On the Nonlinear Neutral Conformable Fractional Integral-Differential Equation

Rui Li,Wei Jiang,Sen Wang,Jiale Sheng

doi:10.4236/am.2020.1110069

Abstract

In this paper, we investigate the nonlinear neutral fractional integral-differential equation involving conformable fractional derivative and integral. First of all, we give the form of the solution by lemma. Furthermore, existence results for the solution and sufficient conditions for uniqueness solution are given by the Leray-Schauder nonlinear alternative and Banach contraction mapping principle. Finally, an example is provided to show the application of results.

Highlights

The theory of fractional calculus has played a major role in control theory, fluid dynamics, biological systems, economics and other fields [1] [2] [3]
Compared with Riemann-Liouville and Caputo type fractional derivatives, the conformable fractional derivative satisfies the Leibniz rule and chain rule, and can be converted to classical derivative [5]. This is of great help to study fractional differential equations
In the past few years, the conformable fractional derivative has been used in the field of fractional newtonian mechanics, heat equation, biology and so on, and the results are abundant

Summary

Introduction

The theory of fractional calculus has played a major role in control theory, fluid dynamics, biological systems, economics and other fields [1] [2] [3]. Li, Liu and Jiang gave sufficient conditions of the existence of positive solutions for a class of nonlinear fractional differential equations with caputo derivative [9]. In 2009, the existence and uniqueness of solutions of the Caputo fractional neutral differential equations with unbounded delays were discussed [13]. In 2011, Li and Zhang considered the Caputo fractional neutral integral-differential equations with unbounded delay, which used the fixed point theorem to study the existence of mild solutions of equations [14]. Abbas gave the existence of solutions and uniqueness of solution for fractional neutral integro-differential equations by the Hadamard fractional derivative of order α ∈ (0,1) and the Riemann-Liouville integral [16].

Preliminaries

Main Results

An Illustrative Example

Conclusion