Analytic solutions of iterative functional equations. (English) Zbl 1006.39022
The authors consider the functional equation
\[
G( z,f(z),f^{2}(H_2(z,f(z))),\dots ,f^{n} ( H_n(z,f(z),\dots ,f^{n-1}(z)))=0
\]
with respect to the function \(f(z)\) analytic in the disk \(|z|<r\), continuous in \(|z|\leq r\) and satisfying the condition \(|f(z)|\leq r\) for \(|z|\leq r\). The known functions \(G,H_2,\dots ,H_n\) are analytic in the corresponding domains. The authors give some conditions for the equation to have a solution and a unique solution.
Reviewer: Vladimir Mityushev (Paris)
MSC:
39B12 | Iteration theory, iterative and composite equations |
39B32 | Functional equations for complex functions |
30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Keywords:
iterative functional equation; analytic solution; difference quotient; compact convex set; fixed pointReferences:
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