Fixed point theory in probabilistic metric spaces


T. L. Hicks




Most fixed point theorems for Probabilistic Metric spaces ($PM$-spaces) have been proved for the same subclass of $PM$-spaces. It is shown that this subclass ismetrizable. Furthermore, the compatible metric d is related to the distribution functions by \[ d(x,y)<t\quadext{if and ony if}\quad F_{x,y}(t)>1-t. \] This allows an exact translation of the contraction condition, as well as other conditions studied in metric spaces, to $PM$-spaces. Thus, theorems follow immediately from corresponding theorems for metric spaces.