New Existence and Stability Results for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 42.3484 16.7875" width="42.3484pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-92" d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z"/></g><g transform="matrix(.017,0,0,-0.017,5.92,0)"><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><g transform="matrix(.017,0,0,-0.017,17.145,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,23.934,0)"><path id="g113-120" d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"/></g><g transform="matrix(.017,0,0,-0.017,36.155,0)"><path id="g113-94" d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"/></g></svg>-</span>Caputo−Fabrizio Fractional Nonlocal Implicit Problems

Abdo, Mohammed S.; Shammakh, Wafa; Alzumi, Hadeel Z.

doi:https://doi.org/10.1155/2023/6123608

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Volume 2023 | Article ID 6123608 | https://doi.org/10.1155/2023/6123608

New Existence and Stability Results for -Caputo−Fabrizio Fractional Nonlocal Implicit Problems

Mohammed S. Abdo,¹Wafa Shammakh,²and Hadeel Z. Alzumi²

Academic Editor: Watcharaporn Cholamjiak

Received21 Sept 2022

Revised26 Oct 2022

Accepted24 Nov 2022

Published01 Mar 2023

Abstract

In the context of -weighted Caputo−Fabrizio fractional derivatives, we develop and extend the existence and Ulam−Hyers stability results for nonlocal implicit differential equations. The fixed-point theorems due to Banach and Krasnoselskii are the foundation for the proof of existence and uniqueness results. Additionally, the Ulam−Hyers stability demonstrates the assurance of the existence of solutions via Gronwal inequality. Also, we offer an example as an application to explain and validate the acquired results. Finally, in terms of our outcome, we designate a more general problem for the -Caputo−Fabrizio fractional system that includes analogous problems to the problem at hand.

1. Introduction

It is noteworthy that fractional calculus (FC) has received considerable attention from researchers due to its wide variety of applications in several scientific domains. The significant concepts and definitions of FC have been introduced by Osler [1] and Kilbas et al. [2]. Samko et al. [3] and Diethelm and Ford [4] provided some basic history of FC and its applications in engineering and various fields of science.

Many classes of fractional differential equations (FDEs) have been extensively investigated and analyzed in the past decades; for instance, theories involving the existence of unique solutions have been notarized in [5–7] and references therein. Numerical and analytical methods are developed with the aim of solving such equations then tracked as useful in modeling some real-world problems, as shown in [8–10].

The qualitative properties of solutions address an indispensable piece of the theory of FDEs. The beforehand previously mentioned district has been investigated well for classical DEs. Regardless, for FDEs, there are various aspects and viewpoints that require further investigation and analysis. The existence and uniqueness have been intensively studied by using Riemann−Liouville (R-L), Caputo, Hilfer, and other FDs, as shown in [11–16] and references therein.

Generalized FDs and integrals and their applications were discussed by several authors. For example, Kilbas et al. [2] introduced some nice properties of -Reiman−Liouville FD. The -Caputo FD has been defined by Almedia [17]. Then, Sousa and Oliveira proposed another generalization in the Hilfer sense [18].

In the previously mentioned derivatives, there exists a singular kernel. In this manner, as of late, some authors presented some new kinds of FDs in which they have supplanted a singular kernel with a nonsingular kernel, as shown in [19–21]. The nonlocal FDs with nonsingular kernels have been demonstrated as a decent tool to model real-world problems in various areas of engineering and science, as shown in [22–25].

On the other hand, Jarad et al. [26] introduced the concept of weighted FDs. Some recent papers dealt with the theory of existence and Ulam−Hyers (UH) (and Generalized Ulam−Hyers (GUH)) stability of different types of FDEs due to their importance in many areas of exploration; for instance, Shaikh et al. [27] established the existence and uniqueness results of the following CF-type Cauchy problem:

The existence and UH stability results in the following CF fractional implicit equation: and were investigated by Abbas et al. [28] in b-metric spaces and by Salim et al. [29] in Banach spaces, respectively. The equation (2) with was considered by Gul et al. [30].

On the other hand, Abdo et al. [31], considered the following weighted Caputo problem:

In this regard, Al-Refai and Jarrah [32], introduced the concept of the -Caputo−Fabrizio FD, where is a monotone function and is a weight function. They also obtained the uniqueness result of the Cauchy problem. Motivated by studies [31, 32], we consider the following weighted implicit nonlocal FDE:and the following type implicit nonlocal FDE:where , , , , is a given function, and is a - Caputo−Fabrizio FD, and with on .

We pay attention to the topic of the novel weighted operators with another function. As far as we are aware, no studies using the -Caputo−Fabrizio FDs have been published that address the qualitative aspects of the aforementioned problems. Consequently, to enhance and enrich the literature on this new trend, which is extremely restricted right now, we develop and extend the existence, and Ulam–Hyers stability results for problems (4) and (5) based on Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, and Gronwal inequality. Besides, we also give a more general problem as a system, that covers the problems at hand.

Remark 1. (i)If , then problem (5) reduces to problem (4)(ii)If , , and , then problem (5) reduces to the implicit problem (2), as shown in [28, 29](iii)If , , and , then problem (5) reduces to the Cauchy problem (1) without implicit term, as shown in [27](iv)Our current results for the problem (5) stay available on problem (4)(v)With different values of , our current problems cover many problems associated with less general operators; for instance, the operator presented by Caputo and Fabrizio [19]The accessories of this paper are arranged as follows: Section 2 gives some fundamental results about advanced FC. Our key findings for the problems (4) and (5) are obtained in Section 3. A comprehensive example that verifies the validity of the theories is provided in Section 4. Section 5 includes the conclusions of the work.

2. Primitive Results

In this section, we begin by giving some notations and basic nomenclature. Let , , and be the set of real numbers. and denote the set of continuous and absolutely continuous functions, respectively, on , endowed with the usual supremum norm. Let and be the monotone and weight function, respectively, with and on .

Definition 1 (see [32]). Let , and . The left -Caputo−Fabrizio FD is defined aswhere , and is a normalization function satisfying .
The previous operator can be written as

Definition 2 (see [32]). Let , and . The left -Caputo−Fabrizio FI is defined as

Lemma 1 (see [32]). Let . Then,

In particular, if , we have .

Lemma 2 (see [32]). Let , and with . Then, the following FDE:has the unique solution

For our forthcoming analysis, we need Banach’s contraction map [33] and Krasnoselskii’s fixed point theorem [34].

Lemma 3 (Gronwall’s Lemma [35]). Let , and . Let’s assume that functions are continuous if

Then,

3. Main Results

In this section, we give some qualitative analyses of -Caputo−Fabrizio type problems (4) and (5).

Lemma 4. Let , and be continuous with . Then, the following -weighted FDE:has the unique solutionwhere , and .

Proof. Let’s assume that satisfies the first equation of (14). From Corollary 2.1 in [32], the equationimplies thatSo, by the nonlocal condition , we obtainwhich is (15).
Conversely, if satisfies (15), then by Lemma 1, we havewhere . Moreover, .

Hence, we can deduce the next corollary:

Corollary 1. Let , and with . Then, the problem (5) is equivalent towhere with .

We define the operator by

Then, the fixed point of operator is equivalent to the solution of the -type problem (5).

The first result is based on Banach FPT [33].

Theorem 1. assume that:

(Hy₁): There exists such that for each , . (Hy₂): There exists such that and , for . If Then, the -type problem (5) has a unique solution on .

Proof. Let and . Then, by (Hy), we have which impliesSince , and applying the mean value theorem for integrals, we obtainfor some , thenWe suppose , we have from (Hy₂), (24), and (26) thatAs the condition (22), is a contraction and by the Banach’s fixed point theorem, has a unique fixed point which is unique solution of (5).

Next, we give existence results based on the Krasnoselskii fixed point theorem [34].

Theorem 2. Let , and be continuous satisfying (Hy₁) and (Hy₂). In addition, we assume that

(Hy₃): There exist constants with such that for each , . (Hy₄): , for , and . Then, the -type problem (5) has a least one solution if

Proof. From (20), we define the operators byandwhere . Let us define . We chooseFor , we use assumption (Hy₃) to getwhich impliesFor any , we haveDue to (29), we deduce that .
Since and are continuous, we show that is a contraction operator. For each , and for any , we haveHence,From (22), . So, is a contraction.
Next, to prove that is a compact and continuous operator, we provide the following steps: Step 1: is continuous. Let be in such that in . Then, , as , due to continuity of and . Step 2: is uniformly bounded. By (31) we have From (26) and (34), and for any , (38) becomes Thus, is uniformly bounded on . Step 3: is compact. Let , and with . Then,Sincedue to mean value theorem for integrals. Consequently, (40) givesThus, is equicontinuous on . As per preceding steps, is relatively compact on , and by the Arzela−Ascoli theorem, has at least one fixed point. By virtue of Krasnoselskii’s theorem [34], the -weighted problem (5) has a least one solution.

3.1. UH Stability Analysis

In this section, we give the UH stability and generalized UH stability results for the -weighted problem (5).

Definition 3. (5) is UH stable if there exists a such that for each and each solution of the inequalitythere exists a solution of (5), satisfying

Remark 2. satisfies (44) if and only if there exists with(i), ,(ii)For all ,

Lemma 5. Let , and be a solution of (43). Then, satisfieswhereand .

Proof. Let be a solution of (43). It follows from (ii) of Remark 2 thatThen, the solution of problem (48) isOnce more by (i) of Remark 2, we get

Theorem 3. Assume that the conditions of Theorem 1 are fulfilled. Then, the solution of the -weighted problem (5) is UH and generalized UH stable.

Proof. Let be a solution of (43), and be a unique solution of -weighted problem (5)From Corollary 1, we getwhere . Clearly, if , then . Also, by (Hy₁) and (Hy₂) along with (24), we haveFrom (22), we get . Hence, (52) becomesUsing Lemma 5, (Hy₁), and (Hy₂), we haveUsing Gronwall’s Lemma [35], we obtainwhereHence, (5) is UH stable. Moreover, if there exists a nondecreasing function such that . Then, from (56), we havewith , which proves (5) is GUH stable.

4. Example

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

Example 1. consider the following -Caputo−Fabrizio type problem:Clearly, , , , , , , and are positive constants with . Setfor , , and(I)Application of Theorem 1: Note that . Now, let , and . Then, Thus, (Hy₁) and (Hy₂) hold with , and . In addition, the condition (22) holds. Indeed, we have , , and Thus, Theorem 1 shows that (59) has a unique solution on .(II)Application of Theorem 3: For with . It follows from Theorem 3 that (59) is HU and GUH stable.(III)Application of Theorem 2: For , and , we haveConsequently, (Hy₃) and ((Hy₄) hold with and . Also, we haveThus, all the assumptions of Theorem 2 are satisfied. Hence, (59) has a solution on [1, 2].

Remark 3. consider a more general problem as a system that contains a number of problems similar to our current problem (5) as follows:where , is the generalized Caputo−Fabrizio FD of order .

The system (66) can be written aswhere

By using Corollary 1, the system (67) has the following solution:where , and .

Remark 4. Following the methodology of proof used in the preceding parts, we can obtain the same results (Theorems 1–3) for the nonlinear system (67) in view of the formula (69).

5. Conclusions

In this regard, Al-Refai and Jarrah [32] defined the weighted Caputo−Fabrizio FD of the Caputo sense. As an extra contribution to this topic, we developed and extended the existence, uniqueness, and UH stability results for nonlocal implicit equations involving -Caputo−Fabrizio FDs. Our approach has been based on Banach’s and Krasnoselskii’s fixed point theorem. As an application, we have given a convenient example that validates the theoretical results. Finally, in light of our present results, a more general problem for the nonlocal implicit system has been presented that contains similar problems to the problem considered.

In the future direction, it will be interesting to study the current systems under -Atangana−Baleanu of the Caputo sense, introduced recently in [21, 36].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was conducted during our work at Hodeidah University.

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Copyright © 2023 Mohammed S. Abdo 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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