Existence Theorems for Hybrid Fractional Differential Equations with <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 11.3962 13.4804" width="11.3962pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z"/></g></svg>-</span>Weighted Caputo–Fabrizio Derivatives

Alshammari, Mohammad; Alshammari, Saleh; Abdo, Mohammed S.

doi:https://doi.org/10.1155/2023/8843470

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Mathematical Methods for Fractional Differential Equations in Applied Sciences

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Volume 2023 | Article ID 8843470 | https://doi.org/10.1155/2023/8843470

Existence Theorems for Hybrid Fractional Differential Equations with -Weighted Caputo–Fabrizio Derivatives

Mohammad Alshammari,¹Saleh Alshammari,¹and Mohammed S. Abdo^2,3

Academic Editor: Kolade M. Owolabi

Received10 Dec 2022

Revised30 Sept 2023

Accepted03 Oct 2023

Published23 Oct 2023

Abstract

In this study, two classes of hybrid boundary value problems involving -weighted Caputo–Fabrizio fractional derivatives are considered. Based on the properties of the given operator, we construct the hybrid fractional integral equations corresponding to the hybrid fractional differential equations. Then, we establish and extend the existence theory for given problems in the class of continuous functions by Dhage’s fixed point theory. Furthermore, as special cases, we offer further analogous and comparable conclusions. Finally, we give two examples as applications to illustrate and validate the results.

1. Introduction

The theory of fractional calculus has attracted the attention of a remarkable number of researchers from various fields in recent years. The physical meaning of fractional orders is that the dynamical systems of fractional order can be represented by a fractional differential equation (FDE) with a noninteger derivative. These systems are referred described as having fractional dynamics. Undoubtedly, it has been demonstrated that the use of fractional derivatives (FDs) is very beneficial for modeling a wide range of problems and natural phenomena in engineering and applied sciences; for example, see renowned monographs by Osler [1], Samko et al. [2], Kilbas et al. [3], and Diethelm and Ford [4]. The literature contains a variety of FD concepts, including those presented by Riemann–Liouville and Caputo [3], which include the singular kernel , .

These FDs play an important role in modeling numerous physical and biological phenomena. In any case, as was referenced in Caputo and Fabrizio [5, 6], certain peculiarities connected with material heterogeneities cannot be well modeled utilizing Riemann–Liouville or Caputo FDs. Because of this reality, the authors in [5] proposed another FD involving the nonsingular kernel , ; then, Losada and Nieto [7] studied some of its properties. The existence and uniqueness of solutions are essential properties of mathematical models [8, 9] and are among the advantages of applied theory. The model must possess these properties in order to be reliable and useful. Existence refers to the fact that the model must describe a well-defined problem, which can be solved within a certain mathematical framework. In other words, the model should not have any ambiguity or inconsistency that would make it impossible to solve. In [10–18], the authors studied the existence of solutions for various types of FDEs involving the Caputo–Fabrizio FD and other fractional operators. For instance, Abbas et al. [11] handled some existence results for Caputo–Fabrizio type implicit FDEs in b-metric spaces. The existence and uniqueness of solutions for the following problem were demonstrated by Shaikh et al. [14]. It is possible to think of hybrid differential equations as quadratic perturbations of nonlinear differential equations. They are of great interest to scholars because they are particular instances of dynamical systems. Dhage and Lakshmikantham [19] provide details on various perturbations for nonlinear differential and integral equations. For additional updates on the availability of hybrid FDEs theory, we refer to [19–21]. For instance, the following hybrid classicalhas been studied by Dhage and Lakshmikantham [19]. Taking on the analogous approach of [19], Zhao et al. in [20] extended the investigation of hybrid (2) to the following Riemann–Liouville type hybrid FDE:

Herzallah and Baleanu [21] discussed the existence results of hybrid FDE (3) using the Caputo FD. Furthermore, various classes of hybrid FDEs subject to different conditions have additionally been concentrated on by many specialists, see [22–27]; e.g., Ali et al. [26] developed an existence analysis for nonlinear hybrid FDEs with -Hilfer FD and hybrid boundary conditions. For a nonlocal hybrid of Caputo fractional integrodifferential equations, Ahmad et al. [22] presented the existence results.

Almeida [28] proposed a general operator so called -Caputo FD when the kernel is and the derivative is . Then, the authors in [29] expanded some of the properties of this operator to include the Laplace transform of it. Regarding this, Jarad et al. [30] developed the idea of weighted FDs with another function. Abdo et al. [31] proved the positive solutions of the following -weighted Caputo problem:

Recently, Al-Rafai and Jarrah [32] extended the concept of weighted FD to the -weighted Caputo–Fabrizio FD, where and are monotone function and weight function, respectively.

Motivated by the abovementioned studies, we discuss the existence of solutions of the following weighted hybrid FDE:and the following -weighted hybrid FDEwhere and , with , , , and denote the weighted Caputo–Fabrizio and -weighted Caputo–Fabrizio FDs, respectively, and with on . Our first contribution focuses on several special cases of problems (5) and (6) connecting the weighted function to another monotone function. Then, using Dhage’s fixed-point theory and some added conditions of functions , we show the existence of solutions to (5) and (6) under -weighted Caputo–Fabrizio FDs or the diverse hybrid cases. Specifically, our results support, extend, and enhance those found in [31–33].

Remark 1. This work can be a generalization of some of the studied problems in the literature, for example,(i)In problem (5), if we choose , , and , then we obtain the following problem: which has been studied by Salim et al. [33].(ii)If we choose , , , and , then our problem (6) reduces to problem (4), which was considered by Abdo et al. [31].

Remark 2. (1)If , then problem (6) reduces to problem (5)(2)If , , , and , then problem (6) reduces to problem (7), see [33](3)If , , , , , and , then problem (6) reduces to problem (1), see [14](4)Many problems with less general operators with various values of and , such as the one proposed by Caputo and Fabrizio in [5] are part of our current problemsThe rest of this work is arranged as follows. Section 2 gives some basic results about generalized Caputo–Fabrizio FD and functional spaces. Our main results of problems (5) and (6) are discussed in Section 3. Two examples that confirm the validity of the main results are provided in Section 4. Finally, we include the conclusions in Section 5.

2. Primitive Results

We start off this section by providing some definitions and fundamental results. Let , . will constantly represent real space. Define the supremum norm in by , and the multiplication in by . Obviously, is a Banach algebra with the norm and multiplication in it. The weight function and the monotone function, respectively, are represented by and with and on .

Definition 3 (see [32]). Let , and . Then, the -weighted Caputo–Fabrizio FD of is given by the following equation:where , and is a normalization function satisfying .
The above-given operator can be written as follows:

Definition 4 (see [32]). Let , and . Then, the -weighted Caputo–Fabrizio fractional integral is defined as follows:

Lemma 5 (see [32]). Let , and . Then,

In particular, if , then .

Lemma 6 (see [32]). Let , and with . Then, the following FDEhas the unique solution

Theorem 7 (Dhage’s fixed point theorem [34]). 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 , (iv) , where . Then, there exists such that .

3. Main Results

Here, we provide some qualitative analyses of two types of Caputo–Fabrizio hybrid problems that are (5) and (6).

Lemma 8. Let be continuous function on with and assume that is increasing in , a.e., for each . Then, the solution of the -weighted hybrid FDEsatisfies the following equation:where , and with .

Proof. Applying the operator of the first equation of (14), we have the following equation:Using Lemma 5, we have the following equation:Comparing (16) and (17), we obtain the following equation:which impliesTaking to both sides of (19), we have the following equation:Applying the boundary condition of (14) and using (20), we obtain the following equation:Hence,Substituting (22) into (19), we obtain (15).
Due to Lemma 8, we can infer the following result:

Corollary 9. Let and with . Then, the solution of (6) satisfies the following equation:where , and as in Lemma 8.

Now, we need the following assumptions on , and . (As1) and are continuous (As2) There exist positive functions and with bounds and , respectively, such that (As3) There exist two functions and be nondecreasing continuous such that (As4) There exists such thatand , where , , and

Theorem 10. Suppose that (As1)–(As4) hold, then the -weighted problem (6) has at least one solution defined on .

Proof. Define the set , where satisfies (As4). Certainly, is a convex, closed, and bounded subset of . By Corollary 9, we define three operators and bySo, we can write the formula (23) in the operator form as follows:Now, we show that , , and fulfill all the assumptions of Theorem 7, through the following claims:

Step 11. and are Lipschitzian on .
For and , we have from (As2) thatwhich impliesThus, are Lipschitzian on with Lipschitz constants and , respectively.

Step 12. is a completely continuous.
In the beginning, we show that is continuous on . Let be a sequence in with . Then, from Lebesgue’s convergence theorem [35], we obtain the following equation:Hence, , for all . Thus, is a continuous on .
Next, let . Then, by (As3), we have the following equation:Since and applying the mean value theorem for integral, we obtain the following equation:for some . It follows thatHence, (33) becomesTherefore, , for all , where given by (As4). This consequence proves that is uniformly bounded set on . Finally, we show that the set is an equicontinuous in .
Let with , and Then,As , the continuity of and imply thatThus, is equicontinuous on . As a result of the Ascoli–Arzelà theorem [4], is a completely continuous.

Step 13. Assumption (iii) of Theorem 7 is satisfied.
Let and such that . Then,which implies

Step 14. Assumption (iv) of Theorem 7 is satisfied, i.e., , where .
Sincewe have the following equation:Thus, all the assumptions of Theorem 7 are satisfied, and hence, the equation has a solution in . As a result, -weighted hybrid problem (6) has a solution on .

3.1. Special Results

In this subsection, we discuss some special cases of problem (6).

Consider in problem (6), we obtain the following weighted hybrid FDE:where all constants and symbols correspond to those in problem (6). Since the next Lemma is a duplication of Lemma 8 with , we shall omit its proof.

Lemma 15. Let be continuous function on with and assume that is increasing in , a.e., for each . Then, the solution to the following weighted hybrid FDEsatisfies the equationwhere , and are as in Lemma 8.

According to Lemma 15, we can define three operators , and by

In addition, we must provide some constants as follows:and

The following existence theorem can be stated without proof.

Theorem 16. Suppose that (As1)–(As4) hold with (47) and (48). Then, the weighted hybrid problem (5) has at least one solution defined on .

Remark 17. Our results for problem (6) are applied for the following special cases: Case 1: if and , then, we have the initial value problem of hybrid FDE: Case 2: if and , then, we have the terminal value problem of hybrid FDE: Case 3: if and , then, we have the antiperiodic the problem of hybrid FDE: Case 4: if we choose , and , then our problems (5) and (6) reduce to the following problems:

4. Examples

Here, we give two examples to demonstrate the attained results.

Example 1. Consider the following -weighted hybrid problem:Clearly, , , , , and SetNote that . For and . Then,Thus, (), (), and () hold with , , and . Then, , , , and . In addition, the condition holds. Indeed, we have , , , and , where andWhenwe have the following equation:From (56) and (58), we obtain the following equation:Hence, . Thus, there exists such that (57) holds. On the other hand, we have the following equation:Using the MATLAB program, we find that the constant satisfies the inequality . Thus, Theorem 10 shows that (53) has a solution on [0, 1].

Example 2. Let and consider the following weighted hybrid problem:Clearly, , , , , , whereObserve that . For , and . Then,Thus, (), (), and () hold with , , and . Then, , , , and . In addition, the condition holds. Indeed, we have , , , and with andThen, we have the following equation:Using the MATLAB program, we find that the constant satisfies the inequality . Thus, Theorem 16 shows that (61) has a solution on .

5. Conclusions

The existence and uniqueness of solutions are among the qualitative properties of mathematical models. These properties are important because they help ensure that the model provides reliable and accurate results and that the results are applicable to a wide range of situations. Without these qualitative properties, a model may not accurately reflect the real-world phenomena it is meant to describe, which can lead to incorrect conclusions and unreliable predictions. In this work, we have successfully analyzed the nonlinear hybrid differential equations by the application of fractional calculus. Specifically, problems (5) and (6) have been considered using the -weighted Caputo–Fabrizio FDs, which incorporate a nonsingular kernel. First, we have provided several special results and various observations for our proposed problems in the frame of -weighted Caputo–Fabrizio FDs, which made our results more generalizable and studyable to a wide range of previously studied and research-worthy problems. Then, through the utilization of Dhage’s fixed point theory for sums of three operators, we have established the existence of solutions to the proposed hybrid problems. Finally, in order to support the theoretical results, we have offered two practical examples. In the future, it will be interesting if the current systems are studied in the frame of -weighted Atangana–Baleanu–Caputo, recently introduced in [36, 37].

Data Availability

No data are used in this work.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The last author (Mohammed S. Abdo ) would like to thank Prince Sultan University for the support through the TAS research lab.

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Copyright © 2023 Mohammad Alshammari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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