Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 327--337 | DOI:10.5890/DNC.2020.06.012

A. Anguraj$^{1}$, K. Ramkumar$^{1}$, E. M. Elsayed$^{2}$,$^{3}$

$^{1}$ Department of Mathematics, PSG College of Arts & Science, Coimbatore, 641 014, India

$^{2}$ Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

$^{3}$ Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Abstract

This article presents the result on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations driven by a fractional Brownian motion. The results are obtained by using the method of successive approximation and Bihari’s inequality.

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. and Informatics, 28, 739-751.

2. [2]	Anguraj, A. and Banupriya, K. (2019), Existence, uniqueness and stability results for impulsive neutral stochastic functional differential equations with infinite delay and Poisson jumps, Discontinuity, Nonlinearity, and Complexity, 8(1), 1-11.

3. [3]	Ren, Y. and Xia, N. (2009), Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput, 210(1), 72-79.

4. [4]	Anguraj, A. and Vinodkumar, A. (2009), Existence, uniqueness and stability results of impulsive stochastic partial neutral functional differential equations with infinite delays, Electron. J. Qual. Theory. Differ. Equ, 67, 1-13.

5. [5]	Sakthivel, R. and Luo, J. (2009), Asymptotic stability of impulsive stochastic differential equations with infinite delays, J. Math. Anal. Appl, 356, 1-6.

6. [6]	Cui, J. and Yan, L. (2012), Successive approximation of neutral stochastic evolution equations with infinite delay and poisson jumps, Appl. Math. Comut, 218, 6776-6784.

7. [7]	Sun, M. and Xu, M. (2017), Exponential stability and interval stability of a class of stochastic hybrid systems driven by both Brownian motion and poisson jumps, Physica A, 487, 58-73.

8. [8]	Jing Cui, Litan Yan and Xicho sun. (2011), Exponential stability for neutral stochastic partial functional differential equations with delays and poisson jumps, Stat. Pro. Letters, 81, 1970-1977.

9. [9]	Mahmudov, N.I. (2006), Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stoch Anal Appl, 24, 79-95.

10. [10]	Kolmogorov, A.N. (1940), The Wiener spiral and some other interesting curves in Hilbert space, Dokl Akad Nauk SSSR, 26, 115-18.

11. [11]	Mandelbrot, B .B. and Ness, J.W.V. (1968), Fractional Brownian motions, Fractional noises and applications, SIAM Rev, 10, 422-37.

12. [12]	Boufoussi, B. and Hajji, S. (2011), Functional differential equations driven by a fractional Brownian motion, Comput Math Appl, 62, 746-54.

13. [13]	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, 1549-58.

14. [14]	Tindel, S. Tudor, C.A. and Viens, F. (2003), Stochastic evolution equations with fractional Brownian motion, Probab Theory Relat Fields, 127, 186-204.

15. [15]	Dung, N.T. (2014), Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying-time delays, J Korean Stat Soc, 43, 599-608.

16. [16]	Nualart, D. (2006), The Mallivavin calculus and related topics, 2nd ed. Berlin: Springer-verlag.

17. [17]	Pazy, A. (1983), Semigroup of linear operators and application to partial differential equations, Springer Verlag: New York.

18. [18]	Caraballo, T. and Liu, K. (1999), Exponential stability of mild solution of stochastic partial differential equations with delays, Stoch. Anal. Appl, 17(6), 743-763.

19. [19]	Boufossi, B. and Hajji, S. (2010), Successive approximation of neutral functional stochastic differential equations with jumps, Stoch. Probab. Lett, 80, 324-332.

20. [20]	Boufossi, B. and Hajji, S. (2010), Successive approximation of neutral functional stochastic differential equations with jumps, Stoch. Probab. Lett, 80, 324-332.

21. [21]	Bihari, I. (1956), A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta. Math. Acad. Sci. Hungar, 7(1), 71-94.

22. [22]	Da Prato, G. and Zabczyk, J. (1992), Stochastic equations in infinite dimensions, Cambridge University Press: Cambridge.