正则函数的修正的Cauchy型积分算子的不动点定理及迭代构造

本文第一部分讨论了正则函数的{\small Cauchy}型积分算子$Tf]$的{\small H\"{o}lder}连续性及此积分算子$Tf]$的范数与$f$的范数之间的关系.第二部分引入了修正的Cauchy型积分算子$\small \widetilde{T}$,首先利用压缩映射原理证明了$\small \widetilde{T}$算子具有不动点,然后给出了其不动点的迭代序列并证明了此序列强收敛于$\small \widetilde{T}$算子的不动点.

The Fixed Point Theorem and the Iterative Approximation of Modified Cauchy Integral Operator of Regular Functions

In the first part of this paper, we discuss the H\"{o}lder continuity of the cauchy integral operator for regular functions and the relation between $\|Tf]\|_{\alpha}$ and $\|f\|_{\alpha}$. In the second part of this paper, we introduce the modified cauchy integral operator $\widetilde{T}$ for regular functions. Firstly, we prove that the operator $\widetilde{T}$ has a unique fixed point by the Banach's Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator $\widetilde{T}$.