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On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions

Author

Listed:
  • Usman Riaz

    (Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa 25000, Pakistan)

  • Akbar Zada

    (Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa 25000, Pakistan)

  • Zeeshan Ali

    (School of Engineering, Monash University Malaysia, Selangor 47500, Malaysia)

  • Ioan-Lucian Popa

    (Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia 510009, Romania)

  • Shahram Rezapour

    (Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404, Taiwan)

  • Sina Etemad

    (Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran)

Abstract

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions.

Suggested Citation

  • Usman Riaz & Akbar Zada & Zeeshan Ali & Ioan-Lucian Popa & Shahram Rezapour & Sina Etemad, 2021. "On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions," Mathematics, MDPI, vol. 9(11), pages 1-22, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1205-:d:562820
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    References listed on IDEAS

    as
    1. Manzoor Ahmad & Jiqiang Jiang & Akbar Zada & Syed Omar Shah & Jiafa Xu, 2020. "Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative," Complexity, Hindawi, vol. 2020, pages 1-11, April.
    2. Usman Riaz & Akbar Zada & Zeeshan Ali & Manzoor Ahmad & Jiafa Xu & Zhengqing Fu, 2019. "Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-20, June.
    3. 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).
    4. Fathalla A. Rihan, 2013. "Numerical Modeling of Fractional-Order Biological Systems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, August.
    5. Zada, Akbar & Ali, Wajid & Park, Choonkil, 2019. "Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 60-65.
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