Neutral functional sequential differential equations with Caputo fractional derivative on time scales

In this paper, we establish the existence and uniqueness of a solution for a class of initial value problems for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and Krasnoselskii fixed point theorem. As applications, two examples are included to show the applicability of our results.


Introduction
For the past decades, fractional differential equations have been of increasing importance due to their diverse applications in different fields, such as control theory, electrochemistry, viscoelasticity, electromagnetism, biology, economics, quantum calculus, etc. (see [2-10, 15, 17, 29-31, 34] and the references therein). On the other hand, the theory of differential equations on time scales has developed very intensively during the last decades (see for example [11, 12, 18-22, 24-27, 33] and the references therein). In 1988, Stefan Hilger [16,28] introduced in his thesis the concept of "calculation of chains of measures" in order to unify the discrete and continuous analysis. So, all results found will be valid in the discrete case and in the continuous case.
In [13], Ahmad and Ntouyas proved some existence results for the following initial value problem of Caputo-Hadamard sequential fractional order neutral functional differential equations: where D p , D q are the Caputo-Hadamard fractional derivatives, 0 < p, q < 1, F, G : In this paper, we generalize the problem considered in [13] to time scales, and we discuss existence and uniqueness of solutions to the following Cauchy problem of Caputo sequential fractional order neutral functional differential equations on time scale T: where c ω , c are the Caputo fractional derivatives, 0 < ω, The present paper is organized as follows. In Sect. 2, some notations are introduced. In Sect. 3, three results for problem (1)-(3) are proved by using the following fixed point theorems: the Banach contraction principle, the nonlinear alternative of Leray-Schauder type, and Krasnoselskii's fixed point theorem. Finally, in the last section, we give two examples to illustrate the applicability of our results.

Preliminaries
In this section, we collect notations, definitions, and results which are needed in the sequel.
Let C(J, R) be the Banach space of all continuous functions from J into R with the norm Also C is endowed with the norm

Some properties of time scale
A time scale T is an arbitrary nonempty closed subset of R (for more details, see [20][21][22][24][25][26][27]). If σ (τ ) > τ , then we say that τ is right-scattered; if ρ(τ ) < τ , then τ is said to be leftscattered. Points that are simultaneously right-scattered and left-scattered are called isolated. If τ < sup T and σ (τ ) = τ , then τ is called right-dense; if τ > inf T and ρ(τ ) = τ , then τ is called left-dense. 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.
provided the limit exists. We call h (τ ) the delta derivative (or Hilger derivative) of h at τ . Moreover, we say that h is delta differentiable on T κ provided h (τ ) exists for all τ ∈ T κ . The function h : T κ → R is then called the (delta) derivative of h on T κ .

Some properties of fractional calculus on time scales
We introduce a new notion of fractional derivative on time scales.
where is the gamma function.
Definition 2.8 (Caputo fractional derivative on time scales) Let T be a time scale, τ ∈ T, 0 < ω < 1, and ζ : T → R. The Caputo -fractional derivative of order ω of ζ is defined by

Lemma 3.2 The function u ∈ E is a solution of the problem
if and only if where α ∈ R. From the condition D u(0) = φ we find that α = φ -(0, ζ (0)). Then We find β = ζ (0), and (6) is proved. The converse follows by direct computation.
Assumptions: We need the following assumptions: (Ax.2) There exists a nonnegative constant μ such that (Ax.5) There exists a constant L > 0 such that We establish our existence results for IVP (1)-(3). The first result is based on the Banach contraction principle.

then there exists a unique solution for IVP (1)-(3) defined on E.
Proof Define the operator : On the other hand, by Lemma 2.6 we deduce By (8), the operator is a contraction. Hence, by Banach's contraction principle, has a unique fixed point, which is a unique solution on [-, T] T of problem (1)-(3).
The second result is based on Leray-Schauder nonlinear alternative. Before the statement, we recall the notion of completely continuous. A bounded linear operator T from Banach space X to Banach space Y is called completely continuous if, for every weakly convergent sequence (x n ) from X, the sequence (Tx n ) is norm-convergent in Y .