Accelerated fixed point approximation for Suzuki-type mappings using HK-iteration: convergence theory and polynomiographic dynamics

Abstract

This study investigates the tracking and convergence characteristics of Suzuki-type generalized nonexpansive operators utilizing a multistage iterative process. We establish both weak and strong convergence theorems for the sequence \(\{j_v\}_{v=1}^{\infty}\) generated by this multi-tiered architecture, investigating its foundational approximation properties within modern fixed-point theory. By using the auxiliary intermediate iterates \(t_v\) and \(u_v\) of the mapping \(\mathcal{M}\), the suggested framework incorporates deeper operator features into the scheme. The algorithm's behavior and efficiency are confirmed by a thorough numerical example, and its stability is confirmed and its unique dynamical patterns and graphical structures are revealed by comparing it with multiple classical iterations using Kalantari's polynomiography.