Fixed Point Theorems for TSR-Contraction Mapping in Probabilistic Metric Spaces

TL;DR

The paper introduces TSR-contraction mappings in probabilistic metric spaces and establishes existence and uniqueness of fixed points under this new contraction principle. It formalizes two variants, TSR contraction and TSR-P contraction, and proves a unique fixed point in complete Menger spaces when the contraction constant is at most $\tfrac{1}{2}$ and a natural monotonicity condition holds. The results are extended to general probabilistic metric spaces with left-continuous $t$-norms, including localized fixed-point results on $t$-open/$t$-closed spheres and a variant ensuring fixed points when $f^m$ is contracting. Collectively, the work broadens fixed-point theory in probabilistic metric spaces by providing new, verifiable sufficient conditions for contraction mappings to have unique fixed points.

Abstract

The concept of fixed point plays a crucial role in various fields of applied mathematics. The aim of this paper is to establish the existence of a unique fixed point of some type of functions which satisfy a new contraction principle, namely, TSR-contraction principle in various types of probabilistic metric spaces. The proposed contraction mapping is different from our traditional definitions of contraction mapping.