In this paper, we introduce the concept of a strongly asymptotically quasi-nonexpansive sequence of mappings in the context of a Hilbert space. Firstly, we prove the weak convergence of a Picard type iterative method to a common fixed point for the sequence as well as the strong convergence when a Halpern type regularization scheme is considered. Among other features, our results are applied to get convergence to a fixed point of iterative procedure of Ishikawa and Halpern–Ishikawa for a lipschitzian asymptotically quasi-nonexpansive mappings (resp. as the Picard type iteration and the Halpern type iteration for the sequence of strongly asymptotically quasi-nonexpansive mappings) and, the convergence of proximal like methods for asymptotically nonexpansive mappings. Finally, we show an example of an asymptotically nonexpansive mappings and compute some of the methods studied in the paper.

中文翻译：

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渐近非扩张和拟非扩张映射的一些迭代和近端相似方法的统一

在本文中，我们介绍了在希尔伯特空间背景下的一个强渐近拟非扩张映射序列的概念。首先，我们证明了Picard型迭代方法对序列的公共不动点的弱收敛性，以及当考虑Halpern型正则化方案时的强收敛性。除其他功能外，我们的结果适用于对Lipschitzian渐近拟非扩张映射（分别为Picard类型迭代和Halpern类型迭代，针对强序列的迭代）收敛到Ishikawa和Halpern–Ishikawa迭代过程的固定点渐近拟非扩张映射），以及渐近非扩张映射的类似近端方法的收敛性。最后，