Theory of Fractional Hybrid Problems in the Frame of <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>Hilfer Fractional Operators

Ali, Saeed M.; Albalawi, Wedad; Abdo, Mohammed S.; Zahran, Heba Y.; Abdel-Aty, Abdel-Haleem

doi:https://doi.org/10.1155/2022/1079214

Journal of Function Spaces

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Numerical Methods for Differential and Integral Equations

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Volume 2022 | Article ID 1079214 | https://doi.org/10.1155/2022/1079214

Theory of Fractional Hybrid Problems in the Frame of -Hilfer Fractional Operators

Saeed M. Ali,¹Wedad Albalawi,²Mohammed S. Abdo,³Heba Y. Zahran,^4,5,6and Abdel-Haleem Abdel-Aty^7,8

Academic Editor: Youssri Hassan Youssri

Received16 Dec 2021

Revised15 Feb 2022

Accepted17 Feb 2022

Published20 Mar 2022

Abstract

In the present manuscript, we develop and extend a qualitative analysis for two classes of boundary value problems for nonlinear hybrid fractional differential equations with hybrid boundary conditions involving a -Hilfer fractional order derivative introduced by Sousa and de Oliveira (2018). First, we derive the equivalent fractional integral equations to the proposed problems from some properties of the -fractional calculus. Next, we establish the existence theorems in the weighted spaces via equivalent fractional integral equations with the help of Dhage’s fixed-point theorem (2004). Besides, for an adequate choice of the kernel , we recover most of all the preceding results on fractional hybrid equations. Finally, two examples are constructed to make our main findings effective.

1. Introduction

Recently, a lot of keen interest in the topic of fractional calculus (FC) has been shown by many researchers and investigators in view of its theoretical development and extensive applications in the applied and natural sciences. Different types of differential and integral operators of arbitrary orders have been introduced by Kilbas et al. [1]. In the same regard, Atangana and Baleanu [2] proposed a new fractional derivative (FD) based on a nonsingular and nonlocal kernel. On the advanced improvement of the FC without a singular kernel of the sinc function, the Yang-Gao-Machado-Baleanu FD was introduced in [3]. Some properties of the FD without a singular kernel were introduced by Lozada and Nieto [4].

Hilfer in [5] proposed a generalization of the Riemann-Liouville fractional derivative (RLFD) and Caputo fractional derivative (CFD) when the author deliberated fractional time evolution in physical phenomena. The author named it a generalized FD, whereas more recently, it was named the Hilfer fractional derivative (HFD). This operator carries two parameters () that may be decreased to the RLFD and CFD definitions if and , respectively. So, such a derivative incorporates between the RLFD and CFD. Some important laws and applications of this operator are obtained in [6, 7] and the references therein.

Initial value problems (IVPs) involving the HFD were investigated by many authors, like Furati et al. [8], Gu and Trujillo [9], and Wang and Zhang [10]. Some existence and uniqueness results of IVPs for -Hilfer-type coupled hybrid equations were obtained in [11]. Boundary value problems (BVPs) with the nonlocal conditions and the HFD were studied in [12]. The authors in [13] investigated the existence and stability of the solution of BVPs for -Hilfer-type fractional integrodifferential equations with boundary conditions (BCs).

-Fractional derivatives (-FDs) have been considered in [1] to be a generalization of RLFD. Some properties of these operators have been given by Agrawal in [14]. These operators are different from the other classical operators because the kernel is shown to be linked to another function . For instance, Almeida [15] gave a new generalization of CFD with some interesting properties. In addition, the authors in [16] introduced a generalized type of the Laplace transform of generalized fractional operators in the frame of both RLFD and CFD.

The new version of the HFD with respect to another function has been introduced by Sousa and de Oliveira [17]. Recently, the investigation of diverse qualitative properties of solutions to several fractional differential equations (FDEs) involving generalized FDs has become the key theme of applied mathematics research. Many interesting results concerning the existence and stability of solutions by using various kinds of fixed-point techniques were formulated; e.g., Abdo et al. [18, 19] investigated various types of the Ulam-Hyers stability for -Hilfer fractional problems with infinite delay and without infinite delay, respectively. The Ulam-Hyers-Rassias stability for FDEs using the -Hilfer operator was discussed by Sousa and de Oliveira [20].

On the other hand, hybrid-type FDEs have attained a considerable saucepan of interest and investigations of several researchers. This category of hybrid FDEs comprises the perturbations of primitive differential equations in various manners. For instance, Dhage and Lakshmikantham [21] considered the following IVPs for hybrid DEs: where and . Zhao et al. [22] studied the following hybrid FDEs with RLFD: where and .

On the other hand, Benchohra et al. [23] considered the following BVP for the Caputo-type FDE: where with and .

Hilal and Kajouni [24] considered the BVP for Caputo-type hybrid FDEs with BC: where with , , and .

However, as far as we could possibly know, no one considered the existence of solution for the hybrid-type BVPs involving HFD with respect to Here, we discuss the existence theorems of BVPs for Hilfer-type FDE with respect to and BCs:

Also, we consider the Hilfer-type hybrid FDE with respect to and hybrid BCs: where and , , and are the HFD and RLFD with respect to , respectively, , and .

Observe that the considered problems (5) and (6) are the first investigations of hybrid FDEs with hybrid BCs involving -HFD. Moreover, we have chosen this operator, besides the fact that it is a global operator and it generalizes more than twenty the freedom of choice of the ordinary differential operator; some of its advantages have been explained in Remark 3. So, we are sure that the obtained results will be a beneficial contribution and an extension of the current results in the literature. We will also refer here to some recent results related to the subject of our study (see [25–31]).

The rest of the work is displayed as follows. In Section 2, we give some advantageous preliminaries related to our work. In Section 3, we derive the equivalent solutions to linear problems corresponding to the proposed problems. Then, we prove the existence of solutions to given problems via Dhage’s fixed-point theorem. Finally, two examples to justify reported results are offered in Section 4.

2. Preliminaries

Let us initially present some imperative definitions and primer ideas related to our work.

Let where, and , and let Consider the functional spaces and as follows: equipped with the norms

Obviously, is the Banach space.

In the next expressions, we will consider to be an increasing function such that for all .

Definition 1 (see [1]). Let , and Then, -RL fractional integral of is defined by

Definition 2 (see [17]). Let , , and . Then, the -HFD of is defined by Specifically, if , we have

Remark 3. The operator is an interpolator of the following FDs: (i)Classical HFD (for , see [5]), Hilfer-Hadamard FD (for , see [32]), and Hilfer-Katugampola FD (for , see [33])(ii)Standard RLFD (for , see [1]) and standard CFD (for , see [1])(iii)Generalized RLFD (for , see [1]) and generalized CFD (for , see [15])(iv)Generalized Liouville (for , see [1]) and generalized Weyl (for , see [34])

Lemma 4 (see [1, 17]). Let , and Then, .

Lemma 5 (see [17]). Let and , where . If , then

Lemma 6 (see [17]). Let and . Then, is bounded in Moreover, we have

Theorem 7 [35]. Let be a nonempty, convex, closed subset of the Banach algebra . Let the operators , , and such that (i) and are Lipschitzian, with 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 pay attention to deriving equivalent solutions to linear problems associated with problems (5) and (6); then, we prove the existence of solution to problems (5) and (6) using Dhage’s fixed-point technique.

3.1. Fractional Integral Equations (FIEs)

The forthcoming results give the equivalent of solution formulas for the proposed problems. For brevity, we set the following symbols:

Lemma 8. Let , where , and is continuous. Then, the function is a solution of the linear fractional BVP: if and only if satisfies the FIE: where

Proof. Let be a solution of (18). We need to prove that is also a solution of (19). By the definition of and Lemma 6, we have
Now, by applying to the first equation in (18) and using Lemma 5, we can write Set Then, Taking the limit , we get From the mixed boundary conditions , we get which implies Then, (21) becomes which shows that formula (19) is satisfied, where Conversely, let satisfy (19) which can be written as (25). In (19), taking the limit as and and then using Lemma 6, we get In another direction, by applying on (19) and using Lemmas 4 and 5, we obtain This finishes the proof.

Lemma 9. Let , where and let and . Then, the function is a solution of the following linear fractional hybird BVP: if and only if satisfies the FIE: where is defined as in Lemma 8.

Proof. Let be a solution of (29). We need to prove that is also a solution of (30). By the definition of and Lemma 6, we have
Now, by applying on the first equation of (29) and using Lemma 5, we can write Set ; it follows that Taking the limit , we get which implies To determine the constant , we use the BC of (29). It was followed by which implies Then, (32) becomes which shows that formula (30) is satisfied.
Conversely, let satisfy (29) which can be written as (37). In (30), taking the limit as and and then using Lemma 6, we get In the same context, we have from (30) that Operate on both sides of (39); then, use Lemmas 4 and 5 to get This finishes the proof.

3.2. Existence Theorems

In this portion, we prove the existence theorems to problems (5) and (6) in the weighted space by means of Dhage’s fixed-point technique.

By a solution of (6), we mean a function such that (1)the function if (2)satisfies the equations in (6)

Before pursuing the main findings, we state the following assumptions:

() Let , and be continuous functions such that , for each .

() There exist two positive functions such that for each and .

() There exist such that

() There exists a constant such that where and

As a result of Lemma 8, we present the next lemma.

Lemma 10. Let , where and are continuous. Then, the nonlinear fractional BVP is equivalent to

As a result of Lemma 9, we present the next lemma.

Lemma 11. Let , , where and are continuous. Then, the nonlinear fractional hybrid BVP is equivalent to

Theorem 12. Suppose that ()–() hold. Then, problem (6) has at least one solution in .

Proof. Set Obviously, is a convex, closed, bounded subset of . By Lemma 11, problem (6) is equivalent to Define the operators and by Then, we can express equation (50) as follows: Now, we will show that , , and meet all the requirements for Theorem 7. This will be accomplished in a series of next steps.
Step 1. are Lipschitzian on .
Let and Then, by (), we have Therefore, is Lipschitzian on with the Lipschitz constant
Similarly, we conclude that is Lipschitzian on with the Lipschitz constant , i.e., Step 2. is completely continuous.
First, we show that is continuous. Let be a sequence such that in . Then, Since is a continuous function with , we have Since is increasing and using (17), we obtain This shows that is continuous on .
Next, we show that is uniformly bounded in . Indeed, for any , we have By (44), then for each
Now, we prove that is an equicontinuous set in
Let and with Then, Since for any and , there exist such that Hence, The continuity of shows that as
This confirms that is an equicontinuous set in As a result of the Arzelà-Ascoli theorem, is completely continuous.
Step 3. for and
Let , and such that . Then, Therefore, which implies Step 4. Condition (iv) in Theorem 7 holds. That is,
From Step 2 and (44), we have Hence, where and . Thus, all the assumptions of Theorem 7 are fulfilled, so the equation has a solution in . As a result, problem (6) has a solution on .

Theorem 13. Suppose that ()–() hold. Then, the -Hilfer problem (5) has at least one solution on

Proof. The proof of this theorem is quite similar to that of Theorem 12 with consideration that and in problem (6). Thus, we omit the details.

Remark 14. (1)The conditions () and () are also correct if we replace the functions with constants(2)Problem (6) reduces to problem (5) when and Consequently, Theorem 12 is applied to problem (5)(3)Problem (5) reduces to problem (3) when , , and (see [23])(4)Problem (6) reduces to problem (4) when , , , and (see [24])(5)In particular, if , then the obtained results correspond to Caputo-type FDE and RL-type FDE, for and , respectively(6)The corresponding fractional problems involving the Hilfer-Katugampola type and Hilfer-Hadamard type appear as a special case of our proposed problems for , , and , respectively

4. Examples

In this part, we construct two examples to explain the main results.

Example 15. Consider the -Hilfer hybrid FDE with hybrid BC: Define , , and by It is easy to show that for any , we have and for each , we have Hence, the hypotheses ()–() hold with , and Then, , , , and Taking , , , , , and , we get , , , and . From the condition (), when , and . Also, ; it follows from (44) that Hence, . Using the MATLAB program, satisfies the inequality Hence, all assumptions of Theorem 12 are satisfied, so problem (67) has at least one solution on

Example 16. As a special case when and , we consider the -Hilfer FDE with BC: Comparing problem (71) with problem (5), we obtain It is obvious that , and in this case, the space is reduced to the space of continuous functions Hence, () holds. It is easy to show that for each , we have Thus, the hypothesis () holds with , and Then, , , , and So, we get , . From the condition (), we have Finally, we need to show that . Indeed, from (44), we have It follows that there exists with such that . Therefore, problem (71) can be applied to Theorem 12.

5. Concluding Remarks

We have acquired further existence results for the solution of BVPs for the -Hilfer problem (6), and the -Hilfer hybrid problem (5) relies on the reduction of proposed problems to FIEs. Dhage’s hybrid fixed-point theorem in the Banach algebra has been applied. The reported results in the current paper are also valid for the hybrid FDEs involving RLFD and CFD, and they are also true for special cases of the function . We confirm that the obtained results of this work are recent and generalize some of the previous results in the literature. More precisely, when taking different values of function and parameter , the studied problems cover several problems involving classical fractional operators such as RLFD, CFD, HFD, Hilfer-Katugampola FD, and Hilfer-Hadamard FD, as mentioned in Remark 14, which have been incorporated into the operators used in our investigation. Using these investigations, other qualitative analyses of the solution such as stability and continuous dependence results can be discussed, and this is what we desire to think about in a future paper.

Data Availability

Data are available upon request.

Conflicts of Interest

The authors declare that they have no conflict regarding the publication of this paper.

Acknowledgments

The authors thank the Research Center for Advanced Materials Science (RCAMS)” at King Khalid University, Saudi Arabia, for funding this work under the grant number KKU/RCAMS/G013-21. The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education, in Saudi Arabia, for funding this research work through the project number IFP-KKU-2020/9. This work was funded under the Princess Nourah bint Abdulrahman University Researchers Supporting Project number PNURSP2022R157, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

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