Existence Results for a Class of Coupled Hilfer Fractional Pantograph Differential Equations with Nonlocal Integral Boundary Value Conditions

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.


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][2][3][4][5][6][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][9][10][11][12][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 α and type β ∈ ½0, 1, and we can find the Riemann-Liouville fractional derivative when β = 0, and the Caputo fractional derivative when β = 1. Fractional differen-tial equations involving the Hilfer fractional derivative have many applications, see [15][16][17][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][25][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 H D α 1 ,β 1 , H D α 2 ,β 2 are the Hilfer fractional derivatives of order α 1 and α 2 , 1 < α 1 , α 2 < 2 and parameter β 1 , β 2 , 0 < β 1 , β 2 < 1, respectively, f 1 , f 2 : ½a, b × ℝ × ℝ × ℝ → ℝ are two continuous functions; I δ 1 , I δ 2 are the Riemann-Liouville fractional integrals of order δ 1 and δ 2 , respectively, a ≥ 0, A 1 , A 2 , B 1 , B 2 , C 1 , C 2 ∈ ℝ, and 0 < λ 1 , λ 2 < 1. 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.

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: Cð½a, b, ℝÞ with a ≥ 0 is the Banach space of all continuous functions from ½a, b to ℝ, Lða, bÞ is the space of Lebesgue integrable functions on a finite closed interval ½a, bðb > aÞ of the real line ℝ, and AC k ½a, b is the space of real-valued functions f ðtÞ which have continuous derivatives up to order k − 1 on ½a, b such that f ðk−1Þ ðtÞ belongs to the space of absolutely continuous functions AC½a, b. Definition 1. (see [8,11]). The Riemann-Liouville fractional integral of order α > 0 of a continuous function f : ½a,∞Þ → ℝ, is defined by provided the right-hand side exists on ða, ∞Þ.
Definition 2 (see [8,11]). The Riemann-Liouville fractional derivative of order α > 0 of a continuous function f , is defined by where n = ½α + 1, ½α denotes the integer part of the real number α, provided the right-hand side is pointwise defined on ða, ∞Þ.
Definition 3 (see [8,11]). The Caputo fractional derivative of order α > 0 of a continuous function f , is defined by provided the right-hand side is pointwise defined on ða, ∞Þ.
The following lemma gives a composition between the Riemann-Liouville fractional integral operator and the Hilfer fractional derivative operator.
Then, the following problem is equivalent to the system of equations: Proof. Let us assume that ðx, yÞ is a solution of problem (9). Applying the fractional integrals I α 1 and I α 2 on both sides of the equations in (9) and using Lemma 6, we obtain Since where for i = 1, 2, c 0i , c 1i are real constants.
Then, we get from the conditions: A 1 xðbÞ + B 1 I δ 1 xðμ 1 Þ = C 1 and A 2 yðbÞ + B 2 I δ 2 yðμ 2 Þ = C 2 ; 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.
In view of Lemma 7, we define the operator T : where 3

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We should note that problem (2) has a solution ðx, yÞ if and only if the operator T has a fixed point.
In what is coming, for convenience, we set the following: For i = 1, 2, 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.
Proof. We transform the boundary value coupled systems (2) into a fixed point problem. Applying the Banach contraction mapping principle, we show that T defined by (16) and (17) has a unique fixed point. We let We first show that TB r 0 ⊂ B r 0 , where B r 0 = fðx, yÞ ∈ X × Y : kðx, yÞk ≤ r 0 g.
For any ðx, yÞ ∈ B r 0 , we have Similarly, we get

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Finally, which implies that TB r 0 ⊂ B r 0 . Next, we show that the operator T is a contraction; we let ðx, yÞ, ð x, yÞ ∈ X × Y; then for t ∈ ½a, b, we have with a similar method, we also get Finally, we can obtain And since, 3ðL f 1 Ω 1 + L f 2 Ω 2 Þ < 1, then the operator T is a contraction. Therefore, we conclude by Banach's contraction mapping principle that T has a fixed point which is the unique solution ðx * , y * Þ of problem (2). The proof is completed.
Next, we present a result based on Krasnoselskii's fixed point theorem.
For all ðx, yÞ, ð x, yÞ ∈ X × Y, and for t ∈ ½a, b, we have Similarly, It is easy to see, using (29), that T 2 is a contraction mapping.
The continuity of the functions f 1 and f 2 implies the continuity of the operator T 1 . In addition, T 1 is uniformly bounded on B ε 0 as