Global optimization problem and probabilistic distance

Abstract

In this article, the probabilistic metric distance between two disjoint sets is utilised to define the essential criteria for the existence and uniqueness of the best proximity point, which takes into account the global optimization problem. In order to solve this problem, we pretend that we are trying to obtain the optimal approximation to the solution of a fixed point equation. Here, we introduce two types of probabilistic proximal contraction mappings and use a geometric property called Ω-property in the context of probabilistic metric spaces. We also obtain some consequences for self-mappings, which give the fixed point results. Some examples are provided to validate the findings. As an application, we obtain the solution to a second-order boundary value problem using a minimum t-norm in the context of probabilistic metric spaces.