标题： 关于广义方程的Josephy-Halley方法 显示英文标题

标题： On Josephy-Halley method for generalized equations

摘要： 我们将经典的三阶 Halley 迭代推广到广义方程的形式 \[ 0 \in f(x) + F(x), \]，其中 \(f\colon X\longrightarrow Y\)在 Banach 空间上二次连续 Fréchet 可微，并且 \(F\colon X\tto Y\)是具有闭图的集值映射。 基于预测-校正框架，我们的方法首先求解一个部分线性化的包含问题以生成预测值 \(u_{k+1}\)，然后在 Halley 型校正步中引入二阶信息以得到 \(x_{k+1}\)。 在参考解处线性化的一致正则性和\(f''\)的 Hölder 连续性条件下，我们证明了迭代序列局部以阶\(2+p\)收敛（当\(p=1\)时为三次收敛）。 此外，通过构造一个合适的标量主要函数，我们得出了半局部 Kantorovich 型条件，这些条件保证了从初始猜测的一个明确邻域出发的算法定义良好且具有 R-三次收敛性。 数值实验——包括一维和二维测试问题——验证了理论上的收敛速度，并展示了 Josephy-Halley 方法相对于其 Josephy-Newton 对应方法的效率。 显示英文摘要

摘要： We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where \(f\colon X\longrightarrow Y\) is twice continuously Fr\'echet-differentiable on Banach spaces and \(F\colon X\tto Y\) is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor \(u_{k+1}\), then incorporates second-order information in a Halley-type corrector step to obtain \(x_{k+1}\). Under metric regularity of the linearization at a reference solution and H\"older continuity of \(f''\), we prove that the iterates converge locally with order \(2+p\) (cubically when \(p=1\)). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart.