Discontinuity, Nonlinearity, and Complexity
Existence, Uniqueness and Stability of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays Driven by a Fractional Brownian Motion
Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 327337  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
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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]  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, 739751. 

[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), 111. 

[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), 7279. 

[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, 113. 

[5]  Sakthivel, R. and Luo, J. (2009), Asymptotic stability of impulsive stochastic differential equations with infinite delays, J. Math. Anal. Appl, 356, 16. 

[6]  Cui, J. and Yan, L. (2012), Successive approximation of neutral stochastic evolution equations with infinite delay and poisson jumps, Appl. Math. Comut, 218, 67766784. 

[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, 5873. 

[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, 19701977. 

[9]  Mahmudov, N.I. (2006), Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stoch Anal Appl, 24, 7995. 

[10]  Kolmogorov, A.N. (1940), The Wiener spiral and some other interesting curves in Hilbert space, Dokl Akad Nauk SSSR, 26, 11518. 

[11]  Mandelbrot, B .B. and Ness, J.W.V. (1968), Fractional Brownian motions, Fractional noises and applications, SIAM Rev, 10, 42237. 

[12]  Boufoussi, B. and Hajji, S. (2011), Functional differential equations driven by a fractional Brownian motion, Comput Math Appl, 62, 74654. 

[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, 154958. 

[14]  Tindel, S. Tudor, C.A. and Viens, F. (2003), Stochastic evolution equations with fractional Brownian motion, Probab Theory Relat Fields, 127, 186204. 

[15]  Dung, N.T. (2014), Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varyingtime delays, J Korean Stat Soc, 43, 599608. 

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

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

[18]  Caraballo, T. and Liu, K. (1999), Exponential stability of mild solution of stochastic partial differential equations with delays, Stoch. Anal. Appl, 17(6), 743763. 

[19]  Boufossi, B. and Hajji, S. (2010), Successive approximation of neutral functional stochastic differential equations with jumps, Stoch. Probab. Lett, 80, 324332. 

[20]  Boufossi, B. and Hajji, S. (2010), Successive approximation of neutral functional stochastic differential equations with jumps, Stoch. Probab. Lett, 80, 324332. 

[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), 7194. 

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