Minimization Theorem and Fixed-Point Theorem in Probabilistic Metric Space

Abstract

This paper is devoted to the study of the basic theory and applications of probabilistic metric spaces (PM-space). In this paper, the topological structure and properties for PM-space are considered. The conditions of metrization and the form of metric functions for PM-spaces, Menger PM-spaces and probabilistic normed linear spaces (PN-space) are given, and the characterizations of various probabilistically bounded sets are presented. As applications, we utilize these results obtained in this paper to study the linear operator theory and fixed-point theory on PM-spaces. In this paper, we prove minimization theorem in the generating space of quasi-probabilistic metric space. Also, we prove common fixed-point theorem for the space which satisfies the minimization theorem.