Cauchy Sequences in Fuzzy Metric Spaces and Fixed Point Theorems

In this paper, contractive mappings of \CM{} type in fuzzy metric spaces are studied. A class \Psi_۱ of gauge functions \psi:(۰,۱]\to(۰,۱] such that, for any r\in(۰,۱), there exists \rho\in(r,۱) such that ۱-r> \tau >۱-\rho implies \psi(\tau)\geq ۱-r, is introduced, and it is shown that fuzzy \psi-contractive mappings are fuzzy contractive mappings of \CM{} type. A characterization of Cauchy sequences in fuzzy metric spaces is presented, and it is utilized to establish fixed point theorems. Examples are given to support the results. Our results cover those of Mihet (Fuzzy \psi-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. ۱۵۹(۲۰۰۸) ۷۳۹-۷۴۴), Wardowski (Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst. ۲۲۲(۲۰۱۳) ۱۰۸-۱۱۴) and others.

کلیدواژه ها:

Fuzzy metric spaces ، Cauchy sequences ، Fixed point theorems ، Contractive mappings ، Gauge functions