Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives

Usman Riaz,Zhengqing Fu,Zeeshan Ali,Akbar Zada,Manzoor Ahmad,Jiafa Xu

doi:10.1155/2019/5093572

Abstract

This work is committed to establishing the assumptions essential for at least one and unique solution of a switched coupled system of impulsive fractional differential equations having derivative of Hadamard type. Using Krasnoselskii’s fixed point theorem, the existence, as well as uniqueness results, is obtained. Along with this, different kinds of Hyers–Ulam stability are discussed. For supporting the theory, example is provided.

Highlights

Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals and derivatives of arbitrary order
Fractional differential equations (FDEs) serve as an excellent tool for the description of hereditary properties of various materials and processes [4]
Coupled systems of FDEs have been investigated by many authors

Summary

Introduction

Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals and derivatives of arbitrary order. Many researchers have done valuable work on the same task and interesting results were formed for linear and nonlinear integral and differential equations; for details see [43, 44] This stability analysis is very useful in many applications, such as numerical analysis and optimization, where finding the exact solution is quite difficult. Wang et al [55] discussed the existence, blowing-up solutions, and Ulam–Hyers stability of fractional differential equations with Hadamard derivative by using some classical methods. The notations p(t+i ), q(t+j ) are right limits and p(t−i ), q(t−j ) are left limits; Ii, Ĩi, Ij, Ĩj : R 󳨀→ R are continuous functions; HDα, HDβ are the Hadamard derivative operators of order α and β, respectively.

Preliminaries

Existence Results

Ulam Stability Analysis

Example

Conclusion