On a functional-differential equation related to Golomb's self-described sequence
Résumé
The functional-differential equation $f'(t)=1/f(f(t))$ is closely related to Golomb's self-described sequence, denoted F. We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for eveery number $p >= 0$ there is exactly one increasing solution with $p$ as a fixed point. We also show that in general an initial condition doesn't determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely many times. In fact we conjecture that the difference of two increasing solutions behave very similarly as the error term $E(n)$ in the asymptotic expression $F(n) = \phi^{2-\phi}n^{\phi-1} + E(n)$(where $\phi$ is the golden number). || l'équation différentielle-fonctionnelle $f'(t)=1/f(f(t))$ a des liens étroits avec la suite auto-décrite de Golomb, notée $F$. Nous décrivvons les solutions croissantes de cette équation. Nous montrons qu'une telle solution possède néces