₀), 𝑘∈ℕ₀ are defined as in Definition 4. Then ℒℎ_𝑘⋅,𝑡₀𝑧,𝑡₀=1𝑧^𝑘+1 (28) for those regressive 𝑧∈ℂ satisfying lim_𝑡→∞ℎ_𝑘𝑡,𝑡₀𝑒_⊖𝑧𝑡,𝑡₀=0. (29)

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

In this section, inspired by property of ℎ_𝑘(⋅,𝑡₀) in Theorem 25 for 𝑘∈ℕ₀, we define fractional Δ-power functions ℎ_𝛼(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 generalized Δ-power function ℎ_𝛼(𝑡,𝑡₀) on time scales ℎ_𝛼𝑡,𝑡₀=ℒ⁻¹1𝑧^𝛼+1(𝑡) (30) to those suitable regressive 𝑧∈ℂ⧵{0} such that ℒ⁻¹ exist for 𝛼∈ℝ, 𝑡≥𝑡₀. Fractional generalized Δ-power function ℎ_𝛼(𝑡,𝑠) on time scales is defined as the shift of ℎ_𝛼(𝑡,𝑡₀); that is, ℎ_𝛼(𝑡,𝑠)=ℎ_𝛼⋅,𝑡₀(𝑡,𝑠)𝑡≥𝑠≥𝑡₀. (31)

Applying the initial value theorem of Laplace transform (see, e.g., [15, Theorem 1.3]), for 𝛼>0, we have ℎ_𝛼𝑡₀,𝑡₀=lim_𝑧→∞1𝑧⋅𝑧^𝛼+1=0. (32)

Theorem 27. For 𝛼,𝛽∈ℝ, one has ℎ_𝛼⋅,𝑡₀∗ℎ_𝛽⋅,𝑡₀(𝑡)=ℎ_𝛼+𝛽+1𝑡,𝑡₀. (33)

Proof. According to convolution theorem, ℒℎ_𝛼⋅,𝑡₀∗ℎ_𝛽⋅,𝑡₀(𝑡)𝑧,𝑡₀=ℒℎ_𝛼𝑡,𝑡₀𝑧,𝑡₀ℒℎ_𝛽𝑡,𝑡₀𝑧,𝑡₀=1𝑧^𝛼+1⋅1𝑧^𝛽+1=1𝑧^𝛼+𝛽+2=ℒℎ_𝛼+𝛽+1𝑡,𝑡₀𝑧,𝑡₀. (34) By the uniqueness of inverse transform for Laplace transform, we obtain ℎ_𝛼⋅,𝑡₀∗ℎ_𝛽⋅,𝑡₀(𝑡)=^𝑡_𝑡0ℎ_𝛼⋅,𝑡₀(𝑡,𝜎(𝜏))ℎ_𝛽(𝜏,𝑠)Δ𝜏=ℎ_𝛼+𝛽+1𝑡,𝑡₀. (35)

Moreover, if we take 𝛼=0, then 1∗ℎ_𝛽𝑡,𝑡₀=^𝑡_𝑡0ℎ_𝛽(𝜏,𝑠)Δ𝜏=ℎ_𝛽+1𝑡,𝑡₀. (36) That is, ℎ^Δ_𝛽+1𝑡,𝑡₀=ℎ_𝛽𝑡,𝑡₀. (37)

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, 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 𝐼⁰_Δ,𝑡0𝑓(𝑡)=𝑓(𝑡) and 𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=ℎ_𝛼−1⋅,𝑡₀(𝑡)=∗𝑓_𝑡𝑡0ℎ_𝛼−1⋅,𝑡₀((𝜏𝑓(𝜏)=𝑡,𝜎))Δ𝜏_𝑡𝑡0ℎ_𝛼−1(𝑡,𝜎(𝜏))𝑓(𝜏)Δ𝜏, (38) for 𝛼>0, 𝑡>𝑡₀.
When 𝕋=ℝ, ℎ_𝛼−1(𝑥,𝑎)=(𝑥−𝑎)^𝛼−1/Γ(𝛼), according to Definition 17, ℎ_𝛼−1()=()⋅,𝑎)(𝑥,𝑡𝑥−𝑡^𝛼−1Γ(𝛼) (39) satisfy 𝜕ℎ_𝛼−1(⋅,𝑎)(𝑥,𝑡)𝜕ℎ𝜕𝑥=−^𝛼−1(⋅,𝑎)(𝑥,𝑡)ℎ𝜕𝑡_𝛼−1(⋅,𝑎)(𝑥,𝑎)=ℎ_𝛼−1(𝑥,𝑎). (40)
When 𝕋=ℤ, ℎ_𝛼−1(𝑥,𝑎)=(𝑥−𝑎)^(𝛼−1)/Γ(𝛼), according to Definition 17, ℎ_𝛼−1()=()⋅,𝑎)(𝑥,𝑡𝑥−𝑡^(𝛼−1)Γ(𝛼) (41) satisfy ℎ_𝛼−1(⋅,𝑎)(𝑥+1,𝑡+1)−ℎ_𝛼−1()ℎ⋅,𝑎)(𝑥,𝑡+1=−_𝛼−1(⋅,𝑎)(𝑥,𝑡+1)+ℎ_𝛼−1(),ℎ⋅,𝑎)(𝑥,𝑡_𝛼−1(⋅,𝑎)(𝑥,𝑎)=ℎ_𝛼−1(𝑥,𝑎). (42)
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 𝛼 is defined by 𝐼^𝛼_𝑎+𝑓(𝑥)1∶=Γ(𝛼)^𝑥_𝑎1(𝑥−𝑡)^1−𝛼𝑓(𝑡)𝑑𝑡(𝑥>𝑎). (43)

Example 30. When 𝕋=ℤ, Consider the following.(1)The 𝑛th integral of 𝑓 is defined by _𝑎Δ^−𝑛𝑥(𝑡)=1Γ(𝑛)^𝑡−𝑛_𝑠=𝑎(𝑡−𝜎(𝑠))^(𝑛−1)𝑥(𝑠). (44) Here 𝑡−𝑛≥𝑎, 𝜎(𝑠)=𝑠+1. Note that power function (𝑡−𝜎(𝑠))^(𝑛−1)/Γ(𝑛) vanishes at 𝑠≥𝑡−𝑛+1. So _𝑎Δ^−𝑛𝑥(𝑡)=1Γ(𝑛)^𝑡−1_𝑠=𝑎(𝑡−𝜎(𝑠))^(𝑛−1)𝑥(𝑠)=1Γ(𝑛)^𝑡−𝑛_𝑠=𝑎(𝑡−𝜎(𝑠))^(𝑛−1)𝑥(𝑠),for𝑡−𝑛≥𝑎. (45)(2)The 𝛾th fractional sum of 𝑓 is defined by _𝑎Δ^−𝛾𝑥(𝑡)=1Γ𝛾^𝑡−1_𝑠=𝑎(𝑡−𝜎(𝑠))^(𝛾−1)𝑥(𝑠),for𝑡≥𝑎−−𝛾. (46)

Definition 31. Let 𝛼≥0, 𝑚=[𝛼]+1, and 𝑓∶𝕋→ℝ. For 𝑠,𝑡∈𝕋^𝑘𝑚 with 𝑠<𝑡, the Riemann-Liouville fractional Δ-derivative of order 𝛼 is defined by the expression 𝐷^𝛼_Δ,𝑠𝑓(𝑡)∶=𝐷^𝑚_Δ𝐼^𝑚−𝛼_Δ,𝑠𝑓(𝑡), (47) if it exists.

Throughout this paper, we denote 𝑓^Δ𝑛=𝐷^𝑛_Δ𝑓=𝐷^𝑛_Δ,𝑠𝑓, 𝑛∈ℕ, and, for 𝛼<0,   𝐷^𝛼_Δ,𝑠𝑓(𝑡) means 𝐼^−𝛼_Δ,𝑠𝑓(𝑡) and 𝐼^𝛼_Δ,𝑠𝑓(𝑡) means 𝐷^−𝛼_Δ,𝑠𝑓(𝑡).

Finally, we present the definition of Δ-Mittag-Leffler function which is an important tool for solving fractional difference equation.

Definition 32. Δ-Mittag-Leffler function is defined by _Δ𝐹_𝛼,𝛽𝜆,𝑡,𝑡₀=^∞_𝑗=0𝜆^𝑗ℎ_{𝑗𝛼+𝛽−1}𝑡,𝑡₀, (48) provided that the right series is convergent, where 𝛼,𝛽>0, 𝜆∈ℝ.

Example 33. When 0<𝜆<1, for any 𝑗,  |ℎ_{𝑗𝛼+𝛽−1}(𝑡,𝑡₀)|<𝑀, we can obtain that the series ∑^∞_𝑗=0𝜆^𝑗ℎ_{𝑗𝛼+𝛽−1}(𝑡,𝑡₀) is convergent.

Example 34. When 𝕋=ℝ, ∑^∞_𝑗=0𝜆^𝑗ℎ_{𝑗𝛼+𝛽−1}(𝑡,𝑡₀)=∑^∞_𝑗=0𝜆^𝑗((𝑡−𝑡₀)^{𝑗𝛼+𝛽−1}/Γ(𝑗𝛼+𝛽)). Since ∑^∞_𝑗=0𝜆^𝑗((𝑡−𝑡₀)^{𝑗𝛼+𝛽−1}/Γ(𝑗𝛼+𝛽)) is convergent for any 𝑡≥𝑡₀, ∑^∞_𝑗=0𝜆^𝑗ℎ_{𝑗𝛼+𝛽−1}(𝑡,𝑡₀) is convergent. That is, _Δ𝐹_𝛼,𝛽(𝜆,𝑡,𝑡₀) is defined as 𝑡≥𝑡₀.

As to the Laplace transform of Δ-Mittag-Leffler function, we have the following theorem.

Theorem 35. The Laplace transform of Δ-Mittag-Leffler function is ℒ_Δ𝐹_𝛼,𝛽𝜆,𝑡,𝑡₀𝑧,𝑡₀=𝑧^𝛼−𝛽𝑧^𝛼|𝜆|<|𝑧|−𝜆^𝛼. (49)

Proof. According to the definition of Laplace transform, it is obtained that ℒ_Δ𝐹_𝛼,𝛽𝜆,𝑡,𝑡₀𝑧,𝑡₀=^∞_𝑡0Δ𝐹_𝛼,𝛽𝜆,𝑡,𝑡₀⋅𝑒^𝜎_⊖𝑧𝑡,𝑡₀=Δ𝑡_{∞𝑡0∞𝑗=0}𝜆^𝑗ℎ_{𝑗𝛼+𝛽−1}𝑡,𝑡₀⋅𝑒^𝜎_⊖𝑧𝑡,𝑡₀=Δ𝑡_∞𝑗=0𝜆^𝑗∞_𝑡0ℎ_{𝑗𝛼+𝛽−1}𝑡,𝑡₀⋅𝑒^𝜎_⊖𝑧𝑡,𝑡₀=Δ𝑡_∞𝑗=0𝜆^𝑗ℒℎ_{𝑗𝛼+𝛽−1}𝑡,𝑡₀𝑧,𝑡₀=^∞_𝑗=0𝜆^𝑗1𝑧^{𝑗𝛼+𝛽}=𝑧^∞−𝛽_𝑗=0𝜆^𝑗𝑧^−𝛼𝑗=𝑧^𝛼−𝛽𝑧^𝛼|𝜆|<|𝑧|−𝜆^𝛼. (50)

By differentiating 𝑘 times with respect to 𝜆 on both sides of the formula in Theorem 35, we get the following result: ℒ𝜕^𝑘𝜕𝜆^𝑘Δ𝐹_𝛼,𝛽𝜆,𝑡,𝑡₀𝑧,𝑡₀=𝑘!𝑧^𝛼−𝛽(𝑧^𝛼−𝜆)^𝑘+1. (51)

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

In this section, we mainly give the properties of fractional Δ-integral and Δ-derivative on time scales which are needed in the following sections.

Theorem 36. Let 𝛼>0, 𝑚=[𝑎]+1, and 𝛽∈ℝ. Then (1)𝐼^𝛼_Δ,𝑡0ℎ_𝛽−1𝑡,𝑡₀=ℎ_{𝛽+𝛼−1}𝑡,𝑡₀,(2)𝐷^𝛼_Δ,𝑡0ℎ_𝛽−1𝑡,𝑡₀=ℎ_{𝛽−𝛼−1}𝑡,𝑡₀. (52)

Proof. (1) According to Definition 28 and Theorem 27, we have 𝐼^𝛼_Δ,𝑡0ℎ_𝛽−1𝑡,𝑡₀=ℎ_𝛼−1⋅,𝑡₀∗ℎ_𝛽−1⋅,𝑡₀(𝑡)=ℎ_{𝛼+𝛽−1}𝑡,𝑡₀. (53)
(2) By Definition 31, it is obtained that 𝐷^𝛼_Δ,𝑡0ℎ_𝛽−1𝑡,𝑡₀=𝐷^𝑚_Δ𝐼^𝑚−𝛼_Δ,𝑡0ℎ_𝛽−1𝑡,𝑡₀=𝐷^𝑚ℎ_{𝑚+𝛽−𝛼−1}𝑡,𝑡₀. (54) Then 𝐷^𝑚ℎ_{𝑚+𝛽−𝛼−1}𝑡,𝑡₀=ℎ_{𝛽−𝛼−1}𝑡,𝑡₀. (55)

In particular, if 𝛽=1, 𝛼>0, then the Riemann-Liouville fractional Δ-derivatives of a constant are, in general, not equal to zero: 𝐷^𝛼_Δ,𝑡01=ℎ_−𝛼𝑡,𝑡₀(0<𝛼<1). (56)

On the other hand, for 𝑗=1,2,…,𝑚, 𝐷^𝛼_Δ,𝑡0ℎ_𝛼−𝑗(𝑡,𝑠)=0. (57) In fact, 𝐷^𝛼_Δ,𝑡0ℎ_𝛼−𝑗(𝑡,𝑠)=𝐷^𝑚_Δ𝐼^𝑚−𝛼_Δ,𝑡0ℎ_𝛼−𝑗(𝑡,𝑠)=𝐷^𝑚_Δℎ_𝑚−𝑗(𝑡,𝑠)=0. (58) From Theorem 36, we derive the following result in [3] when 𝕋=ℝ.

Corollary 37 (see [3]). If 𝛼≥0 and 𝛽>0, then 𝐼^𝛼_𝑎+(𝑡−𝑎)^𝛽−1(𝑥)=Γ𝛽Γ()𝛽+𝛼𝑥−𝑎^{𝛽+𝛼−1}(),𝐷𝛼>0_𝛼𝑎+(𝑡−𝑎)^𝛽−1(𝑥)=Γ𝛽Γ()𝛽−𝛼𝑥−𝑎^{𝛽−𝛼−1}(𝛼≥0). (59) In particular, if 𝛽=1 and 𝛼≥0, then the Riemann-Liouville fractional derivatives of a constant are, in general, not equal to zero: 𝐷^𝛼_𝑎+1(𝑥)=(𝑥−𝑎)^−𝛼Γ()().1−𝛼0<𝛼<1 (60) On the other hand, for 𝑗=1,2,…,[𝛼]+1, 𝐷^𝛼_𝑎+(𝑡−𝑎)^𝛼−𝑗(𝑥)=0. (61)

As to the fractional sum and difference, we have the following result, which is an improvement of Lemma 3.1 in [1].

Corollary 38. Let 𝜇∈ℝ⧵{…,−2,−1}. Then (1)Δ_𝑎^−𝜈(𝑡−𝑎)^(𝜇)=𝜇^(−𝜈)(𝑡−𝑎)^(𝑢+𝜐)[],(2)Δ,𝑓𝑜𝑟𝑡≥𝑎−−𝜈_𝜈𝑎(𝑡−𝑎)^(𝜇)=𝜇^(𝜈)(𝑡−𝑎)^{(𝑢−𝜐)},𝑓𝑜𝑟𝑡≥𝑎+1. (62)

Lemma 39 (Taylor’s formula). Let 𝑛∈ℕ. Suppose that the function 𝑓 is 𝑛 times differentiable on 𝕋^𝑘𝑛. Let 𝛼∈𝕋^𝑘𝑛−1, 𝑡∈𝕋, and 𝑡>𝛼. Then one has 𝑓(𝑡)=^𝑛_𝑘=0ℎ_𝑘(𝑡,𝛼)𝑓^Δ𝑘(𝛼)+^𝑡_𝛼ℎ_𝑛(𝑡,𝜎(𝜏))𝑓^Δ𝑛+1(𝜏)Δ𝜏. (63)

Proof. Let 𝑔(𝑡)∶=𝑓^Δ𝑛+1(𝑡). Then 𝑓 solves the initial value problem 𝑥^Δ𝑛+1=𝑔(𝑡),𝑥^Δ𝑘(𝛼)=𝑓^Δ𝑘(𝛼),0≤𝑘≤𝑛. (64)
Note that the Cauchy function for 𝑦^Δ𝑛+1=0 is 𝑦(𝑡,𝑠)=ℎ_𝑛(𝑡,𝜎(𝑠)). By the variation of constants formula in Theorem 7, 𝑓(𝑡)=𝑢(𝑡)+^𝑡_𝛼𝑦(𝑡,𝜎(𝜏))𝑔(𝜏)Δ𝜏, (65) where 𝑢 solves the initial value problem 𝑢^Δ𝑛+1=0,𝑢^Δ𝑚(𝛼)=𝑓^Δ𝑚(𝛼),0≤𝑚≤𝑛. (66) To validate the claim that 𝑢(𝑡)=∑^𝑛_𝑘=0ℎ_𝑘(𝑡,𝛼)𝑓^Δ𝑘(𝛼), set 𝜔(𝑡)∶=^𝑛_𝑘=0ℎ_𝑘(𝑡,𝛼)𝑓^Δ𝑘(𝛼). (67) By the properties of ℎ_𝑘(𝑡,𝛼), 𝜔^Δ𝑛+1(𝑡)=0. We have moreover that 𝜔^Δ𝑚(𝑡)=^𝑛_𝑘=𝑚ℎ_𝑘−𝑚(𝑡,𝛼)𝑓^Δ𝑘(𝛼), (68) so that 𝜔^Δ𝑚(𝛼)=^𝑛_𝑘=𝑚ℎ_𝑘−𝑚(𝛼,𝛼)𝑓^Δ𝑘(𝛼)=𝑓^Δ𝑚(𝛼), (69) for 0≤𝑚≤𝑛. We consequently have that 𝜔 also solves (66), whence 𝑢≡𝜔 by uniqueness.

Lemma 40. (1) For 𝛼>0, 𝑛=[𝛼]+1, let 𝑓 be a function which is 𝑛 times Δ-differentiable on 𝕋^𝑘𝑛 with 𝑓^Δ𝑛 rd-continuous over 𝕋, and it is valid that 𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘+𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑛+𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡). (70)
(2) For 𝛼≥0, 𝑛=[𝛼]+1, let 𝑓 be a function which is 𝑛 times Δ-differentiable on 𝕋^𝑘𝑛 with 𝑓^Δ𝑛 rd-continuous over 𝕋 and 𝐷^𝛼_Δ,𝑡0𝑓 exists almost on 𝕋, and it is valid that 𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘−𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑛−𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡). (71)

Proof. By Taylor’s formula 𝑓(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+^𝑡_𝑡0ℎ_𝑛−1(𝑡,𝜎(𝜏))𝑓^Δ𝑛(𝜏)Δ𝜏, (72) we have 𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘+𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_𝛼−1⋅,𝑡₀∗ℎ_𝑛−1⋅,𝑡₀∗𝑓^Δ𝑛(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘+𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_𝛼−1⋅,𝑡₀∗ℎ_𝑛−1⋅,𝑡₀∗𝑓^Δ𝑛(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘+𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝛼+𝑛−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡). (73) Besides, 𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑚_Δ𝐼^𝑚−𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑚_Δ^𝑛−1_𝑘=0ℎ_{𝑘+𝑚−𝛼}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑚+𝑛−𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡=_{𝑛−1𝑘=0})ℎ_𝑘−𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑛−𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡), (74) where 𝐷^𝑚_Δℎ_{𝑚+𝑛−𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡)=ℎ_{𝑛−𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡)+^𝑚_𝑘=0ℎ_{𝑚+𝑛−𝛼−1−𝑘}𝑡₀,𝑡₀𝑔^{Δ𝑚−𝑘}=ℎ_{𝑛−𝛼−1}⋅,𝑡₀∗𝑓^Δ𝑛(𝑡). (75)

When 𝕋=ℝ, there is the following corollary.

Corollary 41 (see [3]). Let 𝛼≥0 and 𝑛=[𝛼]+1. If 𝑦(𝑥)∈𝐴𝐶^𝑛[𝑎,𝑏], then the fractional derivative 𝐷^𝛼_𝑎+𝑦 exists almost everywhere on [𝑎,𝑏] and can be represented in the form 𝐷^𝛼_𝑎+𝑦(𝑥)=^𝑛−1_𝑘=0𝑦^(𝑘)(𝑎)Γ()()1+𝑘−𝛼𝑥−𝑎^𝑘−𝛼+1Γ(𝑛−𝛼)^𝑥_𝑎𝑦^(𝑛)(𝑡)𝑑𝑡(𝑥−𝑡)^{𝛼−𝑛+1}. (76)  

Theorem 42. For 𝛼>0 and 𝛽>0, then (𝐼^𝛼_Δ,𝑡0𝐼^𝛽_Δ,𝑡0𝑓)(𝑡)=(𝐼^𝛼+𝛽_Δ,𝑡0𝑓)(𝑡).

Proof. According to Definition 28, Theorems 20 and 27, 𝐼^𝛼_Δ,𝑡0𝐼^𝛽_Δ,𝑡0𝑓(𝑡)=ℎ_𝛼−1⋅,𝑡₀∗ℎ_𝛽−1⋅,𝑡₀(𝑡)=ℎ∗𝑓_𝛼−1⋅,𝑡₀∗ℎ_𝛽−1⋅,𝑡₀(𝑡)=ℎ∗𝑓_{𝛼+𝛽−1}⋅,𝑡₀∗𝑓(𝑡)=𝐼^𝛼+𝛽_Δ,𝑡0𝑓(𝑡). (77)

Theorem 43. For 𝛼>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 (1)𝐷^𝑛_Δ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑛+𝛼_Δ,𝑡0𝑓(𝑡),(2)𝐷^𝑛_Δ𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼−𝑛_Δ,𝑡0𝑓(𝑡). (78)

Proof. (1) Suppose that 𝑓 is a function which is 𝑠 times Δ-differentiable on 𝕋^𝑘𝑠 with 𝑓^Δ𝑠 rd-continuous over 𝕋. By Lemma 40(2), 𝐷^𝑛_Δ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑛_Δ^𝑠−1_𝑘=0ℎ_𝑘−𝛼𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑠−1−𝛼}⋅,𝑡₀∗𝑓^Δ𝑠(𝑡=_{𝑠−1𝑘=0})ℎ_{𝑘−𝛼−𝑛}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑠−1−𝛼−𝑛}⋅,𝑡₀∗𝑓^Δ𝑠(𝑡)=𝐷^𝑛+𝛼_Δ,𝑡0𝑓(𝑡). (79) By a similar way, we can get (2).

When 𝕋=ℤ, we have the following corollary.

Corollary 44. Let 𝑓∶ℕ_𝑎→ℝ be given. For any 𝑘∈ℕ₀ and 𝜇>0with 𝑀−1<𝜇≤𝑀, one has Δ^𝑘Δ_𝑎^−𝜇𝑓(𝑡)=Δ_𝑎^𝑘−𝜇𝑓(𝑡),Δ,𝑓𝑜𝑟𝑡≥𝑎−−𝜇_𝑘Δ^𝜇_𝑎𝑓(𝑡)=Δ_𝑎^𝑘+𝜇𝑓(𝑡),𝑓𝑜𝑟𝑡≥𝑎+1. (80)

Theorem 45. For 𝛼>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 (1)𝐷^𝑛+𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝛼_Δ,𝑡0𝐷^𝑛_Δ,𝑡0𝑓(𝑡)+^𝑛−1_𝑘=0ℎ_{𝑘−𝛼−𝑛}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀,(2)𝐷^𝑛−𝛼_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼_Δ,𝑡0𝐷^𝑛_Δ,𝑡0𝑓(𝑡)+^𝑛−1_𝑘=0ℎ_{𝑘+𝛼−𝑛}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀. (81)

Proof. (1) In the proof of Theorem 43(1), if we take 𝑠=𝑛+𝑚, then we have 𝐷^𝑛_Δ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑛+𝛼_Δ,𝑡0𝑓(𝑡)=^{𝑛+𝑚−1}_𝑘=0ℎ_{𝑘−𝛼−𝑛}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝑠−1−𝛼−𝑛}⋅,𝑡₀∗𝑓^Δ𝑠(𝑡). (82) As 𝐷^𝑛_Δ𝑓(𝑡) is 𝑚 times Δ-differentiable on 𝕋^{𝑘𝑚+𝑛}, we have 𝐷^𝛼_Δ,𝑡0𝐷^𝑛_Δ𝑓(𝑡)=^𝑚−1_𝑘=0ℎ_𝑘−𝛼𝑡,𝑡₀𝑓^Δ𝑛+𝑘𝑡₀+ℎ_{−𝛼+𝑚−1}⋅,𝑡₀∗𝑓^Δ𝑠(𝑡). (83) 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 𝐷^𝑛_Δ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝛼_Δ,𝑡0𝐷^𝑛_Δ𝑓(𝑡+_{𝑛−1𝑘=0})ℎ_{−𝛼−𝑛+𝑘}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀. (84) Thus 𝐷^𝑛_Δ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝛼_Δ,𝑡0𝐷^𝑛_Δ𝑓(𝑡) (85) is valid if and only if 𝑓^Δ𝑘𝑡₀=0,𝑘=0,…,𝑛−1. (86)
(2) Similarly, we have 𝐼^𝛼_Δ,𝑡0𝐷^𝑛_Δ𝑓(𝑡)=^𝑚−1_𝑘=0ℎ_𝑘+𝛼𝑡,𝑡₀𝑓^Δ𝑘+𝑛𝑡₀+ℎ_{𝛼+𝑚−1}⋅,𝑡₀∗𝑓^Δ𝑚+𝑛(𝑡),𝐷^𝑛_Δ𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼−𝑛_Δ,𝑡0𝑓(𝑡)=^{𝑛+𝑚−1}_𝑘=0ℎ_{𝑘+𝛼−𝑛}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀+ℎ_{𝛼+𝑚−1}⋅,𝑡₀∗𝑓^Δ𝑚+𝑛(𝑡). (87) Therefore 𝐷^𝑛_Δ𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼_Δ,𝑡0𝐷^𝑛_Δ𝑓(𝑡)+^𝑛−1_𝑘=0ℎ_{𝑘+𝛼−𝑛}𝑡,𝑡₀𝑓^Δ𝑘𝑡₀. (88) Provided that 𝑓 is at least 𝑛 times Δ-differentiable with the highest order derivative rd-continuous over 𝕋.
Thus 𝐷^𝑛_Δ𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼_Δ,𝑡0𝐷^𝑛_Δ𝑓(𝑡) (89) is valid if and only if 𝑓^Δ𝑘𝑡₀=0,𝑘=0,…,𝑛−1. (90)

In particular, there are corollaries for 𝕋=ℝ and for 𝕋=ℤ.

Corollary 46 (see [3]). Let 𝛼>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 𝑓_𝑚−𝛼∈𝐴𝐶^𝑚([𝑎,𝑏]). Then one has the following index rule: 𝐷^𝛼_𝑎+𝐷^𝛽_𝑎+𝑓(𝑥)=𝐷^𝛼+𝛽_𝑎+𝑓(𝑥)−^𝑚_𝑗=1𝐷^𝛽−𝑗_𝑎+𝑓()()𝑎+𝑥−𝑎^{−𝑗−𝛼}Γ.1−𝑗−𝛼 (91)

Corollary 47. Let 𝑓∶ℕ_𝑎→ℝ be given. For any 𝑘∈ℕ₀ and 𝜈>0, Δ_𝑎^−𝜈Δ^𝑘𝑓(𝑡)=Δ_𝑎^𝑘−𝜐𝑓(𝑡)−^𝑘−1_𝑗=0Δ^𝑗𝑓(𝑎)Γ()𝜐−𝑘+𝑗+1𝑡−𝑎^{(𝜐−𝑘+𝑗)},𝑓𝑜𝑟𝑡≥𝑎−[−𝜈]. (92)

Theorem 48. Let 𝑓 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: (1)𝐷^𝛽_Δ,𝑡0𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼−𝛽_Δ,𝑡0𝑓(𝑡),(2)𝐼^𝛼_Δ,𝑡0𝐷^𝛽_Δ,𝑡0𝑓(𝑡)=𝐷^𝛽−𝛼_Δ,𝑡0𝑓(𝑡)−^𝑀_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝐷^𝛽−𝑘_Δ,𝑡0𝑓𝑡₀. (93)

Proof. According to Theorem 42 and (24), we have 𝐷^𝛽_Δ,𝑡0𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑀_Δ𝐼^𝑀−𝛽_Δ,𝑡0𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝐷^𝑀_Δℎ_{𝑀+𝛼−𝛽−1}⋅,𝑡₀(𝑡)=ℎ∗𝑔_{𝛼−𝛽−1}⋅,𝑡₀𝑡)=𝐼∗𝑔(_{𝛼−𝛽Δ,𝑡0}𝑓(𝑡). (94) In addition, 𝐼^𝛼_Δ,𝑡0𝐷^𝛽_Δ,𝑡0𝑓(𝑡)=𝐼^𝛼_Δ,𝑡0𝐷^𝑀_Δ𝐼^𝑀−𝛽_Δ,𝑡0𝑓(𝑡)=𝐷^𝑀_Δ𝐼^𝛼_Δ,𝑡0𝐼^𝑀−𝛽_Δ,𝑡0𝑓(𝑡)−^𝑀−1_𝑘=0ℎ_{𝛼−𝑀+𝑘}𝑡,𝑡₀𝐷^{𝑘+𝛽−𝑀}_Δ,𝑡0𝑓𝑡₀=𝐼^𝛼−𝛽_Δ,𝑡0𝑓(𝑡)−^𝑀_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝐷^𝛽−𝑘_Δ,𝑡0𝑓𝑡₀. (95)

As a direct corollary of Theorem 48, we get Lemma 2.5 in [3].

Corollary 49 (see [3]). Let 𝑅(𝛼)>0, 𝑛=[𝑅(𝛼)]+1, and let 𝑓_𝑛−𝛼(𝑥)=(𝐼^𝑛−𝛼_𝑎+𝑓)(𝑥). If 𝑓(𝑥)∈𝐿₁(𝑎,𝑏) and 𝑓_𝑛−𝛼(𝑥)∈𝐴𝐶^𝑛[𝑎,𝑏], then 𝐼^𝛼_𝑎+𝐷^𝛼_𝑎+𝑓(𝑥)=𝑓(𝑥)−^𝑛_𝑗=1𝑓^{(𝑛−𝑗)}_𝑛−𝛼(𝑎)Γ()𝛼−𝑗+1𝑥−𝑎^𝛼−𝑗 (96) 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. Let 𝑓∶ℕ_𝑎→ℝ be given. For any 𝑘∈ℕ₀ and 𝜈,𝜇>0 with 𝑀−1<𝜇≤𝑀, Δ_𝑎^−𝜈Δ^𝜇_𝑎𝑓(𝑡)=Δ_𝑎^𝜇−𝜐𝑓(𝑡)−^𝑀−1_𝑗=0Δ_𝑎^{𝑗−(𝑀−𝜇)}𝑓(𝑎)Γ()𝜐−𝑘+𝑗+1𝑡−𝑎^{(𝜐−𝑀+𝑗)}. (97)

Next, we will give the Laplace transform of fractional integral and derivative on time scales.

Theorem 51. (1) Let 𝛼>0 and 𝑓∶𝕋→ℝ be locally Δ-integrable. For 𝑠,𝑡∈𝕋 with 𝑡>𝑡₀, one has ℒ𝐼^𝛼_Δ,𝑡0𝑓(𝑡)𝑧,𝑡₀=1𝑧^𝛼ℒ𝑓(𝑡)𝑧,𝑡₀. (98)
(2) Let 𝛼>0 and 𝑓∶𝕋→ℝ be locally Δ-integrable. For 𝑡₀,𝑡∈𝕋^𝑘𝑚 with 𝑡>𝑠, one has ℒ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)𝑧,𝑡₀=𝑧^𝛼ℒ𝑓(𝑡)𝑧,𝑡₀−^𝑚_𝑗=1𝑧^𝑗−1𝐷^𝛼−𝑗_Δ,𝑡0𝑓𝑡₀. (99)

Proof. (1) According to Definition 28 and convolution theorem, we have ℒ𝐼^𝛼_Δ,𝑡0𝑓(𝑡)𝑧,𝑡₀=ℒℎ_𝛼−1⋅,𝑡₀∗𝑓(𝑡)𝑧,𝑡₀=ℒℎ_𝛼−1𝑡,𝑡₀𝑧,𝑡₀ℒ𝑓(𝑡)𝑧,𝑡₀=1𝑧^𝛼ℒ𝑓(𝑡)𝑧,𝑡₀. (100)
(2) By Definition 31 and (26) and taking the Laplace transform of fractional integral into account, we get ℒ𝐷^𝛼_Δ,𝑡0𝑓(𝑡)𝑧,𝑡₀=ℒ𝐷^𝑚_Δ,𝑡0𝐼^𝑚−𝛼_Δ,𝑡0𝑓(𝑡)𝑧,𝑡₀=𝑧^𝑚ℒ𝐼^𝑚−𝛼_Δ,𝑡0𝑓(𝑡)𝑧,𝑡₀−^𝑚−1_𝑗=0𝑧^{𝑚−𝑗−1}𝐷^𝑗𝐼^𝑚−𝛼_Δ,𝑡0𝑓𝑡₀=𝑧^𝑚1𝑧^𝑚−𝛼ℒ𝑓(𝑡)𝑧,𝑡₀−^𝑚−1_𝑗=0𝑧^{𝑚−𝑗−1}𝐷^{𝑗−𝑚+𝛼}_Δ,𝑡0𝑓𝑡₀=𝑧^𝛼ℒ𝑓(𝑡)𝑧,𝑡₀−^𝑚_𝑗=1𝑧^𝑗−1𝐷^𝛼−𝑗_Δ,𝑡0𝑓𝑡₀. (101)

5. Cauchy Type Problem with Riemann-Liouville Fractional Derivative

In this section, we consider Cauchy type problem with Riemann-Liouville fractional derivative 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)(𝑡),𝐷=𝑓𝑡,𝑦)(𝛼>0_{𝛼−𝑘Δ,𝑡0}𝑦𝑡₀=𝑏_𝑘(𝑘=1,…,𝑚=−[−𝛼]). (102)(103) In the space 𝐿^𝛼_Δ[𝑎,𝑏) defined for 𝛼>0 by 𝐿^𝛼_Δ[𝑎,𝑏)∶=𝑦∈𝐿_Δ[𝑎,𝑏)∶𝐷^𝛼_Δ,𝑡0∈𝐿_Δ[𝑎,𝑏). (104) Here 𝐿_Δ[𝑎,𝑏)∶=𝐿_Δ,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 if 𝑦(𝑡)∈𝐿_Δ[𝑎,𝑏) satisfies one of these relations, then it also satisfies the other.

Theorem 52. Let 𝛼>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)∈𝐿_Δ[𝑎,𝑏) for any 𝑦∈𝐺. If 𝑦(𝑡)∈𝐿_Δ[𝑎,𝑏), then Cauchy type problem (102) and (103) is equivalent to 𝑦(𝑡)=^𝑚_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝑏_𝑘+𝐼^𝛼_Δ,𝑡0𝑓𝑡,𝑦(𝑡). (105)

Proof. First we prove the necessity. We apply 𝐼^𝛼_Δ,𝑡0 to both sides of (102) and get by Theorem 48𝐼^𝛼_Δ,𝑡0𝐷^𝛼_Δ,𝑡0𝑦(𝑡)=𝑦(𝑡)−^𝑚_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝐷^𝛼−𝑘_Δ,𝑡0𝑦𝑡₀=ℎ_𝛼−1⋅,𝑡₀∗𝑓(⋅,⋅)(𝑡).=𝐼^𝛼_Δ,𝑡0𝑓𝑡,𝑦(𝑡). (106) Thus 𝑦(𝑡)=^𝑚_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝑏_𝑘+𝐼^𝛼_Δ,𝑡0𝑓𝑡,𝑦(𝑡). (107) Now we prove the sufficiency. Applying the operator 𝐷^𝛼_Δ,𝑡0 to both sides of (105) and by (57) and Theorem 48(1), we have 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)=^𝑚_𝑘=1𝐷^𝛼_Δ,𝑡0ℎ_𝛼−𝑘𝑡,𝑡₀𝑏_𝑘+𝐷^𝛼_Δ,𝑡0𝐼^𝛼_Δ,𝑡0𝑓(𝑡(𝑡).𝑡,𝑦)=𝑓𝑡,𝑦 (108) Now we show that the relations in (103) also hold. For this, we apply the operators 𝐷^𝛼−𝑘_Δ,𝑡0  (𝑘=1,…,𝑚) to both sides of (105): 𝐷^𝛼−𝑘_Δ,𝑡0𝑦(𝑡)=^𝑚_𝑗=1𝐷^𝛼−𝑘_Δ,𝑡0ℎ_𝛼−𝑗𝑡,𝑡₀𝑏_𝑗+𝐷^𝛼−𝑘_Δ,𝑡0𝐼^𝛼_Δ,𝑡0𝑓(𝑡)=𝑡,𝑦_𝑚𝑗=1𝐷^𝑚−𝑘_Δ,𝑡0𝐼^𝑚−𝛼_Δ,𝑡0ℎ_𝛼−𝑗𝑡,𝑡₀𝑏_𝑗+𝐼^𝑘_Δ,𝑡0𝑓(𝑡)=𝑡,𝑦_𝑚𝑗=1𝐷^𝑚−𝑘_Δ,𝑡0ℎ_𝑚−𝑗𝑡,𝑡₀𝑏_𝑗+𝐼^𝑘_Δ,𝑡0𝑓𝑡,𝑦(𝑡). (109) Since 𝐷^𝑚−𝑘_Δ,𝑡0ℎ_𝑚−𝑗𝑡,𝑡₀=ℎ0,𝑘<𝑗,1,𝑘=𝑗,_𝑘−𝑗𝑡,𝑡₀𝐷,𝑘>𝑗,_{𝛼−𝑘Δ,𝑡0}𝑦𝑡₀=𝑏_𝑘(𝑘=1,…,𝑚). (110)

In the following, we bring in Lipschitzian-type condition: 𝑓𝑡,𝑦₁(𝑡)−𝑓𝑡,𝑦₂(𝑡)≤𝐴𝑦₁(𝑡)−𝑦₂(𝑡), (111) 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 Theorem 52 be valid, let 𝑓(𝑡,𝑦) satisfy the Lipschitzian condition (111), and |𝑓(𝑡,𝑦)|≤𝑀, _Δ𝐹_𝛼,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 𝐿^𝛼_Δ[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: 𝑦_𝑙(𝑡)=𝑦₀(𝑡)+𝐼^𝛼_Δ,𝑡0𝑓𝑡,𝑦_𝑙−1(𝑡)(𝑙=1,2,…,), (112) where 𝑦₀(𝑡)=^𝑚_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝑏_𝑘. (113) We obtain, by induction that, 𝑦_𝑙(𝑡)−𝑦_𝑙−1(𝑡)≤𝑀𝐴^𝑙−1ℎ_𝑙𝛼𝑡,𝑡₀. (114) In fact, for 𝑙=1, as |𝑓(𝑡,𝑦(𝑡))|≤𝑀, we have 𝑦₁(𝑡)−𝑦₀(𝑡)≤𝑀𝐼^𝛼_Δ,𝑡0ℎ₀𝑡,𝑡₀=𝑀ℎ_𝛼𝑡,𝑡₀. (115) If 𝑦_𝑙−1(𝑡)−𝑦_𝑙−2(𝑡)≤𝑀𝐴^𝑙−2ℎ_{(𝑙−1)𝛼}𝑡,𝑡₀, (116) then 𝑦_𝑙(𝑡)−𝑦_𝑙−1(𝑡)≤𝐴𝐼^𝛼_Δ,𝑡0𝑦_𝑙−1−𝑦_𝑙−2𝑡)𝐼(≤𝐴^𝛼_Δ,𝑡0𝑀𝐴^𝑙−2ℎ_{(𝑙−1)𝛼}⋅,𝑡₀(𝑡)=𝑀𝐴^𝑙−1𝐼^𝛼_Δ,𝑡0ℎ_{(𝑙−1)𝛼}⋅,𝑡₀(𝑡)=𝑀𝐴^𝑙−1ℎ_𝑙𝛼𝑡,𝑡₀. (117) Let 𝑦(𝑡)=lim_𝑙→∞𝑦_𝑙(𝑡)−𝑦₀(𝑡)=^∞_𝑙=1𝑦_𝑙(𝑡)−𝑦_𝑙−1(𝑡), (118) and we have ^∞_𝑙=1𝑦_𝑙(𝑡)−𝑦_𝑙−1(𝑡)≤^∞_𝑙=1𝑀𝐴^𝑙−1ℎ_𝑙𝛼𝑡,𝑡₀=𝑀𝐴^∞_𝑙=1𝐴^𝑙ℎ_𝑙𝛼𝑡,𝑡₀<𝑀𝐴^∞_𝑙=1𝐴^𝑙ℎ_𝑙𝛼𝑏,𝑡₀=𝑀𝐴^Δ𝐹_𝛼,1𝐴,𝑏,𝑡₀−1. (119)
By Weierstrass discriminance, we obtain 𝑦_𝑙(𝑡) convergent uniformly. Next we will show the uniqueness. Assume that 𝑧(𝑡) is another solution to (105); that is, 𝑧(𝑡)=𝑦₀(𝑡)+𝐼^𝛼_Δ,𝑡0𝑓(𝑡,𝑧(𝑡)). (120) As 𝑓(𝑡)𝑦𝑡,𝑦≤𝑀,₀(𝑡)−𝑧(𝑡)≤𝐼^𝛼_Δ,𝑡0𝑓(𝑡,𝑧(𝑡))≤𝑀ℎ_𝛼𝑡,𝑡₀. (121) If 𝑦_𝑙−1(𝑡)−𝑧(𝑡)≤𝑀𝐴^𝑙−1ℎ_𝑙𝛼𝑡,𝑡₀, (122) then 𝑦_𝑙(𝑡)−𝑧(𝑡)≤𝐼^𝛼_Δ,𝑡0𝑓⋅,𝑦_𝑙−1(⋅((⋅𝑡)≤𝐼)−𝑓⋅,𝑧))(_𝛼Δ,𝑡0𝐴𝑦_𝑙−1(𝑡)≤𝐼−𝑧_𝛼Δ,𝑡0𝐴𝑀𝐴^𝑙−1ℎ_𝑙𝛼⋅,𝑡₀(𝑡)≤𝑀𝐴^𝑙𝐼^𝛼_Δ,𝑡0ℎ_𝑙𝛼⋅,𝑡₀(𝑡)=𝑀𝐴^𝑙ℎ_(𝑙+1)𝛼𝑡,𝑡₀. (123) By mathematical induction, we have 𝑦_𝑙(𝑡)−𝑧(𝑡)≤𝑀𝐴^𝑙ℎ_(𝑙+1)𝛼𝑡,𝑡₀ (124) 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: 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝐷^𝛼1_Δ,𝑡0𝑦(𝑡),…,𝐷^𝛼𝑙_Δ,𝑡0𝑦(𝑡)0=𝛼₀≤𝛼₁≤⋯≤𝛼_𝑙,𝐷<𝛼_{𝛼−𝑘Δ,𝑡0}𝑦𝑡₀=𝑏_𝑘(𝑘=1,…,𝑛,𝑛=[𝛼]+1). (125)(126)

Theorem 54. Let 𝑓∶[𝑎,𝑏)_𝕋×𝐺→ℝ be a function such that 𝑓(𝑡,𝑦,𝑦₁,…,𝑦_𝑙)∈𝐿_Δ[𝑎,𝑏) for any (𝑦,𝑦₁,…,𝑦_𝑙)∈𝐺. If 𝑦(𝑡)∈𝐿_Δ[𝑎,𝑏), then 𝑦(𝑡) satisfies a.e. the relations (126) and (125) if and only if it satisfies a.e. the integral equation 𝑦(𝑡)=^𝑛_𝑘=1ℎ_𝛼−𝑘𝑡,𝑡₀𝑏_𝑘+𝐼^𝛼_Δ,𝑡0𝑓𝜏,𝑦(𝜏),𝐷^𝛼1_Δ,𝑡0𝑦(𝜏),…,𝐷^𝛼𝑙_Δ,𝑡0𝑦(𝜏)(𝑡). (127)

Assume that 𝑓 satisfies generalized Lipschitzian condition 𝑓𝑡,𝑦,𝑦₁,…,𝑦_𝑙−𝑓𝑡,𝑧,𝑧₁,…,𝑧_𝑙≤𝐴^𝑙_𝑗=1𝑦_𝑗−𝑧_𝑗. (128)

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 𝑓(𝑡,𝑦,𝑦₁,…,𝑦_𝑙) 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: 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)(𝑡,𝐷=𝑓𝑡,𝑦)_𝛼Δ,𝑡0𝑦𝑡₀=𝜂, (129) where 0<𝛼<1.

Using Theorem 52, we have 𝑦(𝑡)=𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐼^𝛼_Δ,𝑡0𝑓𝜏,𝑦(𝜏)(𝑡). (130) Suppose that 𝑧(𝑡) is the solution to the initial value problem: 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)(𝑡,𝐷=𝑓𝑡,𝑦)_𝛼Δ,𝑡0𝑦𝑡₀=𝜂. (131) We can derive the dependency of the solution upon the initial value.

Theorem 56. Let 𝕋∶=[𝑎,𝑏]_𝕋, 𝑡₀,𝑡∈𝕋^𝑘 and suppose that 𝑓 satisfy the Lipschitz condition; that is, 𝑓(𝑡,𝑧(𝑡))−𝑓𝑡,𝑦(𝑡)≤𝐴𝑧(𝑡)−𝑦(𝑡). (132) Then one has 𝑧(𝑡)−𝑦(𝑡)≤𝜂−𝜂_Δ𝐹_𝛼,𝛼𝐴,𝑏,𝑡₀. (133)

Proof. By the proof of Theorem 53, we know that 𝑦(𝑡)=lim_𝑚→∞𝑦_𝑚(𝑡), 𝑧(𝑡)=lim_𝑚→∞𝑧_𝑚(𝑡), where𝑦₀(𝑡)=𝜂ℎ_𝛼−1𝑡,𝑡₀,𝑦_𝑚(𝑡)=𝑦₀(𝑡)+𝐼^𝛼_Δ,𝑡0𝑓𝜏,𝑦_𝑚−1(𝜏)(𝑡),𝑧₀(𝑡)=𝜂ℎ_𝛼−1𝑡,𝑡₀,𝑧_𝑚(𝑡)=𝑧₀(𝑡)+𝐼^𝛼_Δ,𝑡0𝑓𝜏,𝑧_𝑚−1(𝜏)(𝑡). (134) Using the Lipschitz condition, we have 𝑧₁(𝑡)−𝑦₁(𝑡≤)𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐼^𝛼_Δ,𝑡0𝑓𝜏,𝑧₀(𝜏)−𝑓𝜏,𝑦₀(𝜏)𝑡)≤(𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐼^𝛼_Δ,𝑡0𝐴𝑧₀(𝜏)−𝑦₀(𝜏𝑡)=)(𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝜂−𝜂𝐴𝐼^𝛼_Δ,𝑡0ℎ_𝛼−1𝜏,𝑡₀𝑡)=(𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝜂−𝜂𝐴ℎ_2𝛼−1𝑡,𝑡₀=𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐴ℎ_2𝛼−1𝑡,𝑡₀. (135) Suppose that 𝑧_𝑚−1(𝑡)−𝑦_𝑚−1(𝑡≤)𝜂−𝜂^𝑚−1_𝑗=0𝐴^𝑗ℎ_{𝑗𝛼+𝛼−1}𝑡,𝑡₀. (136) Then 𝑧_𝑚(𝑡)−𝑦_𝑚(𝑡≤)𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐼^𝛼_Δ,𝑡0𝑓𝜏,𝑧_𝑚−1(𝜏)−𝑓𝜏,𝑦_𝑚−1(𝜏)𝑡)≤(𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐼^𝛼_Δ,𝑡0𝐴𝑧_𝑚−1(𝜏)−𝑦_𝑚−1(𝜏(𝑡)≤)𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐼^𝛼_Δ,𝑡0𝐴𝜂−𝜂^𝑚−1_𝑗=0𝐴^𝑗ℎ_{𝑗𝛼+𝛼−1}𝜏,𝑡₀(𝑡)=𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝐴𝜂−𝜂^𝑚−1_𝑗=0𝐴^𝑗𝐼^𝛼_Δ,𝑡0ℎ_{𝑗𝛼+𝛼−1}𝜏,𝑡₀(𝑡)=𝜂−𝜂ℎ_𝛼−1𝑡,𝑡₀+𝜂−𝜂^𝑚−1_𝑗=0𝐴^𝑗+1ℎ_{(𝑗+1)𝛼+𝛼−1}𝑡,𝑡₀=𝜂−𝜂^𝑚_𝑗=0𝐴^𝑗ℎ_{𝑗𝛼+𝛼−1}𝑡,𝑡₀. (137) According to mathematical induction, we have 𝑧_𝑚(𝑡)−𝑦_𝑚(𝑡)≤𝜂−𝜂^𝑚_𝑗=0𝐴^𝑗ℎ_{𝑗𝛼+𝛼−1}𝑡,𝑡₀. (138) Taking the limit 𝑚→∞, we obtain that 𝑧(𝑡)−𝑦(𝑡)≤𝜂−𝜂^∞_𝑗=0𝐴^𝑗ℎ_{𝑗𝛼+𝛼−1}𝑡,𝑡₀≤𝜂−𝜂_Δ𝐹_𝛼,𝛼𝐴,𝑏,𝑡₀. (139)

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 ^𝑚_𝑘=1𝐴_𝑘𝐷^𝛼𝑘_Δ,𝑡0𝑦(𝑡)+𝐴₀𝑦(𝑡)=0𝑡>𝑡₀;𝑚∈ℕ;0<𝛼₁<⋯<𝛼_𝑚 (140) with the Liouville fractional derivatives 𝐷^𝛼𝑘_Δ,𝑡0𝑦(𝑘=1,…,𝑚). Here 𝐴_𝑘∈ℝ(𝑘=0,…,𝑚) are real constants, and, generally speaking, we can take 𝐴_𝑚=1.

In order to solve the equation, we need the following Laplace transform formula: ℒ𝐷^𝛼_Δ,𝑡0𝑦(𝑡)𝑧,𝑡₀=𝑧^𝛼ℒ𝑦(𝑡)𝑧,𝑡₀−^𝑙_𝑗=1𝑑_𝑗𝑧^𝑗−1(),𝑑𝑙−1<𝛼≤𝑙∈ℕ_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦𝑡₀𝑗=1,…,𝑙. (141)(142) First, we derive explicit solutions to (140) with 𝑚=1: 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝑦(𝑡)=0𝑡>𝑡₀;𝑙−1<𝛼≤𝑙;𝑙∈ℕ;𝜆∈ℝ. (143) In order to prove our result, we need the following definition and lemma.

Definition 57. The function 𝑊_𝛼(𝑡) is defined by 𝑊_𝛼(𝑡)=det𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗(𝑡)^𝑛_𝑘,𝑗=1(𝑛=[𝛼]+1,𝑎≤𝑡≤𝑏). (144)

Lemma 58. The solutions 𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡) are linearly independent if and only if  𝑊_𝛼(𝑡^∗)≠0 at some point 𝑡^∗∈[𝑎,𝑏].

Proof. We first prove sufficiency. If, to the contrary, 𝑦_𝑗(𝑡)(𝑗=1,2,…,𝑛) are linearly dependent on Ω, then there exist 𝑛 constants {𝑐_𝑗}^𝑛_𝑗=1, not all zero, such that 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗(𝑡)^𝑛_𝑘,𝑗=1𝑐₁𝑐₂⋮𝑐_𝑛≡0 (145) holds, and thus 𝑊_𝛼(𝑡)≡0 which leads to a contradiction. Therefore, if 𝑊_𝛼(𝑡^∗)≠0 at some point 𝑡^∗∈Ω, then 𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡) are linearly independent. Now we prove the necessity. Suppose, to the contrary, for 𝑡∈Ω,  𝑊_𝛼(𝑡)=0. Consider 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗𝑡^∗𝑛_𝑘,𝑗=1𝐶=0, (146)where 𝑡^∗∈Ω, 𝐶=^𝑐1𝑐2_⋮𝑐𝑛. As 𝑊_𝛼(𝑡^∗)=0, the equations have nontrivial solution 𝑐_𝑗(𝑗=1,2,…,𝑛). Now we construct a function using these constants: 𝑦(𝑡)=^𝑛_𝑗=1𝑐_𝑗𝑦_𝑗(𝑡), (147) and we get that 𝑦(𝑡) is a solution. From (146), we obtain that 𝑦(𝑡) satisfies initial value condition 𝐷^𝛼−𝑘_Δ,𝑡0𝑦𝑡^∗=0,𝑘=1,…,𝑛. (148) However, 𝑦(𝑡)=0 is also a solution to equation satisfying the initial value condition. By the uniqueness of solution, we have ^𝑛_𝑗=1𝑐_𝑗𝑦_𝑗(𝑡)=0, (149) and, thus 𝑦_𝑗(𝑡)(𝑗=1,2,…,𝑛) are linearly dependant, which leads to a contradiction. Thus, if the solutions 𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡) are linearly independent, then 𝑊_𝛼(𝑡^∗)≠0 at some point 𝑡^∗∈Ω.

There hold the following statements.

Theorem 59. Let 𝑙−1<𝛼≤𝑙(𝑙∈ℕ) and 𝜆∈ℝ. Then the functions 𝑦_𝑗(𝑡)=_Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀=^∞_𝑘=0𝜆^𝑘ℎ_{𝑘𝛼+𝛼−𝑗}𝑡,𝑡₀𝑗=1,…,𝑙 (150) yield the fundamental system of solutions to (143). Moreover, 𝑦_𝑗(𝑡), 𝑗=1,2,…,𝑙, satisfy 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗𝑡₀,𝐷=0𝑘,𝑗=1,…,𝑙;𝑘≠𝑗_{𝛼−𝑘Δ,𝑡0}𝑦_𝑘𝑡₀=1(𝑘=1,…,𝑙). (151)

Proof. Applying the Laplace transform to (143) and taking (141) into account, we have ℒ𝑦(𝑡)𝑧,𝑡₀=^𝑙_𝑗=1𝑑_𝑗𝑧^𝑗−1𝑧^𝛼,−𝜆 (152) where 𝑑_𝑗(𝑗=1,…,𝑙) are given by (142).
Formula (49) with 𝛽=𝛼+1−𝑗 yields ℒ_Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀𝑧,𝑡₀=𝑧^𝑗−1𝑧^𝛼|𝜆|<|𝑧|−𝜆^𝛼. (153) Thus, from (152), we derive the following solution to (143): 𝑦(𝑡)=^𝑙_𝑗=1𝑑_𝑗𝑦_𝑗(𝑡),𝑦_𝑗(𝑡)=_Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀. (154) It is easily verified that the functions 𝑦_𝑗(𝑡) are solutions to (143): 𝐷^𝛼_Δ,𝑡0Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀=𝜆_Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀𝑗=1,…,𝑙. (155) In fact, 𝐷^𝛼_Δ,𝑡0Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀=𝐷^𝛼_Δ,𝑡0^∞_𝑘=0𝜆^𝑘ℎ_{𝑘𝛼+𝛼−𝑗}𝑡,𝑡₀=^∞_𝑘=0𝜆^𝑘ℎ_{𝑘𝛼−𝑗}𝑡,𝑡₀=^∞_𝑘=−1𝜆^𝑘+1ℎ_{(𝑘+1)𝛼−𝑗}𝑡,𝑡₀=𝜆⁰ℎ_−𝑗𝑡,𝑡₀+^∞_𝑘=0𝜆^𝑘+1ℎ_{(𝑘+1)𝛼−𝑗}𝑡,𝑡₀=𝜆^∞_𝑘=0𝜆^𝑘ℎ_{𝑘𝛼+𝛼−𝑗}𝑡,𝑡₀=𝜆_Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀. (156) Moreover, 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗(𝑡)=𝐷^𝛼−𝑘_Δ,𝑡0^∞_𝑠=0𝜆^𝑠ℎ_{𝑠𝛼+𝛼−𝑗}𝑡,𝑡₀=^∞_𝑠=0𝜆^𝑠ℎ_{𝑠𝛼+𝑘−𝑗}𝑡,𝑡₀. (157) It follows from (157) that 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗𝑡₀,𝐷=0𝑘,𝑗=1,…,𝑙;𝑘>𝑗_{𝛼−𝑘Δ,𝑡0}𝑦_𝑘𝑡₀=1(𝑘=1,…,𝑙). (158) If 𝑘<𝑗, then 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗(𝑡)=^∞_𝑠=1𝜆^𝑠ℎ_{𝑠𝛼+𝑘−𝑗}𝑡,𝑡₀=^∞_𝑠=0𝜆^𝑠+1ℎ_{𝑠𝛼+𝛼+𝑘−𝑗}𝑡,𝑡₀, (159) and since 𝛼+𝑘−𝑗≥𝛼+1−𝑙>0 for any 𝑘,𝑗=1,…,𝑙, the following relations hold: 𝐷^𝛼−𝑘_Δ,𝑡0𝑦_𝑗𝑡₀=0𝑘,𝑗=1,…,𝑙;𝑘<𝑗. (160)
By (158) and (160), 𝑊_𝛼(𝑡₀)=1. Then 𝑦_𝑗(𝑡), 𝑗=1,…,𝑙, yield the fundamental system of solutions to (143).

Corollary 60. Consider that 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝑦(𝑡)=0𝑡>𝑡₀;0<𝛼≤1;𝜆∈ℝ (161) has its solution given by 𝑦(𝑡)=_Δ𝐹_𝛼,𝛼𝜆,𝑡,𝑡₀, (162) while 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝑦(𝑡)=0𝑡>𝑡₀;1<𝛼≤2;𝜆∈ℝ (163) has the fundamental system of solutions given by 𝑦₁(𝑡)=_Δ𝐹_𝛼,𝛼𝜆,𝑡,𝑡₀,𝑦₂(𝑡)=_Δ𝐹_{𝛼,𝛼−1}𝜆,𝑡,𝑡₀. (164)

Next we derive the explicit solutions to (140) with 𝑚=2 of the form 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝐷^𝛽_Δ,𝑡0𝑦(𝑡)−𝜇𝑦(𝑡)=0,𝑡>𝑡₀;𝑙−1<𝛼≤𝑙;𝑙∈ℕ;0<𝛽<𝛼 (165) with 𝜆,𝜇∈ℝ.

Theorem 61. Let 𝑙−1<𝛼≤𝑙(𝑙∈ℕ), 0<𝛽<𝛼, and 𝜆,𝜇∈ℝ. Then the functions 𝑦_𝑗(𝑡)=^∞_𝑘=0𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘Δ𝐹_{𝛼−𝛽,𝛼+𝑘𝛽+1−𝑗}𝜆,𝑡,𝑡₀ (166) 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 𝑦_𝑗(𝑡), 𝑗=1,2,…,𝑙, in (166) satisfy (151).

Proof. Let 𝑚−1<𝛽≤𝑚(𝑚≤𝑙;𝑚∈ℕ). Applying the Laplace transform to (165) and using (141) as in (152), we obtain ℒ𝑦(𝑡)𝑧,𝑡₀=^𝑙_𝑗=1𝑑_𝑗𝑧^𝑗−1𝑧^𝛼−𝜆𝑧^𝛽,−𝜇 (167) where 𝑑_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦(𝑡₀)−𝜆𝐷^𝛽−𝑗_Δ,𝑡0𝑦(𝑡₀)(𝑗=1,…,𝑚), 𝑑_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦(𝑡₀)(𝑗=𝑚+1,…,𝑙).
For 𝑧∈ℂ and |𝜇𝑧^−𝛽/(𝑧^𝛼−𝛽−𝜆)|<1, we have 1𝑧^𝛼−𝜆𝑧^𝛽=𝑧−𝜇𝑧^{−𝛽𝛼−𝛽}⋅1−𝜆1−𝜇𝑧^−𝛽/𝑧^𝛼−𝛽=−𝜆_∞𝑘=0𝜇^𝑘𝑧^{−𝛽−𝑘𝛽}𝑧^𝛼−𝛽−𝜆^𝑘+1, (168) and hence (167) has the following representation: ℒ𝑦(𝑡)𝑧,𝑡₀=^𝑙_𝑗=1𝑑_𝑗^∞_𝑘=0𝜇^𝑘𝑧^{𝑗−1−𝛽−𝑘𝛽}𝑧^𝛼−𝛽−𝜆^𝑘+1. (169) By (51), for 𝑧∈ℂ and |𝜆𝑧^𝛽−𝛼|<1, we have 𝑧^{𝑗−1−𝛽−𝑘𝛽}𝑧^𝛼−𝛽−𝜆^𝑘+1=𝑧^{(𝛼−𝛽)−(𝛼+𝑘𝛽+1−𝑗)}𝑧^𝛼−𝛽−𝜆^𝑘+1=1ℒ𝜕𝑘!𝜕𝜆^𝑘𝑘Δ𝐹_{𝛼−𝛽,𝛼+𝑘𝛽+1−𝑗}𝜆,𝑡,𝑡₀𝑧,𝑡₀. (170) From (169) and (170), we derive the solution to (165) as 𝑦(𝑡)=^𝑙_𝑗=1𝑑_𝑗𝑦_𝑗(𝑡), (171) where 𝑦_𝑗(𝑡)(𝑗=1,…,𝑙) are given by (166). For 𝑞,𝑗=1,…,𝑙, the direct evaluation yields 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡)=𝐷^𝛼−𝑞_Δ,𝑡0^∞_𝑘=0𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘Δ𝐹_{𝛼−𝛽,𝛼+𝑘𝛽+1−𝑗}𝜆,𝑡,𝑡₀=^∞_𝑘=0𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘𝐷^𝛼−𝑞_Δ,𝑡0^∞_𝑠=0𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝛼+𝑘𝛽−𝑗}𝑡,𝑡₀=^∞_𝑘=0𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘∞_𝑠=0𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝑘𝛽+𝑞−𝑗}𝑡,𝑡₀. (172) For 𝑞>𝑗, 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡₀)=0, and for 𝑞=𝑗, 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡₀)=1. Thus we have 𝑊_𝛼(𝑡₀)=1. It follows from Lemma 58 that the functions 𝑦_𝑗(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: 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡)=𝐷^𝛼−𝑞_Δ,𝑡0ℎ_𝛼−𝑗𝑡,𝑡₀+^∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝑞−𝑗}𝑡,𝑡₀+^∞_𝑘=1𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘ℎ_{𝑘𝛽+𝑞−𝑗}𝑡,𝑡₀+^∞_𝑘=1𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝑘𝛽+𝑞−𝑗}𝑡,𝑡₀=^∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝑞−𝑗}𝑡,𝑡₀+^∞_𝑘=1𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘ℎ_{𝑘𝛽+𝑞−𝑗}𝑡,𝑡₀+^∞_𝑘=1𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝑘𝛽+𝑞−𝑗}𝑡,𝑡₀. (173) 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 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡₀)=0. Thus the relations in (151) are valid. The proof is finished.

Corollary 62. Consider that 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝐷^𝛽_Δ,𝑡0𝑦(𝑡)=0𝑡>𝑡₀;𝑙−1<𝛼≤𝑙;𝑙∈ℕ;0<𝛽<𝛼 (174) has its fundamental system of solution given by 𝑦_𝑗(𝑡)=_Δ𝐹_{𝛼−𝛽,𝛼+1−𝑗}𝜆,𝑡,𝑡₀𝑗=1,…,𝑙. (175)

Finally, we find the explicit solutions to (140) with any 𝑚∈ℕ⧵{1,2}. It is convenient to rewrite (140) in the form 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝐷^𝛽_Δ,𝑡0𝑦(𝑡)−^𝑚−2_𝑘=0𝐴_𝑘𝐷^𝛼𝑘_Δ,𝑡0𝑦(𝑡)=0𝑡>𝑡₀;𝑚∈ℕ⧵{1,2};0=𝛼₀<𝛼₁<⋯<𝛼_𝑚−2<𝛽<𝛼;𝜆,𝐴₀,…,𝐴_𝑚−2∈ℝ. (176)

Theorem 63. Let 𝑚∈ℕ⧵{1,2}, 𝑙−1<𝛼≤𝑙(𝑙∈ℕ), let 𝛽 and 𝛼₁,…,𝛼_𝑚−2 be such that 𝛼>𝛽>𝛼_𝑚−2>⋯>𝛼₁>𝛼₀=0, and let 𝜆,𝐴₀,…,𝐴_𝑚−2∈ℝ. Then the functions 𝑦_𝑗(𝑡)=^∞_{𝑛=0𝑘0+⋯+𝑘𝑚−2=𝑛}1𝑘₀!⋯𝑘_𝑚−2!×^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈𝜕^𝑛𝜕𝜆^𝑛Δ𝐹_{𝛼−𝛽,𝛼+1−𝑗+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈}𝜆,𝑡,𝑡₀ (177) yield the fundamental system of solutions to (176) provided that the series in (177) are convergent. The inner sum is taken over all 𝑘₀,…,𝑘_𝑚−2∈ℕ₀ such that 𝑘₀+⋯+𝑘_𝑚−2=𝑛. Moreover, if 𝛼+1−𝑙>𝛽>𝛼_𝑚−2+𝑙−1, then 𝑦_𝑗(𝑡), 𝑗=1,2,…,𝑙, in (177) satisfy (151).

Proof. Let 𝑙_𝑚−1−1<𝛽≤𝑙_𝑚−1, 𝑙_𝑘−1<𝛼_𝑘<𝑙_𝑘(𝑘=1,…,𝑚−2;0≤𝑙₁≤⋯≤𝑙_𝑚−1≤𝑙). Applying the Laplace transform to (176) and using (141) as in (167), we obtain ℒ𝑦(𝑡)𝑧,𝑡₀=^𝑙_𝑗=1𝑑_𝑗𝑧^𝑗−1𝑧^𝛼−𝜆𝑧^𝛽−∑^𝑚−2_𝑘=0𝐴_𝑘𝑧^𝛼𝑘, (178) where 𝑑_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦𝑡₀−𝜆𝐷^𝛽−𝑗_Δ,𝑡0𝑦𝑡₀−^𝑚−2_𝑘=1𝐴_𝑘𝐷^{𝛼𝑘−𝑗}_Δ,𝑡0𝑦𝑡₀𝑗=1,…,𝑙₁,𝑑_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦𝑡₀−𝜆𝐷^𝛽−𝑗_Δ,𝑡0𝑦𝑡₀−^𝑚−2_𝑘=2𝐴_𝑘𝐷^{𝛼𝑘−𝑗}_Δ,𝑡0𝑦𝑡₀𝑗=𝑙₁+1…,𝑙₂,⋮𝑑_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦𝑡₀−𝜆𝐷^𝛽−𝑗_Δ,𝑡0𝑦𝑡₀𝑗=𝑙_𝑚−2+1,…,𝑙_𝑚−1,𝑑_𝑗=𝐷^𝛼−𝑗_Δ,𝑡0𝑦𝑡₀𝑗=𝑙_𝑚−1+1,…,𝑙. (179) Here ∑^𝑛_𝑘=𝑚𝐴_𝑘∶=0(𝑚>𝑛). For 𝑧∈ℂ and |∑^𝑚−2_𝑘=0𝐴_𝑘𝑧^{𝛼𝑘−𝛽}/(𝑧^𝛼−𝛽−𝜆)|<1, we have 1𝑧^𝛼−𝜆𝑧^𝛽−∑^𝑚−2_𝑘=0𝐴_𝑘𝑧^𝛼𝑘=𝑧^−𝛽𝑧^𝛼−𝛽⋅1−𝜆1−∑^𝑚−2_𝑘=0𝐴_𝑘𝑧^{𝛼𝑘−𝛽}/𝑧^𝛼−𝛽=−𝜆_∞𝑛=0𝑧^−𝛽𝑧^𝛼−𝛽−𝜆^{𝑛+1𝑚−2}_𝑘=0𝐴_𝑘𝑧^{𝛼𝑘−𝛽𝑛}=^∞_{𝑛=0𝑘0+⋯+𝑘𝑚−2=𝑛}𝑛!𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈𝑧^{−𝛽−∑𝑚−2𝜈=0(𝛽−𝛼𝜈)𝑘𝜈}𝑧^𝛼−𝛽−𝜆^𝑛+1, (180) if we also take into account the following relation: 𝑥₀+⋯+𝑥_𝑚−2^𝑛=_{𝑘0+⋯+𝑘𝑚−2=𝑛}𝑛!𝑘₀!⋅⋅⋅𝑘_𝑚−2!^𝑚−2_𝜈=0𝑥^𝑘𝜈_𝜐, (181) where the summation is taken over all 𝑘₀,…,𝑘_𝑚−2∈ℕ₀ such that 𝑘₀+⋯+𝑘_𝑚−2=𝑛.
In addition, for 𝑧∈ℂ and |𝜆𝑧^𝛽−𝛼|<1, we have 𝑧^{𝑗−1−𝛽−∑𝑚−2𝜈=0(𝛽−𝛼𝜈)𝑘𝜈}𝑧^𝛼−𝛽−𝜆^𝑛+1=𝑧^{(𝛼−𝛽)−(𝛼+1−𝑗+∑𝑚−2𝜈=0(𝛽−𝛼𝜈)𝑘𝜈)}𝑧^𝛼−𝛽−𝜆^𝑛+1=1ℒ𝑛!^Δ,𝑡0𝜕^𝑛𝜕𝜆^𝑛∇𝐹_{𝛼−𝛽,𝛼+1−𝑗+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈}𝜆;𝑡,𝑡₀(𝑧). (182) From (178), (180), and (182), we derive the solution to (176), as 𝑦(𝑡)=^𝑙_𝑗=1𝑑_𝑗𝑦_𝑗(𝑡), (183) which shows that arbitrary solution 𝑦(𝑡) can be expressed by 𝑦_𝑗(𝑡), 𝑗=1,…,𝑙, where 𝑦_𝑗(𝑡)(𝑗=1,…,𝑙) are given by (177). For 𝑞, 𝑗=1,…,𝑙, the direct evaluation yields 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡)=𝐷^𝛼−𝑞_Δ,𝑡0^∞_{𝑛=0𝑘0+⋅⋅⋅+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈×𝜕^𝑛𝜕𝜆^𝑛Δ𝐹_{𝛼−𝛽,𝛼+1−𝑗+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈}𝜆;𝑡,𝑡₀=^∞_{𝑛=0𝑘0+⋅⋅⋅+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈×𝜕^𝑛𝜕𝜆^𝑛𝐷^𝛼−𝑞_Δ,𝑡0^∞_𝑠=0𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝛼−𝑗+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈}𝑡,𝑡₀=^∞_{𝑛=0𝑘0+⋅⋅⋅+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈×𝜕^𝑛𝜕𝜆^𝑛∞_𝑠=0𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈+𝑞−𝑗}𝑡,𝑡₀. (184) For 𝑞>𝑗, 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡₀)=0, and for 𝑞=𝑗, 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡₀)=1. Thus we have 𝑊_𝛼(𝑡₀)=1. It follows from Lemma 58 that the functions 𝑦_𝑗(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: 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡)=𝐷^𝛼−𝑞_Δ,𝑡0ℎ_𝛼−𝑗𝑡,𝑡₀+^∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈+𝑞−𝑗}𝑡,𝑡₀+^∞_{𝑛=1𝑘0+⋯+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈𝜕^𝑛𝜕𝜆^𝑛ℎ_∑^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈+𝑞−𝑗}𝑡,𝑡₀+^∞_{𝑛=1𝑘0+⋯+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈𝜕^𝑛𝜕𝜆^𝑛∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈+𝑞−𝑗}𝑡,𝑡₀=^∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+𝑞−𝑗}𝑡,𝑡₀+^∞_{𝑛=1𝑘0+⋯+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈𝜕^𝑛𝜕𝜆^𝑛ℎ_∑^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈+𝑞−𝑗}𝑡,𝑡₀+^∞_{𝑛=1𝑘0+⋯+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜈=0𝐴_𝜈^𝑘𝜈𝜕^𝑛𝜕𝜆^𝑛∞_𝑠=1𝜆^𝑠ℎ_{𝑠(𝛼−𝛽)+∑}^𝑚−2_{𝜈=0(𝛽−𝛼𝜈)𝑘𝜈+𝑞−𝑗}𝑡,𝑡₀. (185) If 𝛼+1−𝑙>𝛽>𝛼_𝑚−2+𝑙−1, then 𝑠(𝛼−𝛽)+𝑞−𝑗≥(𝛼−𝛽)+1−𝑙>0 for 𝑛=0,𝑞, 𝑗=1,…,𝑙,𝑠∈ℕ⁺, and ∑^𝑚−2_𝜈=0(𝛽−𝛼_𝜈)𝑘_𝜈+𝑞−𝑗≥𝛽−𝛼_𝑖+1−𝑙≥𝛽−𝛼_𝑚−2+1−𝑙>0 for 𝑠=0, 𝑞, 𝑗=1,…,𝑙,𝑛∈ℕ⁺. Besides, we also have 𝑠(𝛼−𝛽)+∑^𝑚−2_𝜈=0(𝛽−𝛼_𝜈)𝑘_𝜈+𝑞−𝑗 ≥(𝛼−𝛽) +𝛽 −𝛼_𝑖+1 −𝑙 ≥(𝛼−𝛽)+𝛽−𝛼_𝑚−2+1−𝑙>0 for 𝑞,𝑗=1,…,𝑙,𝑠,𝑛∈ℕ⁺. These imply that 𝐷^𝛼−𝑞_Δ,𝑡0𝑦_𝑗(𝑡₀)=0. Thus the relations in (151) are valid. The result follows.

8. 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 ^𝑚_𝑘=1𝐴_𝑘𝐷^𝛼𝑘_Δ,𝑡0𝑦(𝑡)+𝐴₀𝑦(𝑡)=𝑓(𝑡)𝑡>𝑡₀;0<𝛼₁<⋯<𝛼_𝑚;𝑚∈ℕ (186) with real 𝐴_𝑘∈ℝ(𝑘=0,…,𝑚).

By (141)-(142), for suitable functions 𝑦, the Laplace transform of 𝐷^𝛼_Δ,𝑡0𝑦 is given by ℒ𝐷^𝛼_Δ,𝑡0𝑦(𝑡)𝑧,𝑡₀=𝑧^𝛼ℒ𝑦(𝑡)𝑧,𝑡₀. (187) Applying the Laplace transform to (186) and taking (187) into account, we have 𝐴₀+^𝑚_𝑘=1𝐴_𝑘𝑧^𝛼𝑘ℒ𝑦(𝑡)𝑧,𝑡₀=ℒ𝑓(𝑡)𝑧,𝑡₀. (188) Using the inverse Laplace transform ℒ_Δ⁻¹ from here, we obtain a particular solution to (186) in the form 𝑦(𝑡)=ℒ⁻¹ℒ𝑓(𝑡)𝑧,𝑡₀𝐴₀+∑^𝑚_𝑘=1𝐴_𝑘𝑧^𝛼𝑘(𝑡). (189)

Using the Laplace convolution formula ℒ𝑓∗𝑔𝑧,𝑡₀=ℒ𝑓𝑧,𝑡₀ℒ𝑔𝑧,𝑡₀, (190) we can introduce the Laplace fractional analog of the Green function as follows: 𝐺_{𝛼1,…,𝛼𝑚}(𝑡)=ℒ⁻¹1𝑃_{𝛼1,…,𝛼𝑚}(𝑧)𝑡),𝑃_{𝛼1,…,𝛼𝑚}((𝑧)=𝐴₀+^𝑚_𝑘=1𝐴_𝑘𝑧^𝛼𝑘, (191) and we can express a particular solution of (152) in the form of the Laplace convolution 𝐺_{𝛼1,…,𝛼𝑚}(𝑡) and ℎ_𝑟(𝑡,𝑡₀): 𝑦(𝑡)=𝑓∗𝐺_{𝛼1,…,𝛼𝑚}(𝑡). (192) Generally speaking, we can consider (186) with 𝐴_𝑚=1. First we derive a particular solution to (186) with 𝑚=1 in the form 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝑦(𝑡)=𝑓(𝑡)𝑡>𝑡₀;𝛼>0. (193)

Theorem 64. Let 𝛼>0, 𝜆∈ℝ. Then (193) is solvable, and its particular solution has the form 𝑦(𝑡)=𝑓∗_Δ𝐹_𝛼,𝛼𝜆,𝑠,𝑡₀(𝑡) (194) 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 𝐺_𝛼(𝑡)=ℒ⁻¹1𝑧^𝛼𝑡)=−𝜆(_Δ𝐹_𝛼,𝛼𝜆,𝑡,𝑡₀. (195) Thus (192), with 𝐺_{𝛼1,…,𝛼𝑚}(𝑡)=𝐺_𝛼(𝑡), yields (194). Theorem is proved.

Next we derive a particular solution to (186) with 𝑚=2 of the form 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝐷^𝛽_Δ,𝑡0𝑦(𝑡)−𝜇𝑦(𝑡)=𝑓(𝑡),𝑡>𝑡₀;𝛼>𝛽>0. (196)

Theorem 65. Let 𝛼>𝛽>0, 𝜆,𝜇∈ℝ. Then (196) is solvable, and its particular solution has the form 𝑦(𝑡)=𝑓∗𝐺_{𝛼,𝛽;𝜆,𝜇}(𝑠𝑡)𝐺)(,_{𝛼,𝛽;𝜆,𝜇}(𝑡)=^∞_𝑘=0𝜇^𝑘𝜕𝑘!𝜕𝜆^𝑘𝑘Δ𝐹_{𝛼−𝛽,𝛼+𝑘𝛽}𝜆,𝑡,𝑡₀ (197)(198) 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 𝐺_𝛼,𝛽(𝑡)=ℒ⁻¹1𝑧^𝛼−𝜆𝑧^𝛽𝑡)−𝜇(. (199) According to (168) for 𝑧∈ℂ and |𝜇𝑧^−𝛽/(𝑧^𝛼−𝛽−𝜆)|<1, we have 𝐺_{𝛼,𝛽;𝜆,𝜇}(𝑡)=ℒ^−1∞_𝑛=0𝜇^𝑛𝑧^{−𝛽−𝑛𝛽}𝑧^𝛼−𝛽−𝜆^𝑛+1(𝑡). (200) In addition, for 𝑧∈ℂ and |𝜆𝑧^𝛽−𝛼|<1, we have 𝑧^{−𝛽−𝑛𝛽}𝑧^𝛼−𝛽−𝜆^𝑛+1=1ℒ𝜕𝑛!𝜕𝜆^𝑛𝑛Δ𝐹_{𝛼−𝛽,𝛼+𝑛𝛽}𝜆,𝑡,𝑡₀𝑧,𝑡₀, (201) and hence (200) takes the following form: 𝐺_{𝛼,𝛽;𝜆,𝜇}(𝑡)=^∞_𝑛=0𝜇^𝑛𝜕𝑛!𝜕𝜆^𝑛𝑛Δ𝐹_{𝛼−𝛽,𝛼+𝑛𝛽}𝜆,𝑡,𝑡₀. (202) Thus the result in (197) follows from (192) with 𝐺_{𝛼1,…,𝛼𝑚}(𝑡)=𝐺_{𝛼,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 𝐷^𝛼_Δ,𝑡0𝑦(𝑡)−𝜆𝐷^𝛽_Δ,𝑡0𝑦(𝑡)−^𝑚−2_𝑘=1𝐴_𝑘𝐷^𝛼𝑘_Δ,𝑡0𝑦(𝑡)−𝐴₀𝑦(𝑡)=ℎ_𝑟𝑡,𝑡₀𝑡>𝑡₀ (203) with 𝑚∈ℕ⧵{1,2}, 0<𝛼₁<⋯<𝛼_𝑚−2<𝛽<𝛼, and 𝜆,𝐴₀,…,𝐴_𝑚−2∈ℝ.

Theorem 66. Let 𝑚∈ℕ⧵{1,2}, 0=𝛼₀<𝛼₁<⋯<𝛼_𝑚−2<𝛽<𝛼, and let 𝜆,𝐴₀,…,𝐴_𝑚−2∈ℝ. Then (203) is solvable, and its particular solution has the form 𝑦(𝑡)=𝑓∗𝐺_{𝛼1,…,𝛼𝑚−2,𝛽,𝛼;𝜆}(𝑠𝑡)𝐺)(,_{𝛼1,…,𝛼𝑚−2,𝛽,𝛼;𝜆}(𝑡)=^∞_𝑛=0∑_{𝑘0+⋯+𝑘𝑚−2=𝑛}1𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜐=0𝐴_𝜐^𝑘𝜐𝜕^𝑛𝜕𝜆^𝑛Δ𝐹_{𝛼−𝛽,𝛼+∑}^𝑚−2_{𝜐=0(𝛽−𝛼𝜐)𝑘𝜐}𝜆,𝑡,𝑡₀ (204)(205) provided that the series (205) and integral in (204) are convergent. The inner sum is taken over all 𝑘₀,…,𝑘_𝑚−2 such that 𝑘₀+⋯+𝑘_𝑚−2=𝑛.

Proof. Equation (203) is the same equation as (186) with 𝛼_𝑚=𝛼, 𝛼_𝑚−1=𝛽, 𝐴_𝑚=1,…,𝐴_𝑚−1=−𝜆, and with −𝐴_𝑘 instead of 𝐴_𝑘 for 𝑘=0,…,𝑚−2. Since 𝛼₀=0, (191) takes the form 𝐺_{𝛼1,…,𝛼𝑚−2,𝛽,𝛼;𝜆}(𝑡)=ℒ⁻¹1𝑧^𝛼−𝜆𝛼^𝛽−∑^𝑚−2_𝑘=0𝐴_𝑘𝑧^𝛼𝑘(𝑡). (206) For 𝑧∈ℂ and |∑^𝑚−2_𝑘=0𝐴_𝑘𝑧^{𝛼𝑘−𝛽}/(𝑧^𝛼−𝛽−𝜆)|<1, in accordance with (180), we have 𝐺_{𝛼1,…,𝛼𝑚−2,𝛽,𝛼;𝜆}(𝑡)=ℒ^−1∞_{𝑛=0𝑘0+⋯+𝑘𝑚−2=𝑛}𝑛!𝑘₀!⋅⋅⋅𝑘_𝑚−2!×^𝑚−2_𝜐=0𝐴_𝜐^𝑘𝜐𝑧^{−𝛽−∑𝑚−2𝜐=0(𝛽−𝛼𝜐)𝑘𝜐}𝑧^𝛼−𝛽−𝜆^𝑛+1(𝑡). (207) For 𝑧∈ℂ and |𝜆𝑧^𝛽−𝛼|<1, we have 𝑧^{−𝛽−∑𝑚−2𝜐=0(𝛽−𝛼𝜐)𝑘𝜐}𝑧^𝛼−𝛽−𝜆^𝑛+1=1ℒ𝜕𝑛!𝜕𝜆^𝑛𝑛Δ𝐹_{𝛼−𝛽,𝛼+∑}^𝑚−2_{𝜐=0(𝛽−𝛼𝜐)𝑘𝜐}𝜆,𝑡,𝑡₀𝑧,𝑡₀. (208) 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. Let 𝑙−1<𝛼≤𝑙(𝑙∈ℕ), 𝜆∈ℝ. Then (193) is solvable, and its general solution is given by 𝑦(𝑡)=^𝑙_𝑗=1𝑐_𝑗Δ𝐹_{𝛼,𝛼+1−𝑗}𝜆,𝑡,𝑡₀+𝑓(𝑡), (209) where 𝑐_𝑗(𝑗=1,…,𝑙) are arbitrary real constants.

Theorem 68. Let 𝑙−1<𝛼≤𝑙(𝑙∈ℕ), 0<𝛽<𝛼, 𝜆,𝜇∈ℝ. Then (196) is solvable, and its general solution has the form 𝑦(𝑡)=^𝑙_𝑗=1𝑐_𝑗^∞_𝑛=0𝜇^𝑛𝜕𝑛!𝜕𝜆^𝑛𝑛Δ𝐹_{𝛼−𝛽,𝛼+𝑛𝛽+1−𝑗}𝜆,𝑡,𝑡₀+𝑓∗𝐺_{𝛼,𝛽,𝜆,𝜇}(𝑠)(𝑡), (210) where 𝐺_{𝛼,𝛽,𝜆,𝜇}(𝑡) is given by (198) and 𝑐_𝑗(𝑗=1,…,𝑙) are arbitrary real constants.

Theorem 69. Let 𝑚∈ℕ⧵{1,2}, 𝑙−1<𝛼≤𝑙(𝑙∈ℕ), let 𝛽 and 𝛼₁,𝛼₂,…,𝛼_𝑚−2 be such that 𝛼>𝛽>𝛼_𝑚−2>⋯>𝛼₁>𝛼₀=0 and 𝛼−𝑙+1≥𝛽, and let 𝜆,𝐴₀,…,𝐴_𝑚−2∈ℝ. Then (203) is solvable, and its general solution is given by 𝑦(𝑡)=^𝑙_𝑗=1𝑐_𝑗^∞_{𝑛=0𝑘0+⋯+𝑘𝑚−2=𝑟}1𝑘₀!⋯𝑘_𝑚−2!×^𝑚−2_𝜐=0𝐴_𝜐^𝑘𝜐⋅𝜕^𝑛𝜕𝜆^𝑛Δ𝐹_{𝛼−𝛽,𝛼+1−𝑗+∑}^𝑚−2_{𝜐=0(𝛽−𝛼𝜐)𝑘𝜐}𝜆,𝑡,𝑡₀+𝑓∗𝐺_{𝛼1,…,𝛼𝑚−2,𝛽,𝛼;𝜆}(𝑠)(𝑡), (211) where 𝐺_{𝛼1,…,𝛼𝑚−2,𝛽,𝛼;𝜆}(𝑡) is given by (205) and 𝑐_𝑗(𝑗=1,…,𝑙) are arbitrary real constant.

9. Conclusions

In this paper, we first give a generalized definition of fractional Δ-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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly to the writing of this paper. All authors read and approved the final paper.

Acknowledgments

First, the authors are very grateful to the referees for their careful reading of the paper, and lots of valuable comments and suggestions, which greatly improve this manuscript. Next, this work was supported by the National Natural Science Foundation of China (11171286) and by Jiangsu Province Colleges and Universities Graduate Scientific Research Innovative Program (CXZZ12-0974).