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Attraction, stability and robustness for stochastic functional differential equations with infinite delay. (English) Zbl 1241.60028

The authors established a LaSalle theorem for a stochastic functional differential equation with infinite delay. The main result is given in Theorem 1, which contains existence and uniqueness of the global solution, and attraction and boundedness of this solution. They assume some Lipschitz condition on the coefficient functions more general than the linear growth assumption. The proof is based on the standard truncation technique [X. Mao, Exponential stability of stochastic differential equations. New York: Marcel Dekker (1994; Zbl 0806.60044), Theorem 3.2.2]. This result extends some other results given by X. Mao [Stochastic differential equations and their applications. Chichester: Horwood Publishing (1997; Zbl 0892.60057)], Y. Ren and N. Xia [Appl. Math. Comput. 214, No. 2, 457–461 (2009; Zbl 1221.34222)], R.Z. Khas’minskiĭ [Stochastic stability of differential equations. Rockville, Maryland, USA: Sijthoff & Noordhoff (1980; Zbl 0441.60060)], X. Mao and M. J. Rasias [Stochastic Anal. Appl. 23, No. 5, 1045–1069 (2005; Zbl 1082.60055)], Y. Shen, Q. Luo and X. Mao [J. Math. Anal. Appl. 318, No. 1, 134–154 (2006; Zbl 1090.60059)].
Other very important results are given in Theorem 2 and Theorem 3 under a slightly modified assumption than the above mentioned Lipschitz condition. As an interesting application of these results, a scalar stochastic integro-differential equation with infinite delay is considered.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
34K20 Stability theory of functional-differential equations
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[1] Appleby, J. A.D.; Freeman, A., Exponential asymptotic stability of linear Itô-Volterra equations with damped stochastic perturbations, Electronic Journal of Probability, 8, 1-22 (2005) · Zbl 1065.60060
[2] Arnold, L., Stochastic differential equations: theory and applications (1972), Wiley: Wiley New York
[3] Bereketoglu, H.; Győri, I., Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, Journal of Mathematical Analysis and Applications, 210, 279-291 (1997) · Zbl 0880.34072
[4] Friedman, A., Stochastic differential equations and their applications, Vol. 2 (1976), Academic Press: Academic Press New York · Zbl 0323.60057
[5] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[6] Hale, J. K.; Lunel, S. M.V., Introduction to functional differential equations (1993), Springer: Springer Berlin · Zbl 0787.34002
[7] Hu, Y.; Wu, F.; Huang, C., Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay, Automatica, 45, 2577-2584 (2009) · Zbl 1368.93765
[8] Kallenberg, O., Foundations of modern probability (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0892.60001
[9] Khasminskii, R. Z., Stochastic stability of differential equations (1981), Sijthoff and Noordhoff: Sijthoff and Noordhoff Alphen a/d Rijn · Zbl 1259.60058
[10] Kolmanovskii, V. B.; Nosov, V. R., Stability of functional differential equations (1986), Academic Press: Academic Press New York · Zbl 0593.34070
[11] Kuang, Y.; Smith, H. L., Global stability for infinite delay Lotka-Volterra type system, Journal of Differential Equations, 103, 221-246 (1993) · Zbl 0786.34077
[12] LaSalle, J. P., Stability theory of ordinary differential equations, Journal of Differential Equations, 4, 57-65 (1968) · Zbl 0159.12002
[13] Liptser, R. Sh.; Shiryaev, A. N., Theory of martingale (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0654.60035
[14] Liu, Y.; Meng, X.; Wu, F., Some stability criteria of stochastic functional differential equations with infinite delay, Journal of Applied Mathematics and Stochastic Analysis (2010)
[15] Mao, X., Stability of stochastic differential equations with respect to semimartingale (1991), Wiley: Wiley New York · Zbl 0724.60059
[16] Mao, X., Exponential stability of stochastic differential equations (1994), Dekker: Dekker New York · Zbl 0851.93074
[17] Mao, X., Stochastic differential equations and applications (1997), Horwood: Horwood Chichester · Zbl 0874.60050
[18] Mao, X., Stochastic versions of the LaSalle theorem, Journal of Differential Equations, 153, 175-195 (1999) · Zbl 0921.34057
[19] Mao, X., The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Studies, 7, 307-328 (2000) · Zbl 0993.60054
[20] Mao, X., Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Analysis, 47, 4795-4806 (2001) · Zbl 1042.60517
[21] Mao, X., Some contributions to stochastic asymptotic stability and boundedness via multiple Lyapunov functions, Journal of Mathematical Analysis and Applications, 260, 325-340 (2001) · Zbl 0983.60055
[22] Mao, X.; Rassias, M. J., Khasminskii-type theorems for stochastic differential delay equations, Stochastic Analysis and Applications, 23, 1045-1069 (2005) · Zbl 1082.60055
[23] Mao, X.; Riedle, M., Mean square stability of stochastic Volterra integro-differential equations, Systems & Control Letters, 55, 459-465 (2006) · Zbl 1129.34332
[24] Mohammed, S.-E. A., Stochastic functional differential equations (1986), Longman: Longman Harlow, New York · Zbl 0584.60066
[25] Murakami, S.; Naito, T., Fading memory space and stability properties for functional differential equations with infinite delay, Funkcialaj Ekvacioj, 32, 91-105 (1989) · Zbl 0687.34068
[26] Ren, Y.; Xia, N., Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, Applied Mathematics and Computation, 220, 364-372 (2009) · Zbl 1152.34388
[27] Shen, Y.; Luo, Q.; Mao, X., The improved LaSalle-type theorems for stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 318, 134-154 (2006) · Zbl 1090.60059
[28] Wu, F.; Hu, S.; Huang, C., Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay, Systems & Control Letters, 59, 195-202 (2010) · Zbl 1223.93096
[29] Wu, F.; Xu, Y., Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM Journal on Applied Mathematics, 70, 641-657 (2009) · Zbl 1197.34164
[30] Zhou, S.; Wang, Z.; Feng, D., Stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 357, 416-426 (2009) · Zbl 1173.60331
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