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\(G\)-completeness and \(M\)-completeness in fuzzy metric spaces: a note on a common fixed point theorem. (English) Zbl 1224.54072
\(G\)-completeness was introduced by M. Grabiec [Fuzzy Sets Syst. 27, No. 3, 385–389 (1983; Zbl 0664.54032)] in order to obtain a fuzzy version of the Banach contraction principle. In 2008, a common fixed point theorem in \(G\)-complete fuzzy metric spaces under the t-norm Min was proved by S. Kumar [Acta Math. Hung. 118, No. 1–2, 9–28 (2008; Zbl 1164.47058)]: Every \(G\)-complete fuzzy metric space is \(M\)-complete. In this paper, the authors prove that the common fixed point theorem does hold even if \(G\)-completeness of the space is replaced by \(M\)-completeness or if the strongest t-norm Min is replaced with an arbitrary continuous t-norm.
Reviewer: Shou Lin (Ningde)

MSC:
54E70 Probabilistic metric spaces
54H25 Fixed-point and coincidence theorems (topological aspects)
54A40 Fuzzy topology
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