Existence and Uniqueness of Solutions for Coupled Impulsive Fractional Pantograph Differential Equations with Antiperiodic Boundary Conditions

In this paper, we investigate the solutions of coupled fractional pantograph differential equations with instantaneous impulses. The work improves some existing results and contributes toward the development of the fractional differential equation theory. We first provide some definitions that will be used throughout the paper; after that, we give the existence and uniqueness results that are based on Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Two examples are given in the last part to support our study.


Introduction
Fractional differential equations (FDEs) involve fractional derivatives of the form d α /dx α , which are defined for α > 0, where α is not necessarily an integer. They are generalizations of the ordinary differential equations to a random (noninteger) order. These FDEs have attracted considerable interest due to their ability to model complex phenomena. The fractional differential operators are global, and they are used to model several physical phenomena because they give accurate results. For the new readers that are interested in the fractional calculus theory in a more general concept, please see [1][2][3][4][5][6] and the references therein. Our work is concerned with impulsive coupled systems of pantograph FDEs. Impulsive FDEs have found applications in many areas such as business mathematics, management sciences, and population dynamics. Some physical problems have sudden changes and discontinuous jumps. To model these problems, we impose impulsive conditions on the differential equations at discontinuity points; for more details about impulsive fractional differential equations, we give the following references [7][8][9][10][11][12][13].
Many papers have studied impulsive fractional differential equations with antiperiodic boundary conditions, and results on the existence and uniqueness have been given (see [14][15][16][17]). For example, recently, Zuo et al. [18] investigated the existence results for an equation with impulsive and antiperiodic boundary conditions given by c D α x t ð Þ + γx t ð Þ = f t, x t ð Þ, Ax t ð Þ, Bx t ð Þ ð Þ , t ∈ J = 0, 1 ½ , t ≠ t i , i = 1, 2, ⋯, m, These equations are also called equations with proportional delays. Pantographs are special devices mounted on electric trains to collect current from one or several contact wires. They consist of a pantograph head, frame, base, and drive system, and their geometrical shape is variable. But it is recently being used in electric trains. Many researchers have investigated the pantograph differential equations and their properties; see [19][20][21].
Motivated by all the previous works, we consider in this paper coupled impulsive fractional pantograph differential equations with antiperiodic boundary conditions as follows: where c D α 1 and c D α 2 are the Caputo fractional derivatives of orders α 1 and α 2 , respectively, f 1 , i Þ and xðt − i Þ representing the right and left limits of xðtÞ at t = t i , i = 1, ⋯, m, and also Δy |t=t j = yðt + j Þ − yðt − j Þ, with yðt + j Þ and yðt − j Þ representing the right and left limits of yðtÞ at t = t j , j = 1, ⋯, n.
The objective of this paper is to establish the existence and uniqueness results of the solutions of problem (3) by means of Banach's contraction principle and Krasnoselskii's fixed point theorem.
The main contributions of this paper are as follows: (i) We consider a new system of impulsive pantograph fractional differential equations (ii) We consider antiperiodic boundary value conditions with a more general form This paper contributes toward the development of qualitative analysis of impulsive fractional differential equations. This paper is organized as follows: in Section 2, we give some definitions and useful lemmas that will be used throughout the work; after that, in Section 3, we will establish the existence and uniqueness results by means of the fixed point theorems; last but not least, in Section 4, we give two illustrative examples. Consequently, the space PCðJ, XÞ × PCðJ, YÞ is a Banach space with the norm kðx, yÞk = kxk + kyk.

Preliminaries and Lemmas
We note that the space L p ðJ, ℝÞ is a Banach space of Lebesque measurable functions with k:k L p ðJÞ < ∞: Definition 1 (see [1]). The fractional integral of order α with the lower limit zero for a function f is defined as provided the right-hand side is pointwise defined on ½0, ∞Þ, where Γð:Þ denotes the Gamma function.
Definition 2 (see [1]). The Riemann-Liouville derivative of order α with the lower limit zero for a function f is defined as provided the function f is absolutely continuous up to order ðn − 1Þ derivatives, where Γð:Þ denotes the Gamma function.
(2) For any γ > 0 and t 1 , t 2 ∈ J, (3) For any γ > 0 and t 1 , Lemma 6 (see [23]). Let M be a closed, convex, and nonempty subset of a Banach space X, and let F 1 and F 2 be operators such that Lemma 7 (see [24]). Let X be a Banach space, and let J = ½0 , T. Suppose that W ⊂ PCðJ, XÞ satisfies the following conditions: (1) W is a uniformly bounded subset of PCðJ, XÞ x ∈ Wg are relatively compact subsets of X Then, W is a relatively compact subset of PCðJ, XÞ.
Lemma 8 (see [25]). Let f 1 , f 2 → ℝ be two continuous functions. The couple ðx, yÞ given by 3 Advances in Mathematical Physics is a solution of the impulsive problem It follows from Lemma 8, and by using the boundary conditions a 1 xð0Þ + b 1 xð1Þ = 0 and a 2 yð0Þ + b 2 yð1Þ = 0, that the solution of (3) can be expressed as follows: where σ 1 = b 1 /a 1 and σ 2 = b 2 /a 2 .
Remark 10. The expressions of μ 1 and μ 2 are given in the proof.
Proof. We define the operator T : PCðJ, XÞ × PCðJ, YÞ → P CðJ, XÞ × PCðJ, YÞ by where Advances in Mathematical Physics We show now that the operator T has a fixed point, which is a solution of problem (3).
We choose Firstly, we show that TB r ⊂ B r , where B r = fðx, yÞ ∈ P CðJ, XÞ × PCðJ, YÞ: k ðx, yÞk ≤ rg. It follows from the hypotheses above and Lemma 5 that for any ðx, yÞ ∈ B r , we have Similarly, we show that Finally, which implies that TB r ⊂ B r . Next, we show that the operator T is a contraction; we let ðx, yÞ, ð x, yÞ ∈ X × Y; then, for t ∈ J, we have

Advances in Mathematical Physics
With a similar method, we also get Finally, we can obtain And since ðμ 1 + μ 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 (3). The proof is now completed.
Next, we present a result based on Krasnoselskii's fixed point theorem.
Remark 12. The expressions of ε 1 and ε 2 are given in the proof.
We define the operator T by Tðx, yÞðtÞ = ðUðx, yÞðtÞ, Vðx, yÞðtÞÞ for any ðx, yÞ ∈ B r and t ∈ ½a, b, where Advances in Mathematical Physics By splitting the two operators above, we have This upcoming part of the proof requires us to rewrite the operator T as where It follows from ðH 4 Þ and Holder's inequality that for any ðx, yÞ ∈ B r and each t ∈ J, we have Dividing both sides by r and taking the lower limit as r → +∞, we get