A New Three-Step AR-Iteration Scheme with Computational Validation

Abstract

In this paper we introduce a new three-step fixed point iteration scheme, called the AR-iteration (Adaptive Refinement iteration), for approximating fixed points of contraction mappings in Banach spaces. Unlike classical schemes such as Picard, Mann, Ishikawa and Noor iterations, the proposed scheme forms its final update as a convex combination of the mapping evaluated at two intermediate points, giving an additional degree of freedom that can be tuned to reduce the effective contraction factor per iteration. We establish a strong convergence theorem, derive an explicit rate-of-convergence bound showing that the AR-iteration converges at least as fast as the classical Banach contraction rate $k^n$, and prove a $T$-stability result under bounded perturbations. The theoretical findings are validated using three benchmark contraction mappings that are standard in the fixed point literature. Numerical experiments, including convergence tables, parameter sensitivity analysis, and stability testing under random perturbation, confirm that the proposed scheme consistently outperforms Picard, Mann and Ishikawa iterations, and is competitive with the Noor iteration, while offering an additional tunable parameter for further acceleration. All numerical results reported are obtained from direct computation and are reproducible.