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<p>A recent study on nonexpansive monotone operators on partially ordered Banach spaces revealed that the sequence of Cesàro means converges to a fixed point, but also plays an instrumental role in the convergence analysis of the Picard successive approximations. Throughout this paper we provide a new iterative construction of Krasnoselskii type, as alternative to the ergodic iteration, that also supports the analysis of the Picard iterative sequence. Moreover, this study considers a wider class of monotone nonexpansive mappings, namely Reich–Suzuki type nonexpansive operators. Finally, a numerical simulation is provided by polynomiographic methods, so as to confirm the intuitive conclusions about the higher efficiency of the Picard iteration and the Krasnoselskii iteration compared to the ergodic procedure.</p>

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Ergodic theory for monotone Reich–Suzuki type nonexpansive operators

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  • 发表日期：

    2025年2月12日

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  • Andreea Bejenaru,
  • Mihai Postolache

摘要

A recent study on nonexpansive monotone operators on partially ordered Banach spaces revealed that the sequence of Cesàro means converges to a fixed point, but also plays an instrumental role in the convergence analysis of the Picard successive approximations. Throughout this paper we provide a new iterative construction of Krasnoselskii type, as alternative to the ergodic iteration, that also supports the analysis of the Picard iterative sequence. Moreover, this study considers a wider class of monotone nonexpansive mappings, namely Reich–Suzuki type nonexpansive operators. Finally, a numerical simulation is provided by polynomiographic methods, so as to confirm the intuitive conclusions about the higher efficiency of the Picard iteration and the Krasnoselskii iteration compared to the ergodic procedure.