| The Fixed Point Theorem and the Iterative Approximation of Modified Cauchy Integral Operator of Regular Functions |
| Received:July 23, 2007 Revised:April 16, 2008 |
| Key Words:
real Clifford analysis regular function cauchy integral the fixed point mann iteration.
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| Fund Project:the National Natural Science Foundation of China (No.\,10771049; 10771050); the Natural Science Foundation of Hebei Province (No.\,A2007000225) and the Foundation of Hebei Normal University (No.\,L2007Q05); the 11th Five-Year Plan Educational and Scientif |
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| Abstract: |
| 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 $\|T[f]\|_{\alpha}$ 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 $\widetilde{T}$. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.03.019 |
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