Applications of Parametric 2-Metric Spaces to Random Fixed Point Theorems and Pettis Integrability

Authors: Professor Dr. Dhirendra Kumar Singh, Virendra Kumar Saket

Abstract: The study of fixed point theory in generalised metric spaces has become an important topic in nonlinear analysis with applications to areas such as integral equations, optimisation and stochastic processes. We present new extensions of the framework of the parametric 2-metric spaces proposed by Cetkin (2019) in which we relate such a nonlinear geometry to measurable multifunctions and Pettis integrability in Banach spaces. First we recall the basic definitions and topological properties of parametric 2-metric spaces, and point out their role as a nonlinear generalisation of parametric metrics. Thus, we prove theorems on random fixed points and on the integrability of multifunctions in the setting of parametric 2-metric topology. The above results provide a unified framework connecting the theory of fixed points with the measurable selection theory and integrability conditions. We also discuss applications to integral inclusions and stochastic operator equations, showing that parametric 2-metric spaces can be a fruitful area for further research in functional analysis and stochastic settings.