Advertisement

Existence results systems coupled impulsive neutral stochastic functional differential equations with the measure of noncompactness

  • A. BoudaouiEmail author
  • T. Blouhi
Article

Abstract

This paper is devoted to study the existence of solutions for a class of mild solutions for a class of impulsive neutral stochastic functional differential equations driven fractional Brownian motion (fBm) with noncompact semigroup in Hilbert spaces. The new results are obtained by using the Hausdorff measure of noncompactness. The arguments are based upon Mönch’s fixed point theorem. Finally, an example is provided to illustrate the developed theory.

Keywords

Impulsive neutral stochastic functional differential equations Hausdorff measure of noncompactness Mild solution Fractional Generalized Banach space Mönch fixed point Fractional Brownian motion 

Mathematics Subject Classification

35A01 34A37 60H15 60H20 

Notes

References

  1. 1.
    Arthi, G., Park, J.H., Jung, H.Y.: Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion. Commun. Nonlinear Sci. Numer. Simul. 32, 145–157 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Banas, J., Goebel, K.: Measure of Noncompactness in Banach Space. Lecture Notes in Pure and Applied Mathematics. Dekker, New York (1980)zbMATHGoogle Scholar
  3. 3.
    Bainov, D.D., Simeonov, P.S.: Systems with Impulsive Effect. Horwood, Chichester (1989)zbMATHGoogle Scholar
  4. 4.
    Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Boudaoui, A., Caraballo, T., Ouahab, A.: Existence of mild solutions to stochastic delay evolution equations with a fractional Brownian motion and impulses. Stoch. Anal. Appl. 33(2), 244–258 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boudaoui, A., Caraballo, T., Ouahab, A.: Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay. Appl. Anal. 95(9), 2039–2062 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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. Theory Methods Appl. 74(11), 3671–3684 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, Y.K., Anguraj, A., Mallika Arjunan, M.M.: Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Anal. Hybrid Syst. 2, 209–218 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, vol. 152. Cambridge university press, Cambridge (2014)CrossRefzbMATHGoogle Scholar
  10. 10.
    Deng, S., Shu, X.B., Mao, J.: Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point. J. Math. Anal. Appl. 467(1), 398–420 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diop, M.A., Rathinasamy, S., Ndiaye, A.A.: Neutral stochastic integrodifferential equations driven by a fractional Brownian motion with impulsive effects and time-varying delays. Mediterr. J. Math. 13(5), 2425–2442 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac 21(1), 11–41 (1978)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Henderson, J., Ouahab, A., Slimani, M.: Existence results for a semilinear system of discrete equations. Int. J. Differ. Equ. 12, 235–253 (2017)MathSciNetGoogle Scholar
  14. 14.
    Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Springer, Berlin (2006)zbMATHGoogle Scholar
  15. 15.
    Hernández, E., O’Regan, D.: Controllability of Volterra–Fredholm type systems in Banach spaces. J. Franklin Inst. 346(2), 95–101 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hernández, E., Rabello, M., Henríquez, H.R.: Existence of solutions for impulsive partial neutral functional differential equations. J. Math. Anal. Appl. 331, 1135–1158 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hu, L., Ren, Y.: Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays. Acta Appl. Math. 111(3), 303–317 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lakshmikantham, V., Simeonov, P.S.: Theory of impulsive differential equations. World scientific, Singapore (1989)CrossRefzbMATHGoogle Scholar
  20. 20.
    Li, Y., Liu, B.: Existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay. Stoch. Anal. Appl. 25(2), 397–415 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mishura, Y.: Stochastic calculus for fractional Brownian motion and related topics. Lecture Notes in Mathematics, vol. 1929. Springer, Berlin (2008)CrossRefGoogle Scholar
  23. 23.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  24. 24.
    Obukhovski, V., Zecca, P.: Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup. Nonlinear Anal. Theory Methods Appl. 70(9), 3424–3436 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)CrossRefzbMATHGoogle Scholar
  28. 28.
    Tindel, S., Tudor, C.A., Viens, F.: Stochastic evolution equations with fractional Brownian motion. Prob. Theory Relat. Fields 127(2), 186–204 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yan, Z., Lu, F.: Existence results for a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay. J. Appl. Anal. Comput. 5, 329–346 (2015)MathSciNetGoogle Scholar
  30. 30.
    Yu, X., Wang, J.: Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 22, 980–989 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of AdrarAdrarAlgeria
  2. 2.Departement of MathematicsUsto UniversityOranAlgeria

Personalised recommendations