Stability of a Class of Nonlinear Neutral Stochastic Differential Equations with Variable Time Delays

[1] J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint. Search in Google Scholar

[2] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006. Search in Google Scholar

[3] T.A. Burton, Stability by fixed point theory or Liapunov’s theory: a comparison, Fixed Point Theory, 4 (2003), 15-32. Search in Google Scholar

[4] T.A. Burton, Liapunov functionals, fixed points, and stability by Kras- noselskii’s theorem, Nonlinear Stud., 9 (2001), 181-190. Search in Google Scholar

[5] T.A. Burton, Fixed points and stability of a nonconvolution equation, Proc. Amer. Math. Soc., 132 (2004), 3679-3687. 10.1090/S0002-9939-04-07497-0Search in Google Scholar

[6] T.A. Burton and T. Furumochi, Fixed points and problems in stabil- ity theory for ordinary and functional differential equations, Dynamical Systems and Appl., 10 (2001), 89-116. Search in Google Scholar

[7] T.A. Burton and T. Furumochi, Krasnoselskii’s fixed point theorem and stability, Nonlinear Anal. TMA, 49 (2002), 445-454. 10.1016/S0362-546X(01)00111-0Search in Google Scholar

[8] T.A. Burton and B. Zhang, Fixed points and stability of an integral equation: Nonuniquess, Appl. Math. Lett., 17 (2004), 893-846. 10.1016/j.aml.2004.06.015Search in Google Scholar

[9] C.H. Jin and J.W. Luo, Fixed points and stability in neutral differential equations with variable delays, Proc. Amer. Math. Soc., 136 (2008), 909-918. Search in Google Scholar

[10] S.M. Jung, A fixed point approach to the stability of differential equations y′ = F(x, y), Bull. Malays. Math. Sci. Soc., 33(1) (2010), 47-56. Search in Google Scholar

[11] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York, 1991. Search in Google Scholar

[12] V.B. Kolmanovskii and V.R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986. Search in Google Scholar

[13] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equation with Applications, Chapman & Hall/CRC, Boca Raton, 2006. Search in Google Scholar

[14] J.W. Luo, Fixed points and stability of neutral stochastic delay differen- tial equations, J. Math. Anal. Appl., 334 (2007), 431-440. 10.1016/j.jmaa.2006.12.058Search in Google Scholar

[15] J.W. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760. 10.1016/j.jmaa.2007.11.019Search in Google Scholar

[16] J.W. Luo, Stability of stochastic partial differential equations with infinite delays, J. Comput. Appl. Math., 222 (2008), 364-371. 10.1016/j.cam.2007.11.002Search in Google Scholar

[17] J.W. Luo, Fixed points and exponential stability for stochastic Volterra- Levin equations, J. Comput. Appl. Math., 234 (2010), 934-940. 10.1016/j.cam.2010.02.013Search in Google Scholar

[18] J.W. Luo and T. Taniguchi, Fixed points and stability of stochastic neu- tral partial differential equations with infinite delays, Stoch. Anal. Appl., 27 (2009), 1163-1173. 10.1080/07362990903259371Search in Google Scholar

[19] X.R. Mao, Stochastic Differential Equations and their Applications, Hor- wood Publ. House, Chichester, 1997. Search in Google Scholar

[20] Y.N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Math. Comput. Modelling, 40 (2004), 691-700. 10.1016/j.mcm.2004.10.001Search in Google Scholar

[21] M. Wu, N.J. Huang and C.W. Zhao, Stability of half-linear neutral stochastic differential equations with delays, Bull. Austral. Math. Soc. 80 (2009), 369-383. 10.1017/S0004972709000422Search in Google Scholar

[22] M. Wu, N.J. Huang and C.W. Zhao, Fixed points and stability in neutral stochastic differential equations with variable eelays, Fixed Point Theory Appl., 2008 (2008), Article ID 407352. 10.1155/2008/407352Search in Google Scholar

[23] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal., 63 (2005), e233-e242. 10.1016/j.na.2005.02.081Search in Google Scholar

[24] B. Zhang, Contraction mapping and stability in a delay-differential equa- tion, Proc. Dynam. Sys. Appl., 4 (2004), 183-190.Search in Google Scholar