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Journal of Applied Nonlinear Dynamics 8(3) (2019) 407--417 | DOI:10.5890/JAND.2019.09.005

A. Anguraj, K. Ravikumar

Department of Mathematics, PSG College of Arts and Science, Coimbatore-641 014, Tamil Nadu, India

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Abstract

In this article we present the existence, uniqueness and stability of mild solutions for impulsive stochastic functional integro differential equations with non-Lipschitz 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. [1]	Du, B. (2015), Successive approximation of netural functional stochastic differential equations with variable delays, Appl Math. Comput., 268, 609-615.

2. [2]	Taniguchi, T. (1992), Successive approximation to solutions of stochastic differential equations, J. Differ. Equations., 96, 152-169.

3. [3]	Anguraj, A., Kanjanadevi, S., and Trujillo, J.J. (2017), Existence of mild solutions of abstract fractional differential equations with fractional non-instantaneous impulsive conditions, Discontinuity, Nonlinearity, and Complexity, 6(2), 173-183.

4. [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, 815-824.

5. [5]	Joice Nirmala, R. and Balachandran, K. (2016), Controllability of fractional nonlinear systems in banach spaces, Journal of Applied Nonlinear Dynamics, 5(4), 485-494.

6. [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, 413-424.

7. [7]	Diop, M.A., Ezzinbi, K., and Lo, M. (2016), Existence and uniqueness of Mild solutions to neutral stochastic partial functional integro-differential equations with non-Lipschitz co-efficient, International journal of mathematics and mathematical science.

8. [8]	Diop, M.A. and Ezzinbi, K. (2014), Mild solutions to neutral stochastic partial functional integro-differential equations with non-Lipschitz co-efficient, Afrika mathematica.

9. [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), 281-294.

10. [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), 409-420.

11. [11]	Suvinthra, M. and Balachandran, K. (2016), Large deviations for nonlinear ito type stochastic integrodifferential equations, Journal of Applied Nonlinear Dynamics, 6(1), 1-15.

12. [12]	Yang, J. and Zhong, S. (2008), Mean square stability analysis of impulsive stochastic differential equations with delays, J. Comput. Appl. Math., 356, 474-483.

13. [13]	Yang, Z. and Xu, D. (2006), Exponential p-stability of impulsive stochastic differential equations with delays, Physics Letter A, 356, 129-137.

14. [14]	Sakthivel, R. and Luo, J. (2009), Asymptotic stability of nonlinear impulsive stochastic partial differential equations with infinte delays, J. Math. Anal. Appl, 356, 1-6.

15. [15]	Sakthivel, R. and Luo, J. (2009), Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 342, 753-760.

16. [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, 1-13.

17. [17]	Luo, J. and Taniguchi, T. (2009), The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equation driven by Poisson jumps, Stoch. Dyn., 9(1), 627-638.

18. [18]	Boufoussi, B. (2010), Successive approximations of neutral functional stochastic differential equations with Jumps, Stat. prob. lett., 80, 324-332.

19. [19]	Grimmer, R. (1986), Resolvent operator for integral equations in a Banach space, Transactions of the America Mathematical society, 273(1), 333-749.

20. [20]	Desch, W., Grimmer, R., and Schappacher, W. (1984), Some consideration for linear integrodifferential equations, Journal of Mathematical Analysis and Application, 104, 219-234.

21. [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, 71-94.

22. [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, 72-79.

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

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

25. [25]	Sakthivel, R. and Luo, J. (2009), Asymptotic stability of non-linear impulsive stochastic partial differential equations with infinite delays, S. Math. Anal. Appl., 356, 1-6.

26. [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. [27]	Bao, S. and Hou, Z. (2010), Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz co-efficients, Appl. S. comput. Math. appl., 59, 207-214.

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

29. [29]	Govindan, T.E. (2003), Stability of mild solutions of stochastic evolution equations with variable delays, Stochastic Anal. Appl., 21, 1059-1007.