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Cao, Junfei; Yang, Qigui; Huang, Zaitang

Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations. (English) Zbl 1221.60078

Stochastics 83, No. 3, 259-275 (2011).
Summary: The concept of \(p\)-mean almost automorphy for stochastic processes is firstly introduced. Some properties of the \(p\)-mean almost automorphic stochastic processes are further studied. Based on these properties, a class of stochastic functional differential equations given by
\[ dx(t) = (Ax(t) + F(t, x(t), x_t))dt + g(t, x(t), x_t) \circ dW(t) \]
is investigated. Under suitable assumptions, existence, uniqueness and exponential stability of quadratic-mean almost automorphic mild solutions to the equations are discussed by means of semigroups of operators and by the Banach contraction principle. Moreover, the quadratic-mean almost automorphic mild solutions to semilinear stochastic partial functional differential equations, which is illustrated by an example, are in good agreement with the theoretical analysis.

Cited in 23 Documents

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory

Keywords:

existence and uniqueness; quadratic-mean almost automorphic solution; stochastic functional differential equations; exponential stability; Banach contraction principle
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References:

[1] Amerio L., Almost Periodic Functions and Functional Equations (1971) · Zbl 0215.15701
[2] DOI: 10.1016/j.spl.2008.04.008 · Zbl 1156.60046 · doi:10.1016/j.spl.2008.04.008
[3] Bezandry P., Appl. Anal. 117 pp 1– (2007)
[4] Bezandry P., Electron. J. Differ. Equations 117 pp 1– (2007)
[5] DOI: 10.1073/pnas.52.4.907 · Zbl 0134.30102 · doi:10.1073/pnas.52.4.907
[6] DOI: 10.1007/s00233-005-0524-y · Zbl 1086.34052 · doi:10.1007/s00233-005-0524-y
[7] DOI: 10.1016/j.jde.2008.04.001 · Zbl 1166.34021 · doi:10.1016/j.jde.2008.04.001
[8] DOI: 10.1016/j.jde.2008.07.025 · Zbl 1166.34022 · doi:10.1016/j.jde.2008.07.025
[9] DOI: 10.1016/j.na.2008.10.100 · Zbl 1188.34078 · doi:10.1016/j.na.2008.10.100
[10] Corduneanu C., Almost Periodic Functions (1989) · Zbl 0672.42008
[11] DOI: 10.1007/BF02559535 · Zbl 0104.05501 · doi:10.1007/BF02559535
[12] Diagana T., Int. J. Evol. Equ. 1 pp 57– (2005)
[13] Fink A., Lecture Notes in Math. 37, in: Almost Periodic Differential Equations (1974) · Zbl 0325.34039
[14] DOI: 10.1090/S0002-9939-10-10377-3 · Zbl 1202.60109 · doi:10.1090/S0002-9939-10-10377-3
[15] DOI: 10.1155/S0161171279000016 · Zbl 0397.34041 · doi:10.1155/S0161171279000016
[16] DOI: 10.1016/j.jmaa.2007.09.065 · Zbl 1134.43001 · doi:10.1016/j.jmaa.2007.09.065
[17] DOI: 10.1016/j.na.2008.10.046 · Zbl 1172.45002 · doi:10.1016/j.na.2008.10.046
[18] Manthey R., Stochastics 66 pp 37– (1999)
[19] N’Guérékata G., Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces (2001) · doi:10.1007/978-1-4757-4482-8
[20] N’Guérékata G., Ann. Sci. Math. Québec 7 pp 185– (1983)
[21] DOI: 10.1016/j.na.2007.02.012 · Zbl 1140.34399 · doi:10.1016/j.na.2007.02.012
[22] Prato G., Stochastics 23 pp 1– (1987) · Zbl 0634.60053 · doi:10.1080/17442508708833480
[23] Taniguchi T., Stochastics 53 pp 41– (1995)
[24] Taniguchi T., Stochastics 58 pp 191– (1996)
[25] DOI: 10.1016/j.na.2008.08.018 · Zbl 1175.34076 · doi:10.1016/j.na.2008.08.018
[26] Zaidman S., Almost Periodic Functions in Abstract Spaces (1985) · Zbl 0648.42006
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