Existence Results for a Nonlocal Coupled System of Sequential Fractional Differential Equations Involving -Hilfer Fractional Derivatives

Abstract

In this article, we discuss the existence and uniqueness of solutions for a new class of coupled system of sequential fractional differential equations involving -Hilfer fractional derivatives, supplemented with multipoint boundary conditions. We make use of Banach’s fixed point theorem to obtain the uniqueness result and the Leray-Schauder alternative to obtain the existence result. Examples illustrating the main results are also constructed.

1. Introduction

Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders. One of the most important advantages of fractional order models in comparison with integer order ones is that fractional integrals and derivatives are a powerful tool for the description of memory and hereditary properties of some materials. For details and applications, we refer the reader to the texts [1–6]. There are some different definitions of fractional derivatives, from the most popular of Riemann-Liouville and Caputo type fractional derivatives, to the other ones such as Hadamard fractional derivative and the Erdeyl-Kober fractional derivative. A generalization of both Riemann-Liouville and Caputo derivatives was given by Hilfer in [7], which is known as the Hilfer fractional derivative of order and a type Some properties and applications of the Hilfer derivative can be found in [8, 9] and references cited therein.

Initial value problems involving Hilfer fractional derivatives were studied by several authors (see, for example, [10–12]). Nonlocal boundary value problems for Hilfer fractional differential equation have been discussed in [13, 14]. Coupled systems for Hilfer fractional differential equations with nonlocal integral boundary conditions were studied in [15].

The fractional derivative with another function, in the Hilfer sense, called -Hilfer fractional derivative, has been introduced in [16], which unifies several different fractional operators. For some recent results on existence and uniqueness of initial value problems and results on Ulam-Hyers-Rassias stability, see [17–19] and references therein. Recently, in [20], the authors extended the results in [13] to -Hilfer nonlocal implicit fractional boundary value problems. For recent results in -Hilfer fractional derivative, we refer to [21–23] and references cited therein.

In [24], the authors initiated the study of existence and uniqueness of solutions for a new class of boundary value problems of sequential -Hilfer-type fractional differential equations with multipoint boundary conditions of the formwhere is the -Hilfer fractional derivative of order , and parameter , , is a continuous function, and Existence and uniqueness results were proved by using classical fixed point theorems. The Banach’s fixed point theorem was used to obtain the uniqueness result, while nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem are applied to obtain the existence results for the problem (1).

In this paper, we investigate the existence and uniqueness criteria for the solutions of the following nonlocal coupled system of sequential -Hilfer fractional derivative of the formwhere , are the -Hilfer fractional derivatives of orders and , and two parameters , , , given constants , and the points and are continuous functions.

In order to study the problem (2), we convert it into an equivalent fixed point problem and then we use Banach’s fixed point theorem to prove the uniqueness of its solutions, while by applying the Leray-Schauder alternative [25], we obtain the existence result.

The remaining part of the article is structured as follows: Section 3 contains the main results for the problem (2). Examples illustrating the existence and uniqueness results are also included. We recall the related background material in Section 2, in which also we establish a lemma regarding a linear variant of the problem (2).

2. Preliminaries

Here, some notations and definitions of fractional calculus are reminded [1].

Definition 1. The Riemann-Liouville fractional integral of order for 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 and , provided the right-hand side is point-wise defined on .

Definition 3. The Caputo fractional derivative of order of a continuous function is defined bywhere 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 and Caputo fractional derivatives) is defined bywhere , , and .

Remark 5. When , the Hilfer fractional derivative corresponds to the Riemann-Liouville fractional derivativewhile when , the Hilfer fractional derivative corresponds to the Caputo fractional derivativeLet be an increasing function with for all

Definition 6 ([1]). Let and The -Riemann-Liouville fractional integral of order to a function with respect to is defined by

Definition 7 ([16]). Let , , and such that is increasing with for all The -Hilfer fractional derivative of order to a function and type is defined by

Remark 8 ([1]). If , then we have -Riemann-Liouville fractional derivative asand if , we obtain -Caputo fractional derivative by

Lemma 9 ([16]). Let and be constants and be an increasing function with for all . Then, we have

The following lemma contains the compositional property of Riemann-Liouville fractional integral operator with the -Hilfer fractional derivative operator.

Lemma 10 ([16]). Let , , and Then,where

The following lemma deals with a linear variant of the system (2).

Lemma 11. Let , and be given functions. Then, the unique solution of -Hilfer the fractional differential linear systemis given bywhereand it is assumed that

Proof. Assume that is a solution of the nonlocal boundary value problem (15) on . Operating fractional integral on both sides of the first equation in (15) and using Lemma 10, we obtain for

Hence, using the fact that we havewhere and

From the first boundary condition , we can obtain since Then, we get

By a similar way, we obtainwhere is an arbitrary constant.

From the second boundary conditions and , we get the systemwhere

Solving the system (24), we find that

Substituting the value of , in (22) and (23) yields the solution (16) and (17). The converse follows by direct computation. This completes the proof.

3. Main Results

Let us introduce the space endowed with the norm Obviously, is a Banach space. Then, the product space is also a Banach space equipped with norm

In view of Lemma 11, we define an operator bywherewhere

For the sake of computational convenience, we put

Our first result is based on Leray-Schauder alternative ([25] p. 4.).

Lemma 12 (Leray-Schauder alternative). Let be a completely continuous operator (i.e., a map that restricted to any bounded set in is compact). LetThen, either the set is unbounded, or has at least one fixed point.

Theorem 13. Assume that are continuous functions, and there exist real constants and such that Ifwhere are given by (30)-(37); then, the system (2) has at least one solution on

Proof. The operator is continuous, by the continuity of functions and We will show that the operator is completely continuous. Let be bounded set. Then, there exist positive constants such that Then, for any we havewhich implies thatSimilarly, it can be shown thatFrom the above inequalities, it follows that the operator is uniformly bounded, sinceNext, we show that is equicontinuous. Let with Then, we haveAnalogously, we can obtainTherefore, the operator is equicontinuous, and thus, the operator is completely continuous.

Finally, it will be verified that the set is bounded. Let with For any we have

Then,

Hence, we havewhich imply that

Consequently,which proves that is bounded. Thus, the operator , by Lemma 12, has at least one fixed point. Hence, the boundary value problem (2) has at least one solution. The proof is complete.

The uniqueness of solutions of the system (2) is proved in the next theorem, via Banach’s contraction mapping principle.

Theorem 14. Assume that are continuous functions, and there exist positive constants such that for all and we haveThen, the system (2) has a unique solution on provided thatwhere are given by (30)-(37).

Proof. Define and such thatIn the first step, we show that where By the assumption for we haveUsing the above estimates, we obtainHence,In the same way, we can obtain thatIn consequence, it follows thatwhich shows that .

We prove that the operator is a contraction. For and for any we get