This paper considers stochastic functional differential equations (SFDEs) with infinite delay. The main aim is to establish the LaSalle-type theorems to locate limit sets for this class of SFDEs. In comparison with the existing results, this paper gives more general results under the weaker conditions imposed on the Lyapunov function. These results can be used to discuss the asymptotic stability and asymptotic boundedness for SFDEs with infinite delay. In the end, two examples will be given to illustrate applications of our new results established.
Citation: |
[1] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974.
![]() ![]() |
[2] |
A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1976.
doi: 10.1007/978-3-642-11079-5_2.![]() ![]() ![]() |
[3] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7.![]() ![]() ![]() |
[4] |
Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0084432.![]() ![]() ![]() |
[5] |
V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, London, 1986.
![]() ![]() |
[6] |
J. P. LaSalle, Stability theory for ordinary differential equations, Journal of Differential Equations, 4 (1968), 57-65.
doi: 10.1016/0022-0396(68)90048-X.![]() ![]() ![]() |
[7] |
X. Li and X. Mao, The improved lasalle-type theorems for stochastic differential delay equations, Stochastic Analysis and Applications, 30 (2012), 568-589.
doi: 10.1080/07362994.2012.684320.![]() ![]() ![]() |
[8] |
X. Mao, Stochastic versions of the lasalle theorem, Journal of Differential Equations, 153 (1999), 175-195.
doi: 10.1006/jdeq.1998.3552.![]() ![]() ![]() |
[9] |
X. Mao, Lasalle-type theorems for stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 236 (1999), 350-369.
doi: 10.1006/jmaa.1999.6435.![]() ![]() ![]() |
[10] |
X. Mao, A note on the lasalle-type theorems for stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 268 (2002), 125-142.
doi: 10.1006/jmaa.2001.7803.![]() ![]() ![]() |
[11] |
X. Mao, The lasalle-type theorems for stochastic functional differential equations, Nonlinear Studies, 7 (2000), 307-328.
![]() ![]() |
[12] |
X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood, Chichester, 2008.
doi: 10.1016/B978-1-904275-34-3.50013-X.![]() ![]() ![]() |
[13] |
X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Processes and their Applications, 65 (1996), 233-250.
doi: 10.1016/S0304-4149(96)00109-3.![]() ![]() ![]() |
[14] |
S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman (Advanced Publishing Program), Boston, MA, 1984.
![]() ![]() |
[15] |
Y. Shen, Q. Luo and X. Mao, The improved lasalle-type theorems for stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 318 (2006), 134-154.
doi: 10.1016/j.jmaa.2005.05.026.![]() ![]() ![]() |
[16] |
F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 331 (2007), 516-531.
doi: 10.1016/j.jmaa.2006.09.020.![]() ![]() ![]() |
[17] |
F. Wu and S. Hu, The lasalle-type theorem for neutral stochastic functional differential equations with infinite delay, Discrete and Continuous Dynamical Systems, Series A, 32 (2012), 1065-1094.
doi: 10.3934/dcds.2012.32.1065.![]() ![]() ![]() |
[18] |
F. Wu, G. Yin and H. Mei, Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, markov properties, and ergodicity, Journal of Differential Equations, 262 (2017), 1226-1252.
doi: 10.1016/j.jde.2016.10.006.![]() ![]() ![]() |