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Global existence of solutions for stochastic impulsive differential equations. (English) Zbl 1218.60052

Summary: We obtain some results on the global existence of solutions to Itô stochastic impulsive differential equations in \(\mathcal M([0,\infty),\mathbb R^{n})\) which denotes the family of \(\mathbb R^{n}\)-valued stochastic processes \(x\) satisfying \(\sup_{t \in [0,\infty)} \mathbb{E}|x(t)|^{2} < \infty\) under non-Lipschitz coefficients. The Schaefer fixed point theorem is employed to achieve the desired result. An example is provided to illustrate the obtained results.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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[1] Oksendal, B.: Stochastic Differential Equations, Springer, New York, 1995
[2] Liu, J. C.: On the existence and uniqueness of solutions to stochastic differential equations of mixed Brownian and Poissonian. Stochastic Processes and their Applications, 94, 339–354 (2001) · Zbl 1053.60068 · doi:10.1016/S0304-4149(01)00088-6
[3] Ren, Y., Lu, S. P., Xia, N. G.: Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay. J. Comput. Appl. Math., 220, 364–372 (2008) · Zbl 1152.34388 · doi:10.1016/j.cam.2007.08.022
[4] Wei, F. Y., Wang, K.: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J. Math. Anal. Appl., 331, 516–531 (2007) · Zbl 1121.60064 · doi:10.1016/j.jmaa.2006.09.020
[5] Basin, M., Rodkina, A.: On delay-dependent stability for a class of nonlinear stochastic systems with multiple state delays. Nonlinear Anal., 68, 2147–2157 (2008) · Zbl 1154.34044 · doi:10.1016/j.na.2007.01.046
[6] Su, W., Chen, Y.: Global asymptotic stability analysis for neutral stochastic neural networks with time-varying delays. Communications in Nonlinear Science and Numerical Simulation, 14, 1576–1581 (2009) · Zbl 1221.93214 · doi:10.1016/j.cnsns.2008.04.001
[7] Yu, J., Zhang, K., Fei, S.: Further results on mean square exponential stability of uncertain stochastic delayed neural networks. Communications in Nonlinear Science and Numerical Simulation, 14, 1582–1589 (2009) · Zbl 1221.93267 · doi:10.1016/j.cnsns.2008.04.009
[8] Luo, J. W.: Stability of stochastic partial differential equations with infinite delays. J. Comput. Appl. Math., 222, 364–371 (2008) · Zbl 1151.60336 · doi:10.1016/j.cam.2007.11.002
[9] Benchohra, M., Henderson, J., Ntouyas, K.: An existence result for first order impulsive functional differential equations in Banach spaces. J. Comput. Appl. Math., 42, 1303–1310 (2001) · Zbl 1005.34069 · doi:10.1016/S0898-1221(01)00241-3
[10] Guo, D. J.: Existence of solutions for nth order impulsive integro-differential equations in a Banach space. Nonlinear Anal., 47, 741–752 (2001) · Zbl 1042.34584 · doi:10.1016/S0362-546X(01)00219-X
[11] Heikkila, S., Kumpulainen, M., Seikkala, S.: Uniqueness and existence results for implicit impulsive differential equations. Nonlinear Anal., 42, 13–26 (2000) · Zbl 0969.34012 · doi:10.1016/S0362-546X(98)00325-3
[12] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989 · Zbl 0719.34002
[13] Liu, X., Ballinger, G.: Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal., 51, 633–647 (2002) · Zbl 1015.34069 · doi:10.1016/S0362-546X(01)00847-1
[14] Liu, K., Yang, G. W.: Cone-valued Lyapunov functions and stability for impulsive functional differential equations. Nonlinear Anal., 69, 2184–2191 (2008) · Zbl 1151.34063 · doi:10.1016/j.na.2007.07.057
[15] Zhang, Y., Sun, J.: Stability of impulsive functional differential equations. Nonlinear Anal., 68, 3665–3678 (2008) · Zbl 1152.34053 · doi:10.1016/j.na.2007.04.009
[16] Dong, Y., Sun, J., Wu, Q.: H filtering for a class of stochastic Markovian jump systems with impulsive effects. Int. J. Robust Nonlinear Control, 18, 1–13 (2008) · Zbl 1284.93232 · doi:10.1002/rnc.1194
[17] Xu, S. Y., Chen, T.: Robust H fltering for uncertain impulsive stochastic systems under sampled measurements. Automatica, 39, 509–516 (2003) · Zbl 1012.93063 · doi:10.1016/S0005-1098(02)00248-0
[18] Wu, H. J., Sun, J. T.: p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching. Automatica, 42, 1753–1759 (2006) · Zbl 1114.93092 · doi:10.1016/j.automatica.2006.05.009
[19] Yang, J., Zhong, S., Luo, W.: Mean square stability analysis of impulsive stochastic differential equations with delays. J. Comput. Appl. Math., 216, 474–483 (2008) · Zbl 1142.93035 · doi:10.1016/j.cam.2007.05.022
[20] Yang, Z., Xu, D. Y., Xiang, L.: Exponential p-stability of impulsive stochastic differential equations with delays. Physics Letters A, 359, 129–137 (2006) · Zbl 1236.60061 · doi:10.1016/j.physleta.2006.05.090
[21] Liu, B., Liu, X. Z., Liao, X. X.: Existence and uniqueness and stability of solutions for stochastic impulsive systems. Journal of Systems Science&amp; Complexity, 20, 149–158 (2007) · Zbl 1124.93054 · doi:10.1007/s11424-007-9013-6
[22] Hale, J. K.: Theory of Functional Differential Equations, Springer, New York, 1977 · Zbl 0352.34001
[23] Smart, D. R.: Fixed Point Theorem, Cambridge Univ. Press, Cambridge, 1980 · Zbl 0427.47036
[24] Kurenok, V. P.: Existence of global solutions of stochastic differential equations with time dependent coefficients. Doklady Akademii Nauk Belarusi, 44, 30–34 (2000) · Zbl 1156.60316
[25] Gyongy, I., Krylov, N.: Existence of strong solutions for Ito’s stochastic equations via approximations. Probability Theory and Related Fields, 105, 143–158 (1996) · Zbl 0847.60038 · doi:10.1007/BF01203833
[26] Xia, Y. J.: The global existence of sample solutions of stochastic ordinary differential equations. Journal of Guizhou Normal University (Natural Sciences), 20, 13–15 (2002) · Zbl 1005.60073
[27] Cao, G. L.: Existence and pathwise uniqueness of solutions to stochastic differential equations in Banach space. Mathematical Applicata, 19(1), 75–79 (2006) · Zbl 1111.60040
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