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

Abstract

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

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