PreprintarXiv licencearXiv
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.
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BibTeX
@misc{xu2026fuzzy,
title = {{On fuzzy contractions in fuzzy metric spaces}},
author = {Fengshuo Xu and Haitao Ji and Dong Qiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03372},
eprint = {2609.03372},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03372},
}
RIS
TY - UNPB
TI - On fuzzy contractions in fuzzy metric spaces
AU - Fengshuo Xu
AU - Haitao Ji
AU - Dong Qiu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03372
UR - https://arxiv.org/abs/2609.03372
AB - 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.
ER -