New properties of the time-scale fractional operators with application to dynamic equations

We introduce new properties of Riemann–Liouville fractional integral and derivative on time scales. As well as sufficient conditions for existence and uniqueness of solution to an initial value problem for a class differential equations on time scales.


Introduction
Fractional calculus was introduced and developed by Leibniz, Liouville, Riemann, Letnikov, and Grünwald [15]. This branch was applied in physics, natural and social sciences. In recent years, there has been much research activity concerning the Fractional calculus of various dynamic equations. The theory of time scales was introduced by Stefan Hilger in his PhD thesis [22] in 1988, in order to unify and generalize continuous and discrete analysis. For more detailed discussions on the time scale calculus we refer to the books Peterson, 2001, 2003), see [17,18].
In 2016, Benkhettou et al. [21], introduced a concept of fractional derivative of Riemann-Liouville on time scales. Several authors have obtained important results about different subjects on time scales. See for instance M. Rchid et al [19], A. Abdeljawad et al [25], T. Gülsen et al [26].
The main purpose of this paper is to be deduced some new properties of the Riemann-Liouville fractional operator. As applications, we investigate fon existence and uniqueness of solutions some classes fractional dynamic equations.
The paper is organized as follows. In the next sections, we give some definitions and facts of time scale calculus. In Section 3, we establish some new properties of the Riemann-Liouville fractional operator. In Section 4, we investigate some IVPs for some classes fractional dynamic equations. In Section 5, we illustrate our results with examples.
If t is right scattered, then the ∆−derivative is defined by .
A function f : T → R is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit in all left-dense points. The set of rd-continuous functions f : .
Definition 2 (Fractional integral on time scales). [21] Suppose T is a time scale, [a, b] is an interval of T, and h is an integrable function on [a, b]. Let 0 < α < 1. Then the (left) fractional integral of order α of h is defined by where Γ is the gamma function.

Main Results
In this section, we present some of the new properties of the time-scale fractional operators.
The following counter example, to prove that the equality (1) is not always satisfied on time scales. Example 1 (Counter example). We take T = N, a = 1 and h : T → R, h (t) = 1. Let α > 0 and b > 0, then by Definition 2, we have By, the last equality, deduce On the other hand, we have Thus, and T Remark 1. By counter example 1, we conclude that T a I β t • T a I α t = T a I β+α t , for α > 0, β > 0 are not always correct on the time scales, which are proposition 16 in [21]. If you suggest a counterexample to T a D α t • T a I α t = Id and T a I α t • T a D α t = Id, α > 0, you leave to provide exact calculations in Before starting to introduce the properties of the time scale fractional operators, we present a new generalization for the Beta function on time scales.
Definition 4 (Beta function on time scales). We will define the function Remark 2. If T = R, a = 0 and b = 1, then Definition 4 takes the form .
The proof is complete.
Definition 5. Let λ ∈ R and a ∈ R we define the time scales λT and T + a by: λT : = {λt : t ∈ T} , T + a := {t + a : t ∈ T} . Definition 6. Let λ ∈ R, a ∈ R and let be the function v : Remark 3. Let λ ∈ R and a ∈ R, such as λ = 0, then function v λ,a is bijective and the inverse function v −1 λ,a given by , for all s ∈ λ (T + a) . Notation 1. We define the time scales T a,b by: Proposition 4. The Beta function on time scales satisfies the following useful property: , for all r ∈ T a,b . By the chain rule [17], we see that 0,1 (β, α) . The proof is complete.
for α > 0 and β > 0, where β Tu,t 0,1 (β, α) is defined as in Proposition 4. Proof. By Definition 4, we get From Fubini's theorem, we interchange the order of integration to obtain By Proposition 4, we obtain that ∆u.
The proof is complete.

Proof. Form Proposition 5 and Proposition 3, we have
The proof is complete.
Proof. By Definition 2, we have By Pötzsche's chain rule, we have The proof is complete.
Proof. Form Proposition 6, we have The proof is complete.
We consider the following initial value problem: where T a D α t is the Riemann-Liouville fractional derivative operator of order α defined on T. The problem (2) will be studied under the following as- Our main results give necessary and sufficient conditions for the existence and uniqueness of solution to the problem (2).
is a solution of the problem (2) if and only if it is a solution of the following integral equation: Proof. By Definition 2, we have . The proof is complete.
Our first main result is based on the Banach fixed point theorem [2].
for all x, y ∈ R and t ∈ [σ (a) , b] T . Proof. We transform the problem (2) into a fixed point problem. Consider the operator L : C ([σ (a) , b] T , R) → C ([σ (a) , b] T , R) defined by

If
We need to prove that L has a fixed point, which is a unique solution of (2) on [σ (a) , b] T . For that, we show that F is a contraction. Let x, y ∈ C ([σ (a) , b] T , R). For t ∈ [σ (a) , b] T , we have |f (s, x (s)) − f (s, y (s))| ∆s By (4), L is a contraction and thus, by the contraction mapping theorem, we deduce that L has a unique fixed point. This fixed point is the unique solution of (2). Now, we give our second main result guarantees the existence of at least one solution of the problem (2). This result is based on the Schauder's fixed point theorem [2]. Proof. We use Schauder's fixed point theorem to prove that L defined by (5) has a fixed point. The proof is given in several steps.
Step 1: L is continuous. Let x n be a sequence such that |f (s, x n (s)) − f (s, x (s))| ∆s Since f is a continuous function, we have Lx n → Lx in C ([σ (a) , b] T , R).
Step 2: The map L maps bounded sets into bounded sets in C ([σ(a), b] T , R). Indeed, it is enough to show that for any ε there exists a positive constant δ such that, for each x ∈ B (0, ε), we have Lx ∈ B (0, δ). By hypothesis, for each t ∈ [σ (a) , b] T , we get r (s) ϕ (x (s)) ∆s Step 3: The map L maps bounded sets into equicontinuous sets of Similarly, we get As t 1 → t 2 , the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we conclude that L : C ([σ (a) , b] T , R) → C ([σ (a) , b] T , R) is completely continuous. As a consequence of Schauder's fixed point theorem, we conclude that L has a fixed point, which is solution of the problem (2).
Since T a D α t • T a I α t = Id, and T a I α t • T a D α t = Id, for α > 0 are not always correct defined on the time scales. Then, if x is a solution to the problem (6), it has no permanent relationship the solution of integral equation (7).

Example
Remark 5. Let f : [a, b] T → R and a is right-scattered. By Definition 2, we have  Then σ (t) = t + h and µ (t) = 0. We consider the following initial value problem: