American Institute of Mathematical Sciences

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March  2012, 32(3): 1065-1094. doi: 10.3934/dcds.2012.32.1065

The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay

Fuke Wu 1, and  Shigeng Hu 1,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received  September 2010 Revised  December 2010 Published  October 2011

The main aim of this paper is to establish the LaSalle-type theorem to locate limit sets for neutral stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness and the almost sure stability with general decay rate and robustness are obtained. To make our theory more applicable, by the $M$-matrix theory, this paper also examines some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. By specializing the general decay rate as the exponential decay rate and the polynomial decay rate, this paper examines two neutral stochastic integral-differential equations and shows that they are exponentially attractive and polynomially stable, respectively.
Keywords: LaSalle-type theorem, attractor, neutral stochastic functional differential equations, the Itô formula., infinite delay.
Mathematics Subject Classification: Primary: 34K50, 34K40; Secondary: 34D45, 37C7.
Citation: Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065
References:
[1]

J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear Itô-Volterra equations with damped stochastic perturbations, Electron. J. Probab., 8 (2003), 22 pp.  Google Scholar

[2]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,'' Wiley, New York, 1972. Google Scholar

[3]

H. Bao and J. Cao, Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732-1743 doi: 10.1016/j.amc.2009.07.025.  Google Scholar

[4]

H. Bereketoglu and I. Győri, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210 (1997), 279-291. doi: 10.1006/jmaa.1997.5403.  Google Scholar

[5]

A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences,'' SIAM, Philadelphia, PA, 1994. Google Scholar

[6]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, System Control Lett., 48 (2003), 397-406. doi: 10.1016/S0167-6911(02)00293-1.  Google Scholar

[7]

F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, System Control Lett., 57 (2008), 262-270. doi: 10.1016/j.sysconle.2007.09.002.  Google Scholar

[8]

S. Fang and T. Zhang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Related Fields, 132 (2005), 356-390. doi: 10.1007/s00440-004-0398-z.  Google Scholar

[9]

A. Friedman, "Stochastic Differential Equations and their Applications,'' Vol. 2, Academic Press, New York, 1976. Google Scholar

[10]

K. Gopalsamy, "Stability and Oscillation in Delay Differential Equations of Population Dynamics,'' Kluwer Academic, Dordrecht, 1992. Google Scholar

[11]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,'' Springer, Berlin, 1993. Google Scholar

[12]

X. He, The Lyapunov functionals for delay Lotka-Volterra-type models, SIAM J. Appl. Math., 58 (1998), 1222-1236. doi: 10.1137/S0036139995295116.  Google Scholar

[13]

Y. Hu, F. Wu and C. Huang, Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay, Automatica, 45 (2009), 2577-2584. doi: 10.1016/j.automatica.2009.07.007.  Google Scholar

[14]

O. Kallenberg, "Foundations of Modern Probability,'' Springer-Verlag, New York, 1997. Google Scholar

[15]

R. Z. Khasminskii, "Stochastic Stability of Differential Equations,'' Sijthoff and Noordhoff, Alphen a/d Rijn, 1981. Google Scholar

[16]

V. B. Kolmanovskii and V. R. Nosov, "Stability of Functional Differential Equations,'' Academic Press, New York, 1986. Google Scholar

[17]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246. doi: 10.1006/jdeq.1993.1048.  Google Scholar

[18]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'' Academic press, Boston, 1993. Google Scholar

[19]

J. P. LaSalle, "Stability Theory of Ordinary Differential Equations,'' J. Differential Equations, 4 (1968), 57-65. doi: 10.1016/0022-0396(68)90048-X.  Google Scholar

[20]

R. Sh. Liptser and A. N. Shiryaev, "Theory of Martingale,'' Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3.  Google Scholar

[21]

Y. Liu, X. Meng and F. Wu, Some stability criteria of stochastic functional differential equations with infinite delay, J. Appl. Math. Stoch. Anal., 2010. doi: 10.1155/2010/875908.  Google Scholar

[22]

X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingale,'' Wiley, New York, 1991. Google Scholar

[23]

X. Mao, Almost sure polynomial stability for a class of stochastic differential equations, Quart. J. Math. Oxford. Ser. (2), 43 (1992), 339-348.  Google Scholar

[24]

X. Mao, "Exponential Stability of Stochastic Differential Equations,'' Dekker, New York, 1994. Google Scholar

[25]

X. Mao, Exponential stability in mean square of neutral stochastic differential-functional equations, System Control Lett., 26 (1995), 245-251. doi: 10.1016/0167-6911(95)00018-5.  Google Scholar

[26]

X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional-differential equations, SIAM J. Math. Anal., 28 (1997), 389-401. doi: 10.1137/S0036141095290835.  Google Scholar

[27]

X. Mao, "Stochastic Differential Equations and Applications,'' Horwood, Chichester, 1997. Google Scholar

[28]

X. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations, 153 (1999), 175-195. doi: 10.1006/jdeq.1998.3552.  Google Scholar

[29]

X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud., 7 (2000), 307-328.  Google Scholar

[30]

X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Anal., 47 (2001), 4795-4806. doi: 10.1016/S0362-546X(01)00591-0.  Google Scholar

[31]

X. Mao, Some contributions to stochastic asymptotic stability and boundedness via multiple Lyapunov functions, J. Math. Anal. Appl., 260 (2001), 325-340. doi: 10.1006/jmaa.2001.7451.  Google Scholar

[32]

X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142. doi: 10.1006/jmaa.2001.7803.  Google Scholar

[33]

X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069. doi: 10.1080/07362990500118637.  Google Scholar

[34]

X. Mao and M. Riedle, Mean square stability of stochastic Volterra integro-differential equations, System Control Lett., 55 (2006), 459-465. doi: 10.1016/j.sysconle.2005.09.009.  Google Scholar

[35]

S.-E. A. Mohammed, "Stochastic Functional Differential Equations,'' Longman, Harlow/New York, 1986. Google Scholar

[36]

Y. Shen, Q. Luo and X. Mao, The improved LaSalle-type theorems for stochastic functional differential equations, J. Math. Anal. Appl., 318 (2006), 134-154. doi: 10.1016/j.jmaa.2005.05.026.  Google Scholar

[37]

J. Randjelović and S. Janković, On the $p$th moment exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl., 326 (2007), 266-280. Google Scholar

[38]

Y. Ren and N. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math., 220 (2008), 364-372. doi: 10.1016/j.cam.2007.08.022.  Google Scholar

[39]

F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331 (2007), 516-531. doi: 10.1016/j.jmaa.2006.09.020.  Google Scholar

[40]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657. doi: 10.1137/080719194.  Google Scholar

[41]

F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275.  Google Scholar

[42]

F. Wu, S. Hu and C. Huang, Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay, System Control Lett., 59 (2010), 195-202. doi: 10.1016/j.sysconle.2010.01.004.  Google Scholar

[43]

S. Zhou, Z. Wang and D. Feng, Stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 357 (2009), 416-426. doi: 10.1016/j.jmaa.2009.04.015.  Google Scholar

show all references

References:
[1]

J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear Itô-Volterra equations with damped stochastic perturbations, Electron. J. Probab., 8 (2003), 22 pp.  Google Scholar

[2]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,'' Wiley, New York, 1972. Google Scholar

[3]

H. Bao and J. Cao, Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732-1743 doi: 10.1016/j.amc.2009.07.025.  Google Scholar

[4]

H. Bereketoglu and I. Győri, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210 (1997), 279-291. doi: 10.1006/jmaa.1997.5403.  Google Scholar

[5]

A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences,'' SIAM, Philadelphia, PA, 1994. Google Scholar

[6]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, System Control Lett., 48 (2003), 397-406. doi: 10.1016/S0167-6911(02)00293-1.  Google Scholar

[7]

F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, System Control Lett., 57 (2008), 262-270. doi: 10.1016/j.sysconle.2007.09.002.  Google Scholar

[8]

S. Fang and T. Zhang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Related Fields, 132 (2005), 356-390. doi: 10.1007/s00440-004-0398-z.  Google Scholar

[9]

A. Friedman, "Stochastic Differential Equations and their Applications,'' Vol. 2, Academic Press, New York, 1976. Google Scholar

[10]

K. Gopalsamy, "Stability and Oscillation in Delay Differential Equations of Population Dynamics,'' Kluwer Academic, Dordrecht, 1992. Google Scholar

[11]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,'' Springer, Berlin, 1993. Google Scholar

[12]

X. He, The Lyapunov functionals for delay Lotka-Volterra-type models, SIAM J. Appl. Math., 58 (1998), 1222-1236. doi: 10.1137/S0036139995295116.  Google Scholar

[13]

Y. Hu, F. Wu and C. Huang, Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay, Automatica, 45 (2009), 2577-2584. doi: 10.1016/j.automatica.2009.07.007.  Google Scholar

[14]

O. Kallenberg, "Foundations of Modern Probability,'' Springer-Verlag, New York, 1997. Google Scholar

[15]

R. Z. Khasminskii, "Stochastic Stability of Differential Equations,'' Sijthoff and Noordhoff, Alphen a/d Rijn, 1981. Google Scholar

[16]

V. B. Kolmanovskii and V. R. Nosov, "Stability of Functional Differential Equations,'' Academic Press, New York, 1986. Google Scholar

[17]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246. doi: 10.1006/jdeq.1993.1048.  Google Scholar

[18]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'' Academic press, Boston, 1993. Google Scholar

[19]

J. P. LaSalle, "Stability Theory of Ordinary Differential Equations,'' J. Differential Equations, 4 (1968), 57-65. doi: 10.1016/0022-0396(68)90048-X.  Google Scholar

[20]

R. Sh. Liptser and A. N. Shiryaev, "Theory of Martingale,'' Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3.  Google Scholar

[21]

Y. Liu, X. Meng and F. Wu, Some stability criteria of stochastic functional differential equations with infinite delay, J. Appl. Math. Stoch. Anal., 2010. doi: 10.1155/2010/875908.  Google Scholar

[22]

X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingale,'' Wiley, New York, 1991. Google Scholar

[23]

X. Mao, Almost sure polynomial stability for a class of stochastic differential equations, Quart. J. Math. Oxford. Ser. (2), 43 (1992), 339-348.  Google Scholar

[24]

X. Mao, "Exponential Stability of Stochastic Differential Equations,'' Dekker, New York, 1994. Google Scholar

[25]

X. Mao, Exponential stability in mean square of neutral stochastic differential-functional equations, System Control Lett., 26 (1995), 245-251. doi: 10.1016/0167-6911(95)00018-5.  Google Scholar

[26]

X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional-differential equations, SIAM J. Math. Anal., 28 (1997), 389-401. doi: 10.1137/S0036141095290835.  Google Scholar

[27]

X. Mao, "Stochastic Differential Equations and Applications,'' Horwood, Chichester, 1997. Google Scholar

[28]

X. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations, 153 (1999), 175-195. doi: 10.1006/jdeq.1998.3552.  Google Scholar

[29]

X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud., 7 (2000), 307-328.  Google Scholar

[30]

X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Anal., 47 (2001), 4795-4806. doi: 10.1016/S0362-546X(01)00591-0.  Google Scholar

[31]

X. Mao, Some contributions to stochastic asymptotic stability and boundedness via multiple Lyapunov functions, J. Math. Anal. Appl., 260 (2001), 325-340. doi: 10.1006/jmaa.2001.7451.  Google Scholar

[32]

X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142. doi: 10.1006/jmaa.2001.7803.  Google Scholar

[33]

X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069. doi: 10.1080/07362990500118637.  Google Scholar

[34]

X. Mao and M. Riedle, Mean square stability of stochastic Volterra integro-differential equations, System Control Lett., 55 (2006), 459-465. doi: 10.1016/j.sysconle.2005.09.009.  Google Scholar

[35]

S.-E. A. Mohammed, "Stochastic Functional Differential Equations,'' Longman, Harlow/New York, 1986. Google Scholar

[36]

Y. Shen, Q. Luo and X. Mao, The improved LaSalle-type theorems for stochastic functional differential equations, J. Math. Anal. Appl., 318 (2006), 134-154. doi: 10.1016/j.jmaa.2005.05.026.  Google Scholar

[37]

J. Randjelović and S. Janković, On the $p$th moment exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl., 326 (2007), 266-280. Google Scholar

[38]

Y. Ren and N. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math., 220 (2008), 364-372. doi: 10.1016/j.cam.2007.08.022.  Google Scholar

[39]

F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331 (2007), 516-531. doi: 10.1016/j.jmaa.2006.09.020.  Google Scholar

[40]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657. doi: 10.1137/080719194.  Google Scholar

[41]

F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275.  Google Scholar

[42]

F. Wu, S. Hu and C. Huang, Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay, System Control Lett., 59 (2010), 195-202. doi: 10.1016/j.sysconle.2010.01.004.  Google Scholar

[43]

S. Zhou, Z. Wang and D. Feng, Stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 357 (2009), 416-426. doi: 10.1016/j.jmaa.2009.04.015.  Google Scholar

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