A refined multi-step iterative scheme for the approximation of fixed points of nonlinear mappings in Banach space

Abstract

In this paper, we introduce a refined multi-step iterative scheme for approximating fixed points of nonlinear mappings in Banach spaces. The proposed method is designed to enhance convergence speed while maintaining stability and robustness. Under suitable conditions on the control parameters, we establish strong convergence results for contraction mappings and prove that the generated sequence converges to the unique fixed point. Furthermore, stability analysis of the proposed scheme is investigated, and it is shown to be \(T\)-stable under standard assumptions. A comparative study based on the concept of convergence rate demonstrates that the proposed scheme outperforms several classical and modern iterative methods, including Picard, Mann, Ishikawa, and other multi-step methods. To validate the theoretical findings, numerical experiments are carried out on various classes of nonlinear mappings, including rational, monotone, cubic, and logarithmic functions. The results confirm that the proposed method achieves higher accuracy with fewer iterations, thereby offering improved computational efficiency. Consequently, the scheme provides a reliable and effective tool for solving nonlinear operator equations in Banach spaces.