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Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

Karapınar Erdal, Romaguera Salvador, Tirado Pedro

Demonstratio Mathematica

2022

Vol. 55, nr 1

939--951

Involving w-distances we prove a fixed point theorem of Caristi-type in the realm of (non-necessarily T1) quasi-metric spaces. With the help of this result, a characterization of quasi-metric completeness is obtained. Our approach allows us to retrieve several key examples occurring in various fields of mathematics and computer science and that are modeled as non- T1 quasi-metric spaces. As an application, we deduce a characterization of complete G -metric spaces in terms of a weak version of Caristi’s theorem that involves a G-metric version of w-distances.

On statistical convergence in quasi-metric spaces

İlkhan Merve, Kara Emrah Evren

Demonstratio Mathematica

2019

Vol. 52, nr 1

225--236

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.