Study of the Atangana-Baleanu-Caputo type fractional system with a generalized Mittag-Leffler kernel

Abstract: We devote our interest in this work to investigate the sufficient conditions for the existence, uniqueness, and Ulam-Hyers stability of solutions for a new fractional system in the frame of Atangana-Baleanu-Caputo fractional operator with multi-parameters Mittag-Leffler kernels investigated lately by Abdeljawad (Chaos: An Interdisciplinary J. Nonlinear Sci. Vol. 29, no. 2, (2019): 023102). Moreover, the continuous dependence of solution and δ-approximate solutions are analyzed to such a system. Our approach is based on Banach’s and Schaefer’s fixed point theorems and some mathematical techniques. In order to illustrate the validity of our results, an example is given.


Introduction
Fractional differential equations have a profound physical background and rich theoretical connotations and have been particularly eye-catching in recent years. Several-order differential equations refer to equations that contain fractional derivatives or fractional integrals. Fractional order derivatives and integrals have a wide range of applications in many disciplines such as physics, biology, chemistry, etc., such as power with chaotic dynamic behavior systems, dynamics of quasichaotic systems, and complex materials or porous media, random walks with memory, etc. For more information see [1][2][3]. The approximate controllability of the fractional system can be found in [4][5][6][7][8][9][10]. Recently, some researchers have realized the importance of finding new fractional derivatives (FDs) with different singular or nonsingular kernels to meet the need to modeling more real-world problems in different fields of science and engineering. For instance, Caputo and Fabrizio [11] studied a new kind of FDs in the exponential kernel. Atangana and Baleanu (AB) [12] investigated a new type and interesting FD with Mittag-Leffler kernels. Abdeljawad in [13] extended this type for higher arbitrary order and formulated their associated integral operators. But the corresponding integral operators of AB derivative do not have a semigroup property, which makes dealing with them theoretically or mathematically somewhat complicated. Very recently, Abdeljawad in [14,15], introduced a fractional derivative with nonsingular kernel in Atangana-Baleanu-Caputo (ABC) settings with multiparametered Mittag-Leffler (ML) function and study their semigroup properties, its discrete version in [16]. This diversity of FDs has made the topic of fractional calculus attractive and allows researchers to choose the appropriate operator to obtain better results. For some theoretical works on ABC type FDEs, we refer the reader to the series papers [17][18][19][20]. On the other hand side, the study of systems involving FDEs is also important as such systems occur in various problems of applied nature. For some theoretical works on systems of FDEs, we refer to series of papers [21][22][23].
The topic of stability of systems is one of the most important qualitative characteristics of a solution, for more details about the stability of systems see [24][25][26][27].
Abdeljawad et al. [28] studied qualitative analyses of some logistic models in the settings of ABC fractional operators with multi-parameter ML kernels, described as follows: where ABC θ 0 D p,q,v is the generalized left ABC FD of order p ∈ (0, 1] , q, v > 0 and m, n, l > 0. Motivated by the recent advancements of ABC operator, its applications, and by the above works, the aim of the current work is to investigate the existence, uniqueness, stability, and continuous dependence results, and discuss the δ-approximate solutions for a new model in the frame of generalized ABC fractional operators with multi-parameters ML kernels described as follows: where ABC 0 D p,q,v is the generalized ABC FD of order p ∈ (0, 1] , q, v > 0. F k ∈ C ([0, T ] , R + ) and satisfies some conditions described later in our analysis. Many researchers in different fields of science and engineering used ABC FD with one parameter ML kernel, but their corresponding AB integral operators do not have a semigroup property, which makes dealing with them theoretically or mathematically somewhat complicated. Nevertheless, in this work, we use a new operator containing interesting kernels, we believe that the qualitative properties of solutions for FDEs should be studied via this operator. This work aims to investigate some properties of solutions for the proposed model via a nonsingular FD in ABC settings with multi-parameter ML kernel introduced lately by [14,15]. Due to the fractional derivative used in this work have semigroup property and recently proposed, the results obtained in this work are new and open the door for the researchers to study more real-world problems in different fields.
Notice that, the considered system is investigated under the generalized ML law. In the case of the ABC fractional operator, the requirement of the vanishing condition of the right hand side of the dynamic system to fulfill the initial data needs recuperation on the modeled population. However, the nature of the generalized ML kernel will enable the emancipation of any restrictions on the initial data.
The structure of our paper is as follows. In Section 2, we present notations, auxiliary lemmas and some basic definitions that are needed for our analysis. In Section 3, we discuss the existence and uniqueness results for the model (1.1). Ulam-Hyers stability results for the model (1.1) are discussed in Section 4. In Section 5, we study the continuous dependence of solution and δ-approximate solutions for the model (1.1). In Section 6, we provide an example to illustrate the validity of our results. The last section is devoted to concluding remarks about our results.
Then Φ has a fixed point in G.

Existence and uniqueness of solutions
We devoted this section to derive the equivalent fractional integral equations for the model (1.1). First of all, by using fixed point technique and mathematical techniques, we prove the existence and uniqueness of solution for model (1.1).
In view of Lemma 2.8, the equivalent fractional integral of model (1.1) is given as follows Let us consider the continuous operator Φ : G → G defined by . Notice that the model (1.1) has a solution (z 1 , z 2 , ..., z n ) if Φ has a fixed point. To achieve our results, the following hypothesis must be hold.
Proof. First, in view of the continuity of the functions F k , we notice that the operator Φ is continuous. Define a closed ball Now, for (z 1 , z 2 , ..., z n ) ∈ B r , θ ∈ J, then, by (3.3) and k = 1, 2, ......, n, we have .
Then, we have Consequently, we have the following inequalities Inequalities (5.3) can be writting as matrices as followes By simple computations, the above inequality becomes Since ∆ 0. This leads to From the fact It follows that

Conclusions
ABC fractional operators with multi-parameters ML kernels on certain time scales and the integral equations expressed by them are some of the keys in developing fractional calculus. In this work, we have obtained some existence, uniqueness, UH stability results for the fractional system (1.1) in the frame of generalized FD in AB settings containing a multi-parameter ML kernel. As well, the data dependence analysis and δ-approximate solutions of the proposed system are discussed. Our approach is based on some fixed point theorems and mathematical techniques. As an application, one example has been provided in order to illustrate the validity of our results. We realized that, if q 1, then the condition F k (0,z 1 (0), ......, z n (0)) = 0, (k = 1, 2, ..., n) not necessary to guarantee a unique solution.
The considered system has been investigated under the generalized ML law. Observed that in the case of the classical ABC fractional operator, the requirement of the vanishing status condition of the right-hand side of the dynamic system to full the initial data needs recuperation on the modeled population. However, the nature of the generalized ML kernel managed to get rid of any restrictions on the initial data. Due to the fractional operators used in this work have semigroup property and recently proposed, the results obtained here are new and open the door for the researchers to study more realworld problems in different fields. Besides, the results obtained in this work are very significant in developing the theory of fractional analytical dynamics of different biological models.
So in the future, the same analysis can be extended to the system of delay equations under the generalized fractional operator.