A simulation function approach for \(\phi \)-fixed point with an application to the non-linear integral equation

Abstract

This paper introduces the concept of a simulation function and Z-contraction within a Menger probabilistic metric space. Control functions are integrated with Z-contraction to establish several \(\phi \)-fixed point results. Additionally, multiple examples are provided to demonstrate the applicability of the main theorems, including one that does not conform to the Banach contraction principle. Finally, the existence of a solution to a system of integral equations is explored.