THE REVIEW OF ALMOST PERIODIC SOLUTIONS TO A STOCHASTIC DIERENTIAL EQUATION

محورهای موضوعی : فصلنامه ریاضی

Siavosh Ahmadi ¹ (Tehran north branch-Azad University-iran Iran, Islamic Republic of I'm an Assistant Professor of Applied Mathematics at the University of Azad university. I received my doctorate in Fluid mechanics fro pune University. My recent publication include "Tornado Dynamics" in the journal of karaj Azad university.)

تاریخ دریافت : 1395/01/27 تاریخ پذیرش : 1395/01/27 تاریخ انتشار : 1395/01/01

کلید واژه: Stochastic dierential equations, quadratic mean almost periodic, analytic semigroups of linear operators,

چکیده مقاله :

This paper proves the existence and uniqueness of quadratic mean almost periodic mild so-lutions for a class of stochastic dierential equations in a real separable Hilbert space. Themain technique is based upon an appropriate composition theorem combined with the Banachcontraction mapping principle and an analytic semigroup of linear operators.

منابع و مأخذ:

Acquistapace, P., Terreni, B.: A unied approach to abstract linear nonautonomous parabolic equa-

tions. Rend. Sem. Mat. Univ. Padova 78, 47-107 (1987)

Bezandry, P., Diagana, T.: Existence of almost periodic solutions to some stochastic dierential equa-

tions. Appl. Anal. 86, 819-827 (2007)

Bezandry, P., Diagana, T.: Square-mean almost periodic solutions nonautonomous stochastic dier-

ential equations. Electron. J. Dier. Equ. 2007, 1-10 (2007)

REFERENCES 11

Bezandry, P.: Existence of almost periodic solutions to some functional integro- dierential stochastic

evolution equations. Stat. Probab. Lett. 78, 2844- 2849 (2008)

Bezandry, P., Diagana, T.: Existence of S2-almost periodic solutions to a class of nonautonomous

stochastic evolution equations, Electron. J. Qual. Theory Dier. Equ. 35, 1-19 (2008)

Bezandry, P., Diagana, T.: Existence of quadratic-mean almost periodic solutions to some stochastic

hyperbolic dierential equations. Electron. J. Dier. Equ. 2009, 1-14 (2009) 13

Cao, J., Yang, Q., Huang, Z., Liu, Q.: Asymptotically almost periodic solutions of stochastic functional

dierential 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 Innite 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)

Hernandez 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. Dier. Equ. 2004 1-15 (2004)

Hernandez, E., Pelicer, H.L.: Asymptotically almost periodic and almost periodic solutions for partial

neutral dierential equations. Appl. Math. Lett. 18, 1265- 1272 (2005)

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-dierential evolution of volterra type. J.

Integral Equ. 10, 351-379 (1985)

Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE, vol. 16.

Birkhauser, Basel (1995)

N'Guerekata, G.M.: Almost Automorphic Functions and Almost Periodic Functions in Abstract

Spaces. Kluwer Academic Plenum Publishers, New York (2001) 13 Vol. 63 (2013) Almost Periodic

Solutions to a Stochastic Dierential Equation 449

Pazy, A.: Semigroups of Linear Operators and Applications to Partial Equations, in: Applied Math-

ematical Sciences, Vol. 44. Springer, New York (1983)

Ren, Y., Xia, N.: Existence, uniqueness and stability of the solutions to neutral stochastic functional

dierential equations with innite 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 dierential equations with

innite delays. J. Math. Anal. Appl. 356, 1-6 (2009)

Sakthivel, R., Luo, J.: Asymptotic stability of nonlinear impulsive stochastic dierential equations.

Stat. Probab. Lett. 79, 1219-1223 (2009)

Tudor, C.: Almost periodic solutions of ane stochastic evolutions equations. Stoch. Stoch. Rep. 38,

-266 (1992)

Xie, B.: Stochastic dierential equations with non-lopschitz coecients 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 dierential equations. Nonlinear Anal. RWA 11, 3037-3044