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ArticleOriginal scientific text

Title

Injectivity onto a star-shaped set for local homeomorphisms in n-space

Authors 1, 2

Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Udine, via Zanon 6, 33100 Udine, Italy
  2. Dipartimento di Matematica, Pura e Applicata, Università di Padova, via Belzoni 7, 35131 Padova, Italy

Abstract

We provide a number of either necessary and sufficient or only sufficient conditions on a local homeomorphism defined on an open, connected subset of the n-space to be actually a homeomorphism onto a star-shaped set. The unifying idea is the existence of "auxiliary" scalar functions that enjoy special behaviours along the paths that result from lifting the half-lines that radiate from a point in the codomain space. In our main result this special behaviour is monotonicity, and the auxiliary function can be seen as a Lyapunov function for a suitable dynamical system having the lifted paths as trajectories.

Keywords

ENG
global invertibility, local homeomorphisms, star-shaped sets, line-lifting, Lyapunov functions

Bibliography

  1. A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1973), 231-247.
  2. V. I. Arnol'd, Ordinary Differential Equations, 3rd ed., Springer, 1992.
  3. S. Banach and S. Mazur, Über mehrdeutige stetige Abbildungen, Studia Math. 5 (1934), 174-178.
  4. N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Springer, 1970.
  5. F. Browder, Covering spaces, fiber spaces and local homeomorphisms, Duke Math. J. 21 (1954), 329-336.
  6. R. Caccioppoli, Sugli elementi uniti delle trasformazioni funzionali: un teorema di esistenza e di unicità ed alcune sue applicazioni, Rend. Sem. Mat. Univ. Padova 3 (1932), 1-15.
  7. L. M. Drużkowski, The Jacobian Conjecture, Preprint 429, Institute of Mathematics, Jagiellonian University, Kraków, 1990.
  8. P. L. Duren, Univalent Functions, Springer, 1983.
  9. D. Gale and H. Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159 (1965), 81-93.
  10. G. Gorni, A criterion of invertibility in the large for local diffeomorphisms between Banach spaces, Nonlinear Anal. 21 (1993), 43-47.
  11. J. Hadamard, Sur les transformations ponctuelles, Bull. Soc. Math. France 34 (1906), 71-84.
  12. P. Lévy, Sur les fonctions de lignes implicites, ibid. 48 (1920), 13-27.
  13. G. H. Meisters, Inverting polynomial maps of n-space by solving differential equations, in: Fink, Miller, Kliemann (eds.), Delay and Differential Equations, Proceedings in Honour of George Seifert on his retirement, World Sci., 1992, 107-166.
  14. G. H. Meisters and C. Olech, Locally one-to-one mappings and a classical theorem on schlicht functions, Duke Math. J. 30 (1963), 63-80.
  15. G. H. Meisters and C. Olech, Solution of the global asymptotic stability Jacobian conjecture for the polynomial case, in: Analyse Mathématique et Applications, Gauthier-Villars, Paris, 1988, 373-381.
  16. C. Olech, Global diffeomorphism questions and differential equations, in: Qualitative Theory of Differential Equations, Szeged, 1988, Colloq. Math. Soc. János Bolyai 53, North-Holland, 1990, 465-471.
  17. J. M. Ortega and W. C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, 1970.
  18. T. Parthasarathy, On Global Univalence Theorems, Lecture Notes in Math. 977, Springer, 1983.
  19. R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 169-183.
  20. P. J. Rabier, On global diffeomorphisms of Euclidian space, Nonlinear Anal. 21 (1993), 925-947.
  21. M. Rădulescu and S. Rădulescu, Global inversion theorems and applications to differential equations, ibid. 4 (1980), 951-965.
  22. W. C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Amer. Math. Soc. 138 (1969), 183-198.
  23. J. Sotomayor, Inversion of smooth mappings, Z. Angew. Math. Phys. 41 (1990), 306-310.
  24. M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, Houston, Tex., 1970, 1979.
  25. T. Ważewski, Sur l'évaluation du domaine d'existence de fonctions implicites réelles ou complexes, Ann. Soc. Polon. Math. 20 (1947), 81-120.
  26. G. Zampieri, Finding domains of invertibility for smooth functions by means of attraction basins, J. Differential Equations 104 (1993), 11-19.
  27. G. Zampieri, Diffeomorphisms with Banach space domains, Nonlinear Anal. 19 (1992), 923-932.
  28. G. Zampieri and G. Gorni, On the Jacobian conjecture for global asymptotic stability, J. Dynamics Differential Equations 4 (1992), 43-55.
Annales Polonici Mathematici
1994/59/2
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Pages:
171-196
Main language of publication
English
Received
1993-05-06
Published
1994
Exact and natural sciences