## 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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### References:

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