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

Author

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. 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.
    5. 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.
    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.
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