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About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays

Author

Listed:
  • Hristo Kiskinov

    (Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria)

  • Mariyan Milev

    (Department of Mathematics, Physics and Information Technologies, University of Food Technology, 4002 Plovdiv, Bulgaria)

  • Andrey Zahariev

    (Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria)

Abstract

The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well.

Suggested Citation

  • Hristo Kiskinov & Mariyan Milev & Andrey Zahariev, 2022. "About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays," Mathematics, MDPI, vol. 10(23), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4573-:d:992254
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    References listed on IDEAS

    as
    1. Hristo Kiskinov & Ekaterina Madamlieva & Magdalena Veselinova & Andrey Zahariev, 2021. "Existence of Absolutely Continuous Fundamental Matrix of Linear Fractional System with Distributed Delays," Mathematics, MDPI, vol. 9(2), pages 1-18, January.
    2. Ekaterina Madamlieva & Mihail Konstantinov & Marian Milev & Milena Petkova, 2020. "Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay," Mathematics, MDPI, vol. 8(3), pages 1-12, March.
    3. Li, Mengmeng & Wang, JinRong, 2018. "Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 254-265.
    4. Andrey Zahariev & Hristo Kiskinov, 2020. "Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation," Mathematics, MDPI, vol. 8(3), pages 1-18, March.
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    Cited by:

    1. Ekaterina Madamlieva & Marian Milev & Tsvetana Stoyanova, 2023. "On Stability Criteria Induced by the Resolvent Kernel for a Fractional Neutral Linear System with Distributed Delays," Mathematics, MDPI, vol. 11(3), pages 1-21, January.

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