# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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
##### 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 · doi:10.1080/07362999908809633 [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 · doi:10.1080/07362999308809330 [4] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions, (1992) · 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 · doi:10.1080/07362999908809612 [6] Ichikawa, A.: Stability of semilinear stochastic evolution equations, J. math. Anal. appl. 90, 12-44 (1982) · Zbl 0497.93055 · doi:10.1016/0022-247X(82)90041-5 [7] Keck, D. N.; Mckibben, M. A.: Abstract semilinear stochastic Itô – Volterra integrodifferential equations, J. appl. Math. stoch. Anal., 1-22 (2006) [8] Khas’minskii, R.: Stochastic stability of differential equations, (1980) [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 · doi:10.1016/S0304-4149(98)00048-9 [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 · doi:10.1016/S0167-7152(00)00103-6 [12] Liu, K.: Stability of infinite dimensional stochastic differential equations with applications, (2006) [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 · doi:10.1016/j.jmaa.2007.11.019 [14] Mao, X.: Stochastic differential equations and applications, (1997) [15] Mao, X.: Exponential stability of stochastic differential equations, (1994) [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 · doi:10.1016/j.camwa.2007.10.019 [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 · doi:10.1016/j.jmaa.2006.06.029 [18] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · 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 · doi:10.1016/j.jmaa.2006.08.055 [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 · doi:10.1006/jdeq.2001.4073 [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 · doi:10.1016/j.spl.2007.08.003