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A fixed point conjecture for Borsuk continuous set-valued mappings

Dariusz Miklaszewski (2002)

Fundamenta Mathematicae

The main result of this paper is that for n = 3,4,5 and k = n-2, every Borsuk continuous set-valued map of the closed ball in the n-dimensional Euclidean space with values which are one-point sets or sets homeomorphic to the k-sphere has a fixed point. Our approach fails for (k,n) = (1,4). A relevant counterexample (for the homological method, not for the fixed point conjecture) is indicated.

A fixed point theorem for a multivalued non-self mapping

Billy E. Rhoades (1996)

Commentationes Mathematicae Universitatis Carolinae

We prove a fixed point theorem for a multivalued non-self mapping in a metrically convex complete metric space. This result generalizes Theorem 1 of Itoh [2].

A fixed point theorem for branched covering maps of the plane

Alexander Blokh, Lex Oversteegen (2009)

Fundamenta Mathematicae

It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has...

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