Abstract

This paper deals with the existence and uniqueness of solutions for a new class of coupled systems of Hilfer fractional pantograph differential equations with nonlocal integral boundary conditions. First of all, we are going to give some definitions that are necessary for the understanding of the manuscript; second of all, we are going to prove our main results using the fixed point theorems, namely, Banach’s contraction principle and Krasnoselskii’s fixed point theorem; in the end, we are giving two examples to illustrate our results.

1. Introduction

Differential equations play a very important role in the understanding of qualitative features of many phenomenon and processes in different areas and practical fields. A lot of works have been done concerning these equations in the recent years for their importance in applied sciences; for more details about differential equations and their applications, we refer the readers to [1–7].

A more general way to describe natural differential equations is through fractional calculus. Fractional calculus has attracted many researchers recently; this branch of mathematics is used in the modelling of many problems in various fields, like biology, physics, control theory, and economics; for more details, we give the following classical references [8–13].

There are many different definitions of fractional integrals and derivatives in the literature [12]; the most popular definitions are the Riemann-Liouville and the Caputo fractional derivatives. A generalization of these derivatives was introduced by Hilfer in [14], known by the Hilfer fractional derivative of order and type , and we can find the Riemann-Liouville fractional derivative when , and the Caputo fractional derivative when . Fractional differential equations involving the Hilfer fractional derivative have many applications, see [15–18] and the references therein.

On the other hand, another important class of differential equations are called pantograph equations, which are a special class of delay differential equations arising in deterministic situations and are of the form They are also called equations with proportional delays. This class of differential equations was not properly investigated under fractional derivatives. Pantograph is a device used in drawing and scaling. But, recently, this device is being used in electric trains [19, 20]. Many researchers studied the pantograph differential equations and their applications in many sciences such as biology, physics, economics, and electrodynamics. For more details, please see [21, 22].

In [23], the authors studied nonlocal boundary value problems for the Hilfer fractional derivative. Initial value problems involving Hilfer fractional derivatives were studied in [24–26]. Initial value problems for pantograph equations with the Hilfer fractional derivative were studied in [22, 27].

To the best of our knowledge, there is no work involving systems of integral boundary value problems for pantograph equations with the Hilfer fractional derivative. Thus, the objective of this work is to introduce a new class of coupled systems of Hilfer fractional differential pantograph equations with nonlocal integral boundary conditions of the form where , are the Hilfer fractional derivatives of order and , and parameter , , , respectively, are two continuous functions; are the Riemann-Liouville fractional integrals of order and , respectively, , , and .

This paper is organized as follows: we first give some definitions and notions that will be used throughout the work, after that we will establish the existence and uniqueness results by means of the fixed point theorems, and last but not least, we will give some examples that illustrate the results.

2. Preliminaries and Notations

In this section, we introduce some notations and definitions related to fractional calculus that we will use throughout this paper.

We first define the following spaces:

with is the Banach space of all continuous functions from to , is the space of Lebesgue integrable functions on a finite closed interval of the real line , and is the space of real-valued functions which have continuous derivatives up to order on such that belongs to the space of absolutely continuous functions .

Definition 1. (see [8, 11]). The Riemann-Liouville fractional integral of order of a continuous function , is defined by provided the right-hand side exists on .

Definition 2 (see [8, 11]). The Riemann-Liouville fractional derivative of order of a continuous function , is defined by where , denotes the integer part of the real number , provided the right-hand side is pointwise defined on .

Definition 3 (see [8, 11]). The Caputo fractional derivative of order of a continuous function , is defined by provided the right-hand side is pointwise defined on .

Definition 4 (see [14]). The Hilfer fractional derivative of order and parameter of a function is given by where , , , and .

Remark 5. When , the Hilfer fractional derivative becomes the Riemann-Liouville fractional derivative, while when , the Hilfer fractional derivative becomes the Caputo fractional derivative.

The following lemma gives a composition between the Riemann-Liouville fractional integral operator and the Hilfer fractional derivative operator.

Lemma 6 (see [15]). Let , , , , ; then, we have Now, we give a lemma which is the solution of a variant of the integral boundary value coupled systems (2).

Lemma 7. Let , for , , , , and Then, the following problem is equivalent to the system of equations:

Proof. Let us assume that is a solution of problem (9). Applying the fractional integrals and on both sides of the equations in (9) and using Lemma 6, we obtain Since for , , we obtain where for , , are real constants.
Since we have and , we can obtain that .
Then, we get from the conditions: and ; we can find that By substituting the values above in (13), we obtain which is the solution to problem (9).
We get the converse by direct computations. This ends the proof.

3. Main Results

The space endowed with the norm and the space endowed with the norm are two Banach spaces. Moreover, the product space is a Banach space with the norm .

In view of Lemma 7, we define the operator by where

We should note that problem (2) has a solution if and only if the operator has a fixed point.

In what is coming, for convenience, we set the following:

For ,

We are going to prove the existence and uniqueness as well as the existence results for problem (2) by using the Banach contraction principle and Krasnoselskii’s fixed point theorem.

The first result is based on Banach’s fixed point theorem.

Theorem 8. Assume that
For all , there exist such that in addition, if we have where are defined by (18); then, the boundary coupled systems (2) has a unique solution on .

Proof. We transform the boundary value coupled systems (2) into a fixed point problem. Applying the Banach contraction mapping principle, we show that defined by (16) and (17) has a unique fixed point. We let and and choose We first show that , where .
For any , we have Similarly, we get Finally, which implies that .
Next, we show that the operator is a contraction; we let ; then for , we have with a similar method, we also get Finally, we can obtain And since, , then the operator is a contraction.
Therefore, we conclude by Banach’s contraction mapping principle that has a fixed point which is the unique solution of problem (2). The proof is completed.