Nonlocal Hybrid Integro-Differential Equations Involving Atangana–Baleanu Fractional Operators

In this study, we develop a theory for the nonlocal hybrid boundary value problem for the fractional integro-diferential equations featuring Atangana–Baleanu derivatives. Te corresponding hybrid fractional integral equation is presented. Ten, we establish the existence results using Dhage’s hybrid fxed point theorem for a sum of three operators. We also ofer additional exceptional cases and similar outcomes. In order to demonstrate and verify the results, we provide an example as an application.


Introduction
Te fundamental theory of fractional diferential inequalities was developed by Kilbas et al. [1], Samko et al. [2], and Lakshmikantham and Vatsala [3,4] using fractional derivatives (FDs) of the Riemann-Liouville (RL) and Caputo operators. Te investigated fractional inequalities and the comparison fndings were used by Lakshmikantham and Vatsala [3,4] to investigate if there are local, global, and extremal solutions to nonlinear fractional diferential equations (FDEs). Te frst-order hybrid diferential equation (hybrid DE) was pioneered by Dhage and Lakshmikantham [5], who concentrated on the basic discoveries pertaining to the existence and uniqueness of solutions to the following hybrid DE: where F ∈ C(℧ × R, R) and Q ∈ C(℧ × R, R\ 0 { }). Additionally, comparison fndings and solution quality were examined using diferential inequality results from hybrid ODEs. Using a methodology similar to that of [5], Zhao et al. [6] extended the study of hybrid diferential equations to encompass hybrid FDEs with the RL derivative.
On the other hand, Caputo and Fabrizio [16] proposed the FD involving exponential kernel with the aim of removing singular kernels in the classical FD. In [17], Atangana and Baleanu created a new FD in the sense of Caputo that so-called the ABC FD and has the Mittag-Lefer function as its kernel. Te ABC-fractional diferential operator is more appropriate for a better description of realworld occurrences since it is a nonlocal operator with a nonsingular kernel. It can be used in a variety of ways to illustrate various problems, including the spread of dengue fever [18], clinical implications of diabetes and tuberculosis coexistence [19], tumor-immune surveillance mechanism [20], free oscillation of a coupled oscillator [21], smoking models [22], tuberculosis (TB) model [23], and coronavirus [24,25]. For more information on the fundamental developments in the theory of ABC-type nonlinear FDEs, we advise the reader to peruse the references [26][27][28][29][30][31][32].
In this regard, Ahmad et al. [7] considered the following hybrid fractional integro-diferential equations: Te following is ABC-type FDE: which has been studied by Sutar and Kuchhe [33]. Inspired by the works mentioned above, we discuss the existence of solutions to the following nonlocal BVP of hybrid fractional integro-diferential equations: In this work, we concentrate on applying the most recent fractional operators, also referred to as ABC operators, on which researchers are constantly working to increase the number of potential solutions to the problem at hand. Te existence results of the ABC hybrid problem (5) based on Dhage's fxed point theory are established and expanded. When taking into account the hybrid problems and the used operator, our conclusions are entirely unique. Our problem under consideration includes some signifcant special cases that have not yet been researched as in Remark 1. Te arguments made also demonstrate that a variety of unique situations that play a role in our current problem.
Case 4. If we choose P i (9, φ(9)) ≡ 0, and Q(9, φ(9)) ≡ 1, then our problem (5) reduces to the following problem: Tis paper is divided into the following sections: In Section 2, we ofer some defnitions and lemmas to back up our major conclusions. In Section 3, we demonstrate that there are solutions to problem (5) using Dhage's fxed point theory for Banach algebra. An illustration example is given to emphasize the main results. Te work is concluded in Section 4.
Te ABC-FD of order κ for a function φ is given by Further, the ABR-FD is given by where ℵ(κ) > 0 is a normalization function satisfying Defnition 3 (see [17]). Let 0 < κ < 1 and φ be function, then AB integral of order κ is defned by where RL I κ a + is defned by (11).

Main Results
In this section, we prove the existence theorem of the ABCtype nonlocal hybrid problem (5).
, and h ∈ C with h(0) � 0 and P i (0, φ(0)) � 0, for all i � 1, . . . , I. Ten, φ is a solution to the following hybrid linear problem: if and only if where

Journal of Mathematics 3
Proof. Applying AB I κ 0 + to both sides of (19) and using Lemma 3, we have which implies where c 1 , c 2 ∈ R. Taking into account the limits of (22) at ϱ ⟶ 0 and ϱ ⟶ 1, it follows from conditions φ(0) � ρ(φ) and φ(1) � A, respectively, that , Substituting the values of c 1 and c 2 in (22), we get which gives (20) taking into account the assumed value of η A . Te converse follows by direct computation.

□
We can now conclude the following result: Ten, the solution of (5) satisfes the following equation: where η A as in Lemma 4.

Theorem 2. Suppose that (As1)-(As4) hold. Ten, there exists at least one solution for the ABC-type problem (5) on ℧.
Proof. Defne the set D � φ ∈ C: ‖φ‖ C ≤ r , where r satisfes (As3). Certainly, D is a convex, closed, and bounded subset of C. By Corollary 1, we observe that the fxed point problem Oφ � φ is equivalent to the problem (5). Next, we defne three operators O 1 , O 3 : C ⟶ C, and O 2 : D ⟶ C by So, we can write the formula (25) in the operator form as

Journal of Mathematics 5
Let φ, φ∈ C and ϱ ∈℧. Ten, by (As2), we have that implies First, we prove that O 2 is continuous on D. Let φ n n ≥ 1 be a sequence in D with φ n ⟶ φ ∈ D. Ten through Lebesgue's convergence theorem, we have 6 Journal of Mathematics , for all ϱ ∈℧. Tus, O 2 is a continuous on D.
Finally, we show that the set O 2 (D) is an equicontinuous in C.

Journal of Mathematics
which implies As  Let φ ∈ C and y ∈ D be arbitrary elements such that φ � O 1 φO 2 y + O 3 φ. Ten, Claim 4. Assumption (iv) of Teorem 1 is satisfed, i.e., By (As3), we have Tus, all assumptions of Teorem 1 are satisfed; and hence, the equation φ � O 1 φO 2 φ + O 3 φ has a solution in D. As a result, the nonlocal hybrid BVP (5) has a solution on ℧.
Te following is an example. Here, we provide an example to illustrate the obtained results.

Conclusions
In this study, we have expanded and developed the results for the existence of nonlocal hybrid FDEs with ABC FDs. Our strategy is founded on Dhage's hybrid fxed point theorem for sums of three operators. Te observations that were made to support the thoroughness and modernity of our work, where we have presented a number of signifcant special cases and similar results of our current problem that have not yet been addressed in the literature. A practical case as an example to support the theoretical fndings has been provided. It will be fascinating for future systems to be using the psi-ABC fractional operator that was just recently introduced in [36].

Data Availability
No data were used to support this study.

Disclosure
Tis work was conducted during our work at Hodeidah University.