A new four-step iteration scheme for generalized α-nonexpansive multivalued mappings in Banach spaces

In this paper, we introduce a new four-step iterative process for approximating fixed points of generalized alpha-nonexpansive multivalued mappings in uniformly convex Banach spaces. The proposed iteration unifies and extends several classical schemes, including the Mann, Ishikawa and Noor processes, while preserving a simple computational structure. Strong and weak convergence theorems are established under mild conditions such as the Opial property and the demiclosedness of I-PT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I - P_{T}$\end{document}. Unlike contraction-based approaches, the stability of the process is analyzed directly within the generalized alpha-nonexpansive framework, demonstrating robustness under small perturbations. A numerical experiment in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}<^>{2}$\end{document} illustrates the superior performance of the proposed algorithm, achieving faster residual decay and improved asymptotic regularity compared with existing methods. Both theoretical and numerical findings confirm the efficiency and stability of the scheme, suggesting promising applications in nonlinear analysis and optimization problems.