A note on local bases and convergence in fuzzy metric spaces

Abstract

ElsevierGregori Gregori, V.; Miñana Prats, JJ.; Morillas Gómez, S. (2014)

Extended Fuzzy Metrics and Fixed Point Theorems

“…x,y (0) = t>0 M 0 (x, y, t), for all x, y ∈ X. Furthermore, we can deduce the following result proved by Gregori et al in [8]. Proposition 1.…”

“…In this section, we compare τ M 0 -convergent sequences with s-convergent ones, a stronger concept of convergence introduced in [8]. Such comparison is framed in the class of extendable fuzzy metrics.…”

“…Indeed, this is so. First, we notice that the condition itself of being M extendable is used explicitly (Theorem 3.3 [8]), or, in a concealed or relaxed way, in order to obtain fixed point theorems (see Theorem 3.2 of [9] or Theorem 2.4 of [10]). We here go further and we will give a notion of ψ-0-contractive mapping (Definition 7), that is, contractivity assumed only at t = 0.…”

std-Convergence in fuzzy metric spaces

“…For instance, several well-motivated notions of convergence and Cauchyness related to sequences can be found in the literature [2][3][4][9][10][11]13]. In particular, more recently a stronger concept than convergence, called s-convergence, has been used to characterize certain type of fuzzy metric spaces [5]. Then, for a concept of convergence it is natural and interesting to study a concept of Cauchyness, or vice-versa, such that both are pairwise compatible.…”

“…Then, in [4] the authors gave (an appropriate) concept of p-Cauchy sequence and also they initiated the study on the relationship of the concept of p-convergence with local bases. This study has been continued in [6].…”