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Exponential stability for stochastic neutral partial functional differential equations. (English) Zbl 1165.60024
Summary: We consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in {\it T. E. Govindan} [Stochastics 77, 139--154 (2005; Zbl 1115.60064)], {\it K. Liu} and {\it A. Truman} [Statist. Probab. Lett. 50, 273--278 (2000; Zbl 0966.60059)] and {\it T. Taniguchi} [Stoch. Anal. Appl. 16, 965--975 (1998; Zbl 0911.60054); Stochastics 53, 41--52 (1995; Zbl 0854.60051)] are generalized and improved.

MSC:
60H15Stochastic partial differential equations
93E15Stochastic stability
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Full Text: DOI
References:
[1] Caraballo, T.; Real, J.; Taniguchi, T.: The exponential stability of neutral stochastic delay partial differential equations, Discrete contin. Dyn. syst. 18, No. 2 -- 3, 295-313 (2007) · Zbl 1125.60059 · doi:10.3934/dcds.2007.18.295
[2] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions, (1992) · Zbl 0761.60052
[3] Datko, R.: Linear autonomous neutral differential equations in Banach spaces, J. differential equations 25, 258-274 (1977) · Zbl 0402.34066 · doi:10.1016/0022-0396(77)90204-2
[4] Fu, X.; Ezzinbi, K.: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear anal. 54, 215-227 (2003) · Zbl 1034.34096 · doi:10.1016/S0362-546X(03)00047-6
[5] Govindan, T. E.: Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77, 139-154 (2005) · Zbl 1115.60064 · doi:10.1080/10451120512331335181
[6] Hale, J. K.; Meyer, K. R.: A class of functional equations of neutral type, Mem. amer. Math. soc. 76, 1-65 (1967) · Zbl 0179.20501
[7] Hale, J.; Lunel, S. M. Verduyn: Introduction to functional differential equations, (1993) · Zbl 0787.34002
[8] Kolmanovskii, V. B.; Nosov, V. R.: Stability of functional differential equations, (1986) · Zbl 0593.34070
[9] Kolmanovskii, V. B.; Nosov, V. R.: Stability of neutral-type functional differential equations, Nonlinear anal. 6, 873-910 (1982) · Zbl 0494.34052 · doi:10.1016/0362-546X(82)90009-8
[10] Liu, K.: Stability of infinite dimensional stochastic differential equations with applications, Pitman monogr. Surveys pure appl. Math. 135 (2006) · Zbl 1085.60003 · doi:10.1201/9781420034820
[11] Liu, K.: Uniform stability of autonomous linear stochastic functional differential equations in infinite dimensions, Stochastic process. Appl. 115, 1131-1165 (2005) · Zbl 1075.60078 · doi:10.1016/j.spa.2005.02.006
[12] Liu, K.; Xia, X.: On the exponential stability in mean square of neutral stochastic functional differential equations, Systems control lett. 37, 207-215 (1999) · Zbl 0948.93060 · doi:10.1016/S0167-6911(99)00021-3
[13] Liu, K.; Truman, A.: A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. probab. Lett. 50, 273-278 (2000) · Zbl 0966.60059 · doi:10.1016/S0167-7152(00)00103-6
[14] Mao, X. R.: Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. anal. 28, 389-401 (1997) · Zbl 0876.60047 · doi:10.1137/S0036141095290835
[15] Mao, X. R.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057
[16] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Appl. math. Sci. 44 (1983) · Zbl 0516.47023
[17] Randjelović, J.; Janković, S.: On the pth moment exponential stability criteria of neutral stochastic functional differential equations, J. math. Anal. appl. 326, 266-280 (2007) · Zbl 1115.60065 · doi:10.1016/j.jmaa.2006.02.030
[18] Rodkina, A. E.: On existence and uniqueness of solution of stochastic differential equations with heredity, Stochastics 12, 187-200 (1984) · Zbl 0568.60062 · doi:10.1080/17442508408833300
[19] Taniguchi, T.: Almost sure exponential stability for stochastic partial functional differential equations, Stoch. anal. Appl. 16, 965-975 (1998) · Zbl 0911.60054 · doi:10.1080/07362999808809573
[20] Taniguchi, T.: Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53, 41-52 (1995) · Zbl 0854.60051