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p -Moment Mittag–Leffler Stability of Riemann–Liouville Fractional Differential Equations with Random Impulses

Author

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  • Ravi Agarwal

    (Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
    Florida Institute of Technology, Distinguished University Professor of Mathematics, Melbourne, FL 32901, USA)

  • Snezhana Hristova

    (Department of Applied Mathematics and Modeling, University of Plovdiv “Paisii Hilendarski”, 4000 Plovdiv, Bulgaria)

  • Donal O’Regan

    (School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland)

  • Peter Kopanov

    (Department of Applied Mathematics and Modeling, University of Plovdiv “Paisii Hilendarski”, 4000 Plovdiv, Bulgaria)

Abstract

Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the properties of the solutions. We study the case when the time between two consecutive moments of impulses is exponentially distributed. In connection with the application of the Riemann–Liouville fractional derivative in the equation, we define in an appropriate way both the initial condition and the impulsive conditions. We consider the case when the lower limit of the Riemann–Liouville fractional derivative is fixed at the initial time. We define the so called p -moment Mittag–Leffler stability in time of the model. In the case of integer order derivative the introduced type of stability reduces to the p –moment exponential stability. Sufficient conditions for p –moment Mittag–Leffler stability in time are obtained. The argument is based on Lyapunov functions with the help of the defined fractional Dini derivative. The main contributions of the suggested model is connected with the implementation of impulses occurring at random times and the application of the Riemann–Liouville fractional derivative of order between 0 and 1. For this model the p -moment Mittag–Leffler stability in time of the model is defined and studied by Lyapunov functions once one defines in an appropriate way their Dini fractional derivative.

Suggested Citation

  • Ravi Agarwal & Snezhana Hristova & Donal O’Regan & Peter Kopanov, 2020. "p -Moment Mittag–Leffler Stability of Riemann–Liouville Fractional Differential Equations with Random Impulses," Mathematics, MDPI, vol. 8(8), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1379-:d:400083
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    References listed on IDEAS

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    1. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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