IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i23p3642-d1526109.html
   My bibliography  Save this article

Equivalence Between Fractional Differential Problems and Their Corresponding Integral Forms with the Pettis Integral

Author

Listed:
  • Mieczysław Cichoń

    (Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland)

  • Wafa Shammakh

    (Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia)

  • Kinga Cichoń

    (Institute of Mathematics, Faculty of Automatic Control, Robotics and Electrical Engineering, Poznan University of Technology, Piotrowo 3A, 60-965 Poznań, Poland)

  • Hussein A. H. Salem

    (Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria University, Alexandria 5424041, Egypt)

Abstract

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent.

Suggested Citation

  • Mieczysław Cichoń & Wafa Shammakh & Kinga Cichoń & Hussein A. H. Salem, 2024. "Equivalence Between Fractional Differential Problems and Their Corresponding Integral Forms with the Pettis Integral," Mathematics, MDPI, vol. 12(23), pages 1-29, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3642-:d:1526109
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/23/3642/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/23/3642/
    Download Restriction: no
    ---><---

    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.
    2. Mieczysław Cichoń & Hussein A. H. Salem & Wafa Shammakh, 2024. "On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces," Mathematics, MDPI, vol. 12(17), pages 1-23, August.
    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. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    2. Sergei Sitnik, 2023. "Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”," Mathematics, MDPI, vol. 11(15), pages 1-7, August.
    3. Vasily E. Tarasov, 2023. "General Fractional Noether Theorem and Non-Holonomic Action Principle," Mathematics, MDPI, vol. 11(20), pages 1-35, October.
    4. Vassili N. Kolokoltsov & Elina L. Shishkina, 2024. "Fractional Calculus for Non-Discrete Signed Measures," Mathematics, MDPI, vol. 12(18), pages 1-12, September.
    5. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.

    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:12:y:2024:i:23:p:3642-:d:1526109. 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.