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On fixed-point theorems in fuzzy metric spaces. (English) Zbl 0995.54046
The authors have attempted to extend the Banach fixed point theorem to fuzzy contractive mappings on different types of complete fuzzy metric spaces. They have also introduced a uniform structure on the fuzzy metric space introduced in [A. George and P. Veeramani, ibid. 64, No. 3, 395-399 (1994; Zbl 0843.54014)]. The authors have proved fuzzy Banach contraction theorems in the sense of complete fuzzy metric space introduced in [A. George and P. Veeramani, loc. cit.] and in the sense of fuzzy completeness given by M. Grabiec [ibid. 27, No. 3, 385-389 (1988; Zbl 0664.54032)]. They have also introduced a notion called fuzzy contractive sequence and observed that every fuzzy contractive sequence is G-Cauchy (definition of Cauchy sequence as given in Grabiec) and ask the question: Is a fuzzy contractive sequence a Cauchy sequence (in this case the definition of Cauchy sequence should be taken as given in [A. George and P. Veeramani, loc. cit.].

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54A40Fuzzy topology