Measures of noncompactness on \(w\)-distance spaces. (English) Zbl 1462.54026

Summary: The aim of this paper is to provide a new framework for the study of measures of noncompactness in generalized metric spaces. Firstly, we introduce the notion of \(w\)-measure of noncompactness on metric spaces with a \(w\)-distance and extend the diameter and Kuratowski functionals to this setting. At the end, we give a characterization of metric completeness via our main results, providing a new answer to the open question mentioned by I. Arandjelovic [Stavovi o presecanju i njihove primene u nelinearnoj analizi (Serbian). Belgrade: University of Belgrade (PhD Thesis) (1999)].


54E50 Complete metric spaces
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
55M20 Fixed points and coincidences in algebraic topology
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