×

zbMATH — the first resource for mathematics

Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. (English) Zbl 1166.60037
Summary: We study the existence and asymptotic stability in \(p\)th moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay. By employing a fixed point approach, sufficient conditions are derived for achieving the required result. These conditions do not require the monotone decreasing behaviour of the delays.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47, (1990) · Zbl 0723.60074
[2] Caraballo, T.; Liu, K., Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. anal. appl., 17, 743-763, (1999) · Zbl 0943.60050
[3] Caraballo, T.; Real, J., Partial differential equations with delayed random perturbations: existence, uniqueness and stability of solutions, Stoch. anal. appl., 11, 497-511, (1993) · Zbl 0790.60054
[4] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[5] Govindan, T.E., Exponential stability in Mean-square of parabolic quasilinear stochastic delay evolution equations, Stoch. anal. appl., 17, 443-461, (1999) · Zbl 0940.60076
[6] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055
[7] Keck, D.N.; McKibben, M.A., Abstract semilinear stochastic ito – volterra integrodifferential equations, J. appl. math. stoch. anal., 1-22, (2006) · Zbl 1234.60065
[8] Khas’minskii, R., Stochastic stability of differential equations, (1980), Sijthoff & Noordhoff Netherlands · Zbl 0441.60060
[9] Liu, K., Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics, 63, 1-26, (1998) · Zbl 0947.93037
[10] Liu, K.; Mao, X., Exponential stability of non-linear stochastic evolution equations, Stochastic process. appl., 78, 173-193, (1998) · Zbl 0933.60072
[11] 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
[12] Liu, K., Stability of infinite dimensional stochastic differential equations with applications, (2006), Chapman & Hall, CRC London
[13] 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
[14] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichestic, UK · Zbl 0874.60050
[15] Mao, X., Exponential stability of stochastic differential equations, (1994), Marcel Dekker New York · Zbl 0851.93074
[16] Nieto, J.J.; Rodriguez-Lopez, R., Boundary value problems for a class of impulsive functional equations, Comput. math. appl., 55, 2715-2731, (2008) · Zbl 1142.34362
[17] Nieto, J.J.; Rodriguez-Lopez, R., New comparison results for impulsive integro-differential equations and applications, J. math. anal. appl., 328, 1343-1368, (2007) · Zbl 1113.45007
[18] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[19] Taniguchi, T., The exponential stability for stochastic delay partial differential equations, J. math. anal. appl., 331, 191-205, (2007) · Zbl 1125.60063
[20] Taniguchi, T., Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics, 53, 41-52, (1995) · Zbl 0854.60051
[21] 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
[22] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.