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Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays. (English) Zbl 1157.60065

The exponential stability in the \(p\)-th mean and the exponential pathwise stability of mild solutions of SPDEs with delays are investigated in this paper. This is achieved by fixed-point methods, so the author makes the following assumptions: Lipschitz conditions on coefficients, exponential estimation of \(C_0\)-semigroup and certain technical condition connecting constants from previous assumptions. These conditions do not require the monotone decreasing behavior of the delays.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B35 Stability in context of PDEs
93C15 Control/observation systems governed by ordinary differential equations
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References:

[1] J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint; J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint
[2] Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations (2006), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 1090.45002
[3] Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47 (1990) · Zbl 0723.60074
[4] 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
[5] 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
[6] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press · Zbl 0761.60052
[7] 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
[8] Haussmann, U. G., Asymptotic stability of the linear Itô equation in infinite dimensions, J. Math. Anal. Appl., 65, 219-235 (1978) · Zbl 0385.93051
[9] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 90, 12-44 (1982) · Zbl 0497.93055
[10] Jahanipur, Ruhollan, Stability of stochastic delay evolution equations with monotone nonlinearity, Stoch. Anal. Appl., 21, 161-181 (2003) · Zbl 1028.60059
[11] Khas’minskii, R., Stochastic Stability of Differential Equations (1980), Sijthoff & Noordhoff: Sijthoff & Noordhoff Netherlands · Zbl 0441.60060
[12] Liu, K., Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics, 63, 1-26 (1998) · Zbl 0947.93037
[13] Luo, Jiaowan, Fixed points and stability of neutral stochastic delay differential equations, J. Math. Anal. Appl., 334, 431-440 (2007) · Zbl 1160.60020
[14] Mao, X., Exponential stability for stochastic differential delay equations in Hilbert spaces, Chinese Quart. J. Math., 42, 77-85 (1991) · Zbl 0719.60062
[15] Yor, M., Existence et unicité de diffusions à valeurs dans un espace de Hilbert, Ann. Inst. H. Poincaré, 10, 55-88 (1974) · Zbl 0281.60094
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