The object of this paper is to establish a unique common fixed point theorem for six self-mappings satisfying a generalized contractive condition through compatibility of type (β) and weak compatibility in a fuzzy metric space. It significantly generalizes the result of Singh and Jain [The Journal of Fuzzy Mathematics (2006)] and Sharma [Fuzzy Sets and Systems (2002)]. An example has been constructed in support of our main result. All the results presented in this paper are new.
In this paper, we propose the concept of Suzuki type fuzzy Z-contractive mappings, which is a generalization of Fuzzy Z-contractive mappings initiated in the article [S. Shukla, D. Gopal, W. Sintunavarat, A new class of fuzzy contractive mappings and fixed point theorems, Fuzzy Sets and Systems 350 (2018)85-95]. For this type of contractions suitable conditions are framed to ensure the existence of fixed point in G-complete as well as M-complete fuzzy metric spaces. A comprehensive set of examples are furnished to demonstrate the validity of the obtained results.
In this paper, we prove that for a given positive continuous t-norm there is a fuzzy metric space in the sense of George and Veeramani, for which the given t-norm is the strongest one. For the opposite problem, we obtain that there is a fuzzy metric space for which there is no strongest t-norm. As an application of the main results, it is shown that there are infinite non-isometric fuzzy metrics on an infinite set.