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

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.

The original version of this paper, published in Fract. Calc. Appl. Anal. 15 (2012), pp. 304--313, unfortunately contained an error in Corollary 2.2 that was then carried forward to the later parts of the paper. This version contains the corrected form of the document