A System of High-Order Fractional Differential Equations with Integral Boundary Conditions

The existence of a solution for a system of two nonlinear high-order fractional differential equations including the Atangana-Baleanu-Caputo derivative with integral boundary conditions, is proved. Simultaneously, we discuss the existence of a solution by applying the Schauder fixed point theorem and a generalized Darbo fixed point theorem, which involves the concept of measure of noncompactness. The paper also contains some examples that illustrate the application of the main result.

authors study why fractional dynamics is needed in machine learning and optimal randomness. So far, different types of fractional derivatives have been defined, such as the Riemann-Liouville [39], Hadamard [34], Grunwald-Letnikov [46], Weyl [53], Caputo [28] and Caputo-Fabrizio [27] derivative. Although the Caputo-Fabrizio derivative, with its exponential Kernel, aimed to give a better description of the dynamics of systems with memory effect than other classic fractional derivatives, its associated integral was not fractional. To fix this defect, Atangana and Baleanu launched a new fractional operator in the Caputo and Riemann-Liouville sense [18]. Atangana-Baleanu derivative by the nonlocal and nonsingular kernel, Mittag-Leffler function, attracts more interest in applying in different fields. In the field of optimal control, the fractional derivative, in particular the Atangana-Baleanu yields improved results compared to the classical derivative (see [13,19,20,32,36]). In [26], the authors applied successfully the Caputo and Atangana-Baleanu fractional derivatives in data fitting.
Several studies have been devoted to the existence of the solutions of the fractional differential equations via various approaches, see [6,7,12,17,24,40,41,51,52]. Many latest studies in the existence theory focus on the fractional equations with the integral boundary conditions, which improve the classical conditions in the development of mathematical modeling, [11,17,35].
This paper which draws inspiration from the aforementioned works, investigates the existence of a solution to the following high-order fractional boundary value problem in which t ∈ [a, b] and completed by two integral boundary conditions of the form where , ∈ (n − 1, n] for some n ∈ ℕ , , ∈ ℝ , f , g ∶ [a, b] × ℝ 2n × ℝ 2n → ℝ are continuous functions, and ABC a D denotes the order of Atangana-Baleanu fractional derivative in the (left) Caputo sense.
To investigate the existence of the solutions in the space C n−1 ([a, b]) , which is equipped with the measure of noncompactness introduced in [16], we use the Schauder fixed point theorem [50] and the generalized Darbo fixed point theorem [30].
This paper is organized as follows. In section two, the needed preliminaries from the Atangana-Baleanu fractional calculus are stated. In the third section which is the main section, firstly by setting the appropriate solution space, we provide a modified version of the measure of noncompactness on the solution space. So by introducing the main conditions of the problem (1, 2), through some preliminary lemmas and propositions we prove the main result. Also two examples have been studied in order to show the application of the main theorem.

Preliminaries of Fractional Calculus
Firstly, we recall the Sobolev space H 1 (a, b) where a < b and Here u ′ is taken in the weak (distributional) sense; see [25] for more details.
Let us refer to [3,18], from which most of the contents in this section has been adapted.

Lemma 2.2
The high-order ABC-derivative and integral are defined in [4] as follows.

Definition 2.3
Let ∈ (n − 1, n] for some n ∈ ℕ and f be such that f (n−1) ∈ H 1 (a, b) . Then the AB-derivative of -order in the (left) Caputo sense is defined by Moreover, the associated fractional integral of -order is defined by
The following proposition explains the action of the AB-fractional Integral on the ABC-derivative and vice versa.
Proposition 2.5 Let ∈ (n − 1, n] for some n ∈ ℕ and f be such that f n−1 ∈ H 1 (a, b) .

Then we have
Proof The proof of (9) can be found in [4]. The proof of (10) is based on the definition, and one can check that easily. So we omit it for the sake of brevity. ◻

Main Results
In this paper we discuss the existence of solution for system (1)- (2). Due to its boundary conditions we set where ‖g‖ ∞ = max{�g(t)�, t ∈ [a, b]} Let , ∈ (n − 1, n] for some n ∈ ℕ , , ∈ ℝ and , as two real functions on [a, b]. Suppose the system with the following boundary conditions:    , and . Proof In view of Proposition 2.5, by exerting the fractional integral AB a I on (11) under the condition (13), we obtain Similarly, by exerting the fractional integral AB a I on equation (12) with (14), we obtain (17) By differentiating n − 1 times of (21) at t = b and regarding the condition (15), we derive Similarly, from (22), regarding the condition (16), we derive Now, by inserting the formulation of y(s) from (22) into (23), we deduce Similarly, by inserting the formulation of x(s) from (21) into (24), we deduce Obviously, from the system (25) and (26), one can obtain the quantities x (n−1) (a) and y (n−1) (a) . Since x (n−1) (a), y (n−1) (a) are dependent to , , we denote x (n−1) (a) and y (n−1) (a) by s 1 ( , ) and s 2 ( , ) , respectively. Thus, x(t) and y(t) have a form according to (17) and (18), respectively. ◻ We consider two operators  y)) . Hence, due to Corollary 3.2, we derive the following Corollary.
Let us recall, in details a measure of noncompactness on k-times continuously differentiable functions C k ([a, b]) , according to [16].
By the following Lemma, we may compute the measure of noncompactness k on C k ([a, b]) with a more simple formulation.

Lemma 3.5 Every bounded subset of
. Thus, if {g n } ⊂ Y be an arbitrary sequence, and > 0 , by the uniform equicontinuity of , that is convergent and as a result, Cauchy, which means |g n s (t i ) − g n l (t i )| < 3 for i ∈ {1, … , m} , and n s , n l ≥ N . Therefore, for any t ∈ [a, b] , there is i ∈ {1, … , m} , such that |t − t i | < and so for n s , n l ≥ N, ([a, b]). Consequently, it has been proved that any bounded subset of C k ([a, b]) is sequentially relatively compact and, thus, relatively compact in C k−1 ([a, b]) . In the sequel, we set the following conditions on the system (1)- (2).
is a non-decreasing and upper semicontinuous function with respect to it's components and for every t > 0, (t, t) < t. Proof For each > 0 and t 1 , t 2 ∈ [a, b] , with |t 1 − t 2 | < , we have Since the right hand side of the last inequality tends to zero, uniformly, with respect to (x, y) ∈ B r × B r when → 0 , we can find that and |s 2 ( x,y , x,y )| ≤ 2 ( (r, r) + K 0 ); where and in which c 1 , c 2 , d 1 , d 2 are introduced in the Lemma 3.1.
Proof By the definition of the Riemann-Liouville fractional integral of order , which is Thus, by Lemma 3.9 Hence, by Lemma 3.11 Thus, Θ 1 is a self map on B r × B r provided 1 (r, r) + 1 ≤ r . Since (r, r) < r and due to the assumption, 1 < 1 , the desired result will be obtained for r > in which Since lim Proof Let the well defined operator by Θ(x, y) = (Δ 1 (x, y), Θ 2 (x, y)) , where Θ 1 , Θ 2 are introduced in (27) and (28); respectively. For r > 0 introduced in the Proposition 3.12, we have Θ(B r × B r ) ⊂ B r × B r . Let us prove the assertion in two cases. If satisfies the condition of Proposition 3.13, Θ 1 and Θ 2 are equicontinuous on B r × B r , where regarding the Arzela-Ascoli Theorem, Θ 1 , Θ 2 are compact operators on B r × B r . Therefore, by Schauder fixed point theorem [8], Θ has at least one fixed point in B r × B r , and due to the Corollary 3.3, the boundary value system (1)-(2) has at least one solution in B r × B r .
In the case that does not satisfy the condition of Proposition 3.13, since can not be replaced by , which is independent of s we can not necessarily deduce that Θ 1 , Θ 2 are compact operators, whereas by using the concept of measure of noncompactness on C n−1 ([a, b]) and applying a generalization of Schauder fixed point theorem, named generalized Darbo theorem (see [9], Corollary 2.2) we deduce the existence of a fixed point for the operator T. Firstly, we claim that x,y , , 0) → 0 as → 0 uniformly with respect to (x, y) ∈ B r × B r . Therefore, x,y , , 0).