Sur les équations non linéaires itératives

Summary.

Let $m \in {\Bbb N}^*$, $D \subset [0,+\infty[$, and $f: D \to D$. We denote by $f^{[n]}(x)$ the n-th iterate of the function f.¶Let F be a rational function of m variables.¶In this paper, we study functional equations¶¶$f^{[m]}(x) = F(x,f(x),\dots ,f^{[m-1]}(x))$.¶With some conditions on the rational function F, we show that a solution of such a functional equation is necessarily of the form $f(x) = {{ax}\over{1 + bx}}$.