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.

**[1]**Melvin S. Berger,*Nonlinearity and functional analysis*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis; Pure and Applied Mathematics. MR**0488101****[2]**Felix E. Browder,*Nonlinear operators and nonlinear equations of evolution in Banach spaces*, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R. I., 1976, pp. 1–308. MR**0405188****[3]**R. B. Kellogg,*Uniqueness in the Schauder fixed point theorem*, Proc. Amer. Math. Soc.**60**(1976), 207–210 (1977). MR**0423137**, https://doi.org/10.1090/S0002-9939-1976-0423137-6**[4]**S. Smale,*An infinite dimensional version of Sard’s theorem*, Amer. J. Math.**87**(1965), 861–866. MR**0185604**, https://doi.org/10.2307/2373250**[5]**R. D. Nussbaum,*The fixed point index and fixed point theorems for k-set-contractions*, Ph.D. thesis, Univ. of Chicago, 1969.**[6]**Roger D. Nussbaum,*The radius of the essential spectrum*, Duke Math. J.**37**(1970), 473–478. MR**0264434****[7]**Louis A. Talman,*A note on Kellogg’s uniqueness theorem for fixed points*, Proc. Amer. Math. Soc.**69**(1978), no. 2, 248–250. MR**0467416**, https://doi.org/10.1090/S0002-9939-1978-0467416-7**[8]**E. Zeidler,*Vorlesungen über nichlinear Functionalanalysis*. I,*Fixpunktsatze*, Teubner, Leipzig, 1976.

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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