A Comparative Analysis of Classical Contractive Mappings: Sharp Constants, Inclusion Conditions, and Picard Iteration Performance - Syamsuddin Mas'ud (Eigen Mathematics Journal, 2026)
There are some of the classical contractive mappings introduced by Banach, Kannan, Chatterjea, Reich, Hardy--Rogers, and Ćirić. They remain fundamental in fixed point theory. While their theoretical properties are well documented, a detailed and systematic comparison of the convergence performance of Picard iterations among these classes is still lacking in the literature. This study presents a comparative analysis from a numerical perspective on a complete metric space. We check the sharp constants of their contractions, inclusion relationships, establish a sufficient condition for a Chatterjea contraction to be a Reich contraction, and evaluate the practical performance of Picard iteration through simple numerical experiments. We give two concrete examples, a linear map and a quadratic polynomial for this numerical experiment. They are provided to show that the intersection of all six classical classes is not limited to linear mappings. A hierarchical diagram and structural comparison table are also given to support our study. The integration of theoretical results and numerical validation offers a clearer and more practical reference for students and researchers in studying the concepts of fixed point theory.