A uniqueness theorem for fixed points

Smith, H. L.; Stuart, C. A.

doi:10.1090/S0002-9939-1980-0565346-2

A uniqueness theorem for fixed points
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by H. L. Smith and C. A. Stuart PDF

Proc. Amer. Math. Soc. 79 (1980), 237-240 Request permission

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.

References

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The fixed point index and fixed point theorems for k-set-contractions

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