In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard–Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.

中文翻译：

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Banach空间中平均非膨胀映射不动点的逼近

在本文中，我们在 Picard-Mann 混合迭代过程下建立了 Banach 空间中平均非膨胀映射的弱收敛定理和强收敛定理。我们还构建了一个平均非膨胀映射的例子，并表明它超出了非膨胀映射的类别。为了显示我们主要结果的数值准确性，我们展示了本示例的 Picard-Mann 混合迭代过程比所有 Picard、Mann 和 Ishikawa 迭代过程更有效。