A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

Somia Khaldi,Adrian Petrusel,Aiman Mukheimer,Rachid Mecheraoui

doi:10.1155/2020/4237680

Abstract

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

Highlights

Over the last few decades, there has been significant development in the area of ordinary and partial fractional differential equations with boundary integral conditions
Example in last section shows the importance of this new form of assumptions (iii) Results in Section 2 are obtained on the basis of a weak form of the Banach contraction principal, which is stated and proved in this paper (iv) We discuss the Hyers-Ulam-Rassias stability of problems (9) and (10) with respect to a function φ
In this paper, using a fixed-point approach and by means of the ψ-Caputo fractional derivative, we have studied the stability of solutions of a nonlinear fractional mixed VolterraFredholm integro-differential equation with an integral initial condition

Summary

Introduction

Over the last few decades, there has been significant development in the area of ordinary and partial fractional differential equations with boundary integral conditions. Example in last section shows the importance of this new form of assumptions (iii) Results in Section 2 are obtained on the basis of a weak form of the Banach contraction principal, which is stated and proved in this paper (iv) We discuss the Hyers-Ulam-Rassias stability of problems (9) and (10) with respect to a function φ. In this example, we discuss the existence, uniqueness, Ulam-Hyers-Rassias stability, and regularity of solutions for a problem with nonlocal neutral pantograph equation involving a functional derivative and ψ-Caputo fractional derivative

Preliminaries

Existence and Uniqueness Results

Ulam-Hyers-Rassias Stability

Conclusions