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Asymptotic stability of nonlinear impulsive stochastic differential equations. (English) Zbl 1166.60316
Summary: We study the existence and asymptotic stability in \(p\)-th moment of mild solutions of nonlinear impulsive stochastic differential equations. A fixed point approach is employed for achieving the required result.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Appleby, J.A.D., 2008. Fixed points, stability and harmless stochastic perturbations, preprint
[2] Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47, (1990) · Zbl 0723.60074
[3] Caraballo, T.; Liu, K., Exponential stability of mild solutions of stochastic partial differential equations with delays, Stochastic anal. appl., 17, 743-763, (1999) · Zbl 0943.60050
[4] Caraballo, T.; Real, J., Partial differential equations with delayed random perturbations: existence, uniqueness and stability of solutions, Stochastic anal. appl., 11, 497-511, (1993) · Zbl 0790.60054
[5] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[6] Govindan, T.E., Exponential stability in Mean-square of parabolic quasilinear stochastic delay evolution equations, Stochastic anal. appl., 17, 443-461, (1999) · Zbl 0940.60076
[7] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055
[8] Ignatyev, O., Partial asymptotic stability in probability of stochastic differential equations, Statist. probab. lett., 79, 597-601, (2009) · Zbl 1157.60327
[9] Khas’minskii, R., Stochastic stability of differential equation, (1980), Sijthoff & Noordhoff Netherlands · Zbl 0441.60060
[10] Liu, K., Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics, 63, 1-26, (1998) · Zbl 0947.93037
[11] Liu, K.; Mao, X., Exponential stability of non-linear stochastic evolution equations, Stochastic process. appl., 78, 173-193, (1998) · Zbl 0933.60072
[12] Liu, K., Stability of infinite dimensional stochastic differential equations with applications, (2006), Chapman & Hall, CRC London
[13] Liu, K.; Truman, A., A note on almost exponential stability for stochastic partial differential equations, Statist. probab. lett., 50, 273-278, (2000) · Zbl 0966.60059
[14] Luo, J., Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. math. anal. appl., 342, 753-760, (2008) · Zbl 1157.60065
[15] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichestic, UK · Zbl 0874.60050
[16] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[17] Smart, D.R., Fixed point theorems, (1980), Cambridge Univ. Press Cambridge · Zbl 0427.47036
[18] Taniguchi, T., The exponential stability for stochastic delay partial differential equations, J. math. anal. appl., 331, 191-205, (2007) · Zbl 1125.60063
[19] Taniguchi, T.; Liu, K.; Truman, A., Existence uniqueness, and asymptotic behavior of mild solution to stochastic functional differential equations in Hilbert spaces, J. differential equations, 18, 72-91, (2002) · Zbl 1009.34074
[20] Wan, L.; Duan, J., Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. probab. lett., 78, 490-498, (2008) · Zbl 1141.37030
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