Motivated and inspired by the growing contribution with respect to iterative approximations from some researchers in the literature, we design and investigate two types of brand-new semi-implicit viscosity iterative approximation methods for finding the fixed points of nonexpansive operators associated with contraction operators in complete ${\operatorname{CAT}(0)}$ spaces and for solving related variational inequality problems. Under some suitable assumptions, strong convergence theorems of the sequences generated by the approximation iterative methods are devised, and a numerical example and some applications to related variational inequality problems are included to verify the effectiveness and practical utility of the convergence theorems. Our main results presented in this paper do not only improve, extend and refine some corresponding consequences in the literature, but also show that the additional variational inequalities, general variational inequality systems and equilibrium problems can be solved via approximation of the iterative sequences. Finally, we provide an open question for future research.

中文翻译：

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两种新的半隐式粘度迭代，用于逼近与收缩算子和应用程序相关的非膨胀算子的固定点

受文献中一些研究人员对迭代逼近的不断增长的贡献的启发和启发，我们设计和研究了两种全新的半隐式粘度迭代逼近方法，以完全找到与收缩算子相关的非膨胀算子的固定点$ {\ operatorname {CAT}（0）} $空格，用于解决相关的变分不等式问题。在一些合适的假设下，设计了由逼近迭代法生成的序列的强收敛定理，并通过数值例子和对相关变分不等式问题的一些应用来验证收敛定理的有效性和实用性。本文介绍的主要结果不仅可以改善，扩展和完善了文献中的相应结果，但也表明可以通过近似迭代序列来解决其他变分不等式，一般变分不等式系统和平衡问题。最后，我们为将来的研究提供了一个开放的问题。