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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
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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  • [1] M. Berger, Nonlinearity and functional analysis, Academic Press, New York, 1977. MR 0488101 (58:7671)
  • [2] F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., Vol. 18, Amer. Math. Soc., Providence, R.I., 1976. MR 0405188 (53:8982)
  • [3] R. B. Kellogg, Uniqueness in the Schauder fixed point theorem, Proc. Amer. Math. Soc. 60 (1976), 207-210. MR 0423137 (54:11118)
  • [4] S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861-866. MR 0185604 (32:3067)
  • [5] R. D. Nussbaum, The fixed point index and fixed point theorems for k-set-contractions, Ph.D. thesis, Univ. of Chicago, 1969.
  • [6] -, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473-478. MR 0264434 (41:9028)
  • [7] L. A. Talman, A note on Kellogg's uniqueness theorem for fixed points, Proc. Amer. Math. Soc. 69 (1978), 248-250. MR 0467416 (57:7275)
  • [8] E. Zeidler, Vorlesungen über nichlinear Functionalanalysis. I, Fixpunktsatze, Teubner, Leipzig, 1976.

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Keywords: Fixed point, K-set contraction
Article copyright: © Copyright 1980 American Mathematical Society

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