Analytic Solutions of an Iterative Functional Differential Equations Near Regular Points

l=0 |c l | < 1, and xl(z) = x(xl−1(z)) that denote the nth iterate of a map x. In general, F, G are given complex-valued functions of a complex variable. In this paper, analytic solutions of nonlinear iterative functional differential equations are investigated. Existence of locally analytic solutions and their construction is given in the case that all given functions exist regular points. As well as in previous work [6–8], we still reduce this problem to find analytic solutions of a differential-difference equation and a functional differential equation with proportional delay. The existence of analytic solutions for such equation is closely related to the position of an indeterminate constant μ depending on the eigenvalue of the linearization of x at its fixed point 0 in the complex plane. For technical reasons, in [6, 7], only the situation of μ off the unit circle in C and the situation of μ on the circle with the Diophantine condition, “|μ| = 1, μ is not a root of unity, and log(1/|μ −1|) ≤ T log n, n = 2, 3, . . . for some positive constant T”, are discussed. The Diophantine condition requires μ to be far from all roots of unity that the fixed point 0 is irrationally neutral. In this paper, besides the situation that μ is the inside of the unit circle S, we break the restriction of the Diophantine condition and study the situations that the constant μ in (5) (or