Cauchy Sequences in Fuzzy Metric Spaces and Fixed Point Theorems

TL;DR

This paper introduces new classes of contractive mappings in fuzzy metric spaces, characterizes Cauchy sequences, and proves fixed point theorems, extending previous results in fuzzy analysis.

Contribution

It defines a new class of gauge functions and shows fuzzy cpsie-contractive mappings are of ciriec9-Matkowski type, providing broader fixed point results.

Findings

Characterization of Cauchy sequences in fuzzy metric spaces

Fixed point theorems for fuzzy cpsie-contractives

Extension of previous fuzzy fixed point results

Abstract

In this paper, contractive mappings of \'Ciri\'c-Matkowski type in fuzzy metric spaces are studied. A class $\Psi_1$ of gauge functions $\psi:(0,1]\to(0,1]$ such that, for any $r\in(0,1)$, there exists $\rho\in(r,1)$ such that $1-r> \tau >1-\rho$ implies $\psi(\tau)\geq 1-r$, is introduced, and it is shown that fuzzy $\psi$-contractive mappings are fuzzy contractive mappings of \'Ciri\'c-Matkowski 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.\ 159(2008) 739--744), Wardowski (Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst.\ 222(2013) 108--114) and others.