Strong Convergence and Almost I ‐Stability by a Two‐Step Iterative Scheme for Weak Contraction in Banach Space

The present paper develops a new two‐step iterative method, referred to as the P ^∗ iteration, for the approximation of fixed points of weak contractions mappings in Banach spaces. The theoretical analysis establishes the strong convergence of the proposed scheme to the unique fixed point, along with its almost I ‐stability. Further, a comparative study with well‐known iterative methods, including the Picard, Mann, S , Varat, and F ^∗ algorithms demonstrates that the P ^∗ iteration converges at a faster rate. To validate the theoretical findings, numerical examples are provided, along with graphical and numerical illustrations implemented in MATLAB.