In this paper, the isotonicity of the proximity operator and its applications are discussed. We first establish a few new conditions of the mappings such that their proximity operators are isotone with respect to orders induced different minihedral cones. Some properties and examples for these conditions are then introduced. We especially consider the isotonicity of the proximity operator with respect to one order induced by a subdual cone and two orders. To estimate the convergence rate of the iterative algorithms, some other inequality characterizations of the proximity operator with respect to the orders are then proved. As applications, some solvability and approximation theorems for the mixed variational inequality and optimization problems are established by order approaches, in which the mappings need not to be continuous and the solutions are optimal with respect to the orders. By using the isotonicity of the proximity operator with respect to two orders, we overcome the absence of the regularity of the order. The convergence rate of forward–backward algorithms is finally estimated by order approaches.

中文翻译：

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接近算子的等渗性和希尔伯特空间中的混合变分不等式

本文讨论了邻近算子的等渗性及其应用。我们首先建立映射的一些新条件，使得它们的邻近算符相对于由不同微面锥引起的阶次是等调的。然后介绍了这些条件的一些属性和示例。我们特别考虑了邻近算子相对于由亚双锥引起的一个阶和两个阶的等渗性。为了估计迭代算法的收敛速度，然后证明了邻近算子关于阶数的一些其他不等式特征。作为应用，通过阶数方法建立了混合变分不等式和优化问题的一些可解性和逼近定理，其中映射不需要是连续的，并且解决方案对于阶数是最优的。通过使用关于两个阶次的邻近算子的等渗性，我们克服了阶次缺乏规律性的问题。前向-后向算法的收敛速度最终由阶次方法估计。