Probabilistic contraction under a control function

Abstract

Abstract Probabilistic metric spaces are metric structures having uncertainty built within their geometry, which has made them into an appropriate context for modelling many real life problems. Theoretical studies on these structures have also appeared extensively. This paper is intended for some development of fixed point theory in probabilistic metric spaces, which is an active area of contemporary research. We define a new contraction mapping in such spaces and show that the contraction has a unique fixed point if such spaces are G-complete with an arbitrary choice of a continuous t-norm. With a minimum t-norm, the result is further extended in any complete probabilistic metric space. The contraction is defined with the help of a control function which is different from several other control functions used in probabilistic fixed point theory by other authors. The methodology of the proof is new. An illustrative example is given. The present work is a part of probabilistic analysis.