关于求解第二类Volterra积分方程的Picard迭代收敛性的一点注记

A NOTE ON THE CONVERGENCE OF PICARD ITERATION FOR SOLVING VOLTERRA EQUATIONS OF THE SECOND KIND

考虑第二类Volterra积分方程: φ(x)+integral from n=0 to x(K(x,y)φ(y)dy)=f(x),x∈[0,L],(1)其中f(x)∈C([0,L]),核函数 K(x,y)对y可积。

In this note, the convergence of Picard iteration for solving Volterra equations of thesecond kind is discussed. A more general condition of convergence is presented. Some usualconditions of convergence, such as boundedness of kernel function K(x,y), boundedness ofintegral of K(x,y) with respect to y by constant less than l, weak singularities 1/|x-y|αwith α∈(0,1) of the kernel function K(x,y), etc, can be considered as special cases of thecondition presented in this note.