A fixed point theorem for mappings on the ℓ_∞-sum of a metric space and its application

2017/07/18 by Jachymski, Jacek, Maślanka, Łukasz, Strobin, Filip
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.05618

Abstract

The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings f: ℓ_∞(X) → X, where X is a metric space and ℓ_∞(X) is the space of all bounded sequences of elements from~X. Our result generalizes the theorem obtained by Miculescu and Mihail in 2008, who proved a~counterpart of the Banach principle for mappings f:X^m→ X, where X^m is the Cartesian product of m copies of X. We also compare our result with a recent one due to Secelean, who obtained a weaker assertion under less restrictive assumptions. We illustrate our result with several examples and give an application.

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