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.