On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces

Abstract

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p-completeness, it is possible to relax some extra conditions on the space to obtain a fixed point theorem in this framework. To this end, we focus on a fixed point result, proved by Mihet for complete non-Archimedean fuzzy metric spaces (Theorem 1). So, we define a weaker concept than the non-Archimedean fuzzy metric, called t-strong, and we establish an alternative version of Miheţ’s theorem for p-complete t-strong fuzzy metrics (Theorem 2). In addition, an example of t-strong fuzzy metric spaces that are not non-Archimedean is provided.

Vector-Valued Fuzzy Metric Spaces and Fixed Point Theorems

“…Nonetheless, fuzzy metrics show some differences compared with classical ones which still make their study of interest nowadays. On the one hand, they differ in some purely metric topics as completeness or fixed point theory, which are two active topics of research in the literature (see, for instance, the following recent references [5][6][7][8][9][10][11]). On the other hand, fuzzy metrics have been successfully used, compared with their classical counterparts, in engineering problems such as model estimation, modelling multi-agent systems or image filtering (see, for instance, [12][13][14][15][16][17] and references therein).…”