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Analytic solutions of a nonlinear iterative equation near neutral fixed points and poles. (English) Zbl 1030.39025

This is a continuation of earlier papers, particularly by the first author.
Let \(f^m(x)\) indicate the \(m\)th iterate of \(f(x)\). The authors wish to study analytic solutions to the functional equation \[ f^m(x)= G\left( \sum^{m-1}_{k=0} a_kf^k(x)\right) +F(x),\;m\geq 2,\;x\in\mathbb{C}. \] Here the \(a_k\in \mathbb{C}\) and \(G,F,f\) are complex-valued functions of a complex variable. They show that under certain conditions, this functional equation has an analytic solution of the form \(f(x)=\varphi (\varphi^{-1} (x)+\alpha)\) in a neighborhood of the origin where \(\varphi\) is an analytic solution of \[ \varphi(x+m \alpha)=G \left( \sum^{m-1}_{k=0} a_k\varphi (x+k\alpha) \right)+F \bigl(\varphi(x) \bigr). \] The conditions imposed are on the indeterminate constant \(\alpha\), and are of three different sorts, treated separately. In the end of these, the analytic solution \(\varphi\) is only shown to exist in a certain half-plane.
The case where \(G\) and \(F\) have poles of order 1 at the origin is also treated.

MSC:

39B12 Iteration theory, iterative and composite equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B32 Functional equations for complex functions
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