Existence and uniqueness for delay fractional differential equations with mixed fractional derivatives

Ahmed Hallaci1, Hamid Boulares1 and Abdelouaheb Ardjouni2,3,∗ 1 Depatement of Mathematics, Faculty of Sciences, University of 08 Mai 1945 Guelma, P. Box 401, Guelma, 24000, Algeria. 2 Faculty of Sciences and Technology, Department of Mathematics and Informatics, Univ Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria. 3 Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, Univ Annaba, P.O. Box 12, Annaba 23000, Algeria. * Correspondence: abd_ardjouni@yahoo.fr


Introduction
T he fractional differential equations is a hot topic of research due to its various applications in many scientific disciplines such as physics, chemistry, biology, engineering, viscoelasticity, signal processing, electrotechnical, electrochemistry and controllability, see [1][2][3][4][5][6] and the references therein. The neutral fractional differential equations have been studied extensively in the last decades and different technics have been used to solve it, for example, fixed point theorems, upper and lower solution method, spectral theory, etc. For some recent contributions in fractional boundary value problems, we refer [7][8][9] and the references therein. To the best of our knowledge, the use of mixed fractional derivative in neutral fractional differential equations which is an important type of fractional differential equations is still not sufficiently generalized. Our main aim is to solve mixed fractional differential equations.
Benchohra et al. [7], investigated the existence of solutions for the following Riemann-Liouville fractional order functional differential equation with infinite delay Agarwal et al. [8], studied the initial value problem of fractional neutral Caputo fractional derivative and established the existence results of solutions of this problem by using Krasnoselskii's fixed point theorem.
In [9], Ahmad et al. studied the existence and uniqueness of solutions to the following boundary value problem where D α and D β are the Caputo-Hadamard fractional derivatives, 0 < α, β < 1. Motivated and inspired by above mentioned works, in this paper we investigate the existence and uniqueness of solutions for the following initial value problem of the mixed Riemann-Liouville and Caputo fractional functional differential equation with delay where RL D α and C D β are the Riemann Liouville and the Caputo fractional derivatives respectively, 0 < α < 1, Also, by utilizing the contraction mapping principle, we give uniqueness results. The organization of this paper is as follows: In Section 2 we recall some useful preliminaries and present the equivalent fixed point problem corresponding to (1). In Section 3, we discuss the existence and uniqueness of solutions for (1) via fixed point theory. An example is constructed for illustrating the obtained results.

Preliminaries
In this section, we introduce notation, definitions, and preliminary facts that we need in the sequel. By C(I, R) we mean the Banach space of all continuous functions from I into R with the norm Definition 1 ([2,4,6]). The Riemann-Liouville fractional integral of the function u of order α > 0 is defined by 4,6]). The Riemann-Liouville fractional derivative of the function u of order α ∈ (n − 1, n] is defined by 4,6]). The Caputo fractional derivative of the function u of order α ∈ (n − 1, n] is defined by Let α > 0 be a real number, we have following results: The unique solution of the linear fractional differential equation where n = [α] + 1, [α] denotes the integer part of α.

Lemma 3.
(1) is equivalent to the following integral equation Proof. Using Lemma 1, equation one of (1) can be written as Using the condition lim t→0 t 1−α C D β u(t) = 0, we get c 0 = 0. On the other hand, from Lemma 2, one gets Clearly u (0) = φ (0), so we obtain c 1 = φ (0) and because u (0) = 0, we find c 2 = 0, then we get the integral equation Our main results are based on the following Krasnoselskii fixed point theorem and the contraction mapping principle.
Theorem 1 (Krasnoselskii fixed point theorem [10,11]). If M is a nonempty bounded, closed and convex subset of a Banach space E and A and B two operators defined on M with values in E such that i) Ax + By ∈ M, for all x, y ∈ M, ii) A is continuous and compact, iii) B is a contraction. Then there exists z ∈ M such that z = Az + Bz.
Theorem 2 (Contraction mapping principle [10,11]). Let E be a Banach space. If H : E → E is a contraction, then H has a unique fixed point in E.

Main results
We express (2) as where the operators A, Consider the following hypothesis: Proof. We show that the operators A and B defined by (4) and (5) satisfied all hypothesis of Theorem 1.
This means that Au + Bv ∈ B R for any u, v ∈ B R . On the other hand, operator A is continuous from the continuity of f . It remains to prove the compactness of the operator A. Clearly, (7) show that A is uniformly bounded on , which is independent of u and tends to zero as t 2 → t 1 . Thus, the set {Au, u ∈ B R } is equicontinuous and hence it is relatively compact. So, by Ascoli-Arzela theorem, A is compact on B R .
It remains to show that B is a contraction. Let u, v ∈ C([−τ, T], R), then for all t ∈ [−τ, T], we have Proof. We claim that H is contraction mapping, this show that H has a unique fixed point which is the unique solution of (1). To this end, let u, v ∈ C([−τ, T], R), then for all t ∈ [−τ, T], we have Therefore H is a contraction. Thus, the conclusion of Theorem 7 follows by the contraction mapping principle. Now, we give an example to illustrate the usefulness of our main results.