In this paper, from some confusions in examples on the convergence of certain iteration processes in the literature, we introduce the coordinate-convexity to study the convergence of a new three-step iteration process in Banach spaces with directed graphs. We prove some comparison results of the rate of convergence of the proposed iteration process and certain iteration processes for G-contractive mappings. We also prove the weak and strong convergence results for three G-nonexpansive mappings, and give three numerical examples to illustrate the obtained results.

中文翻译：

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一个新的三步迭代过程收敛到具有有向图的 Banach 空间中三个 G-非膨胀映射的公共不动点

在本文中，从文献中某些迭代过程收敛的例子中的一些混淆，我们引入坐标-凸性来研究具有有向图的Banach空间中新的三步迭代过程的收敛性。我们证明了所提出的迭代过程的收敛速度与G-收缩映射的某些迭代过程的一些比较结果。我们还证明了三个G-非膨胀映射的弱收敛和强收敛结果，并给出了三​​个数值例子来说明得到的结果。