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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Impulsive-Integral Inequalities for Attracting and Quasi-Invariant Sets of Neutral Stochastic Integrodifferential Equations with Impulsive Effects

Journal of Applied Nonlinear Dynamics 9(3) (2020) 513--523 | DOI:10.5890/JAND.2020.09.012

K. Ramkumar, A. Anguraj

Department of Mathematics, PSG College of Arts and Science, Coimbatore, 641014, India

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Abstract

In this article, we investigate a class of neutral stochastic integrodifferential equations with impulsive effects. The results are obtained by using the new integral inequalities, the attracting and quasi-invariant sets combined with theories of resolvent operators. Moreover, exponential stability of the mild solution is established with sufficient conditions. An example is provided to illustrate the results of this work.

References

  1. [1]  Anguraj, A. and Vinodkumar, A. (2010), Existence uniqueness and stability of impulsive stochastic partial neutral functional differential equations with infinite delays, J. Appl. Math. Inf, 28(3-4), 739-751.
  2. [2]  Boufoussi, B. and Hajji, S. (2012), Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Stat. Probab. Lett, 82(8), 1549-1558.
  3. [3]  Caraballo, T. and Liu, K. (1999), Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl, 17, 743-763.
  4. [4]  Anguraj, A. and Ravikumar, K. (2017), On existence results of non-linear stochastic partial integrodifferential equations, Int. J. Pure. Appl. Math, 117(11), 417-427.
  5. [5]  Huan, D. and Agarwal, A. (2014), Global attracting and quasi-invariant sets for stochastic Volterra-Levin equations with jumps, Dyn. Contin. Discrete. Impuls. Syst. Ser. A, Math. Anal, 21, 343-353.
  6. [6]  Xu, L. and Long, S. (2012), Attracting and quasi-invariant sets of non-autonomous neutral networks with delays, Neurocomputing, 77(1), 222-228.
  7. [7]  Cheban, D. and Mammana, C. (2004), Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations, Nonlinear Anal, 56, 465-484.
  8. [8]  Xu, D.Y. (2003), Invariant and attracting sets of Volterra difference equations with delays, Comput. Math. Appl, 45, 1311-1317.
  9. [9]  Kolmanovskii, V.B. and Nosov, V.R. (1986), Stability of Functional Differential Equations, Orlando, FL: Academic Press.
  10. [10]  Xu, D.Y. and Zhao, H.Y. (2002), Invariant set and attractivity of nonlinear differential equations with delays, Appl. Math. Lett, 15, 321-325.
  11. [11]  Zhao, H.Y. (2003), Invariant set and attractor of nonautonomous functional differential systems, J. Math. Anal. Appl, 282, 437-443.
  12. [12]  Wang, L.S. and Xu, D.Y. (2003), Asymptotic behavior of a class of reaction-diffusion equations with delays, J. Math. Anal. Appl, 281, 439-453.
  13. [13]  Duan, J.Q., Lu, K.N., and SchmalfuB, B. (2003), Invariant manifolds for stochastic partial differential equations, Ann. Probab, 31, 2109-2135.
  14. [14]  Lu, K.N. and SchmalfuB, B. (2007), Invariant manifolds for stochastic wave equations, J. Diff. Equations, 236, 460-492.
  15. [15]  Xu, D.Y. and Yang, Z.C. (2007), Attracting and invariant sets for a class of impulsive functional differential equations, J. Math. Anal. Appl, 329, 1036-1044.
  16. [16]  Long, S., Teng, L. and Xu, D. (2012), Global attracting set and stability of stochastic neutral partial functional differential equations with impulses, Stat. Probab. Lett, 82, 1699-1709.
  17. [17]  Arthi, G., Park, H.J., and Jung, H.Y. (2016), Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fBm, Commun. Nonlinear Sxi. Number. Simulat, 32, 145-157.
  18. [18]  Grimmer, R. (1982), Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc, 273(1), 333-349.
  19. [19]  Li, Z. (2016), Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm, Neurocomputing, 177, 620-627.
  20. [20]  Da prato, G. and Zabczyk, J. (2014), Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
  21. [21]  Xu, L.G. and Xu, D.Y. (2009), P-attracting and P-invariant sets for a class of impulsive stochastic functional differential equations, Comput. Math. Appl, 57, 54-61.
  22. [22]  Tindel, S., Tudor, C., and Viens, F. (2003), Stochastic evolution equations with fractional Brownian motion, Probab. Theory Relat. Fields, 127(4), 186-204.
  23. [23]  Mao, X. (1997), Stochastic Differential Equations and Applications, Horwood Publishing, Chichester.
  24. [24]  Caraballo, T., Garrido-Atienza, M.J., and Taniguchi, T. (2011), The existence and exponential behavior of solutions to stochastic delay evolution equations with a fBm, Nonlinear Anal, 74(11), 3671-3684.

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