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Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations. (English) Zbl 1243.34006
This paper is concerned with the existence and uniqueness of square-mean pseudo almost automorphic (mild) solutions for a class of fractional stochastic differential equations in a Hilbert space. The authors have used stochastic analysis and fixed point theory to get their results. This paper is well written and it is nice to read. From an applied perspective it is of great significance to introduce stochastic effects in the investigation of fractional differential equations and develop a study of type of periodicity and ergodicity for these equations. Such subject is one of most interesting topics in the theory of evolution equations both due to its theoretical interest as well as due to their concrete applications. We observe that the existence of almost automorphic, pseudo-almost periodic and pseudo-almost automorphic solutions to fractional differential equations has been considered only in a few publications. Some complementary references to this paper are the following:
R. P. Agarwal, B. de Andrade and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations”, Adv. Difference Equ. 2010, Article ID 179750, 25 p. (2010; Zbl 1194.34007).
R. P. Agarwal, B. de Andrade and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations”, Nonlinear Anal., Real World Appl. 11, No. 5, 3532–3554 (2010; Zbl 1248.34004).
C. Cuevas, G. N’Guérékata and A. Sepulveda,“Pseudo almost automorphic solutions to fractional differential and integro-differential equations”, Commun. Appl. Anal. 16, No. 1, 131–152 (2012).
C. Cuevas, H. Soto and A. Sepulveda, “Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations”, Appl. Math. Comput. 218, No. 5, 1735–1745 (2011; Zbl 1246.45012).

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
34F05 Ordinary differential equations and systems with randomness
34G20 Nonlinear differential equations in abstract spaces
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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[1] Bochner, S., A new approach to almost automorphicity, Proc. natl. acad. sci. USA, 48, 2039-2043, (1962) · Zbl 0112.31401
[2] Bezandry, P.H.; Diagana, T., Existence of almost periodic solutions to some stochastic differential equations, Appl. anal., 86, 819-827, (2007) · Zbl 1130.34033
[3] Diagana, T.; Hernndez, E.; Rabello, M., Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay, Math. comput. modelling, 45, 1241-1252, (2007) · Zbl 1133.34042
[4] Diagana, T., Almost periodic solutions to some second-order nonautonomous differential equations, Proc. amer. math. soc., 140, 279-289, (2012) · Zbl 1243.34059
[5] Goldstein, J.A.; N’Guérékata, G.M., Almost automorphic solutions of semilinear evolution equations, Proc. amer. math. soc., 133, 2401-2408, (2005) · Zbl 1073.34073
[6] 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
[7] N’Guérékata, G.M., Topics in almost automorphy, (2005), Springer New York, Boston, Dordrecht, London, Moscow · Zbl 1073.43004
[8] Xiao, T.J.; Liang, J.; Zhang, J., Pseudo almost automorphic solutions to semilinear differential equations in Banach space, Semigroup forum, 76, 518-524, (2008) · Zbl 1154.46023
[9] Xiao, T.J.; Zhu, X.X.; Liang, J., Pseudo almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear anal., 70, 4079-4085, (2009) · Zbl 1175.34076
[10] Diagana, T., Existence of pseudo-almost automorphic solutions to some abstract differential equations with pseudo-almost automorphic coefficients, Nonlinear anal., 70, 3781-3790, (2009) · Zbl 1178.43004
[11] Liang, J.; N’Guérékata, G.M.; Xiao, T.J.; Zhang, J., Some properties of pseudo-almost automorphic functions and applications to abstract differential equations, Nonlinear anal., 70, 2731-2735, (2009) · Zbl 1162.44002
[12] Liu, J.H.; Song, X.Q., Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations, J. funct. anal., 258, 196-207, (2010) · Zbl 1194.47047
[13] Zhao, Z.H.; Chang, Y.K.; Nieto, J.J., Almost automorphic and pseudo almost auto-morphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear anal. TMA, 72, 1886-1894, (2010) · Zbl 1189.34116
[14] Øksendal, B.K., Stochastic differential equations, (2005), Springer Berlin, Heidelberg
[15] Bezandry, P.H.; Diagana, T., Square-Mean almost periodic solutions nonautonomous stochastic differential equations, Electron J. differential equations, 117, 1-10, (2007) · Zbl 1138.60323
[16] 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, Comput. math. appl., 61, 384-391, (2011) · Zbl 1211.60025
[17] Chang, Y.K.; Zhao, Z.H.; NGurkata, 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
[18] Chang, Y.-K.; Zhao, Z.-H.; N’Guérékata, G.M., A new composition theorem for square-Mean almost automorphic functions and applications to stochastic differential equations, Nonlinear anal., 74, 2210-2219, (2011) · Zbl 1217.60043
[19] 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
[20] Chen, Z.; Lin, W., Square-Mean pseudo almost automorphic process and its application to stochastic evolution equations, J. funct. anal., 261, 69-89, (2011) · Zbl 1233.60030
[21] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[22] Ahmed, H.M., On some fractional stochastic integrodifferential equations in Hilbert space, Int. J. math. math. sci., 568078, (2009) · Zbl 1179.60038
[23] El-Borai, M.M.; EI-Said EI-Nadi, K.; Fouad, H.A., On some fractional stochastic delay differential equations, Comput. math. appl., 59, 1165-1170, (2010) · Zbl 1189.60117
[24] El-Borai, M.M.; EI-Said EI-Nadi, K.; Mostafa, O.L.; Ahmed, H.M., Volterra equations with fractional stochastic integrals, Math. probl. eng., 5, 453-468, (2004) · Zbl 1081.45007
[25] M.M. El-Borai, K. EI-Said EI-Nadi, E.G. El-Akabawy, On some fractional evolution equations 59 (2010) 1352-1355. · Zbl 1189.45009
[26] Cui, J.; Yan, L., Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. phys. A: math. theor., 44, 335201, (2011)
[27] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear anal., 69, 3692-3705, (2008) · Zbl 1166.34033
[28] Cuevas, C.; Lizama, C., Almost automorphic mild solutions to a class of fractional differential equations, Appl. math. lett., 21, 1315-1319, (2008) · Zbl 1192.34006
[29] Chen, A.; Chen, F.; Deng, S., On almost automorphic mild solutions for fractional semilinear initial value problems, Comput. math. appl., 59, 1318-1325, (2010) · Zbl 1189.34079
[30] 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
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