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 | References | Similar Articles | Additional Information

Abstract: In a recent paper, R. Kellogg [**3**] showed that if is a completely continuous map of the closure of a bounded, convex, open set *D* in a real Banach space *X*, , 1 is not an eigenvalue of for , and for , then *F* has a unique fixed point in *D*. More recently, L. Talman [**7**] extended this result to *k*-set contractions when . 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 is an eigenvalue of 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0565346-2

Keywords:
Fixed point,
*K*-set contraction

Article copyright:
© Copyright 1980
American Mathematical Society