A uniqueness theorem for fixed points
H. L. Smith, C. A. Stuart
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Source: Crossref
Published: Jun 1, 1980
DOI: 10.1090/s0002-9939-1980-0565346-2
Open original source ↗Source abstract
In a recent paper, R. Kellogg [ 3 ] showed that if F : D ¯ → D ¯ F:\bar D \to \bar 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 ) F \in {C^1}(D) , 1 is not an eigenvalue of F ′ ( x ) F’(x) for x ∈ D x \in D , and F ( x ) ≠ x F(x) \ne x for x ∈ ∂ D x \in \partial D , then F has a unique fixed point in D . More recently, L. Talman [ 7 ] extended this result to k -set contractions when k > 1 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 Talman remain valid provided that the set { x ∈ D : 1 \{ x \in D:1 is an eigenvalue of F ′ ( x ) } F’(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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