Convergence Analysis with Self-Comparative Rate Assessment of a Novel Iterative Method Based on Kirk’s Iteration

This research presents a comprehensive theoretical and computational analysis of the Kirk order-2 iteration method for approximating fixed points of operators that satisfy a weak contractive condition within the framework of a real Banach space. The primary objectives are to establish the strong convergence, stability, and computational efficiency of this iterative scheme. A key contribution of this work is a detailed self-comparative analysis of the convergence rate among six distinct permutations of the iterative scheme’s coefficients. We derive a precise analytical condition that determines which permutation yields a faster convergence rate, providing a theoretical framework for optimizing the algorithm’s performance. Numerical results are presented to validate the theoretical findings and demonstrate the algorithm’s efficiency, confirming that specific permutations of the coefficients can significantly accelerate convergence.