×

zbMATH — the first resource for mathematics

Asymptotically almost periodic solutions of stochastic functional differential equations. (English) Zbl 1230.60058
Summary: We investigate a class of stochastic functional differential equations of the form \[ dx(t)=(Ax(t)+F(t,x(t),x_{t}))dt+G(t,x(t),x_{t})\circ dW(t). \] Our main results concern the existence and exponential stability of quadratic-mean asymptotically almost periodic mild solutions. An example is given to illustrate our results.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fink, A., Almost periodic differential equations, () · Zbl 0325.34039
[2] Arendt, W.; Batty, C., Almost periodic solutions of first and second order Cauchy problems, J. differential equations, 137, 363-383, (1997) · Zbl 0879.34046
[3] Hino, Y.; Murakami, S.; Yoshizawa, T., Almost periodic solutions of abstract functional differential equations with infinite delay, Nonlinear anal., 30, 853-864, (1997) · Zbl 0891.34076
[4] Levitan, B.; Zhikov, V., Almost periodic functions and differential equations, (1982), Cambridge University Press Cambridge · Zbl 0499.43005
[5] Yoshizawa, T., Stability theory and the existence of periodic solutions and almost periodic solutions, () · Zbl 0183.09602
[6] Prüss, J., Evolutionary integral equations and applications, (1993), Birkhäuser Basel · Zbl 0793.45014
[7] Zhang, C., Almost periodic type functions and ergodicity, (2003), Science Press/Kluwer Acad. Publ. Beijing/Dordrecht · Zbl 1068.34001
[8] Engel, K.; Nagel, R., One-parameter semigroups for linear evolution equations, (1999), Springer Berlin
[9] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied mathematical science, vol. 44, (1983), Springer-Verlag Berlin, New York · Zbl 0516.47023
[10] Slutsky, E., Sur LES fonctions aléatoires presque périodiques et sur la decomposition des functions aléatoires, Actualités sceintiques et industrielles, 738, (1938), Herman Paris, pp. 33-55
[11] Udagawa, M., Asymptotic properties of distributions of some functionals of random variable, Rep. statist. appl. res. union jap. sci. eng., 2, 1-98, (1952)
[12] Kawata, T., Almost periodic weakly stationary processes, (), 383-396
[13] Swift, R., Almost periodic harmonizable processes, Georgian math. J., 3, 275-292, (1996) · Zbl 0854.60036
[14] Bezandry, P.; Diagana, T., Existence of almost periodic solutions to some stochastic differential equations, Appl. anal., 117, 1-10, (2007)
[15] Huang, Z.; Yang, Q., Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay, Chaos solitons fract., 42, 773-780, (2009) · Zbl 1198.60024
[16] Christopher, T.; Baker, H.; Buckwar, E., Exponential stability in pth Mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. comput. appl. math., 184, 404-427, (2005) · Zbl 1081.65011
[17] Bao, H.; Cao, J., Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. math. comput., 215, 1732-1743, (2009) · Zbl 1195.34123
[18] Ren, Y.; Xia, N., A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay, Appl. math. comput., 214, 457-461, (2009) · Zbl 1221.34222
[19] Luo, J., A note on exponential stability in pth Mean of solutions of stochastic delay differential equations, J. comput. appl. math., 198, 143-148, (2007) · Zbl 1110.65009
[20] Ren, Y.; Xia, N., Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. math. comput., 210, 72-79, (2009) · Zbl 1167.34389
[21] Ruess, W.; Summers, W., Asymptotic almost periodicity and motions of semigroups of operators, Linear algebra appl., 84, 335-351, (1986) · Zbl 0616.47047
[22] Ruess, W.; Vu, Q., Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. differ. eqs., 122, 282-301, (1995) · Zbl 0837.34067
[23] Arendt, W.; Batty, C., Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line, Bull. London math. soc., 31, 291-304, (1999) · Zbl 0952.34048
[24] Minh, N.; N’Guérékata, G.; Yuan, R., Lectures on the asymptotically behavior of solutions of differential equations, (2008), Science Publishers, Inc
[25] Liu, Q.; Yuan, R., Asymptotic behavior of solutions to abstract functional differential equations, J. math. anal. appl., 356, 405-417, (2009) · Zbl 1209.34094
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.