Kantorovich 定理关于求解强正则广义方程的牛顿方法
Kantorovich's theorem on Newton's method for solving strongly regular generalized equation
AI总结：
本文研究了求解广义方程 $ f(x) +F(x) \ni 0 $ 的牛顿方法，在强正则性假设下，证明该方法在合适邻域内二次收敛，统一了牛顿方法理论中的部分结果。
摘要：
本文考虑了求解形式为 $ f(x) +F(x) \ni 0 $ 的广义方程的牛顿方法，其中 $ f:Ω\to Y $ 是连续可微映射， $ X $ 和 $ Y $ 是巴拿赫空间， $ Ω\subseteq X $ 是开集， $ F:X \rightrightarrows Y $ 是具有非空闭图的多值映射。在强正则性（由 S.M.Robinson 在 [27] 中引入）和起始点满足 Kantorovich 假设的条件下，证明牛顿方法在合适邻域内二次收敛到唯一解。所呈现的分析基于广义方程的巴拿赫扰动引理和主导技术，允许统一牛顿方法理论中的某些结果。
英文摘要：
In this paper we consider the Newton's method for solving the generalized equation of the form $ f(x) +F(x) \ni 0, $ where $f:Ω\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $Ω\subseteq X$ an open set and $F:X \rightrightarrows Y$ be a set-valued mapping with nonempty closed graph. We show that, under strong regularity of the generalized equation, concept introduced by S.M.Robinson in [27], and starting point satisfying the Kantorovich's assumptions, the Newton's method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. The analysis presented based on Banach Perturbation Lemma for generalized equation and the majorant technique, allow to unify some results pertaining the Newton's method theory.
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