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Syamsuddin Mas'ud
Department of Mathematics, Universitas Negeri Makassar
Mathematics

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A Comparative Analysis of Classical Contractive Mappings: Sharp Constants, Inclusion Conditions, and Picard Iteration Performance Syamsuddin Mas'ud
Eigen Mathematics Journal Vol 9 No 1 (2026): June
Publisher : University of Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29303/emj.v9i1.362

Abstract

There are some of the classical contractive mappings introduced by Banach, Kannan, Chatterjea, Reich, Hardy--Rogers, and \'{C}iri\'{c}. 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.
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