Proposed Nature Of The Cauchy Type Integral Operator. In Clifford Analysis

Hypermonogenic function is a new kind of function in Clifford analysis,and is a gener-alization form of the regular function with one complex variable about higher dimentional space under hyperbolic metric.The integral formula for hypermonogenic function is de-fined as quasi-Cauchy's type integral.In the first part of this paper,on the base of the quasi-Cauchy's type integral and Plemelj formula of the hypermonogenic function,we dis-cuss the Holder continuity of the quasi-Cauchy's type integral operator T[f]:two points on the boundary;one point on the boundary,another point in the region(outside the region);two points in the region(outside the region).And we also discuss the relation between‖T[f]‖αand‖f‖αwhich plays an important role to study the second part.In the second part of this paper,we introduce the modified quasi-Cauchy's type integral operator T.Firstly,we prove the operator T has unique fixed point by the Banach's Contraction Mapping Principle.Secondly,we give the iterative sequence,and show the iterative sequence strongly converge to the fixed point of the operator(?).