Article ID Journal Published Year Pages File Type
4622021 Journal of Mathematical Analysis and Applications 2007 15 Pages PDF
Abstract

In this paper, we are concerned with the existence of analytic solutions of a class of iterative differential equationf′(z)=1K(f1(z))a1(f2(z))a2⋯(fn(z))an, in the complex field CC, where K∈C∖{0}K∈C∖{0}, ai∈Rai∈R, fi(z)fi(z) denotes i  th iterate of f(z)f(z), i=1,2,…,ni=1,2,…,n. The above equation is closely related to a discrete derivatives sequence F′(m)F′(m) (see [Y.-F.S. Pétermann, Jean-Luc Rémy, 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)y(z) of an auxiliary equation of the formx′(z)=Kαx′(αz)(x(αz)a1)(x(α2z)a2)⋯(x(αnz)an),x′(z)=Kαx′(αz)(x(αz))a1(x(α2z))a2⋯(x(αnz))an, invertible analytic solutions of the form f(z)=x(αx−1(z))f(z)=x(αx−1(z)) for the original equation are obtained. We discuss not only the constant α at resonance, i.e. at a root of the unity, but also those α near resonance (near a root of the unity) under the Brjuno condition.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
Authors
, ,