On Atangana–Baleanu-Type Nonlocal Boundary Fractional Differential Equations

Department of Mathematics, Hajjah University, Hajjah, Yemen Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada University, Aurangabad, Maharashtra 431001, India Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen Department of Mathematics, Çankaya University, 06790 Etimesgut, Ankara, Turkey Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan


Introduction
Fractional calculus and its potential applications have become a powerful tool to model many complex phenomena in apparently wide-ranging fields of science and technology [1][2][3][4][5][6][7][8]. To meet the needs of modeling many practical problems in different fields of science and engineering, some researchers have realized the necessity development of the concept of fractional calculus by searching for new fractional derivatives with different singular or nonsingular kernels. From this perspective, new fractional operators have turned into the best effective tool of numerous specialists and researchers with their contribution to physical phenomena and their performance in applying to real-world problems. Until 2015, all fractional derivatives had only singular kernels. erefore, it is difficult to use these singularities to simulate physical phenomena. In 2015, Caputo and Fabrizio [9] studied a new kind of FD in the exponential kernel. Some properties of this new type had been discussed by Losada and Nieto in [10]. A new type and interesting FD with Mittag-Leffler kernels has been investigated by Atangana and Baleanu (A-B) in [11]. Abdeljawad in [12] extended the kind investigated by A-B from order between zero and one to higher arbitrary order and formulated their associated integral operators. Also, he proved some properties such as the existence and uniqueness theorems for two classes of fractional derivative, Riemann type (ABR) and Caputo type (ABC), for initial value problems in higher arbitrary order and proved a Lyapunov-type inequality in the frame of Mittag-Leffler kernels for the ABR fractional boundary value problems of order 2 < α ≤ 3. Abdeljawad and Baleanu, in [13,14], deliberated the discrete versions of those new operators. For some theoretical works on Atangana-Baleanu FDEs, we refer the reader to a series of papers [15][16][17]. For important applications and mathematical modeling of the ABC fractional operator, see [18][19][20][21]. On the contrary, there are some important numerical approaches regarding nonsingular kernels; for example, in [22], via a spectral collocation method based on the shifted Legendre polynomials with extending the unknown functions and their derivatives using the shifted Legendre basis, Tuan et al. solved fractional rheological models and Newell-Whitehead-Segel equations with the nonlocal and nonsingular kernels. Nikan et al. in [23] developed the solution of the two-dimensional timefractional evolution model using a finite difference scheme derived from the radial basis function (RBF-FD) method. Ganji et al. in [24] studied the multivariable-order differential equations (MVODEs) with the nonlocal and nonsingular kernels in the Atangana and Baleanu sense of variable orders. ey used the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices and used them with the collocation method to transfer the original equations to a system of algebraic equations. Firoozjaee et al. in [25] studied the Fokker-Planck equation (FPE) with CF-FD being considered. ey presented a numerical approach that is based on the Ritz method with known basis functions to transform this equation into an optimization problem. Baleanu et al. in [26] showed that four fractional integrodifferential inclusions have solutions. Also, they showed that the dimension of the set of solutions for the second fractional integrodifferential inclusion problem is infinite-dimensional under some different conditions. By using the fractional CF derivative, Aydogan et al. [27] introduced two types of new high-order derivations called CFD and DCF. Also, they studied the existence of solutions for two such types of high-order fractional integrodifferential equations. Guoa et al. in [28,29] studied the existence and HU stability of the FDEs with impulse and fractional Brownian motion under nonlocal conditions based on the semigroups of operators method and Mönch fixed point method.
ere are some researchers in various areas who investigated the existence and Ulam-Hyers (UH) stability of solutions for fractional differential equations (FDEs) of some problems with initial (or boundary, nonlocal) conditions; more details can be found within research series [30][31][32][33][34][35].
Recently, Abdo et al. [36] considered ABC-type pantograph FDEs with nonlocal conditions: In this regard, the ABC-type implicit FDEs had been studied by Abdo et al. [37].
Motivated by the above argumentations, we intend to investigate and develop the sufficient conditions for existence, uniqueness, and UH stability results of two new classes for the ABR and ABC fractional differential problems described as follows: where ABR D p a + , ABC D p a + are the ABR and ABC fractional derivatives of order p ∈ (2, 3] and p ∈ (1, 2], respectively, δ ∈ (0, 1], ζ ∈ (a, b), and f: To the best of our understanding, this is the first work that transacts with ABC and ABR fractional derivatives of a high order, especially with AB integral conditions. Also, we studied existence and uniqueness without using the semigroup property. In consequence, our results will be a useful contribution to the existing literature on these interesting operators. e paper is organized as follows. In Section 2, we present some notations and some preliminary facts which are used throughout the paper. In Section 3, we derive the formula of an equivalent integral equation for ABR-type FDEs (4). Section 4 discusses the existence and uniqueness results for ABR-type FDEs (4) and ABC-type FDEs (5). In Section 4, we discuss the stability results in the frame of UH. Section 5 provides two examples to illustrate the validity of our results. Concluding remarks about our results are in the last section.
Definition 1 (see [11]). Let 0 < p ≤ 1. en, the left-sided ABC and ABR fractional derivatives of order p for a function ϑ are defined by respectively, where B(p) is the normalization function such that B(0) � B(1) � 1 and E p is the Mittag-Leffler function defined by e analogous AB fractional integral is given by 2 Journal of Function Spaces Lemma 1 (see [13]). Let 0 < p ≤ 1. If the ABC fractional derivative exists, then we have Definition 2 (see [12]). e relation between the ABR and ABC fractional derivatives is given by Remark 1. By applying AB I p a + ϑ(σ) on both sides of the equation defined in Definition 2 and using Lemma 1, we get Lemma 2 (see [11]). Let ϑ > 0. en, AB I p a + is bounded from Definition 3 (see Definition 3.1 of [12]). Let n < p ≤ n + 1 and ϑ be such that ϑ (n) ∈ H 1 (a, b). Set β � p − n. en, 0 < β ≤ 1, and we define e correspondent AB fractional integral is given by Lemma 3 (see Proposition 3.1 of [12]). For ϑ(σ) defined on J and n < p ≤ n + 1, for some n ∈ N 0 , we have □ Theorem 1 (see [38]). Assume that K is a closed subspace from a Banach space X, and G is a contraction defined as Q: K ⟶ K, i.e., ‖Q(x) − Q(y)‖ ≤ M‖x − y‖ for some 0 < M < 1 and all x, y ∈ K. en, Q has a fixed point in K.
en, the solution of the following linear problem, is given by Lemma 6 (see Example 3.3 of [12]). Let p ∈ (1, 2] and h ∈ C(J, R). en, the solution of the following linear problem, is given by where

Equivalent Integral Equation for Problem (4)
if and only if ϑ satisfies the following fractional integral equation: Proof. Assume that ϑ is the solution to the first equation of (22). en, via Lemma 5, we have Now, by condition (ϑ(a) � 0), we get c 1 � 0, and hence, equation (24) reduces to Taking the operator AB I δ a + on both sides of equation (25), we have Now, replacing σ by ζ, we get By the second condition (ϑ(b) � AB I δ a + ϑ(ζ)) together with equations (25) and (27), we get Substituting c 1 and c 2 in (24), we get (23). Conversely, assume that ϑ satisfies (23). en, by applying the operator ABR D p a + on both sides of (23) and using Lemmas 3 and 4, we get Next, taking σ ⟶ a in (23), we get ϑ(a) � 0. On the contrary, applying AB I δ a + on both sides of (23), we get Replacing σ by ζ, we get By the definition of Λ, we can rewrite (31) as us, the Atangana-Baleanu fractional integral conditions are satisfied.
□ As a result of Lemma 7, we get the following theorem.
en, ABR-type FDEs (4) are equivalent to the following FIE: where , , , Proof. In view of Lemma 7, we have By using definitions of AB I δ a + in the case δ ∈ (0, 1) defined in (8) and AB I p a + in the case p ∈ (2, 3] defined in (18), we can rewrite equation (35) as follows:

Journal of Function Spaces
□ For simplicity, we set

Existence and Uniqueness of Solutions for Problem (4)
In this section, we devote our intention to prove the existence and uniqueness of solutions for ABR-type FDEs (4).

Theorem 4.
Suppose that f: J × R ⟶ R is a continuous function. Moreover, we assume that If

ζ − s)|f(s, ϑ(s)) − f(s, ϑ(s))|ds
From our assumptions, we obtain Hence, Due to Z < 1, we conclude that Ξ is a contraction operator. Hence, eorem 1 implies that Ξ has a unique fixed point in J.

Theorem 5. Assume that the conditions in eorem 4 are satisfied. en, ABR-type FDEs (4) have at least one solution in C(J, R).
Proof. Let us consider the operator Ξ defined in eorem 4. Now, we divide the operator Ξ into two operators Ξ 1 and Ξ 2 such that where Journal of Function Spaces

(50)
Let us consider a closed ball Π φ defined in eorem 4. In order to apply eorem 2, we will divide the proof into the following steps.

Existence of a Unique Solution for
Problem (5) In this section, we devote our intention to proving the existence of a unique solution for ABC-type FDEs (5). R). A function ϑ is a solution of the following ABC problem, if and only if ϑ satisfies the following fractional integral equation: where AB I p a + was defined in (21).
Proof. Assume that ϑ is the solution to the first equation of (57). en, via Lemma 6 and by the same way in Lemma 7, one can prove that the solution of equation (57) where , , , Proof. In view of Lemma 8, we have By using definitions of AB I δ a + in the case δ ∈ (0, 1) defined in (8) and AB I p a + in the case p ∈ (1, 2] defined in (21), we can rewrite equation (61) as follows: then ABC-type FDEs (5) have a unique solution in J.

Ulam-Hyers Stability
In this part, we will discuss two kinds of stability results for ABR-type FDEs (4) and ABC-type FDEs (5), namely, UH and GUH stabilities. Before that, for ε > 0, we consider the following inequality: Definition 4. e problem ABR-type FDEs (4) is Ulam-Hyers stable if there exists a real number C f > 0 such that, for each ε > 0 and for each solution ϑ ∈ C(J, R) of inequality (71), there exists a unique solution ϑ ∈ C(J, R) of ABR-type FDEs (4) with Also, ABR-type FDEs (4) have generalized Ulam-Hyers stability if we can find φ f : Remark 3. Let ϑ ∈ C(J, R) be the solution of inequality (71) if and only if we have a function h ∈ C(J, R) which depends on ϑ such that is a solution to inequality (71), then ϑ satisfies the following inequality: where Proof. In view of Remark 3, we have en, by eorem 3, we get which implies Journal of Function Spaces □ Theorem 8. Suppose that (38) holds. If then ABR-type FDEs (4) are Ulam-Hyers stable.

Conclusion Remarks
In recent interest, the theory of fractional operators in the frame of Atangana and Baleanu is novel and significant; thus, there are some researchers who studied and developed some qualitative properties of solutions of FDEs involving such operators. On the contrary, there are some important numerical approaches regarding nonsingular kernels that are explained in the introduction. For this purpose, the sufficient conditions of the existence and uniqueness of solutions for two classes of nonlinear Atangana-Baleanu FDEs on the interval [a, b], subjected to integral conditions, have been developed and investigated. Furthermore, the stability results through mathematical analysis techniques have been analyzed. Two examples are provided to justify our results.
Observe that our approach used in this work is new because we prove the existence, uniqueness, and UH stability results without using the semigroup property and relies on a minimum number of hypotheses. Our approach was based on the reduction of the proposed problem into the fractional integral equation and using some standard fixed point theorems due to Banach type and Krasnoselskii type. Furthermore, through mathematical analysis techniques, we analyzed the stability results in the UH sense. An example was provided to justify the main results. In fact, our outcomes extended those in [36,37]. e supposed problem with given integroderivative boundary conditions can describe some mathematical models of real and physical processes in which some parameters are frequently acclimated to appropriate circumstances. So, the value of these parameters can change the impacts of fractional integrals and derivatives. e main results are illustrated with a numerical example. In future works, we study the new numerical results regarding this operator with the higher order.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Disclosure
is work was conducted during the work of the first author at Hajjah University (Yemen).