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Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space

Author

Listed:
  • Ahmed Salem

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia)

  • Kholoud N. Alharbi

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
    Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah 51411, Saudi Arabia)

  • Hashim M. Alshehri

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia)

Abstract

In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order 1 < α ≤ 2 and the parameter 0 < ρ < 1 are used to establish our model. The ρ -Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation.

Suggested Citation

  • Ahmed Salem & Kholoud N. Alharbi & Hashim M. Alshehri, 2022. "Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1332-:d:795815
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    References listed on IDEAS

    as
    1. Ahmed Salem & Noorah Mshary, 2020. "On the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation," Advances in Mathematical Physics, Hindawi, vol. 2020, pages 1-11, October.
    2. Bashir Ahmad & Madeaha Alghanmi & Ahmed Alsaedi & Hari M. Srivastava & Sotiris K. Ntouyas, 2019. "The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral," Mathematics, MDPI, vol. 7(6), pages 1-10, June.
    3. Ahmed Salem & Aeshah Al-Dosari & Cristiana J. Silva, 2021. "A Countable System of Fractional Inclusions with Periodic, Almost, and Antiperiodic Boundary Conditions," Complexity, Hindawi, vol. 2021, pages 1-10, June.
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    Citations

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    Cited by:

    1. Ahmed Salem & Hunida Malaikah & Eid Sayed Kamel, 2023. "An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space," Mathematics, MDPI, vol. 11(6), pages 1-17, March.
    2. Ahmed Salem & Lamya Almaghamsi, 2023. "Solvability of Sequential Fractional Differential Equation at Resonance," Mathematics, MDPI, vol. 11(4), pages 1-18, February.
    3. Ahmed Salem & Rawia Babusail, 2022. "Finite-Time Stability in Nonhomogeneous Delay Differential Equations of Fractional Hilfer Type," Mathematics, MDPI, vol. 10(9), pages 1-14, May.
    4. Daliang Zhao, 2023. "Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay," Mathematics, MDPI, vol. 11(19), pages 1-19, September.

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    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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