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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Global surjectivity of submersions via contractibility of the fibers


Author: Patrick J. Rabier
Journal: Trans. Amer. Math. Soc. 347 (1995), 3405-3422
MSC: Primary 58C15
DOI: https://doi.org/10.1090/S0002-9947-1995-1321587-3
MathSciNet review: 1321587
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Abstract: We give a sufficient condition for a $ {C^1}$ submersion $ F:X \to Y$, $ X$ and $ Y$ real Banach spaces, to be surjective with contractible fibers $ {F^{ - 1}}(y)$. Roughly speaking, this condition "interpolates" two well-known but unrelated hypotheses corresponding to the two extreme cases: Hadamard's criterion when $ Y \simeq X$ and $ F$ is a local diffeomorphism, and the Palais-Smale condition when $ Y = \mathbb{R}$. These results may be viewed as a global variant of the implicit function theorem, which unlike the local one does not require split kernels. They are derived from a deformation theorem tailored to fit functionals with a norm-like nondifferentiability.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1321587-3
Article copyright: © Copyright 1995 American Mathematical Society