Study of implicit delay fractional differential equations under anti-periodic boundary conditions

Arshad Ali,Thabet Abdeljawad,Kamal Shah

doi:10.1186/s13662-020-02597-x

Abstract

This research work is related to studying a class of special type delay implicit fractional order differential equations under anti-periodic boundary conditions. With the help of classical fixed point theory due to Schauder and Banach, we derive some results about the existence of at least one solution. Further, we also study some results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam Rassias, and generalized Hyers–Ulam–Rassias stability. We provide some test problems for illustrating our analysis.

Highlights

1 Introduction Differential equations have numerous of applications in many applied fields of sciences
The class of differential equations has remained an interesting area of research
Keeping in mind the applications, researchers are devoted to studying different aspects like existence theory and numerical analysis of the mentioned class of differential equations

Summary

Introduction

Differential equations have numerous of applications in many applied fields of sciences. Various aspects of fractional calculus, such as qualitative theory, stability analysis, optimization, and numerical analysis, have been investigated In this regard a lot of research work can be found in the literature about existence theory. In the last few years the mentioned stabilities have been upgraded for linear and nonlinear fractional order differential equations and their systems (for details, see [19,20,21]). Keeping in mind the applications, researchers are devoted to studying different aspects like existence theory and numerical analysis of the mentioned class of differential equations. Very recently the authors in [32] established qualitative theory for a coupled system of delay fractional order differential equations by using hybrid fixed point theory. Motivated by the above-mentioned work, in this research article we consider the following class of pantograph implicit fractional order differential equations under anti-periodic boundary conditions:. Remark 2 A function z ∈ M is a solution of (6) if there is a function θ(t) ∈ C([0, T], R)

Existence results

Conclusion