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<p>In this paper, we propose a novel unified framework for the Krasnosel’skiĭ–Mann iteration that incorporates perturbed mappings and double inertial steps in uniformly convex Banach spaces. We establish both weak and strong convergence of the generated sequence to a fixed point of the underlying operator. Compared with the iterative schemes developed by Zhao et al. (Inverse Problems, 21, 1791–1799, 2005) and Xu (Inverse Problems, 22, 2021–2034, 2006), our framework achieves faster convergence under more relaxed assumptions and with more flexible parameter settings. We further demonstrate the applicability of the proposed method to variational inequality problems, (multiple-set) split feasibility problems, and split variational inclusion problems. Several numerical experiments are conducted to validate the efficiency and practical performance of the algorithm.</p>

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A Unified Framework for Perturbation-Accelerated Krasnosel’skiĭ-Mann Iteration with Applications to Split Inverse Problems

  • 期刊论文

  • 发表日期：

    2026年3月12日

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  • Yeyu Zhang

摘要

In this paper, we propose a novel unified framework for the Krasnosel’skiĭ–Mann iteration that incorporates perturbed mappings and double inertial steps in uniformly convex Banach spaces. We establish both weak and strong convergence of the generated sequence to a fixed point of the underlying operator. Compared with the iterative schemes developed by Zhao et al. (Inverse Problems, 21, 1791–1799, 2005) and Xu (Inverse Problems, 22, 2021–2034, 2006), our framework achieves faster convergence under more relaxed assumptions and with more flexible parameter settings. We further demonstrate the applicability of the proposed method to variational inequality problems, (multiple-set) split feasibility problems, and split variational inclusion problems. Several numerical experiments are conducted to validate the efficiency and practical performance of the algorithm.