The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus (Corrected Version)

TL;DR

This paper extends classical mean value theorems to fractional calculus using Caputo derivatives and integrals, and applies these results to establish a Nagumo-type uniqueness theorem for fractional differential equations.

Contribution

It introduces generalized mean value theorems for fractional derivatives and integrals, and proves a new uniqueness theorem for fractional initial value problems.

Findings

Generalized mean value theorem with Caputo derivatives

Generalized mean value theorem with fractional integrals

A Nagumo-type uniqueness theorem for fractional differential equations

Abstract

We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz.\ the Riemann-Liouville operator, instead of a classical (first-order) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.