An Application of wt-Distances to Characterize Complete b-Metric Spaces

Abstract

The notion of wt-distance, introduced by Hussain et al. provides a natural generalization to the b-metric framework of the well-known and fruitful concept of w-distance, initiated by Kada et al. Since then, several authors have obtained fixed point theorems for complete b-metric spaces with the help of wt-distances. In this note, we generalize the b-metric version of the celebrated Matkowski fixed point theorem, stated by Czerwik, by replacing the involved b-metric with any wt-distance on the corresponding complete b-metric space. From this result, we derive characterizations of complete b-metric spaces that constitute full generalizations of both a prominent characterization of metric completeness due to Suzuki and Takahashi, and the classical characterization of metric completeness obtained by Hu.