Local invertible analytic solutions for an iterative differential equation related to a discrete derivatives sequence

[摘要] In this paper, we are concerned with the existence of analytic solutions of a class of iterative differential equation (GRAPHICS) in the complex field C, where K epsilon C \ {0}, a(i) epsilon R, f(i) (z) denotes ith iterate of f (z), i = 1, 2,..., n. The above equation is closely related to a discrete derivatives sequence F'(m) (see [Y.-F.S. Petermann, Jean-Luc Remy, Ilan Vardi, Discrete derivative of sequences, Adv. in Appl. Math. 27 (2001) 562-584]). We first give the existence of analytic solutions of the form of power functions for such an equation. Then by constructing a convergent power series solution y(z) of an auxiliary equation of the form x'(z) = K alpha x'(alpha z)(x(alpha z))(a1) (x(alpha(2)z))(a2)... (x (alpha(n)z))(an) , invertible analytic solutions of the form f (z) = x (alpha x (-1) (z)) for the original equation are obtained. We discuss not only the constant alpha at resonance, i.e. at a root of the unity, but also those a near resonance (near a root of the unity) under the Brjuno condition. (c) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-11-01

[关键词] iterative differential equation;discrete derivatives sequence;analytic solution;resonance;diophantine condition;Brjuno condition