Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Smooth robustness of exponential dichotomies


Authors: Luis Barreira and Claudia Valls
Journal: Proc. Amer. Math. Soc. 139 (2011), 999-1012
MSC (2010): Primary 34D09, 34D10, 37D99
DOI: https://doi.org/10.1090/S0002-9939-2010-10531-2
Published electronically: July 29, 2010
MathSciNet review: 2745651
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For sufficiently small $ C^1$-parameterized linear perturbations, we establish the robustness of exponential dichotomies in Banach spaces, with the optimal $ C^1$ dependence of the stable and unstable subspaces on the parameter.


References [Enhancements On Off] (What's this?)

  • 1. L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes in Math. 1926, Springer, 2008. MR 2368551 (2010b:37038)
  • 2. S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), 429-477. MR 1347351 (97a:34121)
  • 3. W. Coppel, Dichotomies and reducibility, J. Differential Equations 3 (1967), 500-521. MR 0223651 (36:6699)
  • 4. Ju. Dalec$ '$kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974. MR 0352639 (50:5126)
  • 5. R. Johnson and G. Sell, Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems, J. Differential Equations 41 (1981), 262-288. MR 630994 (83a:58072)
  • 6. J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517-573. MR 0096985 (20:3466)
  • 7. J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, New York-London, 1966. MR 0212324 (35:3197)
  • 8. R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal. 31 (1998), 559-571. MR 1487846 (99b:34087)
  • 9. K. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system, J. Differential Equations 65 (1986), 321-360. MR 865066 (88a:58078)
  • 10. O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703-728. MR 1545194
  • 11. V. Pliss and G. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations 11 (1999), 471-513. MR 1693858 (2000f:34092)
  • 12. L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl. 314 (2006), 436-454. MR 2185241 (2006h:34117)
  • 13. Y. Yi, A generalized integral manifold theorem, J. Differential Equations 102 (1993), 153-187. MR 1209981 (94c:58148)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34D09, 34D10, 37D99

Retrieve articles in all journals with MSC (2010): 34D09, 34D10, 37D99


Additional Information

Luis Barreira
Affiliation: Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Email: barreira@math.ist.utl.pt

Claudia Valls
Affiliation: Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Email: cvalls@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9939-2010-10531-2
Keywords: Difference equations, parameter dependence, robustness
Received by editor(s): March 19, 2010
Received by editor(s) in revised form: March 26, 2010
Published electronically: July 29, 2010
Additional Notes: The first author was partially supported by FCT through CAMGSD, Lisbon
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society