Discontinuity, Nonlinearity, and Complexity
Existence and Stability Results of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays and Poisson Jumps
Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 245255  DOI:10.5890/DNC.2020.06.006
A. Anguraj, K. Ravikumar
Department of Mathematics, PSG College of Arts & Science, Coimbatore, 641014, India
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Abstract
In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations with infinite delays and poisson jumps. 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 Ramkumar, K. (2017), Exponential stability of nonlinear stochastic partial integrodifferential equations, Int. J. Pure. Appl.Math, 117(13), 283292. 

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

[4]  Anguraj, A., Kanjanadevi, S., and Trujillo.Juan, J. (2017), Existence ofMild Solutions of Abstract Fractional Differential Equations with Fractional NonInstantaneous Impulsive Conditions, Discontinuity, Nonlinearity, and Complexity, 6(2), 173183. 

[5]  Taniguchi, T. (1992), Successive approximation to solutions of stochastic differential equations, J. Differ. Equ., 96, 152169. 

[6]  Suvinthra,M. and Balachandran, K. (2016), Large deviations for nonlinear ito type stochastic integrodifferential equations, Journal of Applied Nonlinear Dynamics, 6(1), 115. 

[7]  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. 

[8]  Sakthivel, R. and Luo, J. (2009), Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett, 79, 12191223. 

[9]  Bouchard, B. and Elie, R. (2008), Discrete time approximation of decoupled forwardbackward SDE with jumps, Stoch. Process. Appl, 118(1), 5375. 

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

[11]  Chen, H. (2015), The existence and exponential stability for neutral stochastic partial differential equationswith infinite delay and poisson jumps, Indian. J. Pure. Appl. Math, 46(2), 197217. 

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

[13]  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. 

[14]  Anguraj, K., Banupriya, K., Dumitru, B., and Vinodkumar, A. (2018), On neutral impulsive stochastic differential equations with poisson jumps, Advances in Difference Equations, 2018:290, 117. 

[15]  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. 

[16]  Diop, M.A., Ezzinbi, K., and Lo, M. (2014), Existence and exponential stability for some stochastic neutral partial functional integrodifferential equations, Random Oper. Stoch. Equ., 22(2), 7383. 

[17]  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. 

[18]  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. 

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

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

[21]  Caraballo, T., Real, J., and Taniguchi, T. (2007), The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Stst, 18, 295313. 

[22]  Grimmer, R. (1982), Resolvent operators for integral equations in Banach space, Trans. Amer.Math. Soc, 273, 333349. 

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

[24]  Hale, J.K. and Kato, J. (1978), Phase space for retarded equations with infinite delay, Funkc. Ekvacioj Ser. Int, 21(1), 1141. 

[25]  Hinto, Y., Murakami, S., and Naito, T. (1991), Functionaldifferential equations with infinite delay, Lecture Notes in Mathematics, SpringerVerlag: Berlin, 1473. 