Existence of Solutions for Fractional Integro-Differential Equations with Non-Local Boundary Conditions

Hamed Bazgir,Bahman Ghazanfari

doi:10.3390/mca23030036

Abstract

In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established by means of contraction mapping principle. Then, we present some examples to illustrate our results.

Highlights

Fractional differential equations arise in many engineering and scientific disciplines such as physics, aerodynamics, polymer rheology, regular variations in thermodynamics, biophysics, blood flow phenomena, electrical circuits, biology, etc
The nonlocal boundary conditions are important in describing some peculiarities happening inside the domain of physical, chemical or other processes [12], while the integral boundary conditions provide the means to assume an arbitrary shaped cross-section of blood vessels in computational fluid dynamics (CFD) studies of blood flow problems [13,14]
Non-local boundary value problems of nonlinear fractional order differential equations have recently been investigated by several researchers

Summary

Introduction

Fractional differential equations arise in many engineering and scientific disciplines such as physics, aerodynamics, polymer rheology, regular variations in thermodynamics, biophysics, blood flow phenomena, electrical circuits, biology, etc. Non-local boundary value problems of nonlinear fractional order differential equations have recently been investigated by several researchers. The domain of study ranges from the theoretical aspects to the analytic and numerical methods for fractional differential equations. Agarwal et al [4] discussed the existence of solutions for a boundary value problem of integro-differential equations of fractional order. With non-local three-point boundary conditions D δ x (t) = 0, D δ+1 x (t) = 0, D δ x (1) − D δ x (η ) = a, where 0 < δ ≤ 1, α − δ > 3, 0 < β < 1, 0 < η < 1. Motivated by the works mentioned, in this paper, we investigate the existence and uniqueness of solutions for the non-linear fractional integro-differential equation c.

Preliminaries

The Main Results

Conclusions