Qualitative Study on Solutions of Piecewise Nonlocal Implicit Fractional Differential Equations

Abdo, Mohammed S.; Idris, Sahar Ahmed; Albalawi, Wedad; Abdel-Aty, Abdel-Haleem; Zakarya, Mohammed; Mahmoud, Emad E.

doi:https://doi.org/10.1155/2023/2127600

Journal of Function Spaces

On this page

Abstract Introduction Examples Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Recent Advances of Fractional Calculus in Applied Science

View this Special Issue

Research Article | Open Access

Volume 2023 | Article ID 2127600 | https://doi.org/10.1155/2023/2127600

Qualitative Study on Solutions of Piecewise Nonlocal Implicit Fractional Differential Equations

Mohammed S. Abdo,¹Sahar Ahmed Idris,²Wedad Albalawi,³Abdel-Haleem Abdel-Aty,⁴Mohammed Zakarya,⁵and Emad E. Mahmoud⁶

Academic Editor: Yusuf Gurefe

Received29 Jul 2022

Revised07 Sept 2022

Accepted05 Oct 2022

Published28 Apr 2023

Abstract

In this paper, we investigate new types of nonlocal implicit problems involving piecewise Caputo fractional operators. The existence and uniqueness results are proved by using some fixed point theorems. Furthermore, we present analogous results involving piecewise Caputo-Fabrizio and Atangana–Baleanu fractional operators. The ensuring of the existence of solutions is shown by Ulam-Hyer’s stability. At last, two examples are given to show and approve our outcomes.

1. Introduction

It merits noticing that fractional calculus (FC) has gotten significant thought from scientists and researchers. It is a result of its wide scope of uses in different fields and disciplines. The crucial concepts and definitions of FC have been presented in [1, 2]. In [3, 4], the authors introduced some fundamental history of fractional calculus and its applications to engineering and different areas of science.

Many classes of fractional differential equations (FDEs) have been intensively investigated in the last decades, for instance, theories involving the existence of unique solutions have been notarized [5–7]. Numerical and analytical methods have been evolving with the target to solve such equations [8–10]. These equations have been tracked as useful in modeling some real-world problems with incredible achievement.

The qualitative properties of solutions represent a very important aspect of the theory of FDEs. The formerly aforesaid region has been studied well for classical differential equations. However, for FDEs, there are many aspects that require further studying and reconnoitering. The attention on the existence and uniqueness has been especially focused by applying Riemann-Liouville (R-L), Caputo, Hilfer, and other FDs, see [11–15].

In this regard, Agarwal et al. [16] investigated the existence of solutions of the following Caputo type FDE:

The basic theory of implicit FDEs with Caputo FD has been investigated by Kucche et al. [17]. Wahash et al. [18] considered the following nonlocal implicit FDEs with -Caputo FD

Problem (2) with has been studied by Benchohra and Bouriah [19].

Motivated by the above works and inspired by [20], we consider the piecewise Caputo implicit FDE (PC-IFDE) of the type: and the following piecewise Caputo nonlocal implicit FDE (PC-NIFDE): where , , and represent the piecewise Caputo FD of order defined by where is a classical derivative on and is standard Caputo FD on .

It is essential to note that the utilization of nonlinear condition in physical issues yields better impact than the initial condition (see [21]).

We pay attention to the topic of the novel piecewise operators. As far as we could possibly know, no outcomes in the literature are addressing the qualitative aspects of the aforesaid problems by using the piecewise FC. Consequently, by conquering this gap, we will examine the existence, uniqueness, and Ulam-Hyers stability results of piecewise Caputo problems (3) and (4) based on the standard fixed point theorems due to Banach-type and Schauder-type. Furthermore, we present similar results containing piecewise Caputo-Fabrizio (PCF) type and piecewise Atangana-Baleanu (PAB) type. An open problem with respect to another function is suggested.

Remark 1. (i)If then problem (4) reduces to the PC-IFDE (3).(ii)If , then problem (4) has been studied by Benchohra and Bouriah [19], Haoues et al. [22], and Abdo et al. [11] for .(iii)Our current results for problem (4) stay available on PC-IFDE (3).The substance of this paper is coordinated as follows: Section 2 presents a few required outcomes and fundamentals about piecewise FC. Our key outcomes for problem (4) are proved in Section 3. Two examples to make sense of the gained outcomes are built in Section 4. Toward the end, we encapsulate our study in the end section.

2. Primitive Results

In this section, we present some concepts of a piecewise FC. Let

Obviously is a Banach space under .

Definition 2 [20]. Let and be a continuous. Then, the piecewise version of RL integral is given by where and

Definition 3 [20]. Let and be a continuous. Then, the piecewise version of Caputo derivative is given by where and

Lemma 4 [20]. Let and Then, the following PC-FDE has the following solution

Lemma 5 [20]. Let and for a given function, . Then, For our aim, we need the Banach fixed-point theorem [23] and the Schauder fixed-point theorem [24].

3. Main Results

In this section, we give some qualitative analyses of the PC-IFDE and PC-NIFDE.

Lemma 6. Let be continuous.
Then, PC-NIFDE (4) is equivalent to where satisfies the functional equation

Proof. Let .
Then, by applying , we obtain

In view of Lemma 5, we have

Case 1. For

Case 2. For

Using the nonlocal condition in both cases we obtain

So, we get (12). On the other hand, let (13) be satisfied. Set

This implies that

Since on and on we obtain and hence

The next assumptions will be applied in the sequel:

() The functions and are continuous with that is a nondecreasing such that

() is continuous and compact with for

() There exist , such that and

() There exists , such that and for

Now, we shall prove the existence theorem for (4) based on Schauder’s theorem.

Theorem 7. Let () and () hold.
Then, piecewise Caputo FNIDE (4) has at least one solution on

Proof. Consider the operator , such that i.e., where with Define the ball where and , with

For any , and by (Assu), we have

Since , we obtain

Hence, the proceed is in the following steps:

Step 1. is bounded.
Case 1. For we have Case 2. For we have From (28) and (29), we conclude that Thus, Since is bounded, then is bounded.

Step 2. is continuous. Let a sequence such that in as Then, for we have For we have where with and Since as and , and are continuous, the Lebesgue dominated convergence theorem gives that

Step 3. is equicontinuous. Let then , we have Let then , we have Since is continuous and compact, (33) and (34) give That means is relatively compact on . So, is completely continuous due to the Arzela–Ascolli theorem. Thus, Schauder’s theorem shows that problem (4) has at least one solution.

Next, we prove the uniqueness theorem for (4) based on Banach’s theorem.

Theorem 8. Let ()-() hold.
If then PC-NIFDE (4) has a unique solution on , where

Proof. Consider and in , then which implies that

Hence, we have two cases:

Case 1. For

Case 2. For

Consequently,

Since , is a contraction. Thus, Banach’s theorem shows that PC-NIFDE (4) has a unique solution that exists on .

3.1. An Analogous Results

In this part, we show some analogous results according to our preceding outcomes.

3.1.1. Piecewise Caputo-Fabrizio NIFDE (PCF-NIFDE)

Consider the following PCF-NIFDE where is the piecewise derivative in the Caputo-Fabrizio sense (see [20]) defined by where and are the classical Caputo-Fabrizio FD (see [25]).

Let based on PCF-NIFDE (42), the results in Theorems 7 and 8 can be presented by where and are a Caputo-Fabrizio integral on (see [25])

3.1.2. Piecewise Atangana-Baleanu NIFDE (PAB-NIFDE)

Consider the following PAB-NIFDE where is the piecewise derivative in the Atangana-Baleanu sense defined by (see [20]) where is the normalization function that satisfies ; and are the classical Atangana-Baleanu FD ([26]).

Based on PAB-NIFDE (45), the results in Theorems 7 and 8 can be presented by where is the Atangana-Baleanu integral on (see [26])

Remark 9. Following the strategy of proof utilized in the previous part, we can get the existence results for nonlinear problems (42) and (45).

3.2. UH Stability Analysis

In this portion, we give the UH Stability of problem (4).

Definition 10. PC-NIFDE (4) is UH stable if there exists a such that for all and each solution of the inequality. there exists a solution of PC-NIFDE (4) that satisfies where and

Remark 11. satisfies inequality (48) if there exist function with (i) (ii)For all

Lemma 12. Let and is a solution of inequality (48). Then, satisfies where and

Proof. Let be a solution of (48).
By part (ii) of Remark 11, we have Then, the solution of problem (52) is

Again by (i) of Remark 11, we obtain

Theorem 13. Under the assumptions of Theorem 8. Then, the solution of PC-NIFDE (4) is HU and GHU stable.

Proof. Let be a solution of inequality (48), and be a unique solution of the following PC-NIFDE.

From Lemma 12, we obtain where and Clearly, if then and Hence, (56) becomes

Using Lemma 12 and (Assu) for , we have

Using classical Gronwall’s Lemma [27], we obtain

For , we have

Using fractional Gronwall’s Lemma [27], we obtain

It follows from (59) and (61) that where

Hence, PC-NIFDE (4) is UH stable in . Moreover, if there exists a nondecreasing function, such that . Then, from (62), we have with which proves PC-NIFDE (4) is GUH stable in

4. Examples

In this portion, we present two examples to illustrate the reported results.

Example 1. Consider the following PC-NIFDE or where , and are positive constants with Set Let , . Then,

Hence, the condition () holds with Also we have

Hence, the condition () holds with . Moreover, the following condition

is satisfied with and Thus, with the assistance of Theorem 8, problem (65) has a unique solution . Further, since and then

and which implies that problem (65) is HU stable.

Example 2. Consider the following PC-NIFDE or where Set for and Let and . Then, Putting and Then, valid for any and . Also, Hence, and hold. Thus, all the assumptions of Theorem 7 are satisfied. Hence, problem (73) has a solution on .

5. Conclusions

Somewhat recently, numerous methodologies have been proposed to portray behaviors of some complex world problems emerging in numerous scholarly fields. One of these problems is the multistep behavior shown by certain problems. In this regard, Atangana and Araz [20] introduced the concept of piecewise derivative. As an extra contribution to this subject, existence, uniqueness, and UH stability results for PC-NIFDE (4) involving a piecewise Caputo FD have been obtained. Our approach to this work has been based on Banach’s and Schaefer’s fixed-point theorem and Gronwall’s Lemma. In light of our current results, the solution form for analogous problems containing piecewise Caputo-Fabrizio and Atangana–Baleanu operators have been presented. Finally, we have created two examples to validate the results obtained.

As an open problem, it will be very interesting to study the present problems on piecewise fractional operators with another function that is more general; precisely, one has to consider in problem (2) with such that where and are -Caputo FD of order introduced by Almeida [28]

Data Availability

No real data were used to support this study. The data used in this study are hypothetical.

Conflicts of Interest

No conflicts of interest are related to this work.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group research project under grant number R.G.P.2/204/43.

References

K. B. Oldham and J. Spanier, “The fractional calculus: theory and applications of differentiation and integration to arbitrary order,” in Mathematics in Science and Engineering, Academic Press, New York, 1974.
View at: Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, 204, Elsevier Science B.V, Amsterdam, 2006.
View at: Google Scholar
A. Loverro, Fractional calculus: History, Definitions and Applications for the Engineer, Rapport Technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, 2004.
R. Hilfer, Three Fold Introduction to Fractional Derivatives, Foundations and Applications, Germany, Anomalous Transport, 2008.
F. Jarad, T. Abdeljawad, and Z. Hammouch, “On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative,” Chaos, Solitons and Fractals, vol. 117, pp. 16–20, 2018.
View at: Publisher Site | Google Scholar
S. Abbas, M. Benchohra, J. R. Graef, and J. Henderson, “Implicit fractional differential and integral equations,” Implicit Fractional Differential and Integral Equations. de Gruyter, vol. 26, 2018.
View at: Publisher Site | Google Scholar
P. Agarwal, M. R. Sidi Ammi, and J. Asad, “Existence and uniqueness results on time scales for fractional nonlocal thermistor problem in the conformable sense,” Advances in Difference Equations, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
Z. Odibat and D. Baleanu, “Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives,” Applied Numerical Mathematics, vol. 156, pp. 94–105, 2020.
View at: Publisher Site | Google Scholar
A. Atangana and J. F. Gómez-Aguilar, “Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu,” Numerical Methods for Partial Differential Equations, vol. 34, no. 5, pp. 1502–1523, 2018.
View at: Publisher Site | Google Scholar
M. Alesemi, N. Iqbal, and M. S. Abdo, “Novel investigation of fractional-order Cauchy-reaction diffusion equation involving Caputo-Fabrizio operator,” Journal of Function Spaces, vol. 2022, Article ID 4284060, 14 pages, 2022.
View at: Publisher Site | Google Scholar
M. S. Abdo, A. G. Ibrahim, and S. K. Panchal, “Nonlinear implicit fractional differential equation involving-Caputo fractional derivative,” In Proceedings of the Jangjeon Mathematical Society, vol. 22, no. 3, pp. 387–400, 2019.
View at: Google Scholar
M. Benchohra, S. Bouriah, and J. J. Nieto, “Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative,” Demonstratio Mathematica, vol. 52, no. 1, pp. 437–450, 2019.
View at: Publisher Site | Google Scholar
A. Salim, M. Benchohra, J. E. Lazreg, J. J. Nieto, and Y. Zhou, “Nonlocal initial value problem for hybrid generalized Hilfer-type fractional implicit differential equations,” Nonautonomous Dynamical Systems, vol. 8, no. 1, pp. 87–100, 2021.
View at: Publisher Site | Google Scholar
Y. Guo, M. Chen, X. B. Shu, and F. Xu, “The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm,” Stochastic Analysis and Applications, vol. 39, no. 4, pp. 643–666, 2021.
View at: Publisher Site | Google Scholar
Y. Guo, X. B. Shu, Y. Li, and F. Xu, “The existence and Hyers–Ulam stability of solution for an impulsive RiemannLiouville fractional neutral functional stochastic differential equation with infinite delay of order ,” Boundary Value Problems, vol. 2019, 18 pages, 2019.
View at: Publisher Site | Google Scholar
R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
View at: Publisher Site | Google Scholar
K. D. Kucche, J. J. Nieto, and V. Venktesh, “Theory of nonlinear implicit fractional differential equations,” Differential Equations Dynamical Systems, vol. 28, no. 1, pp. 1–17, 2020.
View at: Publisher Site | Google Scholar
H. A. Wahash, M. S. Abdo, and S. K. Panchal, “Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with -Caputo fractional derivative,” Journal of Applied Mathematics and Computational Mechanics, vol. 19, no. 1, pp. 89–101, 2020.
View at: Publisher Site | Google Scholar
M. Benchohra and S. Bouriah, “Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order,” Moroccan Journal of Pure and Applied Analysis, vol. 1, no. 1, pp. 1–16, 2015.
View at: Publisher Site | Google Scholar
A. Atangana and S. I. Araz, “New concept in calculus: piecewise differential and integral operators,” Chaos, Solitons and Fractals, vol. 145, article 110638, 2021.
View at: Publisher Site | Google Scholar
A. Bashir and S. Sivasundaram, “Some existence results for fractional integro-differential equations with nonlocal conditions,” Communications in Applied Analysis, vol. 12, pp. 107–112, 2008.
View at: Google Scholar
M. Haoues, A. Ardjouni, and A. Djoudi, “Existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo fractional differential equations,” TJMM, vol. 10, no. 1, pp. 09–13, 2018.
View at: Google Scholar
Y. Zhou, Basic Theory of Fractional Differential Equations, vol. 6, World Scientific, Singapore, 2014.
View at: Publisher Site
A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
View at: Publisher Site
M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Progress in Fractional Differentiation & Applications, vol. 1, no. 2, pp. 73–85, 2015.
View at: Google Scholar
A. Atangana and D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model,” Thermal Science, vol. 20, no. 2, pp. 763–769, 2016.
View at: Publisher Site | Google Scholar
H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.
View at: Publisher Site | Google Scholar
R. Almeida, “A Caputo fractional derivative of a function with respect to another function,” Communications in Nonlinear Science and Numerical Simulation, vol. 44, pp. 460–481, 2017.
View at: Publisher Site | Google Scholar

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

404

Downloads

274

Citations