Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

Wongcharoen, Athasit; Ahmad, Bashir; Ntouyas, Sotiris K.; Tariboon, Jessada

doi:https://doi.org/10.1155/2020/9606428

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2020 | Article ID 9606428 | https://doi.org/10.1155/2020/9606428

Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

Athasit Wongcharoen,¹Bashir Ahmad,²Sotiris K. Ntouyas,^2,3and Jessada Tariboon⁴

Academic Editor: Jorge E. Macias-Diaz

Received06 Feb 2020

Accepted07 Apr 2020

Published04 May 2020

Abstract

We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

1. Introduction

Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders. For details and applications, we refer the reader to the texts in [1–6]. In the literature, there exist several definitions of fractional integrals and derivatives, from the most popular Riemann-Liouville and Caputo-type fractional derivatives to others such as the Hadamard fractional derivative and the Erdeyl-Kober fractional derivative. A generalization of both the Riemann-Liouville and Caputo derivatives was given by Hilfer in [7], which is known as the Hilfer fractional derivative of order and type . One can observe that the Hilfer fractional derivative interpolates between the Riemann-Liouville and Caputo derivatives as it reduces to the Riemann-Liouville and Caputo fractional derivatives for and , respectively. Some properties and applications of the Hilfer derivative can be found in [8, 9] and references cited therein.

One of the important equations governing several phenomena occurring in physical sciences and electrical engineering is the Langevin differential equation, first formulated by Langevin in 1908 [10]. In recent years, several fractional variants of the Langevin equation have been introduced and studied; see, for example, [11–19] and the references cited therein.

Initial value problems involving the Hilfer fractional derivatives were studied by several authors; see for example [20–22]. Nonlocal boundary value problems for the Hilfer fractional differential equation have been discussed in [23]. In [24], the authors proved some results for initial value problems of the Langevin equation with the Hilfer fractional derivative.

Exploring the literature on fractional order boundary value problems, we find that there does not exist any work on boundary value problems of the Langevin equation with the Hilfer fractional derivative. Motivated by this observation, we fill this gap by introducing a new class of boundary value problems of the Hilfer-type Langevin fractional differential equation with three-point nonlocal boundary conditions. In precise terms, we investigate the existence and uniqueness criteria for the solutions of the following nonlocal boundary value problem: where , is the Hilfer fractional derivative of order , and parameter , , , , , and is a continuous function.

In order to study problem (1)–(2), we convert it into an equivalent fixed-point problem and then use Banach’s fixed-point theorem to prove the uniqueness of its solutions. We also obtain two existence results for problem (1)–(2) by applying the nonlinear alternative of the Leray-Schauder type [25] and Krasnoselskii’s fixed-point theorem [26].

As a second problem, we switch onto the multivalued analogue of (1) and (2) by considering the inclusion problem: where is a multivalued map ( is the family of all nonempty subjects of ).

Existence results for problem (3)–(4) with convex and nonconvex valued maps are respectively derived by applying the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem for contractive maps.

The rest of the paper is organized as follows: Section 3 contains the main results for problem (1)–(2), while the existence results for problem (3)–(4) are presented in Section 4. We recall the related background material in Section 2.

2. Preliminaries

In this section, we introduce some notations and definitions of fractional calculus and multivalued analysis and present preliminary results needed in our proofs later [1].

Definition 1. The Riemann-Liouville fractional integral of order for a continuous function is defined by provided that the right-hand side exists on .

Definition 2. The Riemann-Liouville fractional derivative of order of a continuous function is defined by where , denotes the integer part of real number , provided that the right-hand side is point-wise defined on .

Definition 3. The Caputo fractional derivative of order of a continuous function is defined by provided that the right-hand side is point-wise defined on .

Definition 4 (Hilfer fractional derivative [7, 8]). The Hilfer fractional derivative of order and parameter of a function (also known as the generalized Riemann-Liouville fractional derivative) is defined by where , , , and .

Remark 5. When , the Hilfer fractional derivative corresponds to the Riemann-Liouville fractional derivative: while in the definition of the Hilfer fractional derivative corresponds to the Caputo fractional derivative:

In the following lemma, we present the compositional property of the Riemann-Liouville fractional integral operator with the Hilfer fractional derivative operator.

Lemma 6 (see [8]). Let , , , , and . Then

The following lemma deals with a linear variant of boundary value problem (1)–(2).

Lemma 7. Let , , , , , and . Then, the function is a solution of the boundary value problem: if and only if where it is assumed that

Proof. Applying the Riemann-Liouville fractional integral of order to both sides of (12), we obtain by using Lemma 6where is an arbitrary constant and . Applying the Riemann-Liouville fractional integral of order to both sides of (16), we obtain Applying Lemma 6 to (17), we obtain Using in (18), we obtain , and hence we get Next, combining the condition with (19), we have Substituting the value of in (19) yields the solution (14). The converse follows by direct computation. This completes the proof.

3. Existence and Uniqueness Results for Single-Valued Problem (1)–(2)

In view of Lemma 7, we define an operator associated with problem (1)–(2) by where denotes the Banach space of all continuous functions from into with the norm . One can observe that the existence of a fixed point of operator implies the existence of a solution for problem (1)–(2).

For computational convenience, we introduce the following notations:

Now, we present our main results for boundary value problem (1)–(2). Our first existence result is based on the well-known Krasnoselskii’s fixed-point theorem [26].

Theorem 8. Assume that the following conditions hold: (H₁) is a continuous function such that , , with (H₂) , where is given by (23)Then, there exists at least one solution for problems (1) and (2) on

Proof. In order to verify the hypothesis of Krasnoselskii’s fixed-point theorem [26], we split operator defined by (21) into the sum of two operators and on the closed ball with , where and For any , we have This shows that . By using (H₂), it is easy to establish that is a contraction mapping.
Continuity of operator follows from that of . Also, is uniformly bounded on as Now, we prove that operator is compact. Setting , we obtain independently of . Thus, is equicontinuous and hence is relatively compact on . In consequence, it follows from the Arzelá-Ascoli theorem that is compact on . In view of the foregoing arguments, we deduce that the hypothesis of Krasnoselskii’s fixed-point theorem [26] holds true and consequently its conclusion implies that problem (1)–(2) has at least one solution on .

Example 9. Consider the three-point boundary value problem of the Langevin equation with the Hilfer fractional derivative of the form: Here, , , , , , , , , and . Using the given values, it is found that , , , , and . Notice that

Clearly the hypothesis of Theorem 8 holds true and consequently its conclusion implies that the boundary value problem (29) has at least one solution on .

The Leray-Schauder Nonlinear Alternative [25] is used for our next existence result.

Theorem 10. Suppose that (H₂) and the following conditions hold: (H₃) for each , where is a continuous nondecreasing function and (H₄) There exists a constant , such thatwhere are respectively given by (22) and (23).

Then, there exists at least one solution for problem (1)–(2) on .

Proof. Let us verify that operator defined by (21) satisfies the hypothesis of the Leray-Schauder Nonlinear Alternative [25]. In our first step, we establish that operator maps bounded sets (balls) into a bounded set in . For a number , let be a bounded ball in . Then, for , we have and consequently, Next, we will show that maps bounded sets into equicontinuous sets of . Let with and . Then we have Observe that the right-hand side of the above inequality tends to zero independently of as . Thus, the set is equicontinuous. Therefore, the Arzelá-Ascoli theorem applies and hence operator is completely continuous.
Finally, we show that the set of all solutions to equations is bounded for .
Following the computation in the first step, we obtain which yields According to (H₄), there exists satisfying . Introduce a set and notice that is continuous and completely continuous. Then, the choice of implies that there is no , such that for some . In consequence, we deduce by the nonlinear alternative of the Leray-Schauder type [25] that has a fixed-point , which corresponds to a solution of problem (1)–(2). This completes the proof.

Example 11. Consider the three-point boundary value problem of the Langevin equation with the Hilfer fractional derivative of the form: Here , , , , , , , , and . Using the given values, we obtain , , , , and . Next, the nonlinear function is bounded as which satisfies condition H₄ with and . Furthermore, we find that , satisfying H₄ of Theorem 10. Therefore, by applying Theorem 10, the boundary value problem (38) has at least one solution on .

In the following result, we apply Banach’s fixed-point theorem to prove the existence of a unique solution of the problem at hand.

Theorem 12. Assume that (H₅) , for each and .If the constants defined by (22) and (23), respectively, are such that then problem (1)–(2) has a unique solution on .

Proof. Let us first show that defined by (21) satisfies , where with and . For any , we have which implies that .
Next, we let . Then for , we have which implies that . As , is a contraction. Therefore, by Banach’s fixed-point theorem, operator has a fixed point which is indeed a unique solution of problem (1)–(2). The proof is finished.

Example 13. Consider the three-point boundary value problem of the Langevin equation with the Hilfer fractional derivative of the form: Here, , , , , , , , , , and Using the given data, we obtain , , , , , and as Moreover, we have . As all the assumptions of Theorem 12 are satisfied, we therefore deduce by its conclusion that problem (43) has a unique solution on .

4. Existence Results for Multivalued Problems (3) and (4)

Definition 14. A continuous function is said to be a solution of problem (3)–(4) if and there exists a function with , a.e., on such that For each , define the set of selections of by

Lemma 15 (see [27]). Let be a separable Banach space. Let be an -multivalued map and let be a linear continuous mapping from to . Then the operator is a closed graph operator in .

Our first existence result, dealing with the convex valued , is based on the nonlinear alternative of the Leray-Schauder type for (Kakutani) multivalued maps [25] with the assumption that is Carathéodory.

Theorem 16. Suppose that (H₂), (H₄), and the following conditions hold: (A₁) is -Carathéodory, where (A₂) for each , where is a continuous nondecreasing function and Then, there exists at least one solution for problem (3)–(4) on .

Proof. Let us transform problem (3)–(4) into a fixed-point problem by introducing an operator as for and . Notice that the existence of a fixed point of ensures the existence of a solution of problems (1) and (2). This will be achieved by establishing that operator satisfies the hypothesis of the Leray-Schauder nonlinear alternative for the Kakutani maps [25]. We do it in several steps.

Step 1. Since is convex ( has convex values), therefore is convex for each .

Step 2. maps bounded sets (balls) into bounded sets in .

Let be a bounded set in . Then, for each , there exists such that

Then, for , we have

Thus,

Step 3. maps bounded sets into equicontinuous sets of .

Let with and . Then, for each , we obtain

Therefore, is completely continuous by the application of the Arzelá-Ascoli theorem.

Next, we show that is upper semicontinuous by proving that it has a closed graph ([28], Proposition 1.2) as is already shown to be completely continuous.

Step 4. has a closed graph.

Let and . Then, we need to show that . Associated with , there exists such that for each ,

Thus, it suffices to show that there exists , such that for each ,

Let us introduce the linear operator as by

Observe that , as . So, by Lemma 15, is a closed graph operator. Furthermore, we have . Since , we have for some .

Step 5. We show that there exists an open set with for any and all .

Let and . Then, there exists with , such that for , we have

As in the second step, it can be shown that which implies that

Consequently

By H₄, we can find a number with . Define

Notice that operator is compact, upper semicontinuous, and convex valued. By the choice of , we cannot find satisfying for some . Therefore, by the Leray-Schauder nonlinear alternative for the Kakutani maps [25], has a fixed point which is a solution of problem (3)–(4). This completes the proof.

In our next result, we show the existence of solutions for the nonconvex valued case of problem (3)–(4). For that, we need the following assumptions. (B₁) is such that is measurable for each , where (B₂) for almost all and with and for almost all

Recall that is defined by where , , and is a metric space induced from the normed space .

We apply a fixed-point theorem for multivalued maps due to Covitz and Nadler [29]: if is a contraction, then , where .

Theorem 17. Suppose that conditions (B₁) and (B₂) hold and that Then, problem (3)–(4) has at least one solution on .

Proof. In view of (B₁), the set is nonempty for each , and hence has a measurable selection (see Theorem III.6 in [30]). Now, we verify that operator defined by (49) satisfies the hypothesis of Covitz and Nadler’s fixed theorem [29]. In order to establish that for each , let be such that in . Then, and there exists such that, for each Since is compact valued, we pass onto a subsequence (if necessary) to obtain that converges to in . Therefore, , and for each , we have which implies that .
Next, we show that we can find a (defined by (64)) satisfying Let and . Then, there exists such that, for each , By assumption (B₂), we have that . So we can find , such that Define by As the multivalued operator is measurable (Proposition III.4 in [30]), we can find a function which is a measurable selection for . So , and for each , we have .
For each , let us define Thus, Hence By interchanging the roles of and , one can obtain in a similar manner that This shows that is a contraction. So, by Covitz and Nadler’s fixed-point theorem [29], operator has a fixed-point which corresponds to a solution of problem (3)–(4). This completes the proof.

Example 18. Consider the three-point boundary value problem of the Langevin inclusion with the Hilfer fractional derivative of the form: where

Here, , , , , , , , , and . We can find that , , , , and . It is clear that is measurable for all . Now, we see that

By choosing , we have and also we obtain , . Then we find that

Hence, by using Theorem 17, we get that the boundary value problem (75) has at least one solution on .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-61-KNOW-031).

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Copyright © 2020 Athasit Wongcharoen 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.

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