Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay

Ya Wang; Fuke Wu; Xuerong Mao; Enwen Zhu

doi:10.3934/dcdsb.2019182

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January 2020, 25(1): 287-300. doi: 10.3934/dcdsb.2019182

Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay

Ya Wang ^1, , Fuke Wu ^1,, , Xuerong Mao ^2, and Enwen Zhu ^3,

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2.	Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
3.	School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410004, China

¹Corresponding author

Received January 2019 Published January 2020 Early access July 2019

Fund Project: The research was supported in part by the National Natural Science Foundations of China (Grant Nos. 1161101211 and 61873320), and the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship).

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.

Keywords: Stochastic functional differential equations, the LaSalle-type theorem, infinite delay, asymptotic stability, nonnegative semimartingale convergence theorem.

Mathematics Subject Classification: Primary: 34K50, 60H10; Secondary: 34D45, 37C75.

Citation: Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 287-300. doi: 10.3934/dcdsb.2019182

References:

[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.

show all references

References:

[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.

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