Stability Results for Nonlinear Implicit <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.0537 12.533" width="9.0537pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-248" d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z"/></g></svg>-</span>Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

Kaddoura, Issam; Awad, Yahia

doi:https://doi.org/10.1155/2023/5561399

International Journal of Differential Equations

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5561399 | https://doi.org/10.1155/2023/5561399

Stability Results for Nonlinear Implicit -Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions

Issam Kaddoura^1,2and Yahia Awad³

Academic Editor: Patricia J. Y. Wong

Received24 Aug 2023

Revised13 Nov 2023

Accepted13 Dec 2023

Published31 Dec 2023

Abstract

This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit -Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.

1. Introduction

Fractional calculus, which belongs to the realm of mathematical analysis, has emerged as a potent instrument for representing intricate systems and phenomena characterized by memory and nonlocal behavior. Over the past few decades, it has attracted considerable attention from researchers and scientists because of its capacity to capture complex dynamic behaviors that elude traditional integer-order calculus. This mathematical framework has found extensive utility across a range of fields, including physics, engineering, biology, and economics, thereby enriching our comprehension of the underlying dynamics governing complex systems (refer to [1–7] and related works).

The foundation of fractional calculus hinges on the expansion of conventional derivatives and integrals into noninteger orders. Notably, fractional derivatives and integrals have proven indispensable in representing real-world phenomena marked by fractal geometry, anomalous diffusion, and long-range interactions. This transition from integer-order calculus to fractional calculus has laid the groundwork for groundbreaking contributions to the fields of science and engineering. You can explore this further in works such as [5, 8–13] and related references.

A notable advancement within the realm of fractional calculus is the -Caputo fractional derivative. Diverging from the established Riemann–Liouville and Caputo methodologies, the -Caputo derivative introduces a unique kernel incorporating the parameter . This distinctive characteristic sets it apart from classical derivative operators, offering a more adaptable instrument for characterizing intricate systems. Researchers have harnessed the potential of the -Caputo fractional derivative across various domains, including electromagnetics, fluid mechanics, signal processing, and beyond. Explore this development further in works such as [14–25] and related literature.

Furthermore, analyzing stability in fractional order differential equations has assumed paramount importance. A comprehensive comprehension of the stability characteristics of such equations proves vital in forecasting the long-term behavior of dynamic systems governed by fractional calculus. Researchers have extended classical stability concepts to the fractional domain, introducing concepts such as the Ulam–Hyers–Rassias stability types and their extensions. These advances have opened up new avenues for investigating the stability and resilience of fractional order systems under diverse conditions. Delve deeper into this topic through works such as [17, 26–34] and associated references.

In the subsequent discussions, we present a summary of recent research contributions in the field of fractional differential equations (FDEs) and their stability. We start with an overview of notable works.

In [35], Zada et al. studied the existence, uniqueness, and Hyers–Ulam stability results of the following implicit FDE with impulsive condition:where is the Caputo fractional derivative of order , , with , . The functions , , and are continuous functions and .

In [36], the authors considered a class of -Hilfer nonlinear implicit fractional boundary value problems (FBVPs) describing the thermostat control model of the following form:where denotes the -Hilfer fractional derivative operator of order , ∈ (1, 2], , ∈ (0, 1], , , , , , , , , , , , , is the -Riemann–Liouville fractional integral of order , , ∈ (0,1], , and with .

In [17], the authors explored the existence, uniqueness, and Ulam–Hyers type stability for the following nonlinear implicit -Caputo fractional order integro-differential boundary value problem CIFDP:where is the -Caputo fractional derivative of order ∈ (0, 1], , , , and and are constant real numbers.

In [26], Al-Issa et al. developed existence and stability theorems for the implicit fractional order differential problem (ISDP):where is the Caputo fractional derivative of order , with , and . In addition, and are continuous functions with , and is a nondecreasing function with for all .

In [28], the authors investigated the existence and Ulam–Hyers stability of solutions for second-order differential equations with integral boundary conditions:with the following nonlocal boundary conditions:where is a Caputo fractional derivative of order , with , , and are given functions, is a continuous function, , and .

In light of the recent advancements in the field, this research article delves into the stability analysis of nonlinear implicit -Caputo fractional differential equations with fractional integral boundary conditions. We aim to contribute meaningfully to the expanding body of knowledge concerning the -Caputo derivative and its practical applications. Simultaneously, we aim to enrich our comprehension of the stability characteristics inherent in fractional-order differential equations. Our work builds upon the foundational work laid out by previous researchers and extends the utility of -Caputo derivatives to intricate boundary value problems, offering insights into the behavior of complex systems. Therefore, motivated by the preceding discussions, our study delves into the investigation of the existence and uniqueness of solutions for a nonlinear implicit -Caputo fractional order differential problem (ICFDP), characterized by the following equations:where is an increasing function with for all in the interval . The parameters , , and satisfy the conditions and . The operator represents the -Caputo fractional derivative. Our primary findings, which are derived under specific assumptions, are established using the Banach and Krasnoselskii’s fixed point theorems. Furthermore, our investigation encompasses the -Caputo fractional derivative, denoted as . Additionally, we address the topics of Ulam–Hyers stability and the generalized Ulam–Hyers stability.

The article is structured as follows: We initiate our work with an introduction in Section 1. Following that, Section 2 covers notations, definitions, lemmas, and theorems that establish the fundamental basis for our study. In Section 3, we establish the existence and uniqueness of mild solutions for (ICFDPs) (7)–(9) by using the fixed point theorems of Banach and Krasnoselskii. Section 4 delves into Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Additionally, Section 5 provides an illustrative example to showcase the practical application of our main findings. Finally, we conclude the paper in Section 6.

2. Preliminaries

In the subsequent sections, we present various notations, definitions, lemmas, and theorems that hold significance in the progression of our findings within this article.

Definition 1 (see [14]). For any positive real number , the left-sided -Riemann–Liouville fractional integral of order for an integrable function with respect to another function , which is a differentiable increasing function such that for all , is defined as follows:where represents the classical Euler gamma function.

Definition 2 (see [14]). Let be a natural number, and let be two functions such that is an increasing function with for all . In such cases, the left-sided -Caputo fractional derivative of a function of order is defined as follows:where and for and for .

Remark 3. Let , then the differential equation has the following solution:where .

Lemma 4 (see [15]). Let and . Then, and , where .

Definition 5. A fixed point of a mapping , where is a given space, is a point satisfying .

Theorem 6 (Banach fixed point theorem). In a Banach space , if is a nonempty closed subset and is a contraction mapping of into itself, then possesses a unique fixed point.

Theorem 7 (Krasnoselskii’s fixed point theorem). For a Banach space and a bounded closed convex subset of , given mappings and from to with the property for all , if is a contraction and is completely continuous, then there exists a point such that .

3. Existence and Uniqueness of the Solutions

In this section, we demonstrate the presence of mild solutions for the nonlinear implicit -Caputo fractional order differential problems (ICFDPs) (7)–(9), subjected to the following assumptions: : The functions , where , are continuous and possess Lipschitz continuity with constants ∈ [0, 1], obeying the following criterion: : The function is continuous, and there exists a positive function that satisfies the ensuing inequality: for all and . : The function is continuous over the domain , and there exists a positive constant such that : The function is increasing and belongs to the class , with a positive constant such that, for each , : Positive functions and exist in the class such that

By fulfilling these assumptions, we establish the existence of a mild solution to (ICFDPs) (7)–(9).

Remark 8. Given assumption , it follows thatwhich impliesand consequently,Under assumption , we find thatleading to the conclusion thatand furthermore,

Lemma 9. The mild solution of (ICFDPs) (7)–(9) is the solution of the following Volterra integral equation:where is the solution of the following functional integral equation:where is the Green’s function defined bywithand

Proof. Let in equation (7), whereFrom equations (8) and (9), we can obtainSolving equations (30) and (31), and if , then it is obtained thatThen, the solution of (ICFDPs) (7)–(9) is given byUsing the fact that , we get equation (24), and the proof is complete.

Definition 10. A mild solution of the nonlinear implicit ϑ-Caputo fractional order differential problems (ICFDPs) (7)–(9) refers to a function that fulfills the integral equation (24). In this context, represents the solution to the following functional integral equation:for all .

Lemma 11. The function satisfies the following Lipschitz condition:

Proof. For arbitrary and for each , we haveThus,where .
Our first result is based on Banach’s fixed point theorem to obtain the existence of a unique solution of (ICFDPs) (7)–(9).

Theorem 12. Suppose that assumptions – hold, and is the Lipschitz constant as defined in Lemma 11 withsuch that , then (ICFDPs) (7)–(9) have a unique mild solution on .

Proof. Transform (ICFDPs) (7)–(9) into a fixed point problem. Define the operator bywhere satisfies the following implicit functional equation:where and are the functions defined by equations (26) and (28), respectively. We define the ball with radius as follows:whereFirst, we show that the operator is well-defined, i.e., we show that , whereLet . In the following, we show that for each as follows:where , such thatTaking supremum for , we getThen,Thus, equation (44) implies that, for each ,If we take supremum for all , we getHence, .
Second, we show that the operator is a contraction.
Let . Then, for any , we havewhere such thatThen, for any , we haveBut, by assumption , we haveTaking the supremum for all , we getThus,Now, return to equation (52), and by Lemma 11, we haveand taking supremum for , we getNow, if , then operator is a contraction.
Therefore, by Banach’s contraction principle, we deduce that has a unique fixed point , which is a mild solution of (ICFDPs) (7)–(9) on .
In the following, we present our second existence result for the mild solution of (ICFDPs) (7)–(9) based on the Krasnoselskii’s fixed point theorem [1].

Theorem 13. Assume that the assumptions and hold. Ifwhere , then (ICFDPs) (7)–(9) have at least one mild solution on .

Proof. Let the operator be defined in (39). Define the closed diskwithIn addition, define the operators and on byTaking into account that and are defined on , and for any , we haveThe proof is divided into several steps: Step 1: is well defined. Let . Then, for any , we have where such that Taking supremum for all , we have Thus, Moreover, Thus, equation (63) implies that, for each , we obtain that Taking supremum over , we have This proves that for every , where is given in equation (60). Step 2: The operator demonstrates contraction behavior within . It is clear from Lemma 11 that operator is a contraction mapping for . Step 3: The operator exhibits complete continuity (both compactness and continuity) on .First, we establish the continuity of the operator .
Assume that is a sequence such that as in . Then, for every , the following relation holds:where , such thatBy Lemma 11, we havewhere is a Lipschitz constant. Moreover, by the assumption , we haveTaking supremum for all , we getSince , then we get as for each . Consider such that for any , we have and . Thus,For each , the function is integrable on . Then, applying Lebesgue dominated convergence theorem and equation (70), we deduce thatHence, as , and consequently is continuous.
Second, due to the definition of , it is easy to verify that satisfies thatThis proves that is uniformly bounded on .
Third, we prove that maps bounded sets into equicontinuous sets of , i.e., is equicontinuous.
Now, suppose that for every , there exist , and such that and . Then,It is clear that as , the right-hand side of the above inequality tends to zero. Consequently,Hence, the equicontinuity of holds on along with the compactness of operator that is established by the Arzela–Ascoli theorem. This leads to the inference that maintains both continuity and compactness.
Notably, all prerequisites essential for Krasnoselskii’s fixed point theorem are satisfied. This shows that the operator has a fixed point on . Therefore, (ICFDPs) (7)–(9) have a mild solution on . This concludes the proof.

4. Ulam Stability of the Solutions

Consider now the Ulam stability for (ICFDPs) (7)–(9). Let and be a continuous function. We investigate the following inequalities:

Definition 14. (ICFDPs) (7)–(9) are considered Ulam–Hyers stable if there exists a positive real number such that, for every and for any solution satisfying inequality (80), there exists a solution of (ICFDPs) (7)–(9) with the following property:

Definition 15. (ICFDPs) (7)–(9) are said to be generalized Ulam–Hyers stable if there exists a function with , such that for each and each solution satisfying inequality (80), there exists a solution of (ICFDPs) (7)–(9) with

Definition 16. (ICFDPs) (7)–(9) are considered Ulam–Hyers–Rassias stable with respect to if there exists a positive real number such that, for every and for any solution satisfying inequality (81), there exists a solution of the systems (7)–(9) with the following property:

Definition 17. (ICFDPs) (7)–(9) are said to possess generalized Ulam–Hyers–Rassias stability with respect to the function if there exists a positive real constant such that for every solution of the inequality 23, there is a solution of the (ICFDPs) (7)–(9) satisfying the following condition:

4.1. Ulam–Hyers Stability

In the following, we study the Ulam–Hyers stability for (ICFDPs) (7)–(9).

Theorem 18. Assume that the assumptions of Theorem 12 are satisfied. Then, (ICFDPs) (7)–(9) are Ulam–Hyers stable.

Proof. Let and let be a function which satisfies inequality (80), i.e.,and let be the unique solution of (ICFDPs) (7)–(9) which by Lemma 9 is equivalent to the following fractional order integral equation:where is the solution of the following functional integral equation:Applying on both sides of equation (80), we getwhere , such thatThis implies that for each , we haveThus, if we take supremum for all , we getThen, for , we get the following equation:Therefore, (ICFDPs) (7)–(9) are Ulam–Hyers stable. This completes the proof.

Remark 19. If we put , then , which yields that (ICFDPs) (7)–(9) are generalized Ulam–Hyers stable.

4.2. Ulam–Hyers–Rassias Stability

Now, we show that (ICFDPs) (7)–(9) satisfy the Ulam–Hyers–Rassias stable type.

Theorem 20. Assume that assumptions – hold. Then, (ICFDPs) (7)–(9) are Ulam–Hyers–Rassias stable with respect to .

Proof. Let be a mild solution of inequation (82), i.e.,and assume that is a solution of (ICFDPs) (7)–(9), such thatwhere satisfies the following integral equation:Operating by on both sides of inequality (82) and then integrating, we getwhere such thatBut,Hence, in a similar manner as above, we have for each ,Taking supremum for all , we getIf we take , thenTherefore, (ICFDPs) (7)–(9) are Ulam–Hyers–Rassias stable with respect to and with a real constant . This completes the proof.

5. Special Cases and Example

The results we just established concerning the existence of a solution and its stability also hold for special cases. These fractional derivative classes are created by selecting an appropriate value for and taking into account the value of .

In particular, we can deduce some existence results from our approach in the following discussion:(i)When , then the obtained outcomes in the current paper incorporate the investigation of the following implicit fractional order differential problem (ICFDP):(ii)Also, if , , , and , then we have the following implicit fractional-order differential equation which generalized the results studied in [37, 38]:(iii)Putting and in equation (106), we have the following quadratic implicit fractional differential equations with fractional integral boundary conditions: where

Example 1. Consider the following ICFDP:with the following boundary conditions:such thatSettingit is clear that the function is jointly continuous. In fact, for any , and for every , we haveHence, the condition holds with and . On the other hand, we haveHence, the assumption is satisfied with and . This implies that if ,Hence, is Lipschitz with constant .
In addition, the Green’s function is as follows:Then, straightforward calculations with , , , , , , and yield the following conditions:It follows from Theorem 12 that (ICFDPs) (7) and (8) have a unique mild solution on .

6. Conclusion

In our research, we achieved several key outcomes. Firstly, we established a connection between (ICFDPs) (7)–(9) and Volterra integration equation (24). Next, utilizing Banach’s contraction principle and Krasnoselskii’s fixed point theorem, we successfully demonstrated the existence and uniqueness of mild solutions for boundary value problems of implicit fractional order differential equations. Additionally, we verified Ulam–Hayers stability and other related stability types for (ICFDPs) (7)–(9). Notably, we presented a practical numerical example highlighting our findings’ applicability. Furthermore, we emphasized the significance of our results, noting that different variations of and diverse values for in (ICFDPs) (7)–(9) lead to various implicit fractional-order differential equations. In conclusion, our work represents a significant advancement in the field of qualitative analysis of fractional differential equations, introducing a generalized nonlocal boundary condition that investigates Ulam–Hyers stability within the framework of -Caputo fractional derivatives. Future work will delve into exploring coupled systems in greater depth.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors have contributed equally to this paper. The authors reviewed the results and approved the final version of the manuscript.

Acknowledgments

The authors are profoundly thankful to Dr. Shorouk Al-Issa for her valuable comments and support during the article preparation.

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Copyright © 2023 Issam Kaddoura and Yahia Awad. 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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