The Fixed Point and Mann Iteration of a Modified Isotonic Operator
Received:July 30, 2012  Revised:June 04, 2013
Key Word: Clifford analysis   isotonic operator   the fixed point theorem   Mann iteration.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.10771049; 11171349) and the Science Foundation of Hebei Province (Grant No.A2010000346).
 Author Name Affiliation Liping WANG School of Information, Renmin University of China, Beijing 100872, P. R. China College of Mathematics and Information Science, Hebei Normal University, Hebei 050024, P. R. China Zuoliang XU School of Information, Renmin University of China, Beijing 100872, P. R. China
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This paper consists of two parts. In the first part, we discuss the H\"{o}lder continuity of Cauchy-type integral operator $T$ of isotonic functions and the relationship between $\|T[f]\|_{\alpha}$ and $\|f\|_{\alpha}$. In the second part, firstly, we introduce a modified Cauchy-type integral operator $T'$ and demonstrate that the operator $T'$ has a unique fixed point by the Contraction Mapping Principle. Then we give the Mann iterative sequence and prove that the Mann iterative sequence strongly converges to the fixed point of the modified Cauchy-type integral operator $T'$.