The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.

中文翻译：

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渐近非扩张映射的隐式中点过程

渐近非扩张映射的概念是非扩张映射一类的重要概括。隐含中点过程对于求解涉及非线性算子的方程是极其基础的。本文通过隐式中点迭代过程研究了渐近非扩张映射类的收敛性分析。通过使用在非线性映射的不动点的计算中起重要作用的众所周知的迭代算法，建立了渐近非扩张映射类收敛的必要条件。数值例子说明了收敛结果。在宽松的参数条件下，推导了一些算法和强收敛性结果，从而在文献中获得了一些推论。