A global optimality result in probabilistic spaces using control function. (2nd November 2021) - Saha, P.; Guria, S.; Kumar Bhandari, Samir; Choudhury, Binayak S.
ABSTRACT: In this paper we solve the global optimality problem of determining probabilistic distance between two sets. We treat this problem as that of finding an optimal approximate solution of a fixed point equation. It is well studied in the context of metric spaces under the name of best proximity point problem. Our work is in the domain of the extension of this line of research to probabilistic metric spaces. For our purpose we define a probabilistic contraction mapping with the help of a control function. This control function plays a central role in the proof of our main theorem. We also use probabilistic P -property for pairs of subsets which is essentially a geometric property. A well known probabilistic extension of the Banach's contraction mapping principle follows as corollary of our main results. We illustrate our results with an example.