On local diffeomorphisms ofR n that are injective

  • Alexandre Fernandes
  • Carlos Gutierrez
  • Roland Rabanal
Article
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Accepted:

Abstract

There are obtained conditions under which maps fromR n to itself are globally injective. In particular there are proved some partial results related to the Weak Markus-Yamabe Conjecture which states that if a vector field X:R n →R n has the property that, for allp ∈R n , all the eigenvalues ofD X (p) have negative real part, thenX has at most one singularity.

Key Words

Global injectivity Jacobian Conjecture Weak Markus-Yamabe Conjecture 
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References

  1. 1.
    H. Bass, E. Conell andD. Wright,The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse. Bull. of the A.M.S.,7 (1982), 287–330.MATHCrossRefGoogle Scholar
  2. 2.
    M. Berger,Nonlinearity and Fucctional Analysis, Pure and Applied Mathematics74, Academic Press, New York 1977.Google Scholar
  3. 3.
    L.A. Campbell,Unipotent Jacobian matrices and univalent maps. Contemp. Math.264 (2000), 157–177.Google Scholar
  4. 4.
    L.A. Campbell,Characteristic values of the jacobian matrix and global invertibility Annales Polonici Mathematici,76 (2001), No. 1–2, 11–20.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Chamberland andG. Meisters,A mountain pass to the Jacobian Conjecture, Canadian Math. Bull.41 (1998), 442–451.MATHMathSciNetGoogle Scholar
  6. 6.
    A. Fernandes, C. Gutierrez, R. Rabanal,Global asymptotic stability for differentiable vector fields of R 2. To appear in Journal of Differential Equations.Google Scholar
  7. 7.
    S. Nollet andF. Xavier,Global inversion via the Palais-Smale Condition. Discrete and Cont. Dyn. Sys.8, No. 1, (01/2002), 17–28.MATHMathSciNetGoogle Scholar
  8. 8.
    S. Pinchuck,A counterexample to the strong Jacobian conjecture, Math. Z.217 (1994), 1–4.CrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Plastock,Homeomorphisms between Banach spaces. Trans. Amer. Math. Soc.200 (1974), 169–183.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    B. Smyth, F. Xavier,Injectivity of local diffeomorphisms from nearly spectral conditions, J. Diff. Equations 130, (1996), 406–414.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. van den Essen,Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics190, Birkhauser Verlag, Basel, 2000.Google Scholar
  12. 12.
    G. Vidossich,Two remarks on the stability of ordinary differential equations. Nonlinear Analysis, Theory Methods & Applications4, No. 5, 967–974.Google Scholar
  13. 13.
    A. V. Yagzhev,On Keller's problem, Siberian Math. J., bf 21 (1980), 747–754.MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  • Alexandre Fernandes
    • 1
  • Carlos Gutierrez
    • 2
  • Roland Rabanal
    • 2
  1. 1.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrasil
  2. 2.Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrasil

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