Articles

ATTRACTING AND QUASI-INVARIANT SETS OF STOCHASTIC NEUTRAL PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

  • LI Ding-Shi ,
  • XU Dao-Xi
Expand
  • School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China; Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

Received date: 2011-10-31

  Revised date: 2012-04-27

  Online published: 2013-03-20

Supported by

The first author is supported by National Natural Science Foundation of China (11271270 and 11201320).

Abstract

In this article, we investigate a class of stochastic neutral partial functional differ-ential equations. By establishing new integral inequalities, the attracting and quasi-invariant sets of stochastic neutral partial functional differential equations are obtained. The results in [15, 16] are generalized and improved.

Cite this article

LI Ding-Shi , XU Dao-Xi . ATTRACTING AND QUASI-INVARIANT SETS OF STOCHASTIC NEUTRAL PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2013 , 33(2) : 578 -588 . DOI: 10.1016/S0252-9602(13)60021-1

References

[1] Cheban D, Mammana C. Invariant manifolds, global attractors and almost periodic solutions of nonau-tonomous difference equations. Nonlinear Anal, 2004, 56: 465–484

[2] Xu D Y. Asymptotic behavior of nonlinear difference equations with delays. Comput Math Appl, 2001, 42: 393–398

[3] Xu D Y. Invariant and attracting sets of Volterra difference equations with delays. Comput Math Appl, 2003, 45: 1311–1317

[4] Bernfeld S R, Corduneanu C, Ignatyev A O. On the stability of invariant sets of functional differential equations. Nonlinear Anal, 2003, 55: 641–656

[5] Kolmanovskii V B, Nosov V R. Stability of Functional Differential Equations. Orlando, FL: Academic Press, 1986

[6] Liao X X, Luo Q, Zeng Z G. Positive invariant and global exponential attractive sets of neural networks with time-varying delays. Neurocomputing, 2008, 71: 513–518

[7] Xu D Y, Zhao H Y. Invariant set and attractivity of nonlinear differential equations with delays. Appl Math Lett, 2002, 15 : 321–325

[8] Zhao H Y. Invariant set and attractor of nonautonomous functional differential systems. J Math Anal Appl, 2003, 282: 437–443

[9] Wang L S, Xu D Y. Asymptotic behavior of a class of reaction-diffusion equations with delays. J Math Anal Appl, 2003, 281: 439–453

[10] Duan J Q, Lu K N, Schmalfuß B. Invariant manifolds for stochastic partial differential equations. Ann Probab, 2003, 31: 2109–2135

[11] Lu K N, Schmalfuß B. Invariant manifolds for stochastic wave equations. J Differential Equations, 2007, 236: 460–492

[12] Liu K, Xia X, On the exponential stability in mean square of neutral stochastic functional differential equations. Systems Control Lett, 1999, 37: 207–215

[13] Govindan T E. Almost sure exponential stability for stochastic neutral partial functional differential equa-tions. Stochastics, 2005, 77: 139–154

[14] Caraballo T, Real J, Taniguchi T. The exponential stability of neutral stochastic delay partial differential equations. Discrete Contin Dyn Syst, 2007, 18: 295–313

[15] Caraballo T, Liu K. Exponential stability of mild solutions of stochastic partial differential equations with delays. Stoch Anal Appl, 1999, 17: 743–763

[16] Luo J W. Exponential stability for stochastic neutral partial functional differential equations. J Math Anal Appl, 2009, 355: 414–425

[17] Luo J W. Fixed points and stability of neutral stochastic delay differential equations. J Math Anal Appl, 2007, 334: 431–440

[18] Chen H B. Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays. Statist Probab Lett, 2010, 80: 50–56

[19] Bao J H, Hou Z T. Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients. Comput Math Appl, 2010, 59: 207–214

[20] Chen H B. Integral inequality and exponential stability for neutral stochastic partial differential equations with delays. Journal of Inequalities and Applications, 2009: Article ID 297478

[21] Boufoussi B, Hajji S. Successive approximation of neutral functional stochastic differential equations with jumps. Statist Probal Lett, 2010, 80: 324–332

[22] Jiang F, Shen Y. A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients. Comput Math Appl, 2011, 61: 1590–1594

[23] Prato G Da, Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge Univ Press, 1992

[24] Pazy A. Semigroups of linear operators and applications to partial differential equations//Applied Methe-matical Sciences. Vol 44. New York: Springer Verlag, 1983

[25] Ichikawa A. Stability of semilinear stochastic evolution equations. J Math Anal Appl, 1982, 90: 12–44

[26] Wu F, Hu S, Liu Y. Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system. J Math Anal Appl, 2010, 364: 104–118

[27] Xu D Y, Huang Y M, Yang Z G. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete Contin Dyn Syst, 2009, 24: 1005–1023

Outlines

/