Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions

Salem, Ahmed; Alzahrani, Faris; Alnegga, Mohammad

doi:https://doi.org/10.1155/2020/7345658

Mathematical Problems in Engineering

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

Research Article | Open Access

Volume 2020 | Article ID 7345658 | https://doi.org/10.1155/2020/7345658

Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions

Ahmed Salem,¹Faris Alzahrani,¹and Mohammad Alnegga^1,2

Academic Editor: Jose de Jesus Rubio

Received02 Nov 2019

Accepted17 Jan 2020

Published21 Feb 2020

Abstract

This research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. The Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. The existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. This research paper highlights the examples related with theorems that have already been proven.

1. Introduction

Recently, many mathematical fields have been developed rapidly via fractional calculus. Different applications can be described by fractional equations involving fractional derivatives. Fractional calculus was an essential element in many recently published articles, such as a fractional biological population model, a fractional SISR-SI malaria disease model, a fractional Biswas–Milovic model, fractional wave equations, fractional reaction-diffusion equations, and nonlinear fractional shock wave equations. More recent published articles related with fractional calculus can be clearly found in [1–9]. Fractional differential equations have obtained a remarkable reputation among the mathematicians due to rapid development which is applicable in many fields such as mathematics, chemistry, and electronics. For more details, we refer to [10–16]. The coupled systems of fractional differential equations are mainly significant because such systems occur frequently in various scientific applications (see [17–19]).

The Langevin equations (first formulated by Langevin in 1908) have been done with accuracy in order to have a full description of evolution of physical phenomena in fluctuating environment [20]. There is a clear progress on fractional Langevin equations in physics (see [21, 22]). New results on Langevin equations under the variety of boundary value conditions have been published [23–26].

Different forms of fractional integral have been identified and employed in many different applications. Three of the fractional integrals will be used: Riemann–Liouville [10], Atangana and Baleanu [27, 28], and Ntouyas et al. [29, 30].

Recent paper [31] has discussed the existence and uniqueness of solutions obtained from boundary value conditions for nonlinear fractional differential equations for Riemann–Liouville type under the generalized nonlocal integral boundary condition. In addition, the authors in [32] have studied existence and uniqueness of the solution for a certain class of ordinary differential equations of Atangana–Baleanu fractional derivative.

In this paper, we modify the boundary value conditions of coupled systems of Langevin fractional differential equations of Caputo type into new boundary value conditions. So, we deal with the following coupled systems of nonlinear fractional Langevin equations of α and β fractional orders:supplemented by the following:where is the Caputo fractional derivative of order and for . and are Atangana–Baleanu, and Katugampola fractional integrals, respectively. and , for , for , for , and . for and are continuous functions.

From the definitions of fractional integrals mentioned in the next section, it is worth pointing out that the fractional integral of Katugampola is a generalization for Riemann–Liouville fractional integral and Hadamard fractional integral . Also, the fractional integral of Atangana–Baleanu contains the Riemann–Liouville fractional integral and when , we recover the initial function and if , we obtain the ordinary integral. These motivate us to choose these fractional integrals in our boundary conditions. Furthermore, to the extent of our knowledge, this is the first paper that discusses the existence and uniqueness of the solutions to coupled systems of fractional Langevin equations involving the nonlocal integro-multipoint of Atangana–Baleanu type and the nonlocal integral of Katugampola type as boundary value conditions.

The research article has been organized as follows. In the second section, we introduce some main concepts and essential lemma. In the third one, the main results show the existence and uniqueness of solutions to (1) and (2) by O’Regan’s fixed-point theorem, Krasnoselskii’s fixed-point theorem, and Banach’s fixed-point theorem, respectively. Under each one, examples have been considered in order to cover all theorems clearly.

2. Basic Concepts and Relevant Lemmas

We will deduce the main outcomes by the following preliminary concepts in fractional calculus.

Definition 1 (see [33]). For , let , then the Caputo fractional derivative of order β for a continuous function f is defined byprovided the right-hand side exists.

Definition 2 (see [33]). The Riemann–Liouville fractional integral of order ω for a continuous function is given byprovided the integral exists.

Lemma 1 (see [33]). If is a continuous function on , thenwhere , .

Lemma 2 (see [33]). Let . Then,

Definition 3 (see [34]). The Katugampola fractional integral of order β for a function defined on is given by the following formula:provided the right-hand side exists.

Definition 4 (see [35]). The “Atangana–Baleanu” fractional integral is defined byprovided the right-hand side exists whenever and . is called a normalization function satisfying .

Lemma 3. For , the coupled systemsupplemented byhas a solution given bywhereand for

Proof. Clearly, by direct computation, both (11) and (12) are solutions to (1). Conversely, by using Lemma 1, the general solution of (1) can be given asFor i = 1, both first and second boundary conditions, we obtain and . Consequently, . By substituting and in (15), we get (11). Similarly, in case of , the proof is done.

Lemma 4. and , we have

Proof. For each , we haveIf , then for all which means that is increasing on and soIf , then for all and for all whereThese mean that is increasing on and decreasing on and soThe proof is done.

3. Main Results

For each consider is the Banach space of all continuous functions from to all real numbers introduced by the norm . Moreover, product space is a Banach space equipped with .

For convenience, we simplify the following expressions:

Define the operator bywhere

By splitting both (26) and (27), we have

Forthcoming theorems proof need more convenient dialog for the operator T. So, it is required to rewrite it as follows:where

3.1. Existence via O’Regan’s Theorem

The first main result counts on O’Regan’s theorem which proves the existence of the solutions for (1) and (2).

Lemma 5 (see [36]). (O’Regan’s theorem)
Let where be an open subset of a closed and convex set in Banach space X. Assume , where , is bounded, is completely continuous and is so called nonlinear contraction (i.e., there exists a nonnegative nondecreasing function such that and ). Then, either(i) or(ii).

Theorem 1. Let be continuous functions. Assume the following conditions hold:
: there exist nonnegative functions and nondecreasing functions for each such that: there existswhere for each are defined in (21) and (22), respectively.
Then, (1) and (2) have at least one solution on .

Proof. Consider with fix radius aswhere for each , are defined in (21)–(24), respectively.
From (29) which are defined in both (30) and (31) as two separate operators, first of all, we will show that T is uniformly bounded on . Indeed, is bounded since , then we haveBy taking the norm over both sides, we obtainSimilarly,Therefore, is uniformly bounded: is bounded, i.e., , and we haveBy taking the norm over both sides, it yieldsSimilarly, we can deduce thatwhich implies thatThus,Now, we shall show that is completely continuous, , then we haveSimilarly,which shows the independence of the pair and . We conclude that is equicontinuous. By Arzelá–Ascoli theorem, is relatively compact. Thus, is completely continuous.
We shall show that is a contraction mapping. Indeed, and for any , we haveSo, we can writewhich clarifies that is a contraction mapping via . The last step is to show the first case of Lemma 5. By the way of contradiction, so we suppose such that . Then, we have , whereconsequently,equivalently,which clearly contradicts . Hence, the operator T has at least one fixed point . This ensures that (1) and (2) has a solution on

Example 1. Consider the following coupled system of the nonlinear fractional Langevin equation subject to both the nonlocal integro-multipoint of Atangana–Baleanu type and the nonlocal integral of Katugampola type as boundary value conditions.Obviously,From the shown data above, we haveClearly, by applying Theorem 1 we haveHence, our example possess at least one solution on .

3.2. Existence via Krasnoselskii’s Theorem

The second outcome relies on Krasnoselskii’s theorem in order to prove solution existence for equations (1) and (2).

Lemma 6 (see [37]) (Krasnoselskii’s theorem). Let be a bounded, closed, convex, and nonempty subset of Banach space X. Let and be two operators on as the following:(i),(ii) is compact and continuous,(iii) is a contraction mapping.Then, there exists satisfying .

Theorem 2. Assume the following conditions hold

Then, the boundary value problem (1) and (2) has at least one solution on if where is defined in (33).

Proof. Set , andConsideris a subset of Banach space . Our claim is to prove that for any point of , it implies . Indeed, given of arbitrarily, we haveSimilarly,This shows that
Hence, . satisfies the contraction principle as it has been shown in (47). f is a continuous function which helps us to say that is continuous. Moreover, is uniformly bounded on The last step in this proof is to show compactness of the operator Indeed, with , we havewhich is not dependent on the pair and the quantity which ensures that is equicontinuous. So, is relatively compact on . is a compact operator on since satisfies the Arzelá–Ascoli theorem. In conclusion, all terms of Krasnoselskii’s theorem have been applied perfectly. Hence, (1) and (2) possess at least one solution on the given period.

Example 2. Consider the following coupled system of nonlinear fractional Langevin equation subject to both the nonlocal integro-multipoint of Atangana–Baleanu type and the nonlocal integral of Katugampola type as boundary value conditions.Clearly,After calculating, we find , and According to Theorem 2, we see that which indicates that our example has at least one solution on .

3.3. Uniqueness via Banach Fixed-Point Theorem

The last result in this paper is about uniqueness criteria for the solution of (1) and (2), which can be achieved by Banach’s fixed-point theorem.

Theorem 3. Assume that are continuous functions and there exist positive real constants and such that ,

Then (1) and (2) have a unique solution on if whereand for each are defined in (21)–(24).

Proof. Set and and such thatFirst of all, we will show that whereFor all , we haveLikewise,Therefore, we obtainNext, we shall show that the operator T is a contraction operator on . Indeed, for any distinct two pairs , we see thatobviously, it gives usIn a similar technique, we can also haveFrom (70) and (71), we obtainSince the sum of both and is strictly less than one, we can say that the operator T satisfies a contraction criteria. Banach’s fixed-point theorem ensures that the operator T has a unique fixed point. Hence (1) and (2) have a unique solution on .

Example 3. Consider the previous example of the coupled system of nonlinear fractional Langevin equation subject to both the nonlocal integro-multipoint of Atangana–Baleanu type the and nonlocal integral of Katugampola type as boundary value conditions.Clearly,Based on Theorem 3, we can rewrite and as follows:Obviously, We conclude that our example has a unique solution on since .

4. Conclusion

In this research paper, we have proven the existence and uniqueness of solutions for the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary value conditions by selecting . Boundary value conditions have been chosen as two types of fractional integrals as we have shown in (2) for which have never been used together before in any article as far as we know. Existence of solutions have been shown by Krasnoselskii’s theorem and O’Regan’s theorem, and uniqueness solutions have been investigated by Banach’s fixed-point theorem. Examples have been supported in order to demonstrate all theorems very well. Results of this paper are not new in giving configuration, but also provide us new cases related with the choice of the parameters involving in the given problem. For example, the results associated with nonperiodic and nonlocal multipoint nonclassical integral boundary conditions follow by considering of this problem. In case of for all , the results change with respect to the boundary conditions:

Other results can be considered if we take , then the boundary conditions will be

In the future, in the case of obtaining theorems related with the existence and uniqueness of solutions under certain boundary conditions, both how can they be used to prove existence and uniqueness of solutions to the given problem and what are the conditions have to be considered.

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 project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (KEP-8-130-38). The authors, therefore, acknowledge the DSR for technical and financial support.

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