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A Comment on a Controversial Issue: A Generalized Fractional Derivative Cannot Have a Regular Kernel

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Abstract

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

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  1. Emeritus, Polish Academy of Sciences and ul. Bitwy Warszawskiej 1920r 14/52, 02-366, Warszawa, Poland

    Andrzej Hanyga

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  1. Andrzej Hanyga
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Hanyga, A. A Comment on a Controversial Issue: A Generalized Fractional Derivative Cannot Have a Regular Kernel. Fract Calc Appl Anal 23, 211–223 (2020). https://doi.org/10.1515/fca-2020-0008

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  • DOI: https://doi.org/10.1515/fca-2020-0008

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