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Common fixed points of compatible maps in fuzzy metric spaces. (English) Zbl 0985.54009
Several researchers have defined the concept of fuzzy metric space in various ways. In this paper, the authors introduce the concept of compatibility on fuzzy metric space (in the sense of a definition given in [{\it A. George} and the reviewer, ibid. 64, No. 3, 395-399 (1994; Zbl 0843.54014)] and use it to prove common fixed point theorems for compatible mappings. It should be noted that the authors prove two common fixed point theorems for a fuzzy metric space having continuous $t$-norm defined by $a^*b= \min\{a,b\}$, $a,b\in [0,1]$. The authors do not discuss their results for an arbitrary continuous $t$-norm. It is of worth to investigate whether the fixed point theorems proved in this paper are true for arbitrary $t$-norms.

54A40Fuzzy topology
54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
[1] George, A.; Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy sets and systems 64, 395-399 (1994) · Zbl 0843.54014
[2] Jungck, G.: Compatible mappings and common fixed points. Internat. J. Math. math. Sci. 9, 779-791 (1986) · Zbl 0613.54029
[3] Jungck, G.: Compatible mappings and common fixed point (2). Internat. J. Math. math. Sci. 11, No. 2, 285-288 (1988) · Zbl 0647.54035
[4] Kramosil, O.; Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetica 11, 326-334 (1975) · Zbl 0319.54002
[5] Zadeh, L. A.: Fuzzy sets. Inform. and control 8, 338-353 (1965) · Zbl 0139.24606