, Volume 15, Issue 2, pp 304-313
Date: 18 Mar 2012

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

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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 (firstorder) 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.

Dedicated to the memory of my teacher, Professor Dr. Helmut Braß