广义的固定斜率迭代法的Kantorovich型定理
Generalized Kantorovich-type theorem for the Fixed Slope Iterations
AI总结：
本文研究了在导数逆运算被替换成无关x的有界可逆算子B时的扩展牛顿法，并放宽了连续性假设，基于Frechet可微性提出广义Kantorovich定理，扩展了Argyros的结果，对于一般B和Hölder连续导数，得到更弱的半局部收敛条件、更大的唯一性域和更细的误差界。
摘要：
当方程F(x)=0中F(x)的导数的逆运算被替换为无关x的有界可逆算子B时，考虑了扩展的牛顿法修改。将连续性假设放宽为F(x)连续Frechet可微。将Kantorovich主导技术适应于提出并证明Kantorovich定理的广义形式，该定理最初是针对标准修改牛顿法（MNM）中导数Lipschitz连续的情况。在MNM情况下，广义定理被证明扩展了Argyros之前的结果。对于一般的B和Hölder连续导数，所提出的定理导致半局部收敛条件更弱，唯一性域更大，误差界更精细，相比Ahues和Argyros之前的成果。
英文摘要：
The extended modification of the Newton method is considered when the inverse of the derivative (of the operator F(x) in the equation F(x)=0) is replaced by an invertible bounded x-independent operator B. The continuity assumption is relaxed to the requirement that F(x) is continuously Frechet-differentiable. The Kantorovich majorization technique is adapted to formulate and prove the corresponding generalization of the Kantorovich theorem originally stated for the standard modified Newton method (MNM) when the derivative is Lipschitz continuous. In the MNM case, the generalized theorem is shown to extend the existing one due to Argyros. For a generic B and a Holder continuous derivative, the proposed theorem leads to a weaker condition of the semilocal convergence, larger uniqueness domain and finer error bounds compared to the previous results of Ahues and Argyros.
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