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Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations. (English) Zbl 1156.60046
The aim of this paper is to prove the existence and uniqueness of quadratic mean almost periodic solution for an abstract semi-linear integro-differential equation in a real separable Hilbert space . The author follows the ideas of {\it P. H. Bezandry} and {\it T. Diagana} [Appl. Anal. 86, No. 7, 819--827 (2007; Zbl 1130.34033)] for semi-linear stochastic differential equations. As an example, the author applies the results to a stochastic partial differential equation.

60H20Stochastic integral equations
60G05Foundations of stochastic processes
Full Text: DOI
[1] Bezandry, P.; Diagana, T.: Existence of almost periodic solutions to some stochastic differential equations. Appl. anal. 2007, No. 117, 1-10 (2007) · Zbl 1138.60323
[2] Corduneanu, C.: Almost periodic functions. (1989) · Zbl 0672.42008
[3] Ichikawa, A.: Stability of semilinear stochastic evolution equations. J. math. Anal. appl. 90, No. 1, 12-44 (1982) · Zbl 0497.93055
[4] Kannan, D.; Bharucha-Reid, D.: On a stochastic integro-differential evolution of Volterra type. J. integral equations 10, 351-379 (1985) · Zbl 0583.60060
[5] Keck, D.; Mckibben, M.: Functional integro-differential stochastic evolution equations in Hilbert space. J. appl. Math. stoch. Anal. 16, No. 2, 141-161 (2003) · Zbl 1031.60061
[6] Keck, D.; Mckibben, M.: Abstract stochastic integro-differential delay equations. J. appl. Math. stoch. Anal., No. 3, 275-305 (2005) · Zbl 1105.60045
[7] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied mathematical sciences 44 (1983) · Zbl 0516.47023