Existence and Stability for a Coupled Hybrid System of Fractional Differential Equations with Atangana–Baleanu–Caputo Derivative

­e aim of this article is to investigate a coupled hybrid system of fractional dierential equations with the Atangana–Baleanu–Caputo derivative which contains a Mittag–Leer kernel function in its kernel. We rstly apply the Dhage xed point principle to obtain the existence of mild solutions. ­en, we study the Ulam–Hyers stability of the introduced fractional coupled hybrid system. Finally, an example is presented to exhibit the validity of our results.


Introduction
Dhage and Lakshmikantham in 2010 [1] introduced initially rst order hybrid di erential equations (HDEs) and studied some basic results on the existence and uniqueness of solutions. Furthermore, di erential inequalities obtained with respect to HDEs were utilized to examine comparison results and some qualitative properties of the solution. Since the work of [1], many researchers in mathematics and other elds committed to the study of all kinds of HDEs. In particular, some scholars showed that fractional-order hybrid di erential equations described the hereditary and memory properties of biology, chemistry, physics etc., better than integer order HDEs.
On the other hand, thanks to the fact that a large number of practical and real world phenomena in the elds of biology, physics, chemistry, and computer network, can be modeled by coupled systems of di erent types of fractional di erential equations. Here, we sketch some references, but not a list of all references is included, such as Hadamard type [8], Caputo [9] and Riemann-Liouville types [10], and Ψ-Hilfer type [11].
Motivated by this fact and the work referenced above, the intent of this work is to study the existence and stability for a coupled hybrid system of fractional di erential equations with Atangana-Baleanu-Caputo derivative described by the following equation: where ABC 0 D α , ABC 0 D β denote the Atangana-Baleanu-Caputo fractional derivatives of order α and β, respectively, e main contributions of this paper are as follows: (1) e above-given results [2][3][4][5][6][7] consider single fractional-order hybrid differential equations, while we consider coupled fractional-order hybrid systems. (2) e work in [8][9][10][11] investigated the existence of solutions or initial value boundary value problem. However, it is well known that stability is one of the important dynamic behaviors for fractional-order hybrid differential equations. Moreover, we should mention that boundary value problem (1) are rather general and include some common cases such as nonseparated boundary and antiperiodic boundary conditions, and so on, by choosing different values of a i , b i , c i (i � 1, 2). us, it is meaningful to study existence and stability for (1). e remainder of the paper is organized as follows: Section 2 contains some necessary concepts, assumptions, and facts. In Section 3, we prove an existence result and the Ulam-Hyers stability for the problem (1). Finally, an example is constructed to show the correctness of our results. { }. We denote by X � AC([0, T], R) the space of all absolutely continuous functions. Obviously, the product space E � X × X is a Banach space with norm ‖(x, y)‖ E � ‖x‖ + ‖y‖. In the remainder of this paper, we recall the necessary basic notions and properties related to fractional calculus.

Preliminaries
Definition 1 (see [12]). e fractional Riemann-Liouville integral of order α > 0 for a continuous function x on [0, b] is given by the following equation: Definition 2 (see [13,14]). Let x be differentiable on 1]. en, the left Atangana-Baleanu-Caputo derivative (ABC) of x for order α is defined by the following equation: and in the left ABR sense (Riemann-Liouville type) is defined as follows: and the fractional integral associated the above operators is where E α is a parameter Mittag-Leffler function defined by E α (z) ∞ n�0 z n /Γ(nα + 1), N(α) is a normalization function satisfying N(0) � N(1) � 1.

Remark 1.
In works [13,14], it has been verified that . Also, the authors in the work [13] have established that Lemma 1 (see [15]). If Definition 3 (see [16]). An element (x, y) ∈ X × X is said to be a coupled fixed point of a mapping T: Lemma 2 (see [17]). Suppose that S is a nonempty, closed, convex, and bounded subset of the Banach algebra X, S � S × S and P, G: X ⟶ X and F: S ⟶ X are three operators such that, (i) P and G are Lipschitzian with a Lipschitz constants σ and δ, respectively

en, the operator equation T(y, x) � PyFx + Gy admits at least one coupled fixed point in S.
Lemma 3 (see [18]). Let S * be a nonempty, closed, convex, and bounded subset of the Banach space X and let P, G: X ⟶ X and F: S * ⟶ X be three operators such that, (i) P and G are Lipschitzian with a Lipschitz constants en, the operator equation PyFx + Gy � y possesses a solution in S * .

Main Results
To start for verifying the main results, the following assumptions are needed for us in the sequel: In order to study problem (1), we firstly consider the following problem: Similar to Definition 1 in [6], we give a definition of (9) and (x, y) satisfies (9).
An application of Lemma 4 in [7], Lemma 4 in [9], and eorem 3.1 in [6], we derive an equivalent fractional integral equation to (9). In order to make this paper readable, we present the proof.Let (x, y) ∈ E be a solution of (42), (x, y) ∈ E be a solution of problem (1) satisfying Journal of Mathematics Lemma 4. Let 0 < α, β < 1, ψ 1 , ψ 2 ∈ AC([0, T], R) and assume that (H1) holds. en, the solution of problem (9) if and only if it is a solution of the following system: Proof. e main idea of the proof comes from Lemma 4 in [7] and eorem 3.1 in [6]. In the view of Lemma 4 in [7], if (x, y) is a solution of (9), then (x, y) satisfies fractional integral system (10).

Theorem 1. If hypotheses (H1)-(H4) hold and
Proof. We choose R * so that and specify a subset S * of the Banach space X × X by Evidently, S * is a convex, bounded and closed subset of the Banach space E. According to Lemma 4, if (x, y) ∈ S * ⊆X × X is a solution of (1), then it is a solution of the next system For i � 1, 2, we define operators P � (P 1 , x(t), y(t)), en, the coupled hybrid system of (18) transformed into the coupled system of operator equations as follows: (20) is implies that By Lemma 3, we divide our proof into four steps: □ Step 1. P � (P 1 , P 1 ) are Lipschitz operators with constants en it follow from (H2) that us, we operate the supremum norm over t and obtain Along the same lines, one has us, we use the definition of operator P to get erefore, we can confirm that P is also Lipschitzian with Lipschitz constant L P � (L g 1 + L g 2 ).
Step 2. Now, we show that F � (F 1 , F 2 ) is a continuous and compact operator from S * into E . We firstly prove the continuity of F , let (x n , y n ) n∈N be a sequence in S * converging to a point (x, y) ∈ S * . en, Lebesgue dominated convergence theorem yields   Journal of Mathematics Similarly, we prove lim n⟶∞ F 2 x n , y n (t) � F 2 (x, y)(t).
In what follows, the compactness of F is explored on S * . Firstly, we prove the uniform boundedness. Let (x, y) ∈ S * . en using (H4), one has for t ∈ [0, T], Consequently, one has Similarly, we prove Consequently, we get Hence, it yields that F is uniformly bounded. Next we prove that F is equicontinious, Let t 1 , t 2 ∈ [0, T], t 1 < t 2 , any (x, y) ∈ S * , one has Hence, for ε > 0, there exists δ > 0 such that Consequently, we get is implies that F is equicontinious on S * , and so F is relatively compact. In consequence, we apply the Arzelá-Ascoli theorem to show that F is a complete continuous.
Step 3. we show that the item (iii) of Lemma 3 is satisfied.
Step 4. we show that the condition (iv) of Lemma 3 is fulfilled. Since Obviously, for i � 1, 2, L g i � 1/32, K f i (t) � t, M g i � M f i � 1, and taking N(α) � N(β) � 1 as a normalization function, all hypotheses of eorem 2 are satisfied.

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.