Study of Nonlocal Multiorder Implicit Differential Equation Involving Hilfer Fractional Derivative on Unbounded Domains

Thabet, Sabri T. M.; Kedim, Imed

doi:https://doi.org/10.1155/2023/8668325

Journal of Mathematics

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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 8668325 | https://doi.org/10.1155/2023/8668325

Study of Nonlocal Multiorder Implicit Differential Equation Involving Hilfer Fractional Derivative on Unbounded Domains

Sabri T. M. Thabet¹and Imed Kedim²

Academic Editor: Ji Gao

Received27 Jan 2023

Revised04 Mar 2023

Accepted06 Mar 2023

Published01 Apr 2023

Abstract

This paper aims to study the existence and uniqueness of the solution for nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains , in an applicable Banach space by utilizing the Banach contraction principle. Furthermore, we discuss various types of stability such as Ulam–Hyers–Rassias (UHR), Ulam–Hyers (UH), and semi-Ulam–Hyers–Rassias (sUHR) for nonlocal boundary value problem. Absolutely, our results cover that several outcomes have existed in the literature.

1. Introduction

The subject of fractional calculus is an extension of classical calculus, and this means extend an integer order to an arbitrary order. In the field of fractional calculus, there are many definitions of integrals and derivatives. Most of the general public fractional definitions are Caputo and Riemann–Liouville (R-L), which attracted many researcher to extend differential equations of integer orders to fractional such as [1]. Newly, Hilfer has connected the Caputo and R-L derivative by a general formula, and it is known by Hilfer or generalized R-L derivatives [2], which is attracted attention of many authors in the literature, such as [3] studied the Ulam stability and existence results of differential equations in the sense of fractional Hadamard and Hilfer derivatives. Bulavatsky [4] proved the closed solutions to several problems for the anomalous diffusion equation involving Hilfer fractional derivative. The author of this work [5] derived an equivalent form of the Hilfer derivative. Furthermore, the authors in [6] constructed operational calculus of generalized R-L derivatives and used it to solve linear equation of term with constant coefficients via Hilfer derivative. The authors in [7, 8] established existence and uniqueness of solutions for a nonlinear Hilfer fractional problem in weighted spaces by employing various fixed point theorems. Thabet et al. [9] studied an abstract Hilfer fractional integrodifferential equation via technique of measure of noncompactness and established continuous dependence of -approximate solutions by using generalized Gronwall’s inequality. In 2020, Abdo et al. [10] investigated sufficient conditions UH stability and the existence and uniqueness of solutions for -Hilfer fractional integrodifferential equations by Schaefer, Banach, and Schauder theorems with helping of generalized Gronwall’s inequality.

The subject of fixed point theory represents a highly important part of various areas of mathematics and may be regarded a wealth in nonlinear analysis. The fixed point for an adequate map is equivalent to the existence and uniqueness of the solutions for many engineering problems and realistic phenomena. We indicate that some works have been concerned on fixed point theorems and its applications; for example, the authors in [11–13] proved UH, UHR, sUHR, and asymptotic stability as well as the existence and uniqueness of solutions for some fractional problems by topological degree theory and Banach contraction principle. The Banach, Schauder, and Monch fixed point theorems with helping of the measure of noncompactness used to study some qualitative properties and stability of fractional integrodifferential equations [14, 15]. Moreover, several fixed point theorems are employed to treat the qualitative properties of various class systems such as implicit problems [16, 17], hybrid multifractional problem [18], and thermistor fractional problem [19]. In the literature, engineering and physical problems have proved that the dealing with nonlocal initial conditions has a good effect more than the classical initial, and see some papers considered nonlocal conditions [20–23].

In 2018, the authors in [24] studied the existence, uniqueness, and stability of the solution for the following nonlocal implicit differential equation via Hilfer fractional derivative on the bounded interval :

In 2021, the authors of these works [25, 26] established sufficient conditions of the existence and uniqueness solution and discussed UHR stability, UH stability, and sUHR stability for initial Hilfer fractional integrodifferential equations. Ali et al. [27] discussed the stability of pantograph-type implicit fractional differential equations with impulsive conditions. Also, the existence, uniqueness, and UH stabilities result for a coupled Hilfer fractional integrodifferential equation on bounded domains investigated by [28]. Almalahi et al. [29, 30] established qualitative theories for fractional functional differential equation with boundary condition and finite delay as well as a coupled system of hybrid fractional differential equations via Hilfer fractional derivatives. Very recently, Xie et al. [31] investigated some qualitative properties of multiorder differential equations with initial condition involving R-L fractional derivatives of the following form:where .

Inspired by these works [24, 31], this work aims to study the existence and uniqueness results along with stability types of the UH, UHR, and sUHR for the following nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains:where , , are the Hilfer fractional derivatives of order , and types , respectively, such that , and is R-L fractional integral of order , and a function , where is the real Banach space. The novelty and contributions of this paper are to consider a more general problem containing term of generalized fractional derivative in applicable Banach space on unbounded domains, in order to include a lot of outcomes which are existing in the literature, for example, the results in [24, 31].

The organization of this work is as follows: in Section 2, we recall some notations, definitions, and preliminaries. Section 3 establishes the existence and uniqueness of the solution of the problem (3), in an applicable Banach space by utilizing the fixed point theorem. In Section 4, we prove various types of stability by employing the nonlinear analysis topics.

2. Preliminaries

Throughout this section, we present some interesting notations and preliminaries, in order to use it in achieving desired results.

Let the Banach space of continuous functions denoted by , with supremum norm For reaching our goal, we define the following Banach space:where , endowed with the norm

It is easy to prove that be a Banach space as a procedure in the works [32–34].

Definition 1. (see [35]). The R-L fractional integral of order of a function is given by

Definition 2. (see [35]). The R-L fractional derivative of order of a function is given bywhere .

Definition 3. (see [35]). The Caputo fractional derivative of order of a function is given by

Definition 4. (see [2]). The Hilfer fractional derivative of order and type of a function is given by

Remark 1. We observe that the Hilfer fractional derivative returns to the R-L derivative when , i.e., , and returns to the Caputo derivatives when , i.e., .

Lemma 1 (see [36]). If , , and , , then

Lemma 2 (see [36]). Let and . Then,

Lemma 3. Let , and . Then,

Proof. The proof follows from Definition 4, with some calculations.

Definition 5 (see [37]). Let be a nonempty set. Then, is called a generalized metric on , if(i)(ii)(iii)

Theorem 1 (see [37]). Assume that the generalized complete metric space is denoted by , and let the operator is contractive with the Lipschitz constant . If there is a positive integer , where , for some , then the following hold:(i)The sequence converges to a fixed point (ii) is the unique fixed point of in (iii)if , then

3. Existence and Uniqueness of Solution

Let us start this section by deriving the integral solution of the problem (3), as in the following lemma.

Lemma 4. Let a function be continuously differentiable. Then, the solution of the system (3) is equivalent to the following Volterra integral equation:provided that .

Proof. By applying on both sides of (3), then using Lemmas 1 and 3, we getNow, by putting into equation (14) and multiplying by , we getThen,which yields thatwhere . Hence, by substituting equation (17) into equation (14), the proof is finished.□
Next, we need to present the following assumptions which are required to investigate the existence and uniqueness of the solution of the problem (3). Suppose that are continuous functions and the continuously differentiable function such that far all and . There exist the constants , such that , which are verifying the following requirements:where , and .

Theorem 2. Suppose that and are fulfilled. Then, the problem (3) has an one and only one solution on unbounded interval .

Proof. In the light of Lemma 4, we consider the operator defined as follows:Clearly, is itself mapping. Since for any , by using and , we haveAlso, by using Lemma 3, we findIn the following, we establish that is contractive mapping on . For any , and by using and , we getSimilarly,Therefore, we deduce that , which yields that is a contractive operator, since . In view of the Banach contraction principle, has an one and only one fixed point in , which is verifying . Hence, problem (3) has an one and only one solution on unbounded interval .

4. Stability of Solution

In this situation, in order to discuss UHR stability, UH stability, and sUHR stability, we need to introduce applicable metrics and on Banach space . The metric is defined bywhere is a nonnegative increasing continuous function on unbounded interval . Also, the metric is defined bywhere is a nonnegative decreasing continuous function on unbounded interval . We can guarantee that and are metrics on Banach space , as given in the work [38] and references therein.

In the following, we will give the definitions of UHR, UH, and sUHR stability, and then state and prove their theorems.

Definition 6. (see [12]). The solution of the problem (3) is UHR stable if for every continuously differentiable function verifyingwhere is a nonnegative nondecreasing continuous function on unbounded interval , and there is a unique solution of the problem (3), and a constant independent of , whereMoreover, if we replace by , then the solution of the problem (3) is UH stable.

Definition 7. (see [12]). The solution of the problem (3) is sUHR stable, if for every continuously differentiable function verifyingwhere , and there is a unique solution of the problem (3), and a constant independent of for some positive decreasing continuous function on unbounded interval , where

Theorem 3. Suppose that and are fulfilled, be a positive continuous increasing function on unbounded interval , and is continuously differentiable function verifying

Then, there is one and only one solution such thatwhere , which yields that the solution of problem (3) is UHR stable and consequently is UH stable.

Proof. Let be the contractive operator as given in (20).
Now, for , we present from metric and the assumptions thatHence, we obtainIn view of the inequalities (31) and (32), we haveDue to the inequalities (36) and (37), we getBased on of Theorem 1, there is one and only one fixed point such that . As a consequence of of Theorem 1, we can conclude thatAccording to the above conclusions, the solution of problem (3) is UHR stable. Along with this, if , then the solution of problem (3) is UH stable.

Theorem 4. Suppose that and are fulfilled, be a positive decreasing continuous function on unbounded interval , and is continuously differentiable function verifyingwhere . Then, there is one and only one solution , and a constant such thatwhere , which yields that the solution of problem (3) is sUHR stable.

Proof. Similar to Theorem 3, we take the contractive operator as given in (20). From metric and assumptions , we find thatThen, we getDue to the positiveness, continuity of a decreasing function , we findBased on the inequalities (40) and (41), we obtainFrom the inequalities (46) and (47), we haveIn the light of of Theorem 1, there is one and only one fixed point such that . As consequence from of Theorem 1, we can deduce thatAccording to the above conclusions, the solution of the problem (3) is sUHR stable, and the proof is finished.

5. Conclusion

This paper declared that, by convenience Hilfer fractional derivative and Banach contraction principle in an applicable Banach space, the nonlocal multiorder implicit differential equation (3), on unbounded domains , provides existence and uniqueness of the solution as well as the stability of UHR, UH, and sUHR. Our problem is more general and returns to the sense of the Caputo fractional problem when and returns to the sense of the R-L fractional problem when . Moreover, these results were generalized to include many outcomes are existed in the literature, for example, the results in [24, 31]. As a future target, the studied problem with this approach would be more exciting if it was discussed under integral value conditions via Hilfer fractional operators.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have equal contributions. All authors read and approved the final manuscript.

Acknowledgments

This work was conducted during my work at the University of Lahej. The authors express their gratitude to dear unknown referees for their helpful suggestions which improved the final version of this paper.

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Copyright © 2023 Sabri T. M. Thabet and Imed Kedim. 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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