A Three-Step Iterative Scheme for Nonexpansive Mappings: Convergence Analysis and Applications to Convex Optimization

This study focuses on a three-step iterative scheme, referred to as the NIP iteration, for the approximation of fixed points associated with nonexpansive mappings in uniformly convex Banach spaces. Weak convergence is established using Fejér monotonicity, asymptotic regularity and the demiclosedness principle. Strong convergence is proved under uniform convexity, compactness, and Condition (I) of Senter and Dotson. A numerical convergence and computational-cost comparison is developed numerically, showing that the NIP iteration performs better than the Ishikawa, S, Noor, Abbas–Nazir and SP schemes. Numerical experiments for nonlinear nonexpansive mappings validate the theoretical findings. An application to convex optimization via fixed point reformulation is also presented, illustrating the effectiveness of the method.