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On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems

Author

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  • Luís P. Castro

    (CIDMA—Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal
    These authors contributed equally to this work.)

  • Anabela S. Silva

    (CIDMA—Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal
    These authors contributed equally to this work.)

Abstract

This article deals with a class of nonlinear fractional differential equations, with initial conditions, involving the Riemann–Liouville fractional derivative of order α ∈ ( 1 , 2 ) . The main objectives are to obtain conditions for the existence and uniqueness of solutions (within appropriate spaces), and to analyze the stabilities of Ulam–Hyers and Ulam–Hyers–Rassias types. In fact, different conditions for the existence and uniqueness of solutions are obtained based on the analysis of an associated class of fractional integral equations and distinct fixed-point arguments. Additionally, using a Bielecki-type metric and some additional contractive arguments, conditions are also obtained to guarantee Ulam–Hyers and Ulam–Hyers–Rassias stabilities for the problems under analysis. Examples are also included to illustrate the theory.

Suggested Citation

  • Luís P. Castro & Anabela S. Silva, 2023. "On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems," Mathematics, MDPI, vol. 11(2), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:297-:d:1027122
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    References listed on IDEAS

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    1. Meysam Alvan & Rahmat Darzi & Amin Mahmoodi, 2016. "Existence Results for a New Class of Boundary Value Problems of Nonlinear Fractional Differential Equations," Mathematics, MDPI, vol. 4(1), pages 1-10, March.
    2. Sousa, J. Vanterler da C. & Oliveira, D.S. & Capelas de Oliveira, E., 2021. "A note on the mild solutions of Hilfer impulsive fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    3. Kucche, Kishor D. & Mali, Ashwini D. & Fernandez, Arran & Fahad, Hafiz Muhammad, 2022. "On tempered Hilfer fractional derivatives with respect to functions and the associated fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
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    Cited by:

    1. Bukhari, Ayaz Hussain & Raja, Muhammad Asif Zahoor & Alquhayz, Hani & Abdalla, Manal Z.M. & Alhagyan, Mohammed & Gargouri, Ameni & Shoaib, Muhammad, 2023. "Design of intelligent hybrid NAR-GRNN paradigm for fractional order VDP chaotic system in cardiac pacemaker with relaxation oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 175(P2).

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