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.
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29 December 2017
An Erratum to this paper has been published: https://doi.org/10.1515/fca-2017-0082
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Dedicated to the memory of my teacher, Professor Dr. Helmut Braß
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Diethelm, K. The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus. fcaa 15, 304–313 (2012). https://doi.org/10.2478/s13540-012-0022-3
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DOI: https://doi.org/10.2478/s13540-012-0022-3