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Mat. Zametki, 2007, Volume 82, Issue 2, Pages 201–206 (Mi mz3791)  

Existence of Fixed Points for Mappings of Finite Sets

V. I. Danilova, G. A. Koshevoy

a Central Economics and Mathematics Institute, RAS

Abstract: We show that the existence theorem for zeros of a vector field (fixed points of a mapping) holds in the case of a “convex” finite set $X$ and a “continuous” vector field (a self-mapping) directed inwards into the convex hull $\operatorname{co}X$ of $X$. The main goal is to give correct definitions of the notions of “continuity” and “convexity”. We formalize both these notions using a reflexive and symmetric binary relation on $X$, i.e., using a proximity relation. Continuity (we shall say smoothness) is formulated with respect to any proximity relation, and an additional requirement on the proximity (we shall call it the acyclicity condition) transforms $X$ into a “convex” set. If these two requirements are satisfied, then the vector field has a zero (i.e., a fixed point).

Keywords: Brouwer fixed-point theorem, self-mapping, vector field on a finite set, convexity binary relation, proximity relation, acyclic set

DOI: https://doi.org/10.4213/mzm3791

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English version:
Mathematical Notes, 2007, 82:2, 174–179

Bibliographic databases:

UDC: 519.1
Received: 01.03.2005

Citation: V. I. Danilov, G. A. Koshevoy, “Existence of Fixed Points for Mappings of Finite Sets”, Mat. Zametki, 82:2 (2007), 201–206; Math. Notes, 82:2 (2007), 174–179

Citation in format AMSBIB
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