IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i3p626-d1047429.html
   My bibliography  Save this article

On Stability Criteria Induced by the Resolvent Kernel for a Fractional Neutral Linear System with Distributed Delays

Author

Listed:
  • Ekaterina Madamlieva

    (Department of Mathematical Analysis and Differential Equations, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 1756 Sofia, Bulgaria)

  • Marian Milev

    (Department of Mathematics and Physics, University of Food Technology, 4000 Plovdiv, Bulgaria)

  • Tsvetana Stoyanova

    (Department of Management and Administration, University of National and World Economy, 1700 Sofia, Bulgaria)

Abstract

We consider an initial problem (IP) for a linear neutral system with distributed delays and derivatives in Caputo’s sense of incommensurate order, with different kinds of initial functions. In the case when the initial functions are with bounded variation, it is proven that this IP has a unique solution. The Krasnoselskii’s fixed point theorem, a very appropriate tool, is used to prove the existence of solutions in the case of the neutral systems. As a corollary of this result, we obtain the existence and uniqueness of a fundamental matrix for the homogeneous system. In the general case, without additional assumptions of boundedness type, it is established that the existence and uniqueness of a fundamental matrix lead existence and uniqueness of a resolvent kernel and vice versa. Furthermore, an explicit formula describing the relationship between the fundamental matrix and the resolvent kernel is proven in the general case too. On the base of the existence and uniqueness of a resolvent kernel, necessary and sufficient conditions for the stability of the zero solution of the homogeneous system are established. Finally, it is considered a well-known economics model to describe the dynamics of the wealth of nations and comment on the possibilities of application of the obtained results for the considered systems, which include as partial case the considered model.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:626-:d:1047429
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/3/626/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/3/626/
    Download Restriction: no
    ---><---

    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. A. I. Ahmed & T. A. Al-Ahmary & Sagheer Abbas, 2022. "Fractional-Order Chelyshkov Collocation Method for Solving Systems of Fractional Differential Equations," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-17, April.
    4. Hai Zhang & Jinde Cao & Wei Jiang, 2013. "General Solution of Linear Fractional Neutral Differential Difference Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-7, June.
    5. 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.
    6. Ekaterina Madamlieva & Hristo Kiskinov & Milena Petkova & Andrey Zahariev, 2022. "On the Preservation with Respect to Nonlinear Perturbations of the Stability Property for Nonautonomous Linear Neutral Fractional Systems with Distributed Delays," Mathematics, MDPI, vol. 10(15), pages 1-20, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Aydin, Mustafa & Mahmudov, Nazim I., 2022. "On a study for the neutral Caputo fractional multi-delayed differential equations with noncommutative coefficient matrices," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    3. 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.
    4. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:626-:d:1047429. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.