Successive Approximation of Neutral Functional Stochastic Differential Equations in Hilbert Spaces
Annales mathématiques Blaise Pascal, Volume 17 (2010) no. 1, pp. 183-197.

By using successive approximation, we prove existence and uniqueness result for a class of neutral functional stochastic differential equations in Hilbert spaces with non-Lipschitzian coefficients

En utilisant la méthode des approximations successives, nous allons montrer un résultat d’existence et d’unicité, sous des conditions non Lipschitziennes, pour une classe d’équations fonctionelles stochastiques de type neutre dans un espace de Hilbert.

DOI: 10.5802/ambp.282
Classification: 60H20,  34F05,  34G20
Keywords: Semigroup of bounded linear operator, Fractional powers of closed operators, Successive approximation, Mild solution, Cylindrical Q-Wiener process.
Brahim Boufoussi 1; Salah Hajji 1

1 Department of Mathematics Cadi Ayyad University Semlalia Faculty of Sciences 2390 Marrakesh Morocco
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Brahim Boufoussi; Salah Hajji. Successive Approximation of Neutral Functional Stochastic Differential Equations in Hilbert Spaces. Annales mathématiques Blaise Pascal, Volume 17 (2010) no. 1, pp. 183-197. doi : 10.5802/ambp.282. https://ambp.centre-mersenne.org/articles/10.5802/ambp.282/

[1] I. Bihari A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta. Math., Acad. Sci. Hungar, Volume 7 (1956), pp. 71-94 | DOI | MR | Zbl

[2] T. Caraballo; J. Real; T. Taniguchi The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst., Volume 18 (2007) no. 2-3, pp. 295-313 | MR | Zbl

[3] J. DaPrato Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[4] R. Datko Linear autonomous neutral differential equations in Banach spaces, J. Diff. Eqns, Volume 25 (1977), pp. 258-274 | DOI | MR | Zbl

[5] Jerome A. Goldstein Semigroups of linear operators and applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985 | MR | Zbl

[6] T.E. Govindan Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics: An International Journal of Probability and Stochastic Processes, Volume 77 (2005), pp. 139-154 | DOI | MR | Zbl

[7] V. Kolmanovskii; N. Koroleva; T. Maizenberg; X. Mao; A. Matasov Neutral stochastic differential delay equations with Markovian switching, Stochastic Anal. Appl, Volume 21(4) (2003), pp. 819-847 | DOI | MR | Zbl

[8] V.B. Kolmanovskii; V.R. Nosov Stability of functional differential equations, Academic Press, 1986 | MR | Zbl

[9] K. Liu Uniform stability of autonomous linear stochastic fuctional differential equations in infinite dimensions, Stochastic Process. Appl, Volume 115 (2005), pp. 1131-1165 | DOI | MR | Zbl

[10] K. Liu; X. Xia On the exponential stability in mean square of neutral stochastic functional differential equations, Systems Control Lett, Volume 37(4) (1999), pp. 207-215 | DOI | MR | Zbl

[11] N.I. Mahmudov Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stochastic Analysis and Applications, Volume 24 (2006), pp. 79-95 | DOI | MR | Zbl

[12] X. Mao Exponential stability in mean square of neutral stochastic differential functional equations, Systems and Control Letters, Volume 26 (1995), pp. 245-251 | DOI | MR | Zbl

[13] X. Mao Razumikhin-type theorems on exponential stability of neutral stochastic functional-differential equations, SIAM J. Math. Anal, Volume 28(2) (1997), pp. 389-401 | DOI | MR | Zbl

[14] A. Pazy Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983 | MR | Zbl

[15] J. Wu Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences Volume 119, Springer-Verlag, New York, 1996 | MR | Zbl

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