Effect of the continuous t-norm on strict fuzzy contractivity and Cauchyness

Determine whether there exists a strictly fuzzy ψ-contractive sequence that is not Cauchy in a George–Veeramani fuzzy metric space equipped with the product t-norm or the Łukasiewicz t-norm, and thereby determine whether the choice of continuous t-norm affects the implication from strict fuzzy ψ-contractivity to Cauchyness.

Background

The paper constructs a George–Veeramani fuzzy metric space using the minimum t-norm in which the sequence {x_n=n} is both GS-contractive and strictly fuzzy ψ-contractive but is not Cauchy. This provides a negative answer to the corresponding questions for that particular continuous t-norm.

The authors leave unresolved whether analogous counterexamples exist when the product t-norm or the Łukasiewicz t-norm is used. The problem asks whether the t-norm itself can change the validity of the implication that strict fuzzy ψ-contractivity forces Cauchyness.

References

We conclude by proposing two open problems arising naturally from our investigation. Problem 3. Does there exist a strictly fuzzy $\psi$-contractive sequence that is not Cauchy in a GV-fuzzy metric space equipped with the product t-norm or the {\L}ukasiewicz t-norm? In other words, does the choice of the continuous t-norm affect the validity of the statement that strictly fuzzy $\psi$-contractive implies Cauchy?

— On fuzzy contractions in fuzzy metric spaces  (2609.03372 - Xu et al., 3 Sep 2026) in Section 4, Concluding remarks, Problem 3