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On Dynamic Systems in the Frame of Singular Function Dependent Kernel Fractional Derivatives

Author

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  • Thabet Abdeljawad

    (Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia
    Department of Medical Research, China Medical University, Taichung 40402, Taiwan)

  • Fadila Madjidi

    (Department of Mathematics, University of Mohamed Boudiaf-PB 166, M’sila 28000, Algeria)

  • Fahd Jarad

    (Department of Mathematics, Çankaya University, 06790 Etimesgut, Ankara, Turkey)

  • Ndolane Sene

    (Laboratoire Lmdan, Dèpartement de Mathèmatiques de la Dècision, Universitè Cheikh Anta Diop de Dakar, Facultè des Sciences Economiques et Gestion, Dakar Fann BP 5683, Senegal)

Abstract

In this article, we discuss the existence and uniqueness theorem for differential equations in the frame of Caputo fractional derivatives with a singular function dependent kernel. We discuss the Mittag-Leffler bounds of these solutions. Using successive approximation, we find a formula for the solution of a special case. Then, using a modified Laplace transform and the Lyapunov direct method, we prove the Mittag-Leffler stability of the considered system.

Suggested Citation

  • Thabet Abdeljawad & Fadila Madjidi & Fahd Jarad & Ndolane Sene, 2019. "On Dynamic Systems in the Frame of Singular Function Dependent Kernel Fractional Derivatives," Mathematics, MDPI, vol. 7(10), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:946-:d:275532
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    References listed on IDEAS

    as
    1. Shaher Momani & Samir Hadid, 2004. "Lyapunov stability solutions of fractional integrodifferential equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-5, January.
    2. Fahd Jarad & Thabet Abdeljawad & Dumitru Baleanu & Kübra Biçen, 2012. "On the Stability of Some Discrete Fractional Nonautonomous Systems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-9, February.
    3. D. Baleanu & S. J. Sadati & R. Ghaderi & A. Ranjbar & T. Abdeljawad (Maraaba) & F. Jarad, 2010. "Razumikhin Stability Theorem for Fractional Systems with Delay," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-9, June.
    4. S. J. Sadati & D. Baleanu & A. Ranjbar & R. Ghaderi & T. Abdeljawad (Maraaba), 2010. "Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-7, August.
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    Cited by:

    1. Ahmed Salem & Lamya Almaghamsi, 2023. "Solvability of Sequential Fractional Differential Equation at Resonance," Mathematics, MDPI, vol. 11(4), pages 1-18, February.

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