### Abstract

In a recent paper, R. Kellogg [3] showed that if F: D → D is a completely continuous map of the closure of a bounded, convex, open set D in a real Banach space X, F ∈ C^{1}(D), 1 is not an eigenvalue of F^{1}(x) for x ∈ D, andF(x) ≠ x for x ∈ ∂D, then F has a unique fixed point in D. More recently, L. Taiman [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 Taiman remain valid provided that the set {x ∈ D: 1 is an eigenvalue of F^{1}(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.

Original language | English (US) |
---|---|

Pages (from-to) | 237-240 |

Number of pages | 4 |

Journal | Proceedings of the American Mathematical Society |

Volume | 79 |

Issue number | 2 |

DOIs | |

State | Published - 1980 |

### Fingerprint

### Keywords

- Fixed point
- K-set contraction

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*79*(2), 237-240. https://doi.org/10.1090/S0002-9939-1980-0565346-2

**A uniqueness theorem for flxedp oints.** / Smith, Hal; Stuart, C. A.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 79, no. 2, pp. 237-240. https://doi.org/10.1090/S0002-9939-1980-0565346-2

}

TY - JOUR

T1 - A uniqueness theorem for flxedp oints

AU - Smith, Hal

AU - Stuart, C. A.

PY - 1980

Y1 - 1980

N2 - In a recent paper, R. Kellogg [3] showed that if F: D → D is a completely continuous map of the closure of a bounded, convex, open set D in a real Banach space X, F ∈ C1(D), 1 is not an eigenvalue of F1(x) for x ∈ D, andF(x) ≠ x for x ∈ ∂D, then F has a unique fixed point in D. More recently, L. Taiman [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 Taiman remain valid provided that the set {x ∈ D: 1 is an eigenvalue of F1(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.

AB - In a recent paper, R. Kellogg [3] showed that if F: D → D is a completely continuous map of the closure of a bounded, convex, open set D in a real Banach space X, F ∈ C1(D), 1 is not an eigenvalue of F1(x) for x ∈ D, andF(x) ≠ x for x ∈ ∂D, then F has a unique fixed point in D. More recently, L. Taiman [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 Taiman remain valid provided that the set {x ∈ D: 1 is an eigenvalue of F1(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.

KW - Fixed point

KW - K-set contraction

UR - http://www.scopus.com/inward/record.url?scp=84966231975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966231975&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1980-0565346-2

DO - 10.1090/S0002-9939-1980-0565346-2

M3 - Article

VL - 79

SP - 237

EP - 240

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -