Hilfer-Hadamard Nonlocal Integro-Multipoint Fractional Boundary Value Problems

Abstract

This paper is concerned with the existence and uniqueness of solutions for a new class of boundary value problems, consisting by Hilfer-Hadamard fractional differential equations, supplemented with nonlocal integro-multipoint boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

1. Introduction

The fractional calculus has always been an interesting research topic for many years. This is because fractional differential equations describe many real-world process related to memory and hereditary properties of various materials more accurately as compared to classical-order differential equations. Fractional differential equations arise in lots of engineering and clinical disciplines which include biology, physics, chemistry, economics, signal and image processing, and control theory (see the monographs and papers in [1–8]).

Various types of fractional derivatives were introduced among which the following Riemann-Liouville and Caputo derivatives are the most widely used ones. (1)Riemann-Liouville derivative. For , the derivative of is(2)Caputo derivative. For , the derivative of iswhere . Both Riemann-Liouville and Caputo derivatives are defined via fractional integral, the Riemann-Liouville fractional integral, which is defined by

A generalization of derivatives of both Riemann-Liouville and Caputo was given by R. Hilfer in [9], known as the Hilfer fractional derivative of order and a type , which interpolates between the Riemann-Liouville and Caputo derivatives, since it is reduced to the Riemann-Liouville and Caputo fractional derivatives when and , respectively. The Hilfer fractional derivative of order and parameter of a function is defined by where . Some properties and applications of the Hilfer derivative are given in [10, 11] and references cited therein. Initial value problems involving Hilfer fractional derivatives were studied by several authors (see, for example, [12–15] and references therein). Nonlocal boundary value problems for Hilfer fractional derivative were studied in [16, 17].

The Hadamard fractional calculus contains fractional derivative and integral with respect to the logarithmic function while someone say that it is generalized of the derivative , where is an arbitrary order. The Hadamard calculus can be obtained by replacing as , , and in (1)–(3). In the same way, the concept of Hilfer-Hadamard derivative is arrived by the modified definition in (4). Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional differential equations with two point boundary conditions were studied in [18].

In this paper, we study existence and uniqueness of solutions for boundary value problems for Hilfer-Hadamard fractional differential equations with nonlocal integro-multipoint boundary conditions, where is the Hilfer-Hadamard fractional derivative of order and type , , , are given constants, and is a given continuous function, is the Hadamard fractional integral of order , and , , are given points.

Existence and uniqueness results are established by using classical fixed point theorems. We make use of Banach’s fixed point theorem to obtain the uniqueness result, while Schaefer and Krasnoselskii’s fixed point theorem [19] as well as nonlinear alternative of Leray-Schauder type [20] is applied to obtain the existence results for the problem (5).

The paper is constructed as follows: in Section 2, we recall some basic facts needed in our study. The main results are proved in Section 3. Examples illustrating the main results are presented in Section 4.

2. Preliminaries

In this section, some basic definitions, lemmas, and theorems are mentioned.

Definition 1 (Hadamard fractional integral [2]). The Hadamard fractional integral of order for a function is defined as provided the integral exists, where .

Definition 2 (Hadamard fractional derivative [2]). The Hadamard fractional derivative of order , applied to the fuction , is defined as follows: where and denotes the integer part of the real number .

Definition 3 (Hilfer-Hadamard fractional derivative [11]). Let and , . The Hilfer-Hadamard fractional derivative of order and tybe of is defined as where and is the Hadamard fractional integral and derivative defined by (6) and (7), respectively.

The Hilfer-Hadamard fractional derivative may be viewed as interpolating between Hadamard and Caputo-Hadamard fractional derivatives. Indeed, for , this derivative reduces to the Hadamard fractional derivative while for , it leads the Caputo-Hadamard derivative defined by

We recall the following known theorem by Kilbas et al. [2] which will be used in the following.

Theorem 4 ([2]). Let , , , , and . If and , then Since , then the exists for all .

Finally, we will use the following well-known fixed point theorems on Banach space for proving the existence and uniqueness of the solutions to Hilfer-Hadamard fractional boundary value problem (5).

Theorem 5 (Banach’s contraction principle [21]). Let be a Banach space, be closed, and be a contraction (i.e., there exists a constant such that for any ). Then, has a unique fixed point on .

Theorem 6 (Krasnoselskii’s fixed point theorem)[19]. Let be a bounded, closed, convex, and nonempty subset of a Banach space . Let and be the operators satisfying the following conditions: (i) whenever ; (ii) is compact and continuous; and (iii) is a contraction mapping. Then, there exists such that

Theorem 7 (Schaefer fixed point theorem [22]). Let be a completely continuous operator (i.e., a continuous map restricted to any bounded set in is compact). Let . Then, either the set is unbounded or has at least one fixed point.

Theorem 8 (Nonlinear alternative for single-valued maps) ([20]). Let be a Banach space, a closed, convex subset of , an open subset of , and . Suppose that is continuous and compact (that is, is a relatively compact subset of ) map. Then, either (i) has a fixed point in , or(ii)there is (the boundary of in ) and with

3. Main Results

In this section, we prove existence and uniqueness of solutions for nonlinear Hilfer-Hadamard fractional boundary value problem (5). Firstly, we start by proving a basic lemma concerning a linear variant of the boundary value problem (5), which will be used to transform the boundary value problem (5) into an equivalent integral equation. In this case, ; then, we have .

Lemma 9. Let and , where Then, is a solution of the following linear Hilfer-Hadamard fractional differential equation: supplemented with the boundary conditions in (5), if and only if

Proof. By taking the Hadamard fractional integral of order from to on both sides of (12) and using Theorem 4, it follows that Then, we have Equation (15) can be rewritten by where are arbitrary constants. Now, the first boundary condition together with (16) yields , since . Putting in (16), we get Next, the second boundary condition together with (17) yields Substituting the value of in (17), we get equation (13) as desired.
The converse follows by direct computation. The proof is completed.☐

Let us introduce the Banach space endowed with the norm defined by .

In view of Lemma 9, we define an operator , where

In the following, for convenience, we put

We need the following hypotheses in the sequel:

(H₁). There exists a constant such that for all and

(H₂). There exists a continuous nonnegative function such that

(H₃). There exists a real constant such that for all

(H₄). There exist and a continuous nondecreasing function such that

(H₅). There exists a constant such that

3.1. Existence and Uniqueness Result via Banach’s Fixed Point Theorem

We prove an existence and uniqueness result based on Banach’s contraction mapping principle.

Theorem 10. Assume that holds. Then, boundary value problem (5) has a unique solution on provided that , where is defined by (20).

Proof. We will use Banach’s fixed point theorem to prove that , defined by (19), has a unique fixed point. Fixing and using hypothesis , we obtain Choose We divide the proof into two steps.
Step I. We show that , where Let Then, we have Thus, which means that
Step II. To show that the operator is a contraction, let Then, for any , we have Thus, which, in view of , shows that the operator is a contraction. By Theorem 5, we get that the operator has a unique fixed point. Therefore, the problem (5) has a unique solution on The proof is completed.☐

3.2. Existence Result via Krasnoselskii’s Fixed Point Theorem

In this subsection, we prove an existence result based on Krasnoselskii’s fixed point theorem.

Theorem 11. Assume that - hold. Then, the problem (5) has at least one solution on provided that

Proof. By assumption , we can fix , where and consider We split the operator defined by (19) as , where and are given by For any , we have Hence, , which shows that It is easy to prove, using conditions and (32), that the operator is a contraction mapping.
Next, the operator is continuous by the continuity of . Also, is uniformly bounded on , since Finally, we prove the compactness of the operator . For , we have which tends to zero, independently of as . Thus, is equicontinuous. From the Arzelá-Ascoli theorem, we conclude that the operator is compact on . Thus, the hypotheses of Krasnoselskii fixed point theorem are satisfied, and therefore, there exists at least one solution on . The proof is finished.☐

3.3. Existence Result via Schaefer’s Fixed Point Theorem

Our second existence result is based on Schaefer’s fixed point theorem.

Theorem 12. Assume that holds. Then, the boundary value problem (5) has at least one solution on

Proof. We will prove that the operator , defined by (19), has a fixed point, by using Schaefer’s fixed point theorem. We divide the proof into two steps.
Step I. We show that the operator is completely continuous.
We show first that is continuous. Let be a sequence such that in . Then, for each , we have Since is continuous, we get Then Hence, is continuous.
Secondly, we show that the operator maps bounded sets into bounded sets in For a positive number , let be a bounded ball in Then, for , we have and Thirdly, we show that maps bounded sets into equicontinuous sets. Let with and Then, we have which tends to zero, independently of , as . Thus, the Arzelá-Ascoli theorem applies and hence, is completely continuous.
Step II. We show that the set is bounded. Let , then . For any , we have . Then, in view of the hypothesis , as in Step I, we obtain Thus, which shows that the set is bounded. By Theorem 7, we get that the operator has at least one fixed point. Therefore, the boundary value problem (5) has at least one solution on This completes the proof.☐

3.4. Existence Result via Leray-Schauder Nonlinear Alternative

Our final existence result is proved via Leray-Schauder nonlinear alternative.

Theorem 13. Assume that and hold. Then, the boundary value problem (5) has at least one solution on

Proof. As in Theorem 12, we can prove that the operator is completely continuous. We will prove that there exists an open set with for and
Let be such that for some Then, for each , we have Consequently, In view of , there is no solution such that . Let us set The operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 8), we deduce that has a fixed point which is a solution of the boundary value problem (5). The proof is completed.☐

4. Examples

Example 1. Consider the Hilfer-Hadamard nonlocal integro-multipoint fractional boundary value problem of the form

Set constants , , , , , , , , , , , , , , , , and . Then, from these data, we compute that , , and , (i)Let a nonlinear unbounded function be defined by

Now, we can verify that function satisfies the Lipchitz condition with , that is, , for all and Then, we obtain . The conclusion can be gotten from Theorem 10 that the problem (47), with given by (48), has a unique solution on . (ii)If the term in (48) is replaced by , that is,then the nonlinear function is bounded by a function of by which is satisfied a condition . It is easy to check that function in (49) is fulfilled the Lipchitz condition with constant . Since we can deduce that the Hilfer-Hardamard nonlocal integro-multipoint fractional boundary value problem (47), with given by (49), has at least one solution on . In addition, we observe that Theorem 10 cannot be used in this case because (iii)If is a non-Lipschitzian function asto claim the existence of boundary value problem (47), with given by (52), we can find that

Hence, is satisfied with . Therefore, by the benefit of Theorem 12, problem (47) with given by (52) has at least one solution on . (iv)If is a non-Lipschitzian function defined bywe see that holds as by setting and . Then, we obtain and there exists a constant satisfying condition . Thus, using Theorem 13, the boundary value problem (47), with is given in (54), has at least one solution on

5. Conclusion

In this paper, we have presented the existence and uniqueness criteria for solutions for Hilfer-Hadamard fractional differential equation complemented with nonlocal integro-multipoint boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem. Once the fixed point operator is available, we make use of Banach contraction mapping principle to obtain the uniqueness result, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Our results are new in the given configuration and enrich the literature on boundary value problems involving Hilfer-Hadamard fractional differential equations.

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-GOV-B-29.