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)

 
 

 

Nonlinear Fredholm maps of index zero and their singularities


Author: R. A. Plastock
Journal: Proc. Amer. Math. Soc. 68 (1978), 317-322
MSC: Primary 58B15; Secondary 47H15
DOI: https://doi.org/10.1090/S0002-9939-1978-0464283-2
MathSciNet review: 0464283
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ F:X \to Y$ be a $ {C^1}$ Fredholm map of index zero between two Banach spaces. Defining the singular set $ B = \{ x\vert F'(x)$ is not surjective}, we study the local and global effect of B on the map F. In particular it is shown that if $ b \in B$ is isolated in B, then, for $ \dim X$ and $ \dim Y \geqslant 3$, F is a local homeomorphism at b. We then show that if B consists of discrete points, F is a global homeomorphism of X onto Y. A nonlinear partial differential equation is included as an illustration.


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

  • [1] S. Banach and S. Mazur, Über mehrdeutige stetige Abbildungen, Studia Math. 5 (1934), 174-178.
  • [2] M. S. Berger, Nonlinearity and functional analysis, Academic Press, New York, 1977. MR 0488101 (58:7671)
  • [3] R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 169-183. MR 0356122 (50:8593)
  • [4] S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861-866. MR 0185604 (32:3067)
  • [5] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [6] A. J. Tromba, Some theorems on Fredholm maps, Proc. Amer. Math. Soc. 34 (1972), 578-585. MR 0298713 (45:7762)
  • [7] M. S. Berger and R. Plastock, On the singularities of nonlinear Fredholm operators, Bull. Amer. Math. Soc. 83 (1977), 1316-1318. MR 0451279 (56:9566)
  • [8] W. H. Cutler, Negligible subsets of infinite dimensional Fréchet manifolds, Proc. Amer. Math. Soc. 23 (1969), 668-675. MR 0248883 (40:2133)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58B15, 47H15

Retrieve articles in all journals with MSC: 58B15, 47H15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0464283-2
Keywords: Fredholm map of index zero, covering space map, proper
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society