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1994 | 59 | 2 | 171-196
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
59
Numer
2
Strony
171-196
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-05-06
Twórcy
  • Dipartimento di Matematica e Informatica, Università di Udine, via Zanon 6, 33100 Udine, Italy
  • Dipartimento di Matematica, Pura e Applicata, Università di Padova, via Belzoni 7, 35131 Padova, Italy
Bibliografia
  • [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.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv59z2p171bwm
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