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Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

Author

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  • Hossein Fazli

    (State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, China)

  • HongGuang Sun

    (State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, China)

  • Juan J. Nieto

    (Department of Statistics, Mathematical Analysis and Optimization, Institute of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain)

Abstract

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.

Suggested Citation

  • Hossein Fazli & HongGuang Sun & Juan J. Nieto, 2020. "Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited," Mathematics, MDPI, vol. 8(5), pages 1-10, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:743-:d:355267
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    References listed on IDEAS

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    1. Sun, HongGuang & Chen, Wen & Chen, YangQuan, 2009. "Variable-order fractional differential operators in anomalous diffusion modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4586-4592.
    2. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    3. Jean-Philippe Bouchaud & Rama Cont, 1998. "A Langevin approach to stock market fluctuations and crashes," Science & Finance (CFM) working paper archive 500027, Science & Finance, Capital Fund Management.
    4. Hu, D.L. & Chen, W. & Liang, Y.J., 2019. "Inverse Mittag-Leffler stability of structural derivative nonlinear dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 304-308.
    5. Fazli, Hossein & Nieto, Juan J., 2018. "Fractional Langevin equation with anti-periodic boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 332-337.
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    1. B. Radhakrishnan & T. Sathya, 2022. "Controllability of Hilfer Fractional Langevin Dynamical System with Impulse in an Abstract Weighted Space," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 265-281, October.

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