×

On the existence of almost automorphic solutions of nonlinear stochastic Volterra difference equations. (English) Zbl 1274.39004

Summary: In this paper, we introduce a concept of almost automorphy for random sequences. Using the Banach contraction principle, we establish the existence and uniqueness of an almost automorphic solution to some Volterra stochastic difference equation in a Banach space. Our main results extend some known ones in the sense of mean almost automorphy. As an application, almost automorphic solution to a concrete stochastic difference equation is analyzed to illustrate our abstract results.

MSC:

39A10 Additive difference equations
60G07 General theory of stochastic processes
34F05 Ordinary differential equations and systems with randomness
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] D. Araya, R.Castro, and C. Lizama, Almost automorphic solutions of difference equations, Adv, Difference Equ. , 2009, 15pp · Zbl 1177.39002
[2] P. Bezandry and T. Diagana, Almost periodic stochastic processes , Springer, New York, 2011. · Zbl 1237.60002
[3] P. Bezandry, T. Diagana, and S. Elaydi, On the stochastic Beverton-Holt equation with survival rates, J. Difference Eq. Appl. 14, no. 2 (2008), pp. 175-190. · Zbl 1144.39011
[4] S. Bochner, A new approach to almost automorhy, Proc., Natl. Acad, Sci. USA 48 (1962), 2039-2043. · Zbl 0112.31401
[5] C. Corduneanu, Almost periodic functions , 2nd Edition, Chelsea-New York, 1989. · Zbl 0672.42008
[6] C. Cuevas, H. Henriquez, and C. Lizama, On the existence of almost automorphic solutions of Volterra difference equations, J. Difference Equations and Appl. , 10, no. 11 (2012), 1931-1946. · Zbl 1261.39007
[7] T. Diagana, Existence of globally attracting almost automorphic solutions to some nonautonomous higher-order difference equations, Appl. Math. Comput. 219 (2013), 6510-6519. · Zbl 1293.39010
[8] T. Diagana, S. Elaydi, and A-A Yakubu, Population models in almost periodic environments, J. Difference Equations and Appl. , 13, no. 4 (2007), 239-260. · Zbl 1120.39017
[9] T. Diagana and G. N’Guereketa, Almost automorphic solutions to semilinear evolution equations, Funct. Differ. Equ. 13 (2006), 195-206. · Zbl 1102.34044
[10] J. Han and C. Hong, Almost periodic random sequences in probability, J. Math. Anal. Appl. , 336 (2007), 962-974. · Zbl 1127.60029
[11] J. Hong and C. Nunez, The almost periodic type difference equations, Mathl. Comput. Modeling , Vol. 28, No. 12 (1998), pp. 21-31. · Zbl 0992.39003
[12] G. N’Guerekata, Almost automorphic and almost periodic functions in abstract spaces , Kluwer Academic Plenum Publishers, New York, London, Moscou (2001).
[13] G. N’Guerekata, Topics in almost automorphy , Springer, New York, Boston, Dordrecht, London, Moscou, 2005.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.