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Robustness of Hyperbolicity in Delay Equations

  • Luis Barreira
  • Claudia VallsEmail author
Article
  • 9 Downloads

Abstract

We establish the robustness of the notion of an exponential dichotomy for a nonautonomous linear delay equation. We consider the general cases of equations satisfying the Carathéodory conditions and of noninvertible evolution families defined by the linear equations. The stable and unstable spaces of the perturbed equations are obtained as images of linear operators that in their turn are fixed points of appropriate contraction maps.

Keywords

Delay equations Exponential dichotomies Robustness 

Mathematics Subject Classification

Primary 34K06 37D99 

Notes

References

  1. 1.
    Barreira, L., Valls, C.: Stability of Nonautonomous Differential Equations. Lecture Notes in Mathematics, vol. 1926. Springer, Berlin (2008)zbMATHGoogle Scholar
  2. 2.
    Chow, S.-N., Leiva, H.: Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. J. Differ. Equ. 120, 429–477 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coppel, W.: Dichotomies and reducibility. J. Differ. Equ. 3, 500–521 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dalec’kiĭ, J.u., Kreĭn, M.: Stability of Solutions of Differential Equations in Banach space, Translations of Mathematical Monographs, vol. 43. American Mathematical Society, Providence, RI (1974)Google Scholar
  5. 5.
    Hale, J.: Theory of Functional Differential Equations, Applied Mathematical Sciences 3. Springer, Berlin (1977)CrossRefGoogle Scholar
  6. 6.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)zbMATHGoogle Scholar
  7. 7.
    Massera, J., Schäffer, J.: Linear differential equations and functional analysis. I. Ann. Math. (2) 67, 517–573 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Perron, O.: Die stabilitätsfrage bei differentialgleichungen. Math. Z. 32, 703–728 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pliss, V., Sell, G.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 11, 471–513 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Popescu, L.: Exponential dichotomy roughness on Banach spaces. J. Math. Anal. Appl. 314, 436–454 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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