ON NEW APPROACH OF EXISTENCE SOLUTIONS FOR ATANGANA-BALEANU FRACTIONAL NEUTRAL DIFFERENTIAL EQUATIONS WITH DEPENDENCE ON THE LIPSCHITZ FIRST DERIVATIVES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this article, we establish the result on existence and uniqueness of a Atangana-Baleanu fractional neutral differential equations with dependence on the Lipschitz first derivative conditions in Banach space. These results are based on fixed point theorems. Moreover, an example is also provided to illustrate the main results.


INTRODUCTION
Fractional differential equations have appeared in various fields in the past few decades such as chemistry, physics, engineering, control theory, aerodynamics, electrodynamics of complex medium and control of dynamical systems and so on. In consequence, fractional differential equations is obtaining much significance and attention. For details, we refer readers to [12,18,19,23] and references therein. The nonlocal characteristic of the fractional order operators is the main reasons for the popularity of the fractional calculus, which take into account the hereditary properties of several materials and processes.
Many researchers paid more attention to ABC-derivative with several conditions in various spaces in recent years. The AB fractional derivative is familiar to followings nonsingularity as known as nonlocality of the kernel, which acquires the generalized Mittag-Leffler function.
Some of the latest studies on ABC-derivatives such as, Atangana and Koca find the chaos in a simple nonlinear system with AB-fractional derivatives [10]. Jarad et al. investigated a Ordinary Differential Equations in the form of AB-derivative [20]. Ravichandran et al. discussed in details the AB-fractional neutral integro-differntial equations [25].
More precisely Sene discussed Stokes' first problem for heated flat plate with AB-derivative [33]. Owolabi studied the modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative [32]. A substantial deal of research work has been carry through on the application of fractional neutral derivative. Liu et al. discussed a coupled system of nonlinear neutral fractional differential equations [24]. Zhou et al. studied the fractional of neutral differntial equations with infinite delay [36].
In this paper, we are interested in the existence and uniqueness of solutions of the Atangana-Baleanu fractional neutral differential equation in the sense of Caputo to the following abstract form The organization of the paper is as follows: In Section 2, we review some useful properties, definitions, propositions and lemmas of fractional calculus. The existence and uniqueness of solutions for AB-fractional neutral derivative results are proved in Section 3. In the last section is devoted to illustrate an example numerically solved.

PRELIMINARIES
In this section, we presents some definitions, lemmas and proposotions of fractonal calculus, which will be used throughout this paper.
The definition of Riemann-Liouville fractional integral and derivatives are given as follows: • For α > 0, the left R-L fractional integral of order α is given as [20] • For 0 < α < 1, the left R-L fractional derivative of order α is given as [20] (6) • For 0 ≤ α ≤ 1, the Caputo fractional derivative of order α is given as [20] [27,34] and B(α) > 0 is a normalizing function satisfying B(0) = B(1) = 1. The Riemann Atangana-Baleanu fractional derivative of u of order α is defined by The associative fractional integral is defined by where a I α is the left Riemann-Liouville fractional integral given in (5).
) −1 is non-negative and bounded on [a,b) and u(t) is nonnegative and locally integrable [a,b) with .
is relatively compact iff M is uniformly bounded and uniformly equicontinuous.
) Let S be a closed, bounded and convex subset of a real Banach space X and let T 1 and T 2 be operators on S satisfying the following • T 1 is a strict contraction on S, i.e., there exist a l ∈ [a, b) such that • T 2 is continuous on S and T 2 (s) is a relatively compact subset of X.
i.e.,There exist a constant l > 0 such that , u(t) satisfies the following integral equation

EXISTENCE AND UNIQUENESS
In this section, we prove the existence and uniqueness solutions of (3) and (4) is studied with the following assumptions.
for all u, v in Y . L 2 = max t∈D D(t, 0) and L = max{L 1 , L 2 }.
are satisfied for any t ∈ ℜ, and u, v ∈ Y.
To apply the Riemann-Liouville AB fractional derivative to both sides of (15) and utilize that ( AB a D α ( AB a I α u))(t) = u(t). We get Thus, we have Then, the result is acquired by getting from theorem(1) in [7]. Now, we can consider the operator T defined by Then, by A 3 , u ≤ λ we get i.e., Tu(t) ≤ λ . Now to prove uniqueness −[g(t, v(t), Dv(t)) + g(a, v(a), Dv(a)) + AB a I α f (t, v(t), Dv(t))] Hence, the operator Tu(t),t ∈ B λ proved the existence and uniqueness conditions and has a fixed point by Banach contraction principle in Banach spaces X.
Next, we investigate the problem (3) and (4) has a fixed point by utilizing Krasnoselskii's fixed point theroem. Proof. Now, for any λ 0 > 0 and u ∈ B λ 0 , we define two operator T 1 and T 2 on B λ 0 as follows Obviously, u is a solution of (3) and (4) iff the operator T 1 u + T 2 u = u has a solution u ∈ B λ 0 This proof will be given in three steps.
Step 2. T 1 is a contraction on B λ 0 for every u, v ∈ B λ 0 , according to A 4 and (20), we have This shows that T 1 is a contraction.
Step 3. T 2 is completely continuous operator.
First we have to prove that T 2 is continuous on B λ 0 . For every u n , u ⊂ B λ 0 , n = 1, 2, 3.... with lim n→u u n − u = 0, we get lim n→u u n (t) = u(t), for t ∈ [a, b].

Thus by
We can conclude that sup s∈ [a,b] f (t, u n (t), Du n (t)) − f (t, u(t), Du(t)) → 0 as n → ∞ On other hand, for t ∈ [a, b] we can obtain that Hence (T 2 u n )(t) − (T 2 u)(t) → 0 as n → ∞. Therefore T 2 is continuous on B λ 0 . Now, we have to show that T 2 u, u ∈ B λ 0 is relatively compact which is enough to prove that the function T 2 u, u ∈ B λ 0 uniformly bounded and equicontinuous, and ∀ t ∈ [a, b] T 2 u ≤ λ 0 , for any u ∈ B λ 0 , therefore (T 2 u)(t), u ∈ B λ 0 is bounded uniformly. Now, we show that (T 2 u)(t), u ∈ B λ 0 is a equicontinuous.
For any u ∈ B λ 0 and a ≤ t 1 ≤ t 2 ≤ t, we get Therefore, the operator T 2 is a equicontinuous on B λ 0 .
Hence, which implies T 2 is relatively compact on B λ 0 .
Therefore T 2 is satisfies the condition of theorem 2.4 and theorem 2,5, we can conclude that T 2 has a fixed point. Therefore the problem (3) and (4), the operator T has a fixed point u.

EXAMPLE
In this section an example is presented for the existence results to the following problem.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.