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

Abstract

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.

1. Introduction

Fractional differential equations (FDEs) involve fractional derivatives of the form , which are defined for , where 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 [1–6] 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 [7–13].

Many papers have studied impulsive fractional differential equations with antiperiodic boundary conditions, and results on the existence and uniqueness have been given (see [14–17]). For example, recently, Zuo et al. [18] investigated the existence results for an equation with impulsive and antiperiodic boundary conditions given by where is the Caputo fractional derivative of order , , , , the function is in , denotes the jump of at , and and are linear operators. The authors established the existence and uniqueness under some conditions using Banach’s and Krasnoselskii’s fixed point theorems.

On the other hand, in the deterministic situation, there is a very special case of delay differential equations known as the pantograph equations given by

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 [19–21].

Motivated by all the previous works, we consider in this paper coupled impulsive fractional pantograph differential equations with antiperiodic boundary conditions as follows: where and are the Caputo fractional derivatives of orders and , respectively, are two continuous functions, , , with and representing the right and left limits of at , , and also , with and representing the right and left limits of at , .

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.

2. Preliminaries and Lemmas

Let , and , where , and exist, , is a space of continuous real-valued functions on the interval , and .

Similarly, we define , where , and exist, , is a space of continuous real-valued functions on the interval , and .

Then, clearly, and are two Banach spaces with the norms and , respectively.

Consequently, the space is a Banach space with the norm .

We note that the space is a Banach space of Lebesque measurable functions with

Definition 1 (see [1]). The fractional integral of order with the lower limit zero for a function is defined as provided the right-hand side is pointwise defined on , where denotes the Gamma function.

Definition 2 (see [1]). The Riemann-Liouville derivative of order with the lower limit zero for a function is defined as provided the function is absolutely continuous up to order derivatives, where denotes the Gamma function.

Definition 3 (see [1]). The Caputo derivative of order with the lower limit zero for a function is defined as provided the function , where denotes the Gamma function.

Definition 4. A couple is a solution of problem (3) if it satisfies the equations a.e. on , and the conditions and .

Lemma 5 (see [22]). The nonnegative functions and given by have the following properties: (1)For any and ,In addition, we have and . (2)For any and ,(3)For any and such that ,

Lemma 6 (see [23]). Let be a closed, convex, and nonempty subset of a Banach space , and let and be operators such that (1) whenever (2) is compact and continuous(3) is a contraction mappingThen, there exists such that .

Lemma 7 (see [24]). Let be a Banach space, and let . Suppose that satisfies the following conditions: (1) is a uniformly bounded subset of (2) is equicontinuous in , , where and (3)Its -sections , , and are relatively compact subsets of Then, is a relatively compact subset of .

Lemma 8 (see [25]). Let be two continuous functions. The couple given by is a solution of the impulsive problem

It follows from Lemma 8, and by using the boundary conditions and , that the solution of (3) can be expressed as follows: where and .

3. Main Results

Theorem 9. We consider the following hypotheses:
. The functions are continuous, and there exist two constants such that for all .
. and for all , , and .
. We suppose that .
Then, problem (3) has a unique solution .

Remark 10. The expressions of and are given in the proof.

Proof. We define the operator by where We show now that the operator has a fixed point, which is a solution of problem (3).
Let , , , and .
We choose Firstly, we show that , where . It follows from the hypotheses above and Lemma 5 that for any , we have Similarly, we show that 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 (3). The proof is now completed.

Next, we present a result based on Krasnoselskii’s fixed point theorem.

Theorem 11. Assume that the condition and the following additional conditions are satisfied:
. Two functions , , and are nondecreasing functions satisfying the following inequalities: for all .
. We suppose that .
Then, problem (3) has at least one solution.

Remark 12. The expressions of and are given in the proof.

Proof. The set is a closed, bounded, and convex set in for all .

We define the operator by for any and , where