Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

<ns3:p>The fixed-point iteration method is a powerful tool in analysis and its extensions in other branches. It provides a unified structure for proving existence and uniqueness results for various nonlinear problems. This study, conducted within the scope of a real Banach space enriched by uniform convexity, presents an accelerated four-step iterative scheme designed to approximate elements of the fixed-point set corresponding to a mapping called a total asymptotically nonexpansive (briefly, TAN) mapping. The results of weak and strong convergence for the proposed method are established. The results relating to weak convergence require Opial’s condition of the normed space, while the results relating to strong convergence require the addition of the semi-compactness assumption of the mapping under consideration to the previous assumptions. Additionally, numerical simulations are conducted to evaluate the algorithm’s effectiveness and compare it with previously established approaches. These findings extend and refine several recent contributions available in the literature. Lastly, the work includes an application to 2- dimensional equations.</ns3:p>