Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

Muhammad Bahar Ali Khan,Aziz Khan,Thabet Abdeljawad,Gohar Ali,Kamal Shah,Hasib Khan

doi:10.1186/s13662-021-03313-z

Abstract

In this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given.

Highlights

fractional pantograph differential equations (FODEs) have many applications in modeling various real world processes and phenomena
Inspired by the aforementioned work, in this research article, we study the given multiterm problem of FODEs involving delay terms
Definition 4 The problem of delay FODEs (2) is generalized Hyers–Ulam (GHU) stable if there exists β ∈ C(R+, R+), β(0) = 0, and regarding any solution w ∈ X of the inequality (6), there is at most one solution w ∈ X of (2) with w (t) – w(t) ≤ β( ̄), for all t ∈ J

Summary

Introduction

FODEs have many applications in modeling various real world processes and phenomena. in previous several decades it has been given much attention. They have established qualitative results of stability and existence theory as regards the given problem in (1). Definition 3 The delay FODEs (2) is HU stable if there exists CH > 0 such that, for all > 0 and for any solution w ∈ X of the inequality

Results

Conclusion