Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

Ali, Saeed M.; Abdo, Mohammed S.

doi:https://doi.org/10.1155/2022/1879152

Mathematical Problems in Engineering

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

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Fractional-Order Systems: Control Theory and Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 1879152 | https://doi.org/10.1155/2022/1879152

Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

Saeed M. Ali¹and Mohammed S. Abdo²

Academic Editor: Abdellatif Ben Makhlouf

Received20 Jan 2022

Accepted09 Feb 2022

Published30 Mar 2022

Abstract

In this research work, we study two types of fractional boundary value problems for multi-term Langevin systems with generalized Caputo fractional operators of different orders. The existence and uniqueness results are acquired by applying Sadovskii’s and Banach’s fixed point theorems, whereas the guarantee of the existence of solutions is shown by Ulam–Hyer’s stability. Our reported results cover many outcomes as special cases. An example is provided to illustrate and validate our results.

1. Introduction

Fractional order models are more convenient than integer-order ones as fractional derivatives give superb tools for the portrayal of memory and inherited processes. Fractional differential equations (FDEs) have been applied in numerous fields, such as engineering, physical science, chemistry, financial matters, electrodynamics, aerodynamics, and dynamical systems (see [1–8]).

Some new contributions to Langevin FDEs have been investigated (see [9–13]) and the references referred to in that). There are many definitions of fractional integrals and derivatives, e.g., Riemann–Liouville type, Caputo type, Hadamard type, Hilfer type, and Erdelyi–Kober type, etc. With regard to the exciting development of local fractional calculus, it has been applied to deal with numerous different nondifferentiable problems in many applied fields (see [14–16]).

The generalized Riemann–Liouville definition with respect to another function was first presented by Osler [17]. Then, Kilbas et al. [2] presented important characteristics of this operator. The Caputo version called -Caputo fractional derivative has been introduced by Almeida [18]. Some amazing properties and generalized Laplace transform for the same operator were introduced by Jarad and Abdeljawad [19]. This recently defined fractional operator could model more precisely the process utilizing differential kernel. In order to evolve these definitions, special kernels and some kinds of operators are selected to apply on FDEs; for more details, we refer to some recent results associated with this development (see [20–29]).

In 2018, Almeida et al. [30] considered the following -Caputo type FDE with initial conditions:

Ahmad and Nieto [31] considered the following Caputo-type Langevin FDE with boundary conditions:

In 2020, Laadjal et al. [32] studied a Caputo-type multiterm Langevin FDE with boundary conditions of the form:

In this work, we study two classes of generalized Caputo Langevin equations with two various fractional orders. This work is inspired by the recent works of Laadjal et al. [32] and Almeida et al. [30]. Precisely, we consider the following Langevin-type fractional differential problems (FDPs):

where , , , , , , , the symbol denotes the generalized fractional derivative in the Caputo sense of order , and is a given continuous nonlinear function.

Here, we investigate the existence and uniqueness of solutions for FDPs (1.4) and (1.5) involving generalized Caputo fractional derivatives of various orders. Moreover, the guarantee of the existence of solutions is proved by UH stability. The studied results expand and generalize many results by selecting special cases of the function.

The paper is organized as follows: In Section 2, we give some definitions and lemmas that are used in the research paper. Section 3 derives equivalent fractional integral equations to the linear variants of Langevin FDPs (1.4) and (1.5). Section 4 deals with the qualitative analysis of proposed problems. In Section 5, we give an example to substantiate the main outcomes.

2. Preliminaries and Lemmas

We are beginning this portion by endowment with some essential definitions and results required for forthcoming analysis.

We consider the Banach space with the norm .

Let be an integrable function and an increasing function such that , for any .

Definition 1 (see [2]). The -Riemann–Liouville fractional integral of a function of order is described by

Definition 2 (see [2]). The -Riemann–Liouville fractional derivative of a function of order is described bywhere , .

Definition 3 (see [18]). The -Caputo fractional derivative of a function of order is described bywhere = and , .

Lemma 1 (see [2, 18]). Let . Then,(1)(2)(3), for

Lemma 2 (see [18]). If and , then(1)In particular,
if , we haveIf , we haveMoreover, if , then

Lemma 3 (see [18]). Let . Then, the unique solution of the following linear FDP is as follows:is obtained as

Lemma 4. Let . Then, the unique solution of the following linear FDP is as follows:is obtained as

Proof. In view of Lemma 2, we havewhere . By conditions of (14), we get andwhich impliesSubstituting the value of and in (16), we getwhich is identical to (15).

Theorem 1 (see [33]). Let is a Banach space. The map is a -set contraction with , and thus also condensing, if (i) are operators on ; (ii) is contractive; (iii) is compact.
To apply Sadovskii’s and Banach’s fixed point theorems, we will suffice herewith reference to [34, 35].

3. The Linear Variant of FDPs (1.4) and (1.5)

This section deals with the linear variant of FDPs (1.4) and (1.5). For simpleness, we denote and by and , respectively.

Lemma 5. Let and . Then, is a solution of the following linear Langevin-type FDP:if and only if satisfieswhere .

Proof. Applying the operator on both sides of the first (20) and using Lemma 2, we getwhere .
Next, applying the operator on both sides of (22) and using Lemma 2, again, we obtainwhere . In view of (23), we havewhere we used the fact thatwhere . Using the initial conditions of (23), we find that andSubstituting the values of , and in (23), we obtain the solution given by (21), whereThe converse follows by direct calculation. Hence, the proof is achieved.

Lemma 6. Let , and . Then, is a solution of the following linear Langevin-type multiterm FDP:if and only if satisfieswhere .

Proof. Applying the operator on both sides of the first (28) and using Lemma 2, we getwhere .
Next, applying the operator on both sides of (30) and using Lemma 2, again, we obtainwhere . In view of (31), we haveUsing the condition in (32) and and in (31), we find that andwhich impliesSubstituting the values of , and in (31), we obtain the solution given by (29). The converse follows by direct calculation. Hence, the proof is achieved.

4. Qualitative Analysis

4.1. Existence and Uniqueness Results

This subsection proves the existence and uniqueness results for FDPs (4) and (5) by applying Sadovskii’s theorem [34] and Banach’s theorem [35].

Theorem 2 (existence). Let is a continuous function. We assume that
(H1): there exists a function such that , for .
(H2): .
Then, the FDP (1.5) has at least one solution on .

Proof. Let be a closed bounded and convex subset of , where is a fixed constant. By virtue of Lemma 6, we define an operator as follows:for . Let us define two operators byWe observe that , .
In order to show that has a fixed point, we prove that and satisfy the hypotheses of Sadovskii’s theorem. This will be provided in several steps:

Step 1. .
Let us select , where . For any , we havewhich implies that .

Step 2. is compact. We note that is uniformly bounded from Step 1. Let with and . Then, we obtainFrom the continuity of , as . Thus, is equicontinuous. So, by the Arzela–Ascoli theorem, is a relatively compact set.

Step 3. is contractive. Let . Then, we havewhich is contractive since .

Step 4. is condensing.
Due to the fact that is continuous, a is a contraction and is compact, it follows from Lemma 1 that with is a condensing on . From the previous arguments, we conclude through Sadovskii’s theorem that has a fixed point. As a result, FDP (5) has a solution on .
The second result is based on Banach’s fixed point theorem.

Theorem 3 (uniqueness). Let is a continuous function, and there exists such thatIfthen the FDP (5) has a unique solution on , where is defined in Theorem 2.

Proof. We apply Banach’s theorem to prove that defined by (35) has a fixed point. For this end, we show that is a contraction. Let and . Then,which impliesAs , it follows that is a contraction. As a result of Banach’s theorem, there is a unique fixed point such that . Therefore, the FDP (5) has a unique solution on .

Corollary 1 (existence). Let is a continuous function, and there exists a function such that , for . If , where ,
then the FDP (4) has at least one solution on .

Corollary 2 (uniqueness). Let is a continuous function, and there exists such thatIf , where , then the FDP (4) has a unique solution on , where is defined in Corollary 1.

Remark 1. Proofs of Corollaries 1 and 2 can be obtained easily using the same arguments as in the previous theorems. Thus, we omit the details.

Remark 2. If , then the obtained results on FDP (1.5) include the results of Laadjal et al. [32].

Remark 3. The Langevin FDPs (4) and (5) are new to the literature on Langevin-type multiterm FDPs and include many problems, as a special case, for and ; our problems are reduced to Hadamard-type problems and Katugampola problems, respectively.

4.2. UH Stability Analysis

In this subsection, we discuss the UH stability of the considered problem.

Definition 4. FDP (5) is UH stable if there exists a constant such that for each and every solution of the inequalitiesthere exists a solution of FDP (5) that satisfies

Remark 4. satisfies the inequality (45) if and only if there exists a function with(i), (ii)For all

Lemma 7. We suppose that and , and is a solution of the inequality (45). Then, satisfieswhere

Proof. We suppose that is a solution of (45). By Lemma 6 and (ii) of Remark 4, we haveThen, the solution of FDP (50) isAgain by (i) of Remark 4, it is implied that

Theorem 4. Under the hypotheses of Theorem 3, the solution of the FDP (5) is UH stable.

Proof. We suppose that is a solution of the inequality (45) and is a unique solution of FDP (5). From Lemma 6, we obtain , whereClearly, if , , , thenHence, we get .
Using Lemma 7 and due to the fact that , for any , we havewhich impliesFrom (41), we get . It follows thatwhereHence, the FDP (5) is UH stable in .

Example 1. We consider the generalized Caputo-type Langevin FDP:where , , andWe observe thatClearly, all the hypotheses of Theorem 3 hold, and hence, FDP (59) has a unique solution on . Furthermore, we havewhich satisfies the assumption (H1) of Theorem 2. Moreover, we can find thatConsequently, by Theorem 1, the FDP (59) has at least one solution on . Moreover, we haveFrom Theorem 3, the FDP (59) is UH stable on .

5. Conclusions

In this paper, we have given some results dealing with the existence and stability of solutions for two types of fractional boundary value problems for multiterm Langevin equations with generalized Caputo fractional operators of different orders. As an initial step, we obtained the equivalent solutions associated with linear problems by applying the instruments of advanced fractional calculus and characterizing a fixed point problem. Once the fixed point problem is available, the existence and uniqueness theorems are established via Banach’s and Sadovskii’s fixed point techniques, whereas the guarantee of the existence of solutions has been shown by the vigorous techniques, such as UH stability.

We do not apply any significant bearing to the complex transformations, and our outcomes are characteristic of the integral operators’ theory of such kind. Indeed, our methodology is straightforward and can without much of a stretch be applied to an assortment of real-world problems. For the justification of the main results, we have given an example. As a special case, the reported results are new, and we have generalized many results with various values of function.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest related to this work.

Acknowledgments

The authors are grateful for the support provided by their institutions.

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Copyright © 2022 Saeed M. Ali and Mohammed S. Abdo. 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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