Abstract and Applied Analysis

Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

Aimin Liu, Yongjian Liu, and Qun Liu

Full-text: Open access

Abstract

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations d x t = A t x t + F t , x t , x t d t + H ( t , x t , x t ) d W ( t ) . A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup { T t } t 0 is essentially removed, which is generated by the linear densely defined operator A D ( A ) L 2 ( , ) L 2 ( , ) , only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 934534, 11 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425048246

Digital Object Identifier
doi:10.1155/2014/934534

Mathematical Reviews number (MathSciNet)
MR3208574

Zentralblatt MATH identifier
07023344

Citation

Liu, Aimin; Liu, Yongjian; Liu, Qun. Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 934534, 11 pages. doi:10.1155/2014/934534. https://projecteuclid.org/euclid.aaa/1425048246


Export citation

References

  • A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1974.
  • C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, China, 2006.
  • P. H. Bezandry, “Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations,” Statistics & Probability Letters, vol. 78, no. 17, pp. 2844–2849, 2008.
  • E. Slutsky, “Sur les fonctions aléatoires presque périodiques et sur la decomposition des functions aléatoires,” in Actualités Scientifiques et Industrielles, vol. 738, pp. 33–55, Herman, Paris, France, 1938.
  • M. Udagawa, “Asymptotic properties of distributions of some functionals of random variables,” Reports of Statistical Application Research. Union of Japanese Scientists and Engineers, vol. 2, no. 2-3, pp. 1–98, 1952.
  • T. Kawata, “Almost periodic weakly stationary processes,” in Statistics and Probability: Essays in Honor of C. R. Rao, pp. 383–396, North-Holland, New York, NY, USA, 1982.
  • R. J. Swift, “Almost periodic harmonizable processes,” Georgian Mathematical Journal, vol. 3, no. 3, pp. 275–292, 1996.
  • P. H. Bezandry and T. Diagana, “Square-mean almost periodic solution to some nonautonomous stochastic differential equations,” Electronic Journal of Differential Equations, vol. 2007, no. 117, pp. 1–10, 2007.
  • J. Luo, “Stochastically bounded solutions of a nonlinear stochastic differential equations,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 87–93, 2006.
  • W. M. Ruess, “Asymptotic almost periodicity and motions of semigroups of operators,” Linear Algebra and Its Applications, vol. 84, pp. 335–351, 1986.
  • W. M. Ruess and V. Q. Phong, “Asymptotically almost periodic solutions of evolution equations in Banach spaces,” Journal of Differential Equations, vol. 122, no. 2, pp. 282–301, 1995.
  • W. Arendt and C. J. K. Batty, “Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line,” Bulletin of the London Mathematical Society, vol. 31, no. 3, pp. 291–304, 1999.
  • W. Arendt and C. J. K. Batty, “Almost periodic solutions of first- and second-order Cauchy problems,” Journal of Differential Equations, vol. 137, no. 2, pp. 363–383, 1997.
  • J. Cao, Q. Yang, Z. Huang, and Q. Liu, “Asymptotically almost periodic solutions of stochastic functional differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1499–1511, 2011.
  • J. Cao, Q. Yang, and Z. Huang, “On almost periodic mild solutions for stochastic functional differential equations,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 275–286, 2012.
  • H. Huang and J. Cao, “Exponential stability analysis of uncertain stochastic neural networks with multiple delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 2, pp. 646–653, 2007.
  • J. Cao, Q. Yang, and Z. Huang, “Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations,” Stochastics, vol. 83, no. 3, pp. 259–275, 2011.
  • P. H. Bezandry and T. Diagana, “Existence of almost periodic solutions to some stochastic differential equations,” Applicable Analysis, vol. 86, no. 7, pp. 819–827, 2007.
  • P. H. Bezandry and T. Diagana, “Existence of quadratic-mean almost periodic solutions to some stochastic hyperbolic differential equations,” Electronic Journal of Differential Equations, vol. 2009, no. 111, pp. 1–14, 2009.
  • P. H. Bezandry and T. Diagana, “Existence of square-mean almost periodic mild solutions to some nonautonomous stochastic second-order differential equations,” Electronic Journal of Differential Equations, vol. 2010, no. 124, pp. 1–25, 2010.
  • M. Fu and Z. Liu, “Square-mean almost automorphic solutions for some stochastic differential equations,” Proceedings of the American Mathematical Society, vol. 138, no. 10, pp. 3689–3701, 2010.
  • Y. Liu and A. Liu, “Almost periodic solutions for a class of stochastic differential equations,” Journal of Computational and Nonlinear Dynamics, vol. 8, no. 4, Article ID 041002, 6 pages, 2013.
  • S. Elaydi and O. Hájek, “Exponential trichotomy of differential systems,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 362–374, 1988.
  • J. Hong, R. Obaya, and A. Sanz, “Existence of a class of ergodic solutions implies exponential trichotomy,” Applied Mathematics Letters, vol. 12, no. 4, pp. 43–45, 1999.
  • L. Barreira and C. Valls, “Lyapunov functions for trichotomies with growth rates,” Journal of Differential Equations, vol. 248, no. 1, pp. 151–183, 2010. \endinput