迭代数列的收敛性研究

Research on Judging the Convergence of Iterative Sequence

为了判断迭代数列是否收敛,并求解收敛数列的极限,首先,将迭代数列转化为迭代方程;接着,利用压缩映射原理判断迭代方程是否存在不动点;最后,给出在完备及紧的距离空间上,函数存在唯一不动点的条件,从而判断迭代数列是否收敛,得到了判断迭代数列敛散性的若干定理.通过例子,说明了定理在判断迭代数列敛散性方面的有效性,且运用2种证明方法证明了著名的开普勒方程解的存在性和唯一性.

In order to judge whether an iteration sequence is convergent and solve the limit of the convergence sequence, firstly, the iterative sequence is transformed into an iterative equation. Then, the existence of fixed points is determined by the contractive mapping principle. Finally, in the complete and compact space, we obtain the condition that the function has unique fixed point. It can be used to judge whether the iteration sequence is convergent or not. Some theorems for judging the convergence and divergence of iterative sequence are obtained. Some examples are given to show the validity of the theorems in judging the convergence of iterative sequences. Two proof methods are given to prove the existence and uniqueness of the solution of the famous Kepler equation.