Afrika Matematika

, Volume 29, Issue 1–2, pp 233–247 | Cite as

Exponential stability of impulsive neutral stochastic functional differential equation driven by fractional Brownian motion and Poisson point processes



In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven simultaneously by a fractional Brownian motion and a Poisson point processes in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.


Mild solution Impulsive neutral stochastic differential equations Fractional powers of closed operators Fractional Brownian motion Poisson point processes 

Mathematics Subject Classification

60H15 60G22 60J75 


  1. 1.
    Benchohra, M., Ouahab, A.: Impulsive neutral functional differential equations with variable times. Nonlinear Anal. 55(6), 679–693 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic calculus for Fractional brownian motion and Application. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Boufoussi, B., Hajji, S.: Neutral stochastic functional differential equation driven by a fractional Brownian motion in a Hilbert space. Stat. Probab. Lett. 82, 1549–1558 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boufoussi, B., Hajji, S., Lakhel, E.: Functional differential equations in Hilbert spaces driven by a fractional Brownian motion. Afrika Matematika 23(2), 173–194 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Caraballo, T., Garrido-Atienza, M.J., Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74, 3671–3684 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Caraballo, T., Diop, M.A., Ndiaye, A.A.: Asymptotic behavior of neutral stochastic partial functional integro-differential equations driven by a fractional Brownian motion. J. Nonlinear Sci. Appl. 7, 407–421 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dung, T.N.: Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space. Stochastics 87(1), 142–159 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Feyel, D., De la Pradelle, A.: On fractional Brownian processes. Potential Anal. 10, 273–288 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Goldstein, J.A.: Semigroups of linear operators and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1985)Google Scholar
  10. 10.
    Hernandez, E., Keck, D. N., McKibben, M. A.: On a class of measure-dependent stochastic evolution equations driven by fBm. J. Appl Math Stoch Anal., Art ID 69747, p. 26 (2007)Google Scholar
  11. 11.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Nort-Holland/ Kodansha, Amsterdam/New York (1989)MATHGoogle Scholar
  12. 12.
    Lakhel, E.E., McKibben, M.A.: Controllability of Impulsive Neutral Stochastic Functional Integro-Differential Equations Driven by Fractional Brownian Motion. Chapter 8: McKibben, M.A., Webster, M. (eds.) Brownian Motion: Elements, Dynamics, and Applications, pp. 131–148. Nova Science Publishers, New York (2015)Google Scholar
  13. 13.
    Lakhel, E., Hajji, S.: Existence and uniqueness of mild solutions to neutral SFDEs driven by a fractional Brownian motion with non-Lipschitz coefficients. J. Numer. Math. Stoch. 7(1), 14–29 (2015)MathSciNetMATHGoogle Scholar
  14. 14.
    Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations, Series in Modern Appl. Math., vol. 6. World Scientific Publ., Teaneck (1989)Google Scholar
  15. 15.
    Mandelbrot, B., Ness, V.: Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)MATHGoogle Scholar
  17. 17.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefMATHGoogle Scholar
  18. 18.
    Ren, Y., Hu, L., Sakthivel, R.: Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay. J. Comput. Appl. Math. 235(8), 2603–2614 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ren, Y., Cheng, X., Sakthivel, R.: Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm. Appl. Math. Comput. 247, 205–212 (2014)MathSciNetMATHGoogle Scholar
  20. 20.
    Xu, D., Yang, Z., Yang, Z.: Exponential stability of nonlinear impulsive neutral differential equations with delays. Nonlinear Anal. 67(5), 1426–1439 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Brahim Boufoussi
    • 1
  • Salah Hajji
    • 2
  • El Hassan Lakhel
    • 3
  1. 1.Department of Mathematics, Faculty of Sciences SemlaliaCadi Ayyad UniversityMarrakeshMorocco
  2. 2.Department of MathematicsRegional Center for the Professions of Education and TrainingMarrakeshMorocco
  3. 3.National School of Applied SciencesCadi Ayyad UniversitySafiMorocco

Personalised recommendations