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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A uniqueness theorem for fixed points


Authors: H. L. Smith and C. A. Stuart
Journal: Proc. Amer. Math. Soc. 79 (1980), 237-240
MSC: Primary 47H10; Secondary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1980-0565346-2
MathSciNet review: 565346
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Abstract: In a recent paper, R. Kellogg [3] showed that if $ F:\bar D \to \bar D$ is a completely continuous map of the closure of a bounded, convex, open set D in a real Banach space X, $ F \in {C^1}(D)$, 1 is not an eigenvalue of $ F'(x)$ for $ x \in D$, and $ F(x) \ne x$ for $ x \in \partial D$, then F has a unique fixed point in D. More recently, L. Talman [7] extended this result to k-set contractions when $ k < 1$. The main result of this note is to show that, if the dimension of X is larger than one, the result of Kellogg and its extension by Talman remain valid provided that the set $ \{ x \in D:1$ is an eigenvalue of $ F'(x)\} $ has no accumulation points in D, the other assumptions remaining the same. This result is obtained as a corollary of a more general result which gives conditions under which the set of fixed points of F in D is connected.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0565346-2
Keywords: Fixed point, K-set contraction
Article copyright: © Copyright 1980 American Mathematical Society