A note on ${\mathcal {F}}$-metric spaces

Abstract

A new generalization of the metric space notion, named ${\mathcal {F}}$-metric space, was given in [M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl. 20 (2018), no. 3, Art. 128, 20 pp.]. In this paper, we investigate some properties of ${\mathcal {F}}$-metric spaces. A simple proof is given to show that the natural topology induced by an ${\mathcal {F}}$-metric is metrizable. We present a method to construct s-$\hbox {relaxed}_{{p}}$ spaces and, therefore, ${\mathcal {F}}$-metric spaces from bounded metric spaces. We give some results that reveal differences between metric and ${\mathcal {F}}$-metric spaces. In particular, we show that the ordinary open and closed balls in ${\mathcal {F}}$-metric spaces are not necessarily topological open and closed, respectively. This answers a question posed implicitly in the quoted paper. We also show that ${\mathcal {F}}$-metrics are not necessarily jointly continuous functions. Despite some topological differences between metrics and ${\mathcal {F}}$-metrics, we show that the Nadler fixed point theorem and, therefore, the Banach contraction principle in the frame of ${\mathcal {F}}$-metric spaces can be reduced to their original metric versions. This reduction even happens when the Schauder fixed point theorem is investigated in ${\mathcal {F}}$-normed spaces structure. By applying the given technique in this paper, it turns out that some nonlinear ${\mathcal {F}}$-metric contractions and, therefore, the related ${\mathcal {F}}$-metric fixed point results can naturally be reduced to their metric versions. In addition, the same happens for some topological fixed point results.