A uniqueness theorem for flxedp oints

Hal Smith; C. A. Stuart

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

A uniqueness theorem for flxedp oints

Hal Smith, C. A. Stuart

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

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 ∈ 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.

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	https://doi.org/10.1090/S0002-9939-1980-0565346-2
State	Published - Jun 1980

Keywords

Fixed point
K-set contraction

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.1090/S0002-9939-1980-0565346-2

Cite this

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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 ∈ 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.",

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AU - Stuart, C. A.

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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.

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A uniqueness theorem for flxedp oints

Abstract

Keywords

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