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Non-instantaneous Impulses on Random Time in Differential Equations with Ordinary/Fractional Derivatives

  • Ravi Agarwal
  • Snezhana Hristova
  • Donal O’Regan
Chapter

Abstract

In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions of differential equations significantly. The study combines methods of deterministic differential equations and probability theory. Note differential equations with random impulsive moments differs from the study of stochastic differential equations with jumps (see, for example, [105, 127, 128, 129, 130, 131, 134]). We will define and study nonlinear differential equations subject to impulses starting abruptly at some random points and their action continue on intervals with a given finite length. Inspired by queuing theory and the distribution for the waiting time, we study the cases of exponentially distributed random variables, Erlang distributed random variables and Gamma distributed random variables between two consecutive moments of impulses and the intervals where the impulses act are with a constant length.

Bibliography

  1. 25.
    A. Anguraj, A. Vinodkumar, Existence, uniqueness and stability results of random impulsive semilinear differential systems. Nonlinear Anal. Hybrid Syst. 3, 475–483 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 30.
    D. Baleanu, O.G. Mustafa, On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 35.
    J.M. Borwein, O.-Y. Chan, Uniform bounds for the complementary incomplete gamma function. Math. Inequal. Appl. 12(1), 115–121 (2009)zbMATHMathSciNetGoogle Scholar
  4. 65.
    X.L. Hu, Two new inequalities for Gaussian and Gamma distributions. J. Math. Inequal. 4(4), 609–613 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 71.
    A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis (Dover, New York, 1970)zbMATHGoogle Scholar
  6. 82.
    V. Lakshmikantham, S. Leela, J.V. Devi, Theory of Fractional Dynamical Systems (Cambridge Scientific Publishers, 2009)Google Scholar
  7. 105.
    J.M. Sanz-Serna, A.M. Stuart, Ergodicity of dissipative differential equations subject to random impulses. J. Differ. Equ. 155, 262–284 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 125.
    J.R. Wang, M. Feckan, Y. Zhou, Random noninstantaneous impulsive models for studying periodic evolution processes in pharmacotherapy, in Mathematical Modeling and Applications in Nonlinear Dynamics. Nonlinear Systems and Complexity, vol. 14 (Springer, Cham, 2016), pp. 87–107Google Scholar
  9. 127.
    H. Wu, J. Sun, p-moment stability of stochastic differential equations with impulsive jump and Markovian switching. Automatica 42, 1753–1759 (2006)Google Scholar
  10. 131.
    Z. Yan, X. Jia, Existence and controllability results for a new class of impulsive stochastic partial integro-differential inclusions with state-dependent delay. Asian J. Control 19(3), 1–26 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 134.
    J. Yang, S. Zhong, W. Luo, Mean square stability analysis of impulsive stochastic differential equations with delays. J. Comput. Appl. Math. 216(2), 474–483 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ravi Agarwal
    • 1
  • Snezhana Hristova
    • 2
  • Donal O’Regan
    • 3
  1. 1.Department of MathematicsTexas A&M University—KingsvilleKingsvilleUSA
  2. 2.Department of Applied MathematicsPlovdiv UniversityPlovdivBulgaria
  3. 3.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

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