Nonlinear mappings that are globally equivalent to a projection

Plastock, Roy

doi:10.1090/S0002-9947-1983-0678357-0

Nonlinear mappings that are globally equivalent to a projection
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by Roy Plastock

Trans. Amer. Math. Soc. 275 (1983), 373-380

DOI: https://doi.org/10.1090/S0002-9947-1983-0678357-0

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Abstract:

The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse, ${F^ \downarrow }(x)$, to the derivative map ${F^\prime }(x)$ and a global solution to the differential equation ${P^\prime }(t) = {F^ \downarrow }(P(t))(y - {y_0})$. Both these problems are solved and the generalized Hadamard-Levy criterion \[ \int _0^\infty {\inf \limits _{|x| < s} \left ({1/|{F^ \downarrow }(x)|} \right ) ds = \infty } \] is shown to be sufficient for $F$ to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.

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