### 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) |
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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 |

### Keywords

- Fixed point
- K-set contraction

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

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