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Coincidence point theorems in probabilistic and fuzzy metric spaces. (English) Zbl 1118.54024
The authors present some extensions of coincidence point theorems onto different classes of probabilistic metric spaces. More precisely they consider a hybrid pair of single- and multi-valued mappings in Menger and generalized Menger spaces, fuzzy metric spaces and Hicks spaces and they prove coincidence point theorems for such mappings under some generalized contractive conditions. The problem with the paper is that although the authors refer to the book [B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland Series in Probability Applied Mathematics, North-Holland (1983; Zbl 0546.60010)] they have not read it carefully enough. This is why they consider different classes of probabilistic metric spaces in the sense defined in [loc. cit.] instead of formulating more general results for such spaces or Serstnev spaces. They also seem to ignore some results of Istratescu and recent results of O. Hadzić.
MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E70Probabilistic metric spaces
03E72Fuzzy set theory
54A40Fuzzy topology
54C60Set-valued maps (general topology)