Smooth Solutions of a Class of Iterative Functional Differential Equations

Abstract

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 $x^{\prime }\left(t\right)=1/{\mathrm{(c}}_0{x}^{\left[0\right]}\left(t\right)+c_1{x}^{\left[1\right]}\left(t\right)+\cdots +c_m{x}^{\left[m\right]}\left(t\right))$.