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

Karim Guida,Khalid Hilal,Said Melliani,Lahcen Ibnelazyz,Emilio Turco

doi:10.1155/2021/6616899

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.

Highlights

Fractional differential equations (FDEs) involve fractional derivatives of the form dα/dxα, which are defined for α > 0, where α is not necessarily an integer
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
Motivated by all the previous works, we consider in this paper coupled impulsive fractional pantograph differential equations with antiperiodic boundary conditions as follows: 8 >>>>>>>>>>>>< >>>>>>>>>>>>: cDα1 xðtÞ + γ1xðtÞ = f1ðt, xðtÞ, xðλ1tÞ, yðtÞÞ, cDα2 yðtÞ + γ2yðtÞ = f2ðt, xðtÞ, yðtÞ, yðλ2tÞÞ, Δxjt

Summary

Introduction

Fractional differential equations (FDEs) involve fractional derivatives of the form dα/dxα, which are defined for α > 0, where α is not necessarily an integer. 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]). Motivated by all the previous works, we consider in this paper coupled impulsive fractional pantograph differential equations with antiperiodic boundary conditions 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 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

Preliminaries and Lemmas

Main Results

Γðα1Þ ððα1

Examples