Fredholm第一种积分方程Ax=y的表示定理和一次迭代定理

Representation Theorem and One-Iteration Theorem for Fredholm Integral Equation of the First Kind Ax=y

本文给出两个定理.表示定理指出:若具有界L_2核的Fredholm第一种积分方程Ax=y有唯一解(?),则一次迭代定理指出:(?)可由公式(?)=x_0+g_0A*(y-Ax_0)一次迭代求得的充分和必要条件是满足下列条件之一:1.v_0=g_0A*Av_0,v_0=(?)-x_0;2.u_0=g_0AA*u_0,u_0=y-Ax_0;3.g_0=||A*u_0||2/||AA*u_0||2=||u_0||2/||A*u_0||2,u_0=y-Ax_0或g_0=||Av_0||2/||A*Av_0||2=||v_0||2/||Av_0||2。

In this paper, two theorems are presented. The representation theorem states: if the Fredholm integral equation of the first kind Ax=y, with bounded L2 kernel, has a unique solution x, then x= where The One-iteration theorem states: x can be achieved in one-iteration if one of the following conditions is satisfied: