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Fixed points and exponential stability for stochastic Volterra-Levin equations. (English) Zbl 1197.60053

Using the contraction fixed point principle Luo studies the exponential stability of the stochastic Volterra-Levin equation. Conditions are given to ensure that the equation is exponentially stable in mean square and is also almost surely exponentially stable.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K50 Stochastic functional-differential equations
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[1] Burton, T.A., Stability by fixed point theory for functional differential equations, (2006), Dover Publications, Inc. New York · Zbl 1090.45002
[2] Becker, L.C.; Burton, T.A., Stability, fixed points and inverses of delays, Proc. roy. soc. Edinburgh sect. A, 136, 245-275, (2006) · Zbl 1112.34054
[3] Burton, T.A., Fixed points, stability, and exact linearization, Nonlinear anal., 61, 857-870, (2005) · Zbl 1067.34077
[4] Burton, T.A., Fixed points, Volterra equations, and becker’s resolvent, Acta math. hungar., 108, 261-281, (2005) · Zbl 1091.34040
[5] Burton, T.A., Fixed points and stability of a nonconvolution equation, Proc. amer. math. soc., 132, 3679-3687, (2004) · Zbl 1050.34110
[6] Burton, T.A., Perron-type stability theorems for neutral equations, Nonlinear anal., 55, 285-297, (2003) · Zbl 1044.34028
[7] Burton, T.A., Integral equations, implicit functions, and fixed points, Proc. amer. math. soc., 124, 2383-2390, (1996) · Zbl 0873.45003
[8] Burton, T.A.; Furumochi, Tetsuo, Krasnoselskii’s fixed point theorem and stability, Nonlinear anal., 49, 445-454, (2002) · Zbl 1015.34046
[9] Burton, T.A.; Zhang, Bo, Fixed points and stability of an integral equation: nonuniqueness, Appl. math. lett., 17, 839-846, (2004) · Zbl 1066.45002
[10] Furumochi, Tetsuo, Stabilities in FDEs by schauder’s theorem, Nonlinear anal., 63, e217-e224, (2005) · Zbl 1159.39301
[11] Jin, Chuahua; Luo, Jiaowan, Fixed points and stability in neutral differential equations with variable delays, Proc. amer. math. soc., 136, 909-918, (2008) · Zbl 1136.34059
[12] Raffoul, Y.N., Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Math. comput. modelling, 40, 691-700, (2004) · Zbl 1083.34536
[13] Zhang, Bo, Fixed points and stability in differential equations with variable delays, Nonlinear anal., 63, e233-e242, (2005) · Zbl 1159.34348
[14] J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint.
[15] Luo, Jiaowan, Fixed points and stability of neutral stochastic delay differential equations, J. math. anal. appl., 334, 431-440, (2007) · Zbl 1160.60020
[16] Luo, Jiaowan, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. math. anal. appl., 342, 753-760, (2008) · Zbl 1157.60065
[17] Luo, Jiaowan, Stability of stochastic partial differential equations with infinite delays, J. comput. appl. math., 222, 364-371, (2008) · Zbl 1151.60336
[18] Luo, Jiaowan; Taniguchi, Takeshi, Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stoch. anal. appl., 27, 1163-1173, (2009) · Zbl 1177.93094
[19] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[20] Khasminskii, R.Z., Stochastic stability of differential equations, (1981), Sijthoff & Noordhoff Alphen, Rijn · Zbl 1259.60058
[21] Kolmanovskii, V.B.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers Dordrecht
[22] Kushner, H.J., Stochastic stability and control, (1967), Academic Press New York · Zbl 0178.20003
[23] Ladde, G.S.; Lakshmikantham, V., Random differential inequalities, (1980), Academic Press New York · Zbl 0466.60002
[24] Liu, Kai, ()
[25] Mao, Xuerong, Stability of stochastic differential equations with respect to semimartingales, (1991), Longman Scientific and Technical New York · Zbl 0724.60059
[26] Mao, Xuerong, Exponential stability of stochastic differential equations, (1994), Marcel Dekker New York · Zbl 0806.60044
[27] Mao, Xuerong, Stochastic differential equations and their applications, (1997), Horwood Publisher Chichester · Zbl 0892.60057
[28] Mao, Xuerong; Yuan, Chenggui, Stochastic differential equations with Markovian switching, (2006), Imperial College Press London · Zbl 1126.60002
[29] Mohammed, S.-E.A., Stochastic functional differential equations, (1986), Longman Scientific and Technical New York · Zbl 0584.60066
[30] Mao, Xuerong, Almost sure exponential stability of delay equations with damped stochastic perturbation, Stoch. anal. appl., 19, 67-84, (2001) · Zbl 0980.60081
[31] Volterra, V., Sur la théorie mathématique des phénomės héréditaires, J. math. pures appl., 7, 249-298, (1928) · JFM 54.0934.06
[32] Levin, J.J., The asymptotic behavior of Volterra equation, Proc. amer. math. soc., 14, 434-451, (1963)
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