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The subject of fractional calculus (see [

On the other hand, in real applications, it is not always a continuous case, but also a discrete case. So, an useful tool as that time scale is considered. In order to unify differential equations and difference equations, Higer proposed firstly the time scale and built the relevant basic theories (see [

From Theorem

First, we present some preliminaries about time scales in [

A time scale

For

If

A function

The generalized polynomials are the functions

The generalized polynomials are the functions

Let

A subset

According to Theorem 4.13 in [

Let

From Theorem 4.13 in [

The increasing factorial function is defined as

Let

Let

In our discussion, we also need some information about

The function

For

If

If

If

Suppose that (

If

In this section, we first define

From now on, we always assume that

Assume that

The following result is needed frequently.

If

By Theorem

We now will use the Lemma

Assume that

Integration by parts and Lemma

By a similar way, we have

Assume that

By using integration by parts and Lemma

provided that (

Assume that

It follows from (

It is similar to the proof of Theorems 1.5 and 1.3 in [

If the functions

Let

By the uniqueness of inverse Laplace transform and fixing

We define fractional generalized

Applying the initial value theorem of Laplace transform, for

When

Next, in order to define fractional generalized

Let

For a given

Let

For given functions

The convolution is associative; that is,

If

Suppose that

In the following, we will define fractional generalized

Fractional generalized

According to convolution theorem and Definition

In particular, if

Now, we can give definitions of fractional

From now on, we will always denote

Let

Let

Throughout this paper, we denote

In the following, we will give the Laplace transform of fractional

Let

if

if

According to Definition

Finally, we present the definition of

As to the Laplace transform of

The Laplace transform of

From the definition of Laplace transform, it is obtained that

By differentiating

In this section, we mainly give the properties of fractional

Let

In particular, if

On the other hand, for

(1) According to Definition

From Property

If

As to the fractional sum and difference, there is also a similar result in [

Let

Let

By Taylor’s formula,

When

Let

Similarly, for the fractional sum and difference, there is also the following corollary.

For

The semigroup property of the fractional

If

According to Definition

The following assertion shows that the fractional differentiation is an operation inverse to the fractional integration from the left.

If

According to the definition of the fractional

In the following, we will derive the composition relations between fractional

If

The proof is the same with the proof of Property

Let

Since

Let

Applying Laplace transform to

To present the next property, we use the space of function

Let

If

If

Let

Let

Let

According to Property

From Property

Let

It follows from Property

Let

In this section, we consider Cauchy-type problem with Riemann-Liouville fractional nabla derivative

We discuss this Cauchy-type problem in the space

In particular, if

In the following, we prove that Cauchy-type problem and the nonlinear Volterra integral equation are equivalent in the sense that, if

Let

First we prove the necessity. Let

Now, we show that the relations in (

In the following, we establish the existence of a unique solution to the Cauchy-type problem (

Let

Since the Cauchy-type problem (

We define function sequences as follows:

Thus, from Lemma

By Weierstrass discriminance, we obtain

Thus, by (

Next, we consider the generalized Cauchy-type problem as follows:

Let

Assume that

Let the condition of Theorem

We consider fractional differential initial value problem (

Using Theorem

Let

By the proof of Theorem

As a special case, when fractional equation is linear, we can obtain its explicit solutions and we will explain it in the next section.

In this section, we apply the Laplace transform method to derive the fundamental system of solutions to homogeneous equations of the following form:

The Laplace transform method is based on the relation (

In order to prove our result, we also need the following definition and lemma.

The function

The solutions

We first prove sufficiency. If, to the contrary,

There holds the following statements.

Let

Applying the Laplace transform to (

Formula (

The following equation:

Next, we derive the explicit solutions to (

Let

Let

For

The following equation form:

Finally, we find the fundamental system of solutions to (

Let

Let

In addition, for

In above section, we applied the Laplace transform method to derive explicit solutions to the homogeneous equations (

By (

Using the Laplace convolution formula

Let

Equation (

Next, we derive a particular solution to (

Let

Equation (

Finally, we find a particular solution to (

Let

Equation (

As in the case of ordinary differential equations, a general solution to the nonhomogeneous equation (

Let

Let