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On the Existence and Uniqueness of Solution to Fractional‐Order Langevin Equation

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  • Ahmed Salem
  • Noorah Mshary

Abstract

In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional‐order Langevin equation involving two distinct fractional orders with unprecedented conditions (three‐point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional‐order Langevin equation. As a result of employing the so‐called Krasnoselskii and Leray‐Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.

Suggested Citation

  • Ahmed Salem & Noorah Mshary, 2020. "On the Existence and Uniqueness of Solution to Fractional‐Order Langevin Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2020(1).
    • Handle: RePEc:wly:jnlamp:v:2020:y:2020:i:1:n:8890575
    DOI: 10.1155/2020/8890575
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    References listed on IDEAS

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    1. Ahmed Salem & Faris Alzahrani & Lamya Almaghamsi, 2019. "Fractional Langevin Equations with Nonlocal Integral Boundary Conditions," Mathematics, MDPI, vol. 7(5), pages 1-10, May.
    2. Ahmed Salem & Faris Alzahrani & Aeshah Al-Dosari, 2020. "Attainability to Solve Fractional Differential Inclusion on the Half Line at Resonance," Complexity, Hindawi, vol. 2020, pages 1-13, July.
    3. 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.
    4. Fazli, Hossein & Nieto, Juan J., 2018. "Fractional Langevin equation with anti-periodic boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 332-337.
    5. Ahmadian, A. & Salahshour, S. & Ali-Akbari, M. & Ismail, F. & Baleanu, D., 2017. "A novel approach to approximate fractional derivative with uncertain conditions," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 68-76.
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