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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the structure of the set of solutions of equations involving $ A$-proper mappings


Author: P. M. Fitzpatrick
Journal: Trans. Amer. Math. Soc. 189 (1974), 107-131
MSC: Primary 47H15
MathSciNet review: 0336475
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Abstract: Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings $ T:D \subset X \to Y$ which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0336475-5
PII: S 0002-9947(1974)0336475-5
Keywords: Complete projection scheme, A-proper, generalized topological degree, continuum, Fréchet derivative, contraction, compact, fixed point, surjective
Article copyright: © Copyright 1974 American Mathematical Society