×

zbMATH — the first resource for mathematics

A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations. (English) Zbl 1217.60043
Summary: We establish a new composition theorem for square-mean almost automorphic functions under conditions which are different from Lipschitz conditions in the literature. We apply this new composition theorem together with Schauder’s fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation in a real separable Hilbert space. Finally, an interesting corollary is given for the sub-linear growth cases.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35B15 Almost and pseudo-almost periodic solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[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] Agarwal, R.P.; Diagana, T.; Hernández, E., Weighted pseudo almost periodic solutions to some partial neutral functional differential equations, J. nonlinear convex anal., 8, 397-415, (2007) · Zbl 1155.35104
[6] Blot, J.; Cieutat, P.; N’Guérékata, G.M., Dependence results on almost periodic and almost automorphic solutions of evolution equations, Electron. J. differential equations, 2010, 1-13, (2010) · Zbl 1201.47076
[7] Abbas, S.; Bahuguna, D., Almost periodic solutions of neutral functional differential equations, Comput. math. appl., 55, 2593-2601, (2008) · Zbl 1142.34367
[8] Li, H.X.; Huang, F.L.; Li, J.Y., Composition of pseudo almost periodic functions and semilinear differential equations, J. math. anal. appl., 255, 436-446, (2001) · Zbl 1047.47030
[9] 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
[10] 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
[11] 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
[12] Diagana, T.; N’Guérékata, G.M., Almost automorphic solutions to semilinear evolution equations, Funct. differ. equ., 13, 195-206, (2006) · Zbl 1102.34044
[13] 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
[14] Chang, Y.K.; Zhao, Z.H.; Nieto, J.J., Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Rev. mat. complut., (2010)
[15] Bezandry, P.; Diagana, T., Existence of almost periodic solutions to some stochastic differential equations, Appl. anal., 86, 819-827, (2007) · Zbl 1130.34033
[16] 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
[17] 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
[18] 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
[19] 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
[20] Tudor, C., Almost periodic solutions of affine stochastic evolutions equations, Stoch. stoch. rep., 38, 251-266, (1992) · Zbl 0752.60049
[21] Tudor, C.A.; Tudor, M., Pseudo almost periodic solutions of some stochastic differential equations, Math. rep. (bucur.), 1, 305-314, (1999) · Zbl 1019.60058
[22] Fu, M.M.; Liu, Z.X., Square-Mean almost automorphic solutions for some stochastic differential equations, Proc. amer. math. soc., 138, 3689-3701, (2010) · Zbl 1202.60109
[23] Chang, Y.K.; Zhao, Z.H.; N’Guérékata, G.M., Square-Mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comp. math. appl., (2010)
[24] Chang, Y.K.; Zhao, Z.H.; N’Guérékata, G.M.; Ma, R., Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear anal. RWA, 12, 1130-1139, (2011) · Zbl 1209.60034
[25] Liang, J.; Zhang, J.; Xiao, T.J., Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. math. anal. appl., 340, 1493-1499, (2008) · Zbl 1134.43001
[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.