On the Existence and Uniqueness of Solutions for Local Fractional Differential Equations

Hossein Jafari,Hassan Jassim,Maysaa Al Qurashi,Dumitru Baleanu

doi:10.3390/e18110420

Abstract

In this manuscript, we prove the existence and uniqueness of solutions for local fractional differential equations (LFDEs) with local fractional derivative operators (LFDOs). By using the contracting mapping theorem (CMT) and increasing and decreasing theorem (IDT), existence and uniqueness results are obtained. Some examples are presented to illustrate the validity of our results.

Highlights

Differential equations (DEs) with fractional order are generalizations of ordinary differential equations to non-integer order
The existence and uniqueness of solutions of differential equations with the Riemann-Liouville fractional derivative and the Caputo fractional derivative using the Schauder fixed point theorem, the lower and upper solution method, the contracting mapping principle and the Leray-Schauder theory have been investigated in some papers [12,13,14,15]
Theorem 1. (CMT): A contracting mapping T defined on a complete generalized Banach space (GBS) ( X, || · ||α ) has a unique fixed point (FP)

Summary

Introduction

Differential equations (DEs) with fractional order are generalizations of ordinary differential equations to non-integer order. The existence and uniqueness of solutions of differential equations with the Riemann-Liouville fractional derivative and the Caputo fractional derivative using the Schauder fixed point theorem, the lower and upper solution method, the contracting mapping principle and the Leray-Schauder theory have been investigated in some papers [12,13,14,15]. Very recently in [16], the author studied the existence and uniqueness of solutions of some classes of differential equations with local fractional derivative operators. We are interested in the existence and uniqueness of DEs with LFDOs of the form: 2α α.

Basic Definitions and Preliminaries

Main Results

Applications

Conclusions