On Necessary and Sufficient Conditions for Fixed Point Convergence: A Contractive Iteration Principle

While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization grounded in the iterative contraction principle in complete metric spaces. This generalization establishes both the necessary and sufficient conditions for the existence of a unique fixed point to which all iterative sequences converge, along with an accurate error estimate. Furthermore, we present and prove an additional theorem that characterizes the convergence of all iterative sequences to fixed points that may not be unique. Several examples are provided to illustrate the practical application of these results, including a case where the traditional and well-known generalizations of Banach's theorem, such as those by Banach, Kannan, Chatterjea, Hardy-Rogers, Meir-Keeler, and Guseman, are inapplicable.