Existence and stability of impulsive coupled system of fractional integrodifferential equations

Akbar Zada,Hira Waheed,Xiaoming Wang,Jehad Alzabut

doi:10.1515/dema-2019-0035

Abstract

Abstract In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

Highlights

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order, but with this definition, many interesting questions will arise; for example, if the first derivative of a function gives you the slope of the function, what is the geometrical meaning of half derivative? In half order, which operator must be used twice to obtain the first derivative? The early history of these questions goes back to the birth of fractional calculus in 1695 when Gottfried Wilhelm Leibniz suggested the possibility of fractional derivatives for the first time [1]
We have presented some existence and uniqueness results for an impulsive initial value problem of coupled fractional integrodifferential systems involving the Caputo type fractional derivative
The proof of the existence results is based on the nonlinear alternative of Schaefer’s and Krasnoselskii’s fixed point theorem, while the uniqueness of the solution is proved by applying the Banach contraction principle

Summary

Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order, but with this definition, many interesting questions will arise; for example, if the first derivative of a function gives you the slope of the function, what is the geometrical meaning of half derivative? In half order, which operator must be used twice to obtain the first derivative? The early history of these questions goes back to the birth of fractional calculus in 1695 when Gottfried Wilhelm Leibniz suggested the possibility of fractional derivatives for the first time [1]. For the study of CS of FDEs, we refer the reader to [26,27,28,29,30,31] Another important class of DEs is known as impulsive differential equations (IDEs). This class plays the role of an effective mathematical tools for those evolution processes that are subject to abrupt changes in their states. I0β,T η(t, y(t)), where c D0β,t is the Caputo fractional derivative of order β with lower limit 0 and G : In this manuscript, we study the existence, uniqueness and HU stability results of the implicit FDE with impulsive condition as:.

Existence and uniqueness

Ulam stability results

Illustrative example

Conclusion