Analysis of Existence and Stability Results for Impulsive Fractional Integro-Differential Equations Involving the Atangana–Baleanu–Caputo Derivative under Integral Boundary Conditions

In this study, we consider the existence results of solutions of impulsive Atangana–Baleanu–Caputo ( ABC ) fractional integro-diferential equations with integral boundary conditions. Krasnoselskii’s fxed-point theorem and the Banach contraction principle are used to prove the existence and uniqueness of results. Moreover, we also establish Hyers–Ulam stability for this problem. An example is also presented at the end.


Introduction
Te concept of derivatives and integrals of any arbitrary real or complex order is called fractional calculus, and initially, it was proposed in works by mathematicians such as L'Hôpital, Liouville, Leibniz, Riemann, and Abel. Te nonlocality of fractional derivatives, a characteristic inherent to many complex systems, makes them important for modeling phenomena in numerous disciplines of engineering and science. A signifcant study had already been conducted in this area [1][2][3][4][5][6][7]. Fractional derivatives provide more realistic representations of real-world behavior than ordinary derivatives when dealing with some phenomena because they consider the global evolution of the system rather than just local dynamics. In [8], Podlubny has examined the methods and applications of fractional derivatives and fractional diferential equations (FDEs). Many authors have investigated the applications of FDEs [9][10][11].
Tere has recently been a lot of attention drawn to the quadratic perturbation of nonlinear diferential equations also known as hybrid diferential equations. Due to the inclusion of several dynamic systems as special instances, research on hybrid diferential equations is signifcant. Hilal and Kajouni [12] have discussed boundary value problems (BVPs) for hybrid fractional diferential equations (H-FDEs). In [13], the authors have studied on the experimental applications of hybrid functions to FDEs.
Tere are numerous applications of BVPs within the feld of applied mathematics. For instance, concentration in chemical or biological issues and nonlinear sources produce nonlinear difusion and the theory of thermal ignition of gases. In addition, BVPs with integral boundary conditions have numerous contributions of mathematical modeling to the heat conduction process, hemic conduction process, and hydrodynamics issues. Many authors have investigated FDEs with boundary conditions [14][15][16][17][18][19][20][21][22][23].
Te Atangana-Baleanu derivative is being explored to develop a model that depicts the behavior of conventional viscoelastic materials, thermal media, and other materials. Te suggested mechanism is capable of representing material heterogeneities as well as some media or structures at various scales. In [24,25], the authors have discussed the existing results on the Atangana-Baleanu-Caputo derivative. Te entire description of memory inside structures and media with various scales, which cannot be represented by conventional fractional derivatives or those of the Caputo-Fabrizio type, is made possible by new kernel's nonlocality. Additionally, we think that Atangana-Baleanu derivatives can be useful in the investigation of some materials' microstructural behavior, particularly in cases where nonlocal exchanges are involved because they play a key role in establishing the material's physical states. To have a clear picture, in [26], Algahtani has compared the Caputo-Fabrizio derivative and Atangana-Baleanu with the fractional order Allen-Cahn model.
Te Mittag-Lefer function has long been recognized as being extremely benefcial in fractional calculus. Additionally, it has had a substantial impact on the defnitions of other fractional diferential integrals. It has been studied by many authors [27][28][29][30][31]. To put it simply, fractional diferential models and systems generalize ordinary and partial diferential systems. Generalization is caused by the FDEs of noninteger (fractional) order, and their nonlocality always aids in simulating nonlocal interactions in nature. Blood fows, natural structures such as heartbeats, control theory, hypothetical physical science, mechanical frameworks, designing, population dynamics, biotechnology, medical science, economics, and various other real-world things are examples of things that change quickly. Ulam discovered that the question of whether a proposition's claim genuinely obeys, or generally holds, if the hypothesis is slightly altered, is a prevalent and important one in many domains and has drawn the attention of many scholars. Tat is referred to as the Hyers-Ulam stability problem [32][33][34][35][36][37][38].
In [39], Rozi and others examined the following BVPs under the ABC fractional derivative: where H, F, G: J × R ⟶ R. Tey investigated the existence and uniqueness of the aforementioned problem and also proved the U-H type of stability.
In [40], Devi and Kumar studied the existence and uniqueness of results for integro FDEs with the Atangana-Baleanu fractional derivative: where Based on the aforementioned work, we consider the BVP for impulsive Atangana-Baleanu-Caputo (ABC) fractional integro-diferential equations: 2 Mathematical Problems in Engineering Te structure of the study is as follows. In Section 2, we recall some fundamental defnitions, notations, and preliminary facts. Section 3 is focused on the existing results for the fractional integro-diferential systems with the ABC derivative. Section 4 is dedicated to establishing the results of Hyers-Ulam stability. Te application of our theoretical results is given in Section 5.

Preliminaries
We examine several fundamental fndings that are applied to our major study in this part.
Defnition 2 (see [25]). Te equivalent integral for υ is written as where the R − L fractional integral is denoted as I ℘ .
Lemma 1 (see [31]). We assume the problem as Given the conditions under which the RHS disappears at time t � 0, the solution is provided by Here, we indicate the Banach space Ten, A has at least one fxed point.

Main Results
Here, we look into the prerequisites for the existence of a solution to problem (3).
Lemma 2 (see [39]). We assume the linear BVPs with nonlinear integral boundary conditions, and if κ ∈ L(J ′ ), we obtain Ten, the solution ϰ ∈ AC(J ′ ) is given by Proof. By Lemma 1, we can instantly get the result of (9).

Corollary 1. According to Lemma 2, the solution of problem (3) is provided by
Mathematical Problems in Engineering To gain the interconnected results, the hypothesis must be true: and and N V � max t∈J′ ‖V(t, 0, 0, 0)‖, where ϰ, ϰ, y, y∈ Ξ.

Mathematical Problems in Engineering
Hence, we obtain Terefore, Υ is a contraction mapping, and the solution of problem (3) is unique. □ Theorem . By hypotheses (A1) to (A6), we get Ten, there is at least one solution for problem (3). We defne the operators as follows: Proof.

Mathematical Problems in Engineering
We have where F is therefore a contraction.
Step 3: to show equicontinuity, we take t k−1 < t k ∈ J ′ , and we have Obviously, from (33), we see that t 1 ⟶ t 2 , and the RHS of the inequality implies zero Te operator Gϰ is uniformly continuous, and also, G(D) ⊂ D is compact. By the Arzel a � -Ascoli theorem, G is completely continuous. Hence, given problem (3) has at least one solution.

Stability Results
Tis section focuses on showing the results of Hyers-Ulam stability for problem (3).