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Journal of Mathematics
Volume 2014 (2014), Article ID 989526, 9 pages
http://dx.doi.org/10.1155/2014/989526
Research Article

Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under -Dichotomies

School of Mathematics, Jia Ying University, Meizhou, Guangdong 514015, China

Received 20 May 2014; Accepted 14 September 2014; Published 27 October 2014

Academic Editor: Alfred Peris

Copyright © 2014 Lijun Pan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. O. Perron, “Die Stabilitätsfrage bei Differentialgleichungen,” Mathematische Zeitschrift, vol. 32, no. 1, pp. 703–728, 1930. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems. I,” Journal of Differential Equations, vol. 15, pp. 429–458, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems. II,” Journal of Differential Equations, vol. 22, no. 2, pp. 478–496, 1976. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems. III,” Journal of Differential Equations, vol. 22, no. 2, pp. 497–522, 1976. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. R. J. Sacker and G. R. Sell, “Dichotomies for linear evolutionary equations in Banach spaces,” Journal of Differential Equations, vol. 113, no. 1, pp. 17–67, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, vol. 1926 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. L. Barreira and Y. B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, vol. 23 of University Lecture Series, American Mathematical Society, Providence, RI, Providence, RI, USA, 2002. View at MathSciNet
  8. J. Massera and J. Schäfer, Linear Differential Equations and Function Spaces, vol. 21 of Pure and Applied Mathematics, Academic Press, 1966.
  9. J. L. Dalec′kiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, vol. 43, American Mathematical Society, 1974. View at MathSciNet
  10. W. A. Coppel, Dichotomies in Stability Theory, Springer, 1978. View at MathSciNet
  11. J. L. Massera and J. J. Schäffer, “Linear differential equations and functional analysis. I,” Annals of Mathematics, vol. 67, no. 3, pp. 517–573, 1958. View at Publisher · View at Google Scholar · View at MathSciNet
  12. L. Barreira and C. Valls, “Stable manifolds for nonautonomous equations without exponential dichotomy,” Journal of Differential Equations, vol. 221, no. 1, pp. 58–90, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. L. Barreira and C. Valls, “Parameter dependence of stable manifolds under nonuniform hyperbolicity,” Journal of Mathematical Analysis and Applications, vol. 358, no. 2, pp. 419–426, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,” Journal of Differential Equations, vol. 120, no. 2, pp. 429–477, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. V. A. Pliss and G. R. Sell, “Robustness of exponential dichotomies in infinite-dimensional dynamical systems,” Journal of Dynamics and Differential Equations, vol. 11, no. 3, pp. 471–513, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. A. J. G. Bento and C. M. Silva, “Generalized nonuniform dichotomies and local stable manifolds,” Journal of Dynamics and Differential Equations, vol. 25, no. 4, pp. 1139–1158, 2013. View at Publisher · View at Google Scholar
  17. L. Barreira and C. Valls, “Polynomial growth rates,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5208–5219, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. L. Barreira, M. Fan, C. Valls, and J. Zhang, “Robustness of nonuniform polynomial dichotomies for difference equations,” Topological Methods in Nonlinear Analysis, vol. 37, no. 2, pp. 357–376, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. A. J. Bento and C. Silva, “Stable manifolds for nonuniform polynomial dichotomies,” Journal of Functional Analysis, vol. 257, no. 1, pp. 122–148, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. L. Barreira and C. Valls, “Growth rates and nonuniform hyperbolicity,” Discrete and Continuous Dynamical Systems, vol. 22, no. 3, pp. 509–528, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. L. Barreira and C. Valls, “Robustness of general dichotomies,” Journal of Functional Analysis, vol. 257, no. 2, pp. 464–484, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. L. Barreira and C. Valls, “Stable invariant manifolds for parabolic dynamics,” Journal of Functional Analysis, vol. 257, no. 4, pp. 1018–1029, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. X. Chang, J. Zhang, and J. Qin, “Robustness of nonuniform (μ,ν)-dichotomies in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 582–594, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. A. Bento and C. Silva, “Nonuniform (μ,ν)-dichotomies and local dynamics of difference equations,” Journal of Dynamics and Differential Equations, vol. 25, pp. 1139–1158, 2013. View at Google Scholar
  25. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, Berlin, Germany, 1993. View at MathSciNet
  26. L. Barreira, M. Fan, C. Valls, and J. Zhang, “Parameter dependence of stable manifolds for delay equations with polynomial dichotomies,” Journal of Dynamics and Differential Equations, vol. 24, no. 1, pp. 101–118, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus