Atangana–Baleanu–Caputo differential equations with mixed delay terms and integral boundary conditions

In this research work, we investigate a class of Atangana–Baleanu–Caputo (ABC) differential equations (DEs) having proportional and discrete delay terms accompanied by integral boundary conditions. We study the two important aspects, that is, the existence of the solution and stability analysis for the problem mentioned above. We confirm the existence of at least one solution via Schaefer's fixed‐point theorem and its uniqueness by Banach's fixed‐point theorem. Moreover, for stability analysis, we use the concept of Hyers–Ulam stability. For application purposes, we apply the main results to a numerical problem for specific values of the parameters involved.


INTRODUCTION
The theory of fractional calculus takes an important place in most branches of applied sciences owing to the nonlocal and global nature of the operators involved. These operators enable us to understand the dynamic behavior of physical phenomena. Its applications are observed in astronautics, bioengineering, mechanical engineering, marine engineering, chemical engineering, and so on. In literature, numerous studies are focused on the applications; see previous studies [1][2][3][4][5][6][7]. The Caputo fractional differential operator is extensively applied in the modeling of many biological and physical problems. Tuan et al. [8] modeled the COVID-19 transmission with Caputo fractional derivative and studied its mathematical analysis. Rezapour et al. [9] studied the SEIR epidemic model for COVID-19 transmission under the Caputo derivative. Using the Caputo fractional differential operator, Deressa et al. [10] modeled a hyper-chaotic Lorenz-Stenflo and studied its qualitative analysis. Similarly, Alam et al. [11] studied a problem of fractional differential equations (FDEs) with multi-point strip boundary conditions via Caputo fractional derivative. In Matar et al. [12], a p-Laplacian non-periodic and nonlinear boundary value problem (BVP) is studied under a generalized Caputo fractional derivative. Similarly, in Etemad et al. [13], a fractal-fractional model of the AH1N1/09 virus and its generalized Caputo version were studied. Baleanu et al. [14] studied a hybrid Caputo fractional-order thermostat model with hybrid boundary value conditions. Many other results of the existence theory of different classes of FDEs are well studied by researchers as in previous studies [15][16][17][18][19].
Caputo and Fabrizio [20] introduced a new concept for fractional differentiation based on exponential decay kernel. Losada and Nieto [21] studied the properties of the definition mentioned above. This concept gained significant attention from researchers in recent years. They used it in various mathematical models. Rezapour et al. [22] carried out mathematical analysis for the anthrax disease model in animals under Caputo-Fabrizio non-integer-order DEs. Mohammadi et al. [23] carried out theoretical results for the hearing loss model due to the Mumps virus with optimal control under non-integer order Caputo-Fabrizio DEs. Baleanu et al. [24] executed the mathematical analysis of HIV-1 infection of CD4 + T-cell under the new differentiation approach. Baleanu et al. [25] studied a fractional-order model of human liver in the Caputo-Fabrizio sense.
Atangana and Baleanu [26] formulated new concepts of fractional derivative. They used the Mittag-Leffler kernel in the modified definitions, which is nonlocal and non-singular. Due to the nonlocal and non-singular properties of the kernel, these definitions have been given to much importance by researchers around the globe. The new concepts of fractional differentiation are being used in several fields like biological sciences, physical sciences, engineering, and technology. Baleanu et al. [27] studied fractional optimal control problems under Mittag-Leffler non-singular kernel. Most recently, Shah et al. [28] pointed out necessary conditions for the existence of a solution of an integral BVP with impulsive behavior involving a non-singular derivative.
One of the most essential classes of differential equations is the class of delay differential equations (DDEs). The DDEs are used to model evolution phenomena in various life and physical sciences. More specifically, these equations have applications in neural networks, epidemiology, physiology, immunology, and population dynamics [29,30]. Similarly, modeling some physical systems in which the current or future state depends on history due to hereditary characteristics usually need the use of DDEs. In economics, the status at time t is a function of that time with some delay and is fundamental in decision-making problems. It is important to note that, in DDEs, the evolution phenomena at a certain instant of time are determined by the past. For some applications of DDEs, we refer the readers to see previous studies [31][32][33][34]. The delay may be discrete, continuous, or proportional. The topic of DDEs and their applications is a recent one in mathematical subjects, which might result in significant advances. Here, we should mention that some phenomena are complex in nature and cannot be accurately modeled with only discrete DDE or proportional DDE. They involve both types of delays in their nature.
On the other hand, stability analysis that has been considered necessary for the qualitative theory of nonlinear problems has gained importance. Various concepts of stability have been used in the literature. The Hyers-Ulam (HU) stability concept has been given too much importance in the last two decades due to its simple procedure. Previous studies [35][36][37][38] considered various classes of FDEs, including impulsive differential equations, where they applied the HU stability approach to carry out stability analysis for the problems mentioned above.
Motivated by the work cited above, we have considered, the following class of DEs under ABC differential operator with two different delay terms, namely, proportional delay and discrete delay: where ABC D is ABC fractional differential operator of order , the nonlinear function,  ∶ [0, 1] ×  3 →  and the functions g, h ∶  →  are continuous. In this problem, the DEs are subjected to integral boundary conditions. This problem falls in the class of integral BVPs, which have a wide range of applications. They describe many phenomena and physical processes in the applied sciences, which naturally arise in the theory of nonlinear diffusion generated by nonlinear sources, thermal ignition of gases, and concentration in chemical or biological problems. We can see problems with integral BCs, which have recently been investigated by researchers in previous studies [39][40][41][42]. The rest of the paper is organized as follows: In Section 2, a Banach space with its respective norm is defined. Also, definitions of ABC derivative and integral are given in mentioned above section. In Section 3, we provide proof of our main results. Here, criteria for the existence of at least one solution and uniqueness are developed. In Section 4, HU stability analysis is carried out for the concerned problem. In Section 5, a numerical problem is given as an example of the application of the main results. In Section 6, some discussion and concluding remarks are given.

PRELIMINARIES
For simplicity purpose, we denote the space of all continuous functions that is [26]). Let ∈ H 1 (a, b), b > a, ∈ (0, 1], the ABC fractional-order derivative of function is defined by [26]). The corresponding AB fractional integral of order > 0 for a function is defined by

Definition 2.2. (Atangana and Baleanu
It is noteworthy that for = 0 and = 1, the initial function and ordinary integral can be obtained, respectively.

Theorem 2.3.
Considering that the right-hand side of the following FODE vanishes at t = 0, then the solution of is given by [43]). Let  be a convex subset of a norm-linear space S with 0 ∈  and let ∶  →  is a completely continuous operator. Then, the set = {w ∈  ∶ w = w; 0 < < 1} is either unbounded or has a fixed point in .

MAIN RESULTS
if w is solution of the following integral equation Proof. We assume that w satisfies problem (2). Then, consider Applying the Atangana-Baleanu fractional integral, for t ∈ [0, 1], we have Differentiating, we obtain Hence, And Similarly, Thus, From which, we get Also, Putting (12) in (13) and then solving it for c 1 , we have Putting this value of c 1 in (12), we have Inserting these values of c 0 , c 1 in (5), we get (3). □ (1) as
To proceed, we transform the problem into fixed-point problem. We define an operator Our results are based on the following assumptions.
There exists constant K, L, M > 0 such that for each t ∈ [0, 1] and for all w,w ∈ , the following relation holds

Theorem 3.2. Problem (1) has at least one solution under assumptions (A 1 )-(A 5 ).
Proof. For proof of this result, we apply Schaefer's fixed-point theorem which completes in the following four steps.
Since w n → w as n → ∞. This implies z w n → z w . Now, let there exists a real constant > 0 such that |z w n (t)| ≤ and |z w (t)| ≤ . Then, This proves that is continuous as required.
Step 2. In this step, it is needed to show, is bounded. We define a closed set By (A 4 ), we have Using the last result and assumption (A 5 ), we get the following result from (17): Hence, is bounded.
Step 3. In the third step, we will prove that maps bounded sets into equi-continuous sets of . Let x 1 , x 2 ∈ [0, 1] such that x 1 < x 2 . As in Step 2, let be a bounded set of . Now, for w ∈ , we have Simplifying further, we get We see that as x 1 → x 2 , the right-hand side of (19) goes to 0. Therefore, applying Arzela-Ascoli theorem, the completely continuity of is obtained.
Step 4. A priori bounds: In this step, we define a set w = {w ∈  ∶ w = w, or some 0 < < 1}. We will show that this set is bounded. Let w ∈ ℧, then w = w, for some 0 < < 1. Hence for any t ∈ [0, 1], from Step 2, we have Therefore, the set w is bounded. Hence, by application of Schaefer's fixed-point theorem, at leat one fixed point for the operator is confirmed.
satisfy, then problem (1) has a unique solution.
Proof. For w,w ∈  and t ∈ [0, 1], we have where z w (t) and zw(t) satisfy the functional equations z w (t) =  (t, w(t), w( t), w(t − (t))), zw(t) =  (t,w(t),w( t), w(t − (t))), respectively. From Step 1 of Theorem 3.2, we have And Hence, using these inequalities and taking maximum, from (22), we obtain the following simplified result: Since therefore, is contraction, and hence, by Banach's fixed-point theorem, has unique fixed point, which confirms the unique solution of the proposed problem (1). Which completes the proof. □

STABILITY ANALYSIS
In this section, we derive sufficient conditions for HU stability analysis of problem (1). We construct the following inequalities. Let for t ∈ [0, 1], > 0, the following inequality holds by B, such that B < 1. Then, from (28), we write This shows that problem (1) is HU stable. □ Corollary 2. If we find a non-decreasing function ∈ C( + ,  + ), such that ( ) = C( ) with (0) = 0. Then, we have Thus, in this case, the problem (1) is GHU stable.

DISCUSSION AND CONCLUDING REMARKS
Various physical phenomena in which the current or future state depends on history can be modeled via DDEs. The delay may be discrete, continuous, or proportional. Some phenomena involve more than one type of delay, which make them more complex. Besides that, the literature has shown that fractional derivatives with non-singular kernels provide a more precise and complete description of complex systems and processes, which may lead to better predictions, control, and optimization. Therefore, in this research work, we have successfully studied a mixed delay differential problem with integral boundary conditions having proportional delay and discrete delay terms in the framework of the ABC derivative. We have established the criteria for at least one solution through Schaefer's fixed-point theorem and its uniqueness through Banach's fixed-point theorem. Additionally, the stability conditions in the sense of HU have been established via classical techniques of nonlinear functional analysis. We have applied our results to a numerical problem, which shows that the obtained theoretical results are valid. It is worth mentioning that the proposed model here can be extended to implicit differential problems and coupled systems in future research.