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Fractional Calculus for Non-Discrete Signed Measures

Author

Listed:
  • Vassili N. Kolokoltsov

    (Faculty of Computation Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center of Fundamental and Applied Mathematics, 119234 Moscow, Russia)

  • Elina L. Shishkina

    (Department of Mathematical and Applied Analysis, Voronezh State University, 394018 Voronezh, Russia
    Department of Applied Mathematics and Computer Modeling, Belgorod State National Research University (BelGU), 308015 Belgorod, Russia)

Abstract

In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann–Stieltjes integral with a measure having a countable number of extrema. This article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives.

Suggested Citation

  • Vassili N. Kolokoltsov & Elina L. Shishkina, 2024. "Fractional Calculus for Non-Discrete Signed Measures," Mathematics, MDPI, vol. 12(18), pages 1-12, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2804-:d:1475425
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    References listed on IDEAS

    as
    1. Mohammed Al-Refai & Yuri Luchko, 2023. "The General Fractional Integrals and Derivatives on a Finite Interval," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
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