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New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

Author

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  • Osman Tunç

    (Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, Van 65080, Turkey)

Abstract

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work.

Suggested Citation

  • Osman Tunç, 2024. "New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations," Mathematics, MDPI, vol. 12(9), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1342-:d:1385011
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    References listed on IDEAS

    as
    1. Sorina Anamaria Ciplea & Nicolaie Lungu & Daniela Marian & Themistocles M. Rassias, 2022. "On Hyers-Ulam-Rassias Stability of a Volterra-Hammerstein Functional Integral Equation," Springer Optimization and Its Applications, in: Nicholas J. Daras & Themistocles M. Rassias (ed.), Approximation and Computation in Science and Engineering, pages 147-156, Springer.
    2. Liu, Kui & Wang, JinRong & Zhou, Yong & O’Regan, Donal, 2020. "Hyers–Ulam stability and existence of solutions for fractional differential equations with Mittag–Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
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