RESEARCH PAPER
THE MEAN VALUE THEOREMS
AND A NAGUMO-TYPE UNIQUENESS THEOREM
FOR CAPUTO’S FRACTIONAL CALCULUS
Kai Diethelm
1,2
Abstract
Dedicated to the memory of my teacher, Professor Dr. Helmut Braß
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.
MSC 2010 : Primary 26A33; Secondary 34A08, 34A12
Key Words and Phrases: Caputo derivative, mean value theorem of
differential calculus, mean value theorem of integral calculus, fractional
differential equation, Nagumo’s theorem, uniqueness of solutions
1. Introduction
In this short note we shall demonstrate that two well known results connected to classical analysis, namely the mean value theorems of differential
and of integral calculus, can be extended to fractional calculus, i.e. they
c 2012 Diogenes Co., Sofia
°
pp. 304–313 , DOI: 10.2478/s13540-012-0022-3