Journal of Applied Nonlinear Dynamics
Existence and Stability Results for Impulsive Stochastic Functional Integrodifferential Equation with Poisson Jumps
Journal of Applied Nonlinear Dynamics 8(3) (2019) 407417  DOI:10.5890/JAND.2019.09.005
A. Anguraj, K. Ravikumar
Department of Mathematics, PSG College of Arts and Science, Coimbatore641 014, Tamil Nadu, India
Download Full Text PDF
Abstract
In this article we present the existence, uniqueness and stability of mild solutions for impulsive stochastic functional integro diﬀerential equations with nonLipschitz condition. The mild solution is obtained by using a resolvent operator in a diﬀerent sense and the results are proved by using the method of successive approximation and Bihari’s inequality.
References

[1]  Du, B. (2015), Successive approximation of netural functional stochastic differential equations with variable delays, Appl Math. Comput., 268, 609615. 

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

[3]  Anguraj, A., Kanjanadevi, S., and Trujillo, J.J. (2017), Existence of mild solutions of abstract fractional differential equations with fractional noninstantaneous impulsive conditions, Discontinuity, Nonlinearity, and Complexity, 6(2), 173183. 

[4]  Liang, J., Liu, J.H., and Xiao, T.J. (2008), Nonlocal problems for integrodifferential equations, Dynamics of Continuous, Discrete and Impulsive Systems, series A, Mathematical analysis, 15, 815824. 

[5]  Joice Nirmala, R. and Balachandran, K. (2016), Controllability of fractional nonlinear systems in banach spaces, Journal of Applied Nonlinear Dynamics, 5(4), 485494. 

[6]  Liu, A. and Den, Y. (2010), On neutral impulsive stochastic integro differential equations with infinite delays via fractional operators, Mathematical and computer modelling, 51, 413424. 

[7]  Diop, M.A., Ezzinbi, K., and Lo, M. (2016), Existence and uniqueness of Mild solutions to neutral stochastic partial functional integrodifferential equations with nonLipschitz coefficient, International journal of mathematics and mathematical science. 

[8]  Diop, M.A. and Ezzinbi, K. (2014), Mild solutions to neutral stochastic partial functional integrodifferential equations with nonLipschitz coefficient, Afrika mathematica. 

[9]  Haseena, A., Suvinthra, M., and Annapoorani, N. (2017), On large deviations of stochastic integrodifferential equations with browian motion, Discontinuity, Nonlinearity, and Complexity, 6(3), 281294. 

[10]  Mabel Lizzy, R., Balachandran, K., and Suvinthra, M. (2017), Controllability of nonlinear stochastic fractional systems with levy noise, Discontinuity, Nonlinearity, and Complexity, 6(3), 409420. 

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

[12]  Yang, J. and Zhong, S. (2008), Mean square stability analysis of impulsive stochastic differential equations with delays, J. Comput. Appl. Math., 356, 474483. 

[13]  Yang, Z. and Xu, D. (2006), Exponential pstability of impulsive stochastic differential equations with delays, Physics Letter A, 356, 129137. 

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

[15]  Sakthivel, R. and Luo, J. (2009), Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 342, 753760. 

[16]  Anguraj, A. and Vinodkumar, A. (2009), Existence, uniqueness and stability results of impulsive stochastic semilinear neutral functional differential equations with infinite delays, Electron. J. Qual. Theory Differ. Eqn., 67, 113. 

[17]  Luo, J. and Taniguchi, T. (2009), The existence and uniqueness for nonLipschitz stochastic neutral delay evolution equation driven by Poisson jumps, Stoch. Dyn., 9(1), 627638. 

[18]  Boufoussi, B. (2010), Successive approximations of neutral functional stochastic differential equations with Jumps, Stat. prob. lett., 80, 324332. 

[19]  Grimmer, R. (1986), Resolvent operator for integral equations in a Banach space, Transactions of the America Mathematical society, 273(1), 333749. 

[20]  Desch, W., Grimmer, R., and Schappacher, W. (1984), Some consideration for linear integrodifferential equations, Journal of Mathematical Analysis and Application, 104, 219234. 

[21]  Bihari, I. (1956), A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations, Acta. Math. Acad. Sci. Hungar., 7, 7194. 

[22]  Ren, Y. and Xia, N. (2009), Existence, uniqueness and stability of the solution to neutral stochastic functional differential equations its infinite delays, Appl. Math. Comput., 210, 7279. 

[23]  Prato, Da. and Zabczyk, J. (2014), Stochastic equations in infinite dimensions, Cambridge University Press. 

[24]  Kolmanovskii, V.B. and Myshkis, A. (1992), Applied theory of functional differential equations, Kluwer Academic Publishers. 

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

[26]  Wu, F. and Yin, U. (2016), Stochastic functional differential equations with infinite delay. Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity, J. Differ. Equations. 

[27]  Bao, S. and Hou, Z. (2010), Existence of mild solutions to stochastic neutral partial functional differential equations with nonLipschitz coefficients, Appl. S. comput. Math. appl., 59, 207214. 

[28]  Daplato, G. and Zabezyk, S. (1992), Stochastic equations in infinite Dimensions, Cambridge university press, Cambridge. 

[29]  Govindan, T.E. (2003), Stability of mild solutions of stochastic evolution equations with variable delays, Stochastic Anal. Appl., 21, 10591007. 