Smooth solutions of a class of iterative functional differential equations (Q437669)

Summary: By Faà di Bruno's formula, using the fixed-point theorems of Schauder and Banach, we study the existence and uniqueness of smooth solutions of an iterative functional differential equation \[ x'(t) = 1/(c_0 x^{[0]}(t) + c_1 x^{[1]}(t) + \cdots + c_m x^{[m]}(t)), \] where \[ x^{[0]} (t)=t,\enskip x^{[1]} (t)=x(t), \enskip x^{[k]} (t)=x(x^{[k-1]}(t)),\enskip k=2,\dots,m. \]