zbMATH — the first resource for mathematics

Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations. (English) Zbl 1209.60034
Summary: We introduce a new concept of Stepanov-like almost automorphy (or \(S^{2}\)-almost automorphy) for stochastic processes. We use the results obtained to investigate the existence and uniqueness of a Stepanov-like almost automorphic mild solution to a class of nonlinear stochastic differential equations in a real separable Hilbert space. Our main results extend some known ones in the sense of square-mean almost automorphy.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Bochner, S., A new approach to almost automorphicity, Proc. natl. acad. sci. USA, 48, 2039-2043, (1962) · Zbl 0112.31401
[2] Bochner, S., Continuous mappings of almost automorphic and almost periodic functions, Proc. natl. acad. sci. USA, 52, 907-910, (1964) · Zbl 0134.30102
[3] N’Guérékata, G.M., Almost automorphic and almost periodic functions in abstract space, (2001), Kluwer Academic Plenum Publishers New York, London, Moscow · Zbl 1001.43001
[4] N’Guérékata, G.M., Topics in almost automorphy, (2005), Springer New York, Boston, Dordrecht, London, Moscow · Zbl 1073.43004
[5] Hernández, E.; Henríquez, H.R., Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. math. anal. appl., 221, 499-522, (1998) · Zbl 0926.35151
[6] 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) · Zbl 0944.34058
[7] Abbas, S.; Bahuguna, D., Almost periodic solutions of neutral functional differential equations, Comput. math. appl., 55, 2593-2601, (2008) · Zbl 1142.34367
[8] 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) · Zbl 1205.34088
[9] Zhao, Z.H.; Chang, Y.K.; Nieto, J.J., Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear anal. TMA, 72, 1886-1894, (2010) · Zbl 1189.34116
[10] N’Guérékata, G.M., Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup forum, 69, 80-86, (2004) · Zbl 1077.47058
[11] Diagana, T.; N’Guérékata, G.M., Almost automorphic solutions to semilinear evolution equations, Funct. differ. equ., 13, 195-206, (2006) · Zbl 1102.34044
[12] Diagana, T.; N’Guérékata, G.M., Almost automorphic solutions to some classes of partial evolution equations, Appl. math. lett., 20, 462-466, (2007) · Zbl 1169.35300
[13] Diagana, T.; N’Guérékata, G.M., Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. anal., 86, 723-733, (2007) · Zbl 1128.43006
[14] N’Guérékata, G.M.; Pankov, A., Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear anal., 68, 2658-2667, (2008) · Zbl 1140.34399
[15] Lee, H.; Alkahby, H., Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear anal., 69, 2158-2166, (2008) · Zbl 1162.34063
[16] Y.K. Chang, Z.H. Zhao, J.J. Nieto, Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Rev. Mat. Complut. (2010), doi:10.1007/s13163-010-0047-2. · Zbl 1232.34087
[17] Bezandry, P.; Diagana, T., Existence of almost periodic solutions to some stochastic differential equations, Appl. anal., 86, 819-827, (2007) · Zbl 1130.34033
[18] Bezandry, P., Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Statist. probab. lett., 78, 2844-2849, (2008) · Zbl 1156.60046
[19] 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) · Zbl 1183.34080
[20] 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., 30, 21-30, (1988), 115 · Zbl 0900.60072
[21] Da Prato, G.; Tudor, C., Periodic and almost periodic solutions for semilinear stochastic evolution equations, Stoch. anal. appl., 13, 13-33, (1995) · Zbl 0816.60062
[22] Tudor, C., Almost periodic solutions of affine stochastic evolutions equations, Stoch. stoch. rep., 38, 251-266, (1992) · Zbl 0752.60049
[23] Tudor, C.A.; Tudor, M., Pseudo almost periodic solutions of some stochastic differential equations, Math. rep. (bucur.), 1, 305-314, (1999) · Zbl 1019.60058
[24] M.M. Fu, Z.X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc. (in press), arXiv:1001.3049v1 [math.DS]. · Zbl 1202.60109
[25] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[26] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.