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
HTML articles powered by AMS MathViewer

by Roy Plastock PDF

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

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.

References

Über mehrdeutige stetige abbildungen

5

M. S. Berger and R. A. Plastock, On the singularities of nonlinear Fredholm operators of positive index, Proc. Amer. Math. Soc. 79 (1980), no. 2, 217–221. MR 565342, DOI 10.1090/S0002-9939-1980-0565342-5
Felix E. Browder, Covering spaces, fibre spaces, and local homeomorphisms, Duke Math. J. 21 (1954), 329–336. MR 62431
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319

Foliations andfibrations

7

Hadamard, Sur les transformations ponctuelles, Bull. Soc. Math. France 34 (1906), 71–84 (French). MR 1504541
P. Lévy, Sur les fonctions de lignes implicites, Bull. Soc. Math. France 48 (1920), 13–27 (French). MR 1504790
J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
Richard S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125–141. MR 116352, DOI 10.1090/S0002-9947-1959-0116352-7
Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
R. A. Plastock, Nonlinear Fredholm maps of index zero and their singularities, Proc. Amer. Math. Soc. 68 (1978), no. 3, 317–322. MR 464283, DOI 10.1090/S0002-9939-1978-0464283-2
Roy Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 169–183. MR 356122, DOI 10.1090/S0002-9947-1974-0356122-6
Marius Rădulescu and Sorin Radulescu, Global inversion theorems and applications to differential equations, Nonlinear Anal. 4 (1980), no. 5, 951–965. MR 586858, DOI 10.1016/0362-546X(80)90007-3
Werner C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Amer. Math. Soc. 138 (1969), 183–198. MR 240644, DOI 10.1090/S0002-9947-1969-0240644-0
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112

Non-linear Fredholm maps and the Leray-Schauder theory

32

Werner C. Rheinboldt, Solution fields of nonlinear equations and continuation methods, SIAM J. Numer. Anal. 17 (1980), no. 2, 221–237. MR 567270, DOI 10.1137/0717020
Melvin S. Berger, Nonlinearity and functional analysis, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR 0488101