实Clifford分析中三类高阶奇异积分及其非线性微分积分方程

Three Kind High Order Singular Integrals and Nonlinear Differentio-Integral Equation in the Real Clifford Analysis

本文第一部分借助于高阶奇异积分的Hadamard主值的思想以及归纳法的思想，在证明了6个引理的基础上讨论实Clifford分析中三类高阶奇异积分的归纳定义，Hadamard主值的存在性，递推公式，计算公式以及高阶奇异积分在Hadamard主值意义下的12个微分公式．受多复交中解析函数积分表示式多样性的影响，本文中采用的算子的积分表达式就与Clifford分析中正则函数常用的积分表达方式不相同，正因为如此，才使得本文中所得到的15个计算公式和微分公式都十分简洁．本文第二部分在引进并证明了Hile引理型的引理的基础上，利用积分方程方法和Schauder不动点原理（压缩映射原理），讨论了实Clifford分析中含有三类高阶奇异积分的非线性微分积分方程解的存在（唯一）性．

In the first part of this paper, by the think of Hadamard principal value andthe think of induction, in view of prove the six lemma, we consider induce define, existence ofHadamard principal value, recurrence formula, compute formula and twelve differefltial formulafor three kind high order singular integrals in real Clifford analysis. Deal of enlightenment frommake varied for integral expression of function in several complex variables, in the paper adoptexpression of integral operators differ from illtegral expression for regular function in real Cliffordanalysis, thus in this paper we give the ten-five compute formulas and differential formulasbecome very simple. In the second part of this paper, in view of design and proof for Hile's typelemma, applying the method of integral equations and Schauder fixed-point theorem (principle ofcontracting mapping), the existence (uniqueness) of solution for the nonlinear differentio-integralequation contain three find high order singular integrals in real Clifford analysis is proved.