函数方程$f^{[m]}={1}/{f}$的实解

The Real Solutions of Functional Equation $f^{[m]}=1/f$

中文关键词:  迭代  迭代方程  $k$-循环  Babbage方程  逐段连续.

英文关键词:iterate  iterative equation  $k$-circuit  Babbage equation  piecewise continuous.

中文摘要:

      研究了函数方程$f^{[m]}=1/f$，分析了逐段连续解的特点，构造性地得到了所有逐段连续的实解．结果推广了[Amer. Math. Monthly 1998, 105(8): 704-717]上的结果，而且得到了不满足Babbage方程的函数没有实的循环迭代根．

英文摘要:

      The authors study the functional equation $f^{[m]}=1/f$ and analyze the features of its piecewise continuous solutions. All the piecewise continuous real solutions are obtained constructively. The results generalize the ones in [2]. Moreover, the conclusion is drawn that there is no circuit iterative roots for those functions not satisfying Babbage equation.