Abstract
In this paper, we study a coupled system involving Hilfer fractional derivatives with nonlocal integral boundary conditions. Existence and uniqueness results are obtained by applying Leray-Schauder alternative, Krasnoselskii’s fixed point theorem, and Banach’s contraction mapping principle. Examples illustrating our results are also presented.
1. Introduction
The theory of fractional differential equations has been widely used in pure mathematics and applications in the fields of physics, biology, and engineering. There are many interesting results for qualitative analysis and applications. We refer the interested reader, in fractional calculus, to the classical reference texts such as [1–7]. In the literature, there exist several different definitions of fractional integrals and derivatives, and the most popular of them are Riemann–Liouville, Caputo, and other less-known such as Hadamard fractional derivative and the Erdeyl–Kober fractional derivative. A generalization of derivatives of both Riemann–Liouville and Caputo was given by Hilfer in [8] aswhere , , , and .
He named it as generalized fractional derivative of order and a type . Many authors call it the Hilfer fractional derivative. We notice that when , the Hilfer fractional derivative corresponds to the Riemann–Liouville fractional derivative:
When , the Hilfer fractional derivative corresponds to the Caputo fractional derivative:
Such derivative interpolates between the Riemann–Liouville and Caputo derivative. Some properties and applications of the Hilfer derivative are given in [9, 10] and the references cited therein.
Initial value problems involving Hilfer fractional derivatives were studied by several authors, see, for example [11–16] and references therein. Nonlocal boundary value problems for Hilfer fractional derivative were studied in [17].
To the best of our knowledge, there is no work carried out on systems of boundary value problems with Hilfer fractional derivative in the literature. This paper come to fill this gap. Thus, the objective of the present work is to introduce a new class of coupled systems of Hilfer-type fractional differential equations with nonlocal integral boundary conditions and develop the existence and uniqueness of solutions. In precise terms, we consider the following coupled system:where and are the Hilfer fractional derivatives of orders and , , and parameters and , respectively, , and , are the Riemann–Liouville fractional integrals of order and , respectively, the points , , , are continuous functions and , , , are given real constants.
The paper is organized as follows. We present our main results in Section 3, by applying Leray–Schauder alternative, Krasnoselskii’s fixed point theorem, and Banach’s contraction mapping principle, while Section 2 contains some preliminary concepts related to our problem. Examples are constructed to illustrate the main results.
2. Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later [2, 5].
Definition 1. The Riemann–Liouville fractional integral of order of a continuous function is defined byprovided the right-hand side exists on .
Definition 2. The Riemann–Liouville fractional derivative of order of a continuous function is defined bywhere , denotes the integer part of real number , provided the right-hand side is pointwise defined on .
Definition 3. The Caputo fractional derivative of order of a continuous function is defined byprovided the right-hand side is pointwise defined on .
Lemma 1 (see [18]). If , , and , then(1)(2)The following lemma deals with a linear variant of problem (4).
Lemma 2. Let , , , , , , , , , , , , , , and
Then, the systemis equivalent to the following integral equations:
Proof. Operating fractional integral on both sides of the first equation in (9) and using Lemma 1, we obtainThen, we have, since ,Then,whereBy a similar way, we obtainBy settingfrom the boundary conditions and , we obtain and . Then, we obtainFrom and , we haveSubstituting the values of and in (18), we obtain solutions (10) and (11). The converse follows by direct computation. This completes the proof.
3. Main Results
Let , , denote the Banach space of all continuous functions from to . The space endowed with the norm is a Banach space. Let with the norm . It is obvious that the product space is Banach space with the norm .
In view of Lemma 2, we define two operators bywherewhere
For computational convenience, we set
Banach’s contraction mapping principle is applied in the first result to prove existence and uniqueness of solutions of system (4).
Theorem 1. Suppose that are continuous functions. In addition, we assume that
There exist constants , such that, for all and ,
Then, system (4) has a unique solution on , if
Proof. Define and such that
Now, we will show that the set , where . For any , , we find that
For , we have
Hence,
Similarly, we haveand hence
Consequently, it follows thatwhich implies . Next, we will show that the operator is a contraction mapping. For any , we obtain
Therefore, we obtain the following inequality:
In addition, we also obtain
As , therefore, is a contraction operator. By Banach’s fixed point theorem, the operator has a unique fixed point, which is the unique solution of (4) on . The proof is completed.
Now, we prove our second existence result via Leray–Schauder alternative.
Lemma 3 (Leray–Schauder alternative, see [19]). Let be a completely continuous operator. LetThen, either the set is unbounded, or has at least one fixed point.
Theorem 2. Assume that there exist real constants for and such that, for any , we have
If and , where are given in (24)–(27), then (4) has at least one solution on .
Proof. By continuity of the functions on , the operator is continuous. We will show that the operator is completely continuous. Let be bounded. Then, there exist positive constants and such thatThen, for any , we havewhich yieldsSimilarly, we obtain thatHence, from the above inequalities, we obtain that the set is uniformly bounded. Next, we are going to prove that the set is equicontinuous. For any and such that , we haveTherefore, we obtainAnalogously, we can obtain the following inequality:Hence, the set is equicontinuous. By applying the Arzelá–Ascoli theorem, the set is relative compact which implies that the operator is completely continuous. Lastly, we shall show that the set is bounded. Let any , then . For any , we haveThen, we obtainwhich imply thatThus, we obtainwhere , which shows that the set is bounded. Therefore, by applying Lemma 3, the operator has at least one fixed point. Therefore, we deduce that problem (4) has at least one solution on . The proof is complete.
The last existence theorem is based on Krasnoselskii’s fixed point theorem.
Lemma 4 (Krasnoselskii’s fixed point theorem, see [20]). Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let be operators such that (i) , where , (ii) is compact and continuous, and (iii) is a contraction mapping. Then, there exists such that .
Theorem 3. Assume that are continuous functions satisfying assumption in Theorem 1. In addition, we suppose and there exist two positive constants such that, for all and ,Ifthen problem (4) has at least one solution on .
Proof. To apply Lemma 4, we decompose the operator into four operators , and asNote that and . Also, observe that the ball is a closed, bounded, and convex subset of the Banach space . Let be a ball, where a constant . Now, we will show that for satisfying condition (i) of Lemma 4. Setting and using condition (55), then we have, as in Theorem 2 thatSimilarly, we can find thatThat yields . To show that the operator is a contraction mapping satisfying condition (iii) of Lemma 4, for , we haveIt follows from (60) and (61) thatwhich is a contraction by inequality in (56). Therefore, condition (iii) of Lemma 4 is satisfied. Next, we will show that the operator satisfies condition (ii) of Lemma 4. By applying the continuity of the functions on , we can conclude that the operator is continuous. For each , one hasand similarlyThen, we obtain the following fact:which implies that the set is uniformly bounded. In the next step, we will show that the set is equicontinuous. For such that and for any , we can prove thatTherefore, we have as . Similarly, we can show that as .
Thus, tends to zero as . Therefore, the set is equicontinuous. By applying the Arzelá–Ascoli theorem, the operator is compact on . By application of Lemma 4, we have that problem (4) has at least one solution on . This completes the proof.
Example 1. Consider the coupled system of Hilfer fractional differential equations with nonlocal integral boundary conditions of the formHere, , , , , , , , , , , and , , , , and , . Then, we can compute constants as , , , , , , and .(i)Let the nonlinear functions and be defined by It is easy to check that and satisfy in Theorem 1 as and . Setting and , we have Therefore, by applying Theorem 1, we deduce that problem (67) with (68)–(69) has a unique solution on .(ii)Given the functions and by we can check that which satisfy conditions in Theorem 2 by , , , , , and . Hence, we find that and . The conclusion of Theorem 2 implies that problem (67) with (71)-(72) has at least one solution on .(iii)Define the functions and by Observe that condition in Theorem 1 is satisfied for nonlinear functions and with Lipschitz constants , , , and . Next, we can find that . Then, Theorem 1 cannot be used to obtain the existence criteria for the investigated problem. However, we calculate that , , and Hence, all assumptions of Theorem 3 hold. Therefore, by Theorem 3, problem (67) with (74)-(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).