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.


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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[1] Abel, N.H., Oeuvres completes, Christiana, II, 36-39, (1981)
[2] Bessis, D.; Marmi, S.; Turchetti, G., On the singularities of divergent majorant series arising from normal form theory, Rend. mat. (7), 9, 645-659, (1989) · Zbl 0723.34037
[3] Bödewadt, U.T., Zur iteration reeller funktionen, Math. Z., 49, 497-516, (1944) · Zbl 0028.35103
[4] Carleson, L.; Gamelin, T.W., Complex dynamics, (1993), Springer-Verlag New York · Zbl 0782.30022
[5] Fort, M.K., The embedding of homeomorphisms in flows, Proc. amer. math. soc., 6, 960-967, (1955) · Zbl 0066.41306
[6] Jarczyk, W., On continuous solutions of the equation of invariant curves, (), 527-542 · Zbl 0816.39004
[7] Jarczyk, W., Series of iterates summing up to the identity function, Grazer math. ber., 334, 153-162, (1997) · Zbl 0910.39004
[8] Jarczyk, W., On an equation of linear iteration, Aequationes math., 51, 303-310, (1996) · Zbl 0872.39010
[9] Kuczma, M., Fractional iteration of differentiable functions, Ann. polon. math., 22, 217-227, (1969) · Zbl 0185.29403
[10] Kuczma, M., Functional equations in a single variable, Monografie mat., 46, (1968), PWN Warszawa · Zbl 0196.16403
[11] Kuczma, M.; Choczewski, B.; Ger, R., Iterative functional equations, () · Zbl 1141.39023
[12] Matkowski, J.; Zhang, W., On the polynomial-like iterative functional equation, (), 145-170 · Zbl 0976.39014
[13] McCarthy, P.J., Ultrafunctions, projective function geometry, and polynomial functional equations, Proc. London math. soc., 53, 321-339, (1986) · Zbl 0587.39001
[14] Mukherjea, A.; Ratti, J.S., On a functional equation involving iterates of a bijection on the unit interval, Nonlinear anal., Nonlinear anal., 31, 459-464, (1998), II · Zbl 0899.39005
[15] Ng, C.T.; Zhang, W., Invariant curves for a planar mapping, J. differ. equations appl., 3, 147-168, (1997) · Zbl 0891.34070
[16] Ratti, J.S.; Lin, Y.F., A functional equation involving f and f−1, Colloq. math., 60/61, 519-523, (1990) · Zbl 0731.39006
[17] Si, J., On the monotonic continuous solutions of some iterated equation, Demonstratio math., 29, 543-547, (1996) · Zbl 0898.39007
[18] Si, J., On analytic solutions of the equation of invariant curves, C. R. math. rep. acad. sci. Canada, 17, 86-104, (1995)
[19] Si, J.; Wang, X.P., Analytic solutions of a polynomial-like iterative functional equation, Demonstratio math., 32, 95-103, (1999) · Zbl 0930.39016
[20] Si, J.; Zhang, W., Analytic solutions of a functional equation for invariant curves, J. math. anal. appl., 259, 83-93, (2001) · Zbl 0992.39019
[21] Siegel, C.L., Vorlesungen über himmelsmechanik, (1956), Springer-Verlag Berlin · Zbl 0098.23601
[22] Tabor, J.; Tabor, J., On a linear iterative equation, Results math., 27, 412-421, (1995) · Zbl 0831.39006
[23] Zhang, W., Discussion on the solutions of the iterated equation ∑_{i=1}nλifi(x)=F(x), Chinese sci. bull., 32, 1444-1451, (1987) · Zbl 0639.39006
[24] Zhang, W., Discussion on the differentiable solutions of the iterated equation ∑_{i=1}nλifi(x)=F(x), Nonlinear anal., 15, 387-398, (1990) · Zbl 0717.39005
[25] Zhang, W., A generic property of globally smooth iterative roots, Sci. China ser. A, 38, 267-272, (1995) · Zbl 0838.39005
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