Abstract
In this paper, we prove the existence and uniqueness of quadratic mean almost periodic mild solutions for a class of stochastic differential equations in a real separable Hilbert space. The main technique is based upon an appropriate composition theorem combined with the Banach contraction mapping principle and an analytic semigroup of linear operators.
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Abbas S., Bahuguna D.: Almost periodic solutions of neutral functional differential equations. Comput. Math. Appl. 55, 2593–2601 (2008)
Alzabut, J.O., Nieto, J.J., Stamov, G.Tr.: Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis. Boundary Value Probl. Article ID 127510, p. 10 (2009)
Acquistapace P., Terreni B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78, 47–107 (1987)
Bezandry P., Diagana T.: Existence of almost periodic solutions to some stochastic differential equations. Appl. Anal. 86, 819–827 (2007)
Bezandry P., Diagana T.: Square-mean almost periodic solutions nonautonomous stochastic differential equations. Electron. J. Differ. Equ. 2007, 1–10 (2007)
Bezandry P.: Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations. Stat. Probab. Lett. 78, 2844–2849 (2008)
Bezandry P., Diagana T.: Existence of S 2-almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. Qual. Theory Differ. Equ. 35, 1–19 (2008)
Bezandry P., Diagana T.: Existence of quadratic-mean almost periodic solutions to some stochastic hyperbolic differential equations. Electron. J. Differ. Equ. 2009, 1–14 (2009)
Cao J., Yang Q., Huang Z., Liu Q.: Asymptotically almost periodic solutions of stochastic functional differential equations. Appl. Math. Comput. 218, 1499–1511 (2011)
Corduneanu C.: Almost Periodic Functions, 2nd edn. Chelsea, New York (1989)
Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Dorogovtsev A.Ya., Ortega O.A.: On the existence of periodic solutions of a stochastic equation in a Hilbert space. Visnik Kiiv. Univ. Ser. Mat. Mekh. 115, 21–30 (1988)
Diagana T., Mahop C.M., N’Guérékata G.M.: Pseudo almost periodic solutions to some semilinear differential equations. Math. Comput. Model. 43, 89–96 (2006)
Diagana T., Mahop C.M., N’Guérékata G.M., Toni B.: Existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications. Nonlinear Anal. 64, 2442–2453 (2006)
Govindan T.E.: On stochastic delay evolution equations with non-lipschitz nonlinearities in Hilbert spaces. Differ. Integral Equ. 22, 157–176 (2009)
Henríquez H.R., Vasquez C.H.: Almost periodic solutions of abstract retarded functional-differential equations with unbounded delay. Acta Appl. Math. 57, 105–132 (1999)
Hernández E.M., Pelicer, M.L., dos Santos J.P.C.: Asymptotically almost periodic and almost periodic solutions for a class of evolution equations. Electron. J. Differ. Equ. 2004 1–15 (2004)
Hernández E., Pelicer H.L.: Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations. Appl. Math. Lett. 18, 1265–1272 (2005)
Hu L., Ren Y.: Existence results for impulsive neutral stochastic functional integrodifferential equations with infinite delay. Acta Appl. Math. 111, 303–317 (2010)
Ichikawa A.: Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl. 90, 12–44 (1982)
Kannan D., Bharucha-Reid D.: On a stochastic integro-differential evolution of volterra type. J. Integral Equ. 10, 351–379 (1985)
Kolmanovskii V.B., Myshkis A.: Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Norwell (1992)
Lin A., Hu L.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions. Comput. Math. Appl. 59, 64–73 (2010)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE, vol. 16. Birkhäuser, Basel (1995)
N’Guérékata G.M.: Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces. Kluwer Academic Plenum Publishers, New York (2001)
Pazy A.: Semigroups of Linear Operators and Applications to Partial Equations, in: Applied Mathematical Sciences, Vol. 44. Springer, New York (1983)
Ren Y., Chen L.: A note on the neutral stochastic functional differential equations with infinite delay and Possion jumps in an abstract space. J. Math. Phys. 50, 082704 (2009)
Ren Y., Xia N.: Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. Appl. Math. Comput. 210, 72–79 (2009)
Sakthivel R., Kim J.-H., Mahmudov N.I.: On controllability of nonlinear stochastic systems. Rep. Math. Phys. 58, 433–443 (2006)
Sakthivel R., Luo J.: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl. 356, 1–6 (2009)
Sakthivel R., Luo J.: Asymptotic stability of nonlinear impulsive stochastic differential equations. Stat. Probab. Lett. 79, 1219–1223 (2009)
Sakthivel R., Nieto J.J., Mahmudov N.I.: Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay. Taiwan. J. Math. 14, 1777–1797 (2010)
Tudor C.: Almost periodic solutions of affine stochastic evolutions equations. Stoch. Stoch. Rep. 38, 251–266 (1992)
Xie B.: Stochastic differential equations with non-lopschitz coefficients in Hilbert spaces. Stoch. Anal. Appl. 26, 408–433 (2008)
Zhao Z.H., Chang Y.K., Li W.S.: Asymptotically almost periodic, almost periodic and pseudo almost periodic mild solutions for neutral differential equations. Nonlinear Anal. RWA 11, 3037–3044 (2010)
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Chang, YK., Ma, R. & Zhao, ZH. Almost Periodic Solutions to a Stochastic Differential Equation in Hilbert Spaces. Results. Math. 63, 435–449 (2013). https://doi.org/10.1007/s00025-011-0207-9
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DOI: https://doi.org/10.1007/s00025-011-0207-9