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

Abstract

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

1. Introduction

The fundamental theory of fractional differential 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. The investigated fractional inequalities and the comparison findings were used by Lakshmikantham and Vatsala [3, 4] to investigate if there are local, global, and extremal solutions to nonlinear fractional differential equations (FDEs). The first-order hybrid differential 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 and . Additionally, comparison findings and solution quality were examined using differential inequality results from hybrid ODEs. Using a methodology similar to that of [5], Zhao et al. [6] extended the study of hybrid differential equations to encompass hybrid FDEs with the RL derivative.where is the RL fractional derivative of order .

Other varieties of hybrid FDEs with various initial and boundary conditions have also been studied by many researchers (see [7–15]).

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–Leffler function as its kernel. The ABC-fractional differential operator is more appropriate for a better description of real-world 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 [26–32].

In this regard, Ahmad et al. [7] considered the following hybrid fractional integro-differential equations:

The 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-differential equations:where , , , and , with for all denotes the Atangana–Baleanu–Caputo FDs, is the Atangana–Baleanu fractional integral of order , , and .

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. The existence results of the ABC hybrid problem (5) based on Dhage’s fixed 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 significant special cases that have not yet been researched as in Remark 1. The arguments made also demonstrate that a variety of unique situations that play a role in our current problem.

Remark 1. Our results for problem (5) are applied for the following special cases:

Case 1. If for all , then we have the following hybrid problem:

Case 2. If for all , then we have the following hybrid problem:

Case 3. If, then we have the neutral hybrid problem.

Case 4. If we choose , and , then our problem (5) reduces to the following problem:

This paper is divided into the following sections: In Section 2, we offer some definitions and lemmas to back up our major conclusions. In Section 3, we demonstrate that there are solutions to problem (5) using Dhage’s fixed point theory for Banach algebra. An illustration example is given to emphasize the main results. The work is concluded in Section 4.

2. Axiomatic Results

In this section, we are rendering some results of fractional calculus. Let us consider

Obviously is the Banach space. Define a multiplication in by , for all .

Definition 1. (see [1]). Let . Then, the RL fractional integral of order of an integrable function is given by

Definition 2 (see [17]). Let and ,. The -FD of order for a function is given byFurther, the -FD is given bywhere is a normalization function satisfying and is the Mittag–Leffler function.

Definition 3 (see [17]). Let and be function, then integral of order is defined bywhere is defined by (11).

Lemma 1 (see [34]). Let and . If exists, then

Definition 4 (see [34]). Let and be a function with . Then, -type FD satisfies , where and .

Lemma 2 (see [34]). Let be a function defined on and , we havefor some , where .

Lemma 3 (see [34]). Let and . Then, the solution to the following linear problem:is given by

Theorem 1 (see [35]). Let be a nonempty, convex, closed subset of a Banach algebra . Let the operators and such that (i) and are Lipschitzian with a Lipschitz constants and , respectively, (ii) is continuous and compact, (iii) for each , and (iv) where . Then, there exists such that .

3. Main Results

In this section, we prove the existence theorem of the ABC-type nonlocal hybrid problem (5).

Lemma 4. Let , , and with and , for all . Then, is a solution to the following hybrid linear problem:if and only ifwhere .

Proof. Applying to both sides of (19) and using Lemma 3, we havewhich implieswhere . Taking into account the limits of (22) at and , it follows from conditions and , respectively, thatSubstituting the values of and in (22), we getwhich gives (20) taking into account the assumed value of . The converse follows by direct computation.

We can now conclude the following result:

Corollary 1. Let and with for all . Then, the solution of (5) satisfies the following equation:where as in Lemma 4.

In the forthcoming analysis, we need the following assumptions: (As1) and , , are continuous and there exist positive functions and , with bounds and , respectively, such that for and . (As2) There exist two functions and that are continuously nondecreasing, such that (As3) There exists a constant such that where , , and (As4) There exists a constant such that

Theorem 2. Suppose that (As1)–(As4) hold. Then, there exists at least one solution for the ABC-type problem (5) on .

Proof. Define the set , where satisfies (As3). Certainly, is a convex, closed, and bounded subset of . By Corollary 1, we observe that the fixed point problem is equivalent to the problem (5). Next, we define three operators , and bySo, we can write the formula (25) in the operator form asNow, we show that , , and fulfill all the assumptions of Theorem 1 through the following claims.

Claim 1. and are Lipschitzian on .
Let and . Then, by (As2), we havethat impliesThus, are Lipschitzian on with Lipschitz constants and , respectively.

Claim 2. is a completely continuous.
First, we prove that is continuous on . Let be a sequence in with . Then through Lebesgue’s convergence theorem, we haveHence, , for all . Thus, is a continuous on .
Next, let . Then by (As3) and (As4), we havefor all .
Therefore, which shows that is uniformly bounded on .
Finally, we show that the set is an equicontinuous in .
Let with and . Then,which impliesAs , . Thus, is equicontinuous on . As a result of the Ascoli–Arzelà theorem, is completely continuous.

Claim 3. Assumption (iii) of Theorem 1 is satisfied.
Let and be arbitrary elements such that . Then,

Claim 4. Assumption (iv) of Theorem 1 is satisfied, i.e., , where .
Hence,By (As3), we haveThus, all assumptions of Theorem 1 are satisfied; and hence, the equation has a solution in . As a result, the nonlocal hybrid BVP (5) has a solution on .

The following is an example.

Here, we provide an example to illustrate the obtained results.

Consider the following nonlocal hybrid BVPwhere , , , , , , , , , , and Set

Note that and . For and [). Then,

Thus, (), (), and () hold with , , , and . Then, , , , and . In addition, the condition holds. Indeed, we have and

Here, we have

On the assumption that for all . Then,

Hence,

On the other hand, we have

Using the MATLAB program, it follows from (48) and (49) that the constant satisfies the inequality . Hence,

Thus, Theorem 2 shows that (42) has a solution on [0, 1].

4. 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 fixed point theorem for sums of three operators. The observations that were made to support the thoroughness and modernity of our work, where we have presented a number of significant 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 findings has been provided. It will be fascinating for future systems to be using the -ABC fractional operator that was just recently introduced in [36].

Data Availability

No data were used to support this study.

Disclosure

This work was conducted during our work at Hodeidah University.

Conflicts of Interest

Authors declare that they have no conflicts of interest regarding the publication of this paper.