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Existence and Uniqueness of Solution for a Fractional Riemann-Liouville Initial Value Problem on Time Scales

We introduce the concept of fractional derivative of Riemann-Liouville on time scales. Fundamental properties of the new operator are proved, as well as an existence and uniqueness result for a fractional initial value problem on an arbitrary time scale.


Introduction
Let T be a time scale, that is, a closed subset of R. We consider the following initial value problem: where T t0 D α t is the (left) Riemann-Liouville fractional derivative operator or order α defined on T, T t0 I 1−α t the (left) Riemann-Liouville fractional integral operator or order 1 − α defined on T, and function f : J × T → R is a right-dense continuous function. Our main results give necessary and sufficient conditions for the existence and uniqueness of solution to problem (1)-(2).

Preliminaries
In this section, we collect notations, definitions, and results, which are needed in the sequel. We use C(J , R) for a Banach space of continuous functions y with the norm y ∞ = sup {|y(t)| : t ∈ J }, where J is an interval. A time scale T is an arbitrary nonempty closed subset of R. The reader interested on the calculus on time scales is referred to the books [8,9]. For a survey, see [2]. Any time scale T is a complete metric space with the distance d(t, s) = |t − s|, t, s ∈ T. Consequently, according to the well-known theory of general metric spaces, we have for T the fundamental concepts such as open balls (intervals), neighborhoods of points, open sets, closed sets, compact sets, etc. In particular, for a given number δ > 0, the δ-neighborhood U δ (t) of a given point t ∈ T is the set of all points s ∈ T such that d(t, s) < δ. We also have, for functions f : T → R, the concepts of limit, continuity, and the properties of continuous functions on a general complete metric space. Roughly speaking, the calculus on time scales begins by introducing and investigating the concept of derivative for functions f : T → R. In the definition of derivative, an important role is played by the so-called jump operators [9]. Definition 1. Let T be a time scale. For t ∈ T we define the forward jump operator σ : T → T by σ(t) := inf{s ∈ T : s > t}, and the backward jump operator ρ : T → T by ρ(t) := sup{s ∈ T : s < t}.
Remark 2. In Definition 1, we put inf ∅ = sup T (i.e., σ(M ) = M if T has a maximum M ) and sup ∅ = inf T (i.e., ρ(m) = m if T has a minimum m), where ∅ denotes the empty set.
If σ(t) > t, then we say that t is right-scattered; if ρ(t) < t, then t is said to be left-scattered. Points that are simultaneously right-scattered and left-scattered are called isolated. If t < sup T and σ(t) = t, then t is called right-dense; if t > inf T and ρ(t) = t, then t is called left-dense. The graininess function µ : The derivative makes use of the set T κ , which is derived from the time scale T as follows: if T has a left-scattered maximum M , then T κ := T \ {M }; otherwise, T κ := T.
Definition 3 (Delta derivative [1]). Assume f : T → R and let t ∈ T κ . We define provided the limit exists. We call f ∆ (t) the delta derivative (or Hilger derivative) of f at t. Moreover, we say that f is delta differentiable on T κ provided f ∆ (t) exists for all t ∈ T κ . The function f ∆ : T κ → R is then called the (delta) derivative of f on T κ .
Definition 4. A function f : T → R is called rd-continuous provided it is continuous at rightdense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions f : T → R is denoted by C rd . Similarly, a function f : T → R is called ld-continuous provided it is continuous at left-dense points in T and its right-sided limits exist (finite) at right-dense points in T. The set of ld-continuous functions f : T → R is denoted by C ld .
We also make use of the classical gamma and beta functions.
Definition 7 (Gamma function). For complex numbers with a positive real part, the gamma function Γ(t) is defined by the following convergent improper integral: Definition 8 (Beta function). The beta function, also called the Euler integral of first kind, is the special function B(x, y) defined by Remark 9. The gamma function satisfies the following useful property: Γ(t + 1) = tΓ(t). The beta function can be expressed through the gamma function by B(x, y) = Γ(x)Γ(y) Γ(x+y) .

Main Results
We introduce a new notion of fractional derivative on time scales. Before that, we define the fractional integral on a time scale T. This is in contrast with [5,6,7], where first a notion of fractional differentiation on time scales is introduced and only after that, with the help of such concept, the fraction integral is defined.
where Γ is the gamma function.
Definition 11 (Riemann-Liouville fractional derivative on time scales). Let T be a time scale, t ∈ T, 0 < α < 1, and h : (3) Remark 12. If T = R, then Definition 11 gives the classical (left) Riemann-Liouville fractional derivative [13]. For different extensions of the fractional derivative to time scales, using the Caputo approach instead of the Riemann-Liouville, see [3,4]. For local approaches to fractional calculus on time scales we refer the reader to [5,6,7]. Here we are only considering left operators. The corresponding right operators are easily obtained by changing the limits of integration in Definitions 10 and 11 from a to t (left of t) into t to b (right of t), as done in the classical fractional calculus [13]. Here we restrict ourselves to the delta approach to time scales. Analogous definitions are, however, trivially obtained for the nabla approach to time scales by using the duality theory of [10].
Along the work, we consider the order α of the fractional derivatives in the real interval (0, 1). We can, however, easily generalize our definition of fractional derivative to any positive real α. Indeed, let α ∈ R + \ N. Then there exists β ∈ (0, 1) such that α = ⌊α⌋ + β, where ⌊α⌋ is the integer part of α, and we can set Definition 13. If −1 < α < 0, then the (Riemann-Liouville) fractional derivative of order α is the fractional integral of order −α, that is, Definition 14. If −1 < α < 0, then the fractional integral of order α is the fractional derivative of order −α, that is,

Properties of the time-scale fractional operators
In this section we prove some fundamental properties of the fractional operators on time scales.
Proposition 15. Let T be a time scale with derivative ∆, and 0 < α < 1. Then, The proof is complete. Proof. By definition,    By setting s = u + r(t − u), r ∈ R, we obtain that The proof is complete.

Proof. By Propositions 15 and 16, we have
The proof is complete. Theorem 20. Let f ∈ C([a, b]) and α > 0. In order that f ∈ T a I α t ([a, b]), it is necessary and sufficient that and Proof. .
From Proposition 16, we have Conversely, assume that f ∈ C([a, b]) satisfies (4) and (5). Then, by Taylor's formula applied to function T a I Let ϕ(t) := ∆ ∆s T a I 1−α t f (t). Note that ϕ ∈ C([a, b]) by (4). Now, by Proposition 16, we have and thus From the uniqueness of solution to Abel's integral equation [12], this implies that f − T a I Proof. By Theorem 20 and Proposition 16, we have: The proof is complete.

Existence of Solutions to Fractional IVPs on Time Scales
In this section we prove existence of solution to the fractional order initial value problem (1) The proof is complete.
Our first result is based on the Banach fixed point theorem [11]. It is easy to see that S is a Banach space with this norm. The subset of S(ρ) and the operator T are defined by S(ρ) = {X ∈ S : X s ≤ ρ} and Then, Since (t − s) α−1 is an increasing monotone function, by using Proposition 6 we can write that Consequently, By considering ρ = Ma α Γ(α+1) , we conclude that T is an operator from S(ρ) to S(ρ). Moreover, x − y ∞ for x, y ∈ S(ρ). If La α Γ(α) ≤ 1, then it is a contraction map. This implies the existence and uniqueness of solution to problem (1)-(2).
Theorem 24. Suppose f : J × R → R is a rd-continuous bounded function such that there exists M > 0 with |f (t, y)| ≤ M for all t ∈ J , y ∈ R. Then problem (1)-(2) has a solution on J .
Proof. We use Schauder's fixed point theorem [11] to prove that T defined by (3) has a fixed point. The proof is given in several steps.
Step 1: T is continuous. Let y n be a sequence such that y n → y in C(J , R). Then, for each t ∈ J , B ρ be a bounded set of C(J , R) as in Step 2, and y ∈ B ρ . Then, 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 T : C(J , R) → C(J , R) is completely continuous.
We complete this step by considering the estimation in Step 2. As a consequence of Schauder's fixed point theorem, we conclude that T has a fixed point, which is solution of problem (1)- (2).