On fuzzy contractions in fuzzy metric spaces

Abstract: The classical Banach principle relies on contractive sequences being Cauchy. Gregori and Sapena (2002) established a fuzzy analogue but required this as an extra hypothesis, leaving it as an open problem. Gregori et al. (2020) later asked whether strictly fuzzy $ψ$-contractive sequences are necessarily Cauchy. We settle both problems negatively via a single counterexample: in a constructed GV-fuzzy metric space, there exists a sequence which is both GS-contractive and strictly fuzzy $ψ$-contractive, yet not Cauchy. This shows that contractivity never implies Cauchyness in general GV-fuzzy metric spaces, and additional structural conditions in fuzzy fixed point theorems are indispensable.