×

Integral inequality and exponential stability for neutral stochastic partial differential equations with delays. (English) Zbl 1188.60034

Summary: The aim of this paper is devoted to obtain some sufficient conditions for the exponential stability in \(p\) \((p\geq 2)\)-moment as well as almost surely exponential stability for mild solution of neutral stochastic partial differential equations with delays by establishing an integral-inequality. Some well-known results are generalized and improved. Finally, an example is given to show the effectiveness of our results.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Caraballo T: Asymptotic exponential stability of stochastic partial differential equations with delay.Stochastics and Stochastics Reports 1990,33(1-2):27-47. · Zbl 0723.60074 · doi:10.1080/17442509008833662
[2] Caraballo T, Liu K: Exponential stability of mild solutions of stochastic partial differential equations with delays.Stochastic Analysis and Applications 1999,17(5):743-763. 10.1080/07362999908809633 · Zbl 0943.60050 · doi:10.1080/07362999908809633
[3] Caraballo T, Real J, Taniguchi T: The exponential stability of neutral stochastic delay partial differential equations.Discrete and Continuous Dynamical Systems. Series A 2007,18(2-3):295-313. · Zbl 1125.60059
[4] Caraballo T, Real J: Partial differential equations with delayed random perturbations: existence, uniqueness, and stability of solutions.Stochastic Analysis and Applications 1993,11(5):497-511. 10.1080/07362999308809330 · Zbl 0790.60054 · doi:10.1080/07362999308809330
[5] Govindan TE: Stability of mild solutions of stochastic evolution equations with variable delay.Stochastic Analysis and Applications 2003,21(5):1059-1077. 10.1081/SAP-120022863 · Zbl 1036.60052 · doi:10.1081/SAP-120022863
[6] Govindan TE: Existence and stability of solutions of stochastic semilinear functional differential equations.Stochastic Analysis and Applications 2002,20(6):1257-1280. 10.1081/SAP-120015832 · Zbl 1066.60055 · doi:10.1081/SAP-120015832
[7] Haussmann UG: Asymptotic stability of the linear Itô equation in infinite dimensions.Journal of Mathematical Analysis and Applications 1978,65(1):219-235. 10.1016/0022-247X(78)90211-1 · Zbl 0385.93051 · doi:10.1016/0022-247X(78)90211-1
[8] Ichikawa A: Stability of semilinear stochastic evolution equations.Journal of Mathematical Analysis and Applications 1982,90(1):12-44. 10.1016/0022-247X(82)90041-5 · Zbl 0497.93055 · doi:10.1016/0022-247X(82)90041-5
[9] Liu K, Truman A: A note on almost sure exponential stability for stochastic partial functional differential equations.Statistics & Probability Letters 2000,50(3):273-278. 10.1016/S0167-7152(00)00103-6 · Zbl 0966.60059 · doi:10.1016/S0167-7152(00)00103-6
[10] Liu K, Mao X: Exponential stability of non-linear stochastic evolution equations.Stochastic Processes and Their Applications 1998,78(2):173-193. 10.1016/S0304-4149(98)00048-9 · Zbl 0933.60072 · doi:10.1016/S0304-4149(98)00048-9
[11] Liu K: Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Volume 135. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2006:xii+298.
[12] Liu, K.; Shi, Y., Razumikhin-type theorems of infinite dimensional stochastic functional differential equations, No. 202, 237-247 (2006), New York, NY, USA · Zbl 1217.60055
[13] Luo J: Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays.Journal of Mathematical Analysis and Applications 2008,342(2):753-760. 10.1016/j.jmaa.2007.11.019 · Zbl 1157.60065 · doi:10.1016/j.jmaa.2007.11.019
[14] Mahmudov NI: Existence and uniqueness results for neutral SDEs in Hilbert spaces.Stochastic Analysis and Applications 2006,24(1):79-95. 10.1080/07362990500397582 · Zbl 1110.60063 · doi:10.1080/07362990500397582
[15] Mao X: Exponential stability for stochastic differential delay equations in Hilbert spaces.The Quarterly Journal of Mathematics 1991,42(165):77-85. · Zbl 0719.60062 · doi:10.1093/qmath/42.1.77
[16] Taniguchi T, Liu K, Truman A: Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces.Journal of Differential Equations 2002,181(1):72-91. 10.1006/jdeq.2001.4073 · Zbl 1009.34074 · doi:10.1006/jdeq.2001.4073
[17] Taniguchi T: Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces.Stochastics and Stochastics Reports 1995,53(1-2):41-52. · Zbl 0854.60051 · doi:10.1080/17442509508833982
[18] Taniguchi T: The exponential stability for stochastic delay partial differential equations.Journal of Mathematical Analysis and Applications 2007,331(1):191-205. 10.1016/j.jmaa.2006.08.055 · Zbl 1125.60063 · doi:10.1016/j.jmaa.2006.08.055
[19] Taniguchi T: Almost sure exponential stability for stochastic partial functional-differential equations.Stochastic Analysis and Applications 1998,16(5):965-975. 10.1080/07362999808809573 · Zbl 0911.60054 · doi:10.1080/07362999808809573
[20] Wan L, Duan J: Exponential stability of non-autonomous stochastic partial differential equations with finite memory.Statistics & Probability Letters 2008,78(5):490-498. 10.1016/j.spl.2007.08.003 · Zbl 1141.37030 · doi:10.1016/j.spl.2007.08.003
[21] Mao X: Stochastic Differential Equations and Applications. Horwood, Chichester, UK; 1997. · Zbl 0892.60057
[22] Burton TA: Stability by Fixed Point Theory for Functional Differential Equations. Dover, Mineola, NY, USA; 2006:xiv+348. · Zbl 1160.34001
[23] Luo J: Fixed points and stability of neutral stochastic delay differential equations.Journal of Mathematical Analysis and Applications 2007,334(1):431-440. 10.1016/j.jmaa.2006.12.058 · Zbl 1160.60020 · doi:10.1016/j.jmaa.2006.12.058
[24] Appleby JAD: Fixed points, stability and harmless stochastic perturbations. preprint preprint · Zbl 1036.60052
[25] Datko R: Linear autonomous neutral differential equations in a Banach space.Journal of Differential Equations 1977,25(2):258-274. 10.1016/0022-0396(77)90204-2 · Zbl 0402.34066 · doi:10.1016/0022-0396(77)90204-2
[26] Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279. · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[27] Luo Q, Mao X, Shen Y: New criteria on exponential stability of neutral stochastic differential delay equations.Systems & Control Letters 2006,55(10):826-834. 10.1016/j.sysconle.2006.04.005 · Zbl 1100.93048 · doi:10.1016/j.sysconle.2006.04.005
[28] Da Prato G, Zabczyk J: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications. Volume 44. Cambridge University Press, Cambridge, UK; 1992:xviii+454. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.