On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

Dongming Nie,Bilal Ahmed,Azmat Ullah Khan Niazi

doi:10.1155/2020/3276873

Abstract

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.

Highlights

Fractional differential equations (FDEs) have boosted considerably due to their application in various fields of sciences, such as engineering, chemistry, mechanics, and physics
Research in different fields including engineering, physics, and biosciences have proved that numerous system structures explain more exactly with the help of FDEs [16,17,18,19,20,21] and FDEs with delay [22,23,24,25,26,27] are more accurate to illustrate the real world problems compared to FDEs without delay
We know that n(t) n0 − g(t0, ψ) + g(t, nt) + (Mtc/Γ(c + 1)), t ∈ I is an upper solution of equation (1) and n(t) ≥ m(t), so the above definition proves that equation (1) has minimum one positive solution

Summary

Introduction

Fractional differential equations (FDEs) have boosted considerably due to their application in various fields of sciences, such as engineering, chemistry, mechanics, and physics. The discussion of the existence and uniqueness of a solution for nonlinear FDEs normally fixed point theory has been used [16, 32, 33] Motivated by these and [34,35,36], in this work, we have discussed existence and uniqueness of solution after applying some sufficient conditions, obtained positive solution, and at the end established local generalized Ulam–Hyers stability and local generalized.

Preliminaries

Main Results

Stability

Example

Graphical Presentation

Conclusions