The notion of a contraction mapping for a probabilistic metric space recently introduced by T. L. Hicks is compared with the notion previously introduced by V. L. Sehgal and A. T. Bharucha-Reid. By means of appropriate examples, it is shown that these two notions are independent. It is further shown that every Hick's contraction on a PM space (S,F,tW) is an ordinary metric contraction with respect to a naturally defined metric on that space; and it is again pointed out that, in Menger spaces under Min and similar t-norms, the contractions of Sehgal and Bharucha-Reid are also ordinary contractions on related metric spaces.
@article{urn:eudml:doc:39008,
title = {Contractions on probabilistic metric spaces: examples and counterexamples.},
journal = {Stochastica},
volume = {12},
year = {1988},
pages = {5-17},
zbl = {0689.60019},
mrnumber = {MR1004655},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39008}
}
Schweizer, Berthold; Sherwood, Howard; Tardiff, Robert M. Contractions on probabilistic metric spaces: examples and counterexamples.. Stochastica, Tome 12 (1988) pp. 5-17. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39008/