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On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping f : S k k there exists a point x S k such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which S k is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of i n f x | | f ( x ) - f ( - x ) | | which were previously known (as far as the author knows) only for f linear (cf. [1]).

On a generalization of a Greguš fixed point theorem

Ljubomir B. Ćirić (2000)

Czechoslovak Mathematical Journal

Let C be a closed convex subset of a complete convex metric space X . In this paper a class of selfmappings on C , which satisfy the nonexpansive type condition ( 2 ) below, is introduced and investigated. The main result is that such mappings have a unique fixed point.

On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces

Sehie Park (1996)

Commentationes Mathematicae Universitatis Carolinae

Let X be a uniformly convex Banach space, D X , f : D X a nonexpansive map, and K a closed bounded subset such that co ¯ K D . If (1) f | K is weakly inward and K is star-shaped or (2) f | K satisfies the Leray-Schauder boundary condition, then f has a fixed point in co ¯ K . This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.

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