# zbMATH — the first resource for mathematics

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:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54A40 Fuzzy topology
Full Text:
##### References:
 [1] Zi-ke, Deng, Fuzzy pseudo metric spaces, J. math. anal. appl., 86, 74-95, (1982) · Zbl 0501.54003 [2] Erceg, M.A., Metric spaces in fuzzy set theory, J. math. anal. appl., 69, 205-230, (1979) · Zbl 0409.54007 [3] George, A.; Veeramani, P., On some results in fuzzy metric spaces, Fuzzy sets and systems, 64, 395-399, (1994) · Zbl 0843.54014 [4] George, A.; Veeramani, P., On some results of analysis for fuzzy metric spaces, Fuzzy sets and systems, 90, 365-368, (1997) · Zbl 0917.54010 [5] Grabiec, M., Fixed points in fuzzy metric spaces, Fuzzy sets and systems, 27, 385-389, (1989) · Zbl 0664.54032 [6] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems, to appear. · Zbl 0985.54007 [7] O. Hadzic, Fixed point theory in probabilistic metric spaces, Serbian Academy of Sciences and Arts, Branch and Novi Sad, University of Novi Sad, Institute of Mathematics, Novi Sad, 1995. [8] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 $$(1984)$$ 215-229. · Zbl 0558.54003 [9] Kramosil, O.; Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326-334, (1975) [10] Pap, E.; Hadzic, O.; Mesiar, R., A fixed point theorem in probabilistic metric spaces and an application, J. math. anal. appl., 202, 433-449, (1996) · Zbl 0855.54043 [11] E. Parau, V. Radu, Some remarks on Tardiff’s fixed point theorem on Menger spaces, Portugal Math. 54, Fasc 4, 1997, 431-440. · Zbl 0898.54031 [12] V. Rafu, Some fixed point theorems in probabilistic metric spaces, Stability problems for stochastic models, Lecture Notes in Mathematics, vol. 1233, Springer Berlin, 1987, pp. 125-133. [13] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. math., 10, 314-334, (1960) · Zbl 0091.29801 [14] Schweizer, B.; Sherwood, H.; Tardiff, R.M., Contractions on probabilistic metric spaces: examples and counterexamples, Stochastica, XII, 1, 5-17, (1988) · Zbl 0689.60019 [15] Sehgal, V.M.; Bharucha-Reid, A.T., Fixed points of contraction mappings of probabilistic metric spaces, Math. systems theory, 6, 97-102, (1972) · Zbl 0244.60004 [16] Tardiff, R.M., Contraction maps on probabilistic metric spaces, J. math. anal. appl., 165, 517-523, (1992) · Zbl 0773.54033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.