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).
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).
Reviewer: Claudio Cuevas (Pernambuco)
MSC:
| 34A08 | Fractional ordinary differential equations |
| 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 |
Keywords:
fractional stochastic differential equation; existence result; square-mean pseudo almost automorphic; mild solutionReferences:
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