Common Fixed Point Theorem for Six Functions on Menger Probabilistic Generalized Metric Space

TL;DR

This work addresses the existence and uniqueness of a common fixed point for six self-mappings in a Menger probabilistic generalized metric space (PGM-space). It introduces a compatibility framework for triple mappings [A,B,C] and proves an existence-uniqueness theorem under a contraction-type inequality involving the distance distribution functions $G_{x,y,z}$ and a continuous t-norm $*$, notably $G_{Ax, By, Cz}(kt) \,\ge\, G_{Sx, Ty, Dz}(t) * G_{Sx, Ax, Dz}(t) * G_{Ax, By, Cz}(t) * G_{Tx, By, Cz}(t) * G_{Sx, Cz, Dz}(2t)$ for all $x,y,z \in X$ and $t>0$, along with $a*a \ge a$. The method constructs a sequence whose successive iterations form a Cauchy sequence in the complete PGM-space and shows the limit is fixed by all six maps through a series of limiting arguments (Step I–VI). The result extends prior four-function common fixed-point theorems and broadens the applicability of fixed-point theory in probabilistic metric settings.

Abstract

The main aim of this paper is to find a unique common fixed point for six functions in a Menger probabilistic generalized metric space. For this purpose, we have defined the compatibility of three functions and established some required theorems.