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:
| [1] | Lanford, O. E., A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (New Series), 6, 427-434 (1982) · Zbl 0487.58017 |
| [2] | Christensen, J. P.R.; Fisher, P., Linear independence of iterates and entire solutions of functional equations, Proc. Amer. Math. Soc., 103, 1120-1124 (1988) · Zbl 0654.30018 |
| [3] | Rice, R. E.; Schweizer, B.; Sklar, A., When is \(f(f(z))= az^2+ bz +c\)?, Amer. Math. Monthly, 87, 252-263 (1980) · Zbl 0441.30033 |
| [4] | Mukherjea, A.; Ratti, J. S., On a functional equation involving iterates of a bijection on the unit interval, Nonlinear Analysis, 7, 899-908 (1983) · Zbl 0518.39005 |
| [5] | Mukherjea, A.; Ratti, J. S., A functional equation involving iterates of a bijection on the unit interval II, Nonlinear Analysis, 31, 3/4, 459-464 (1998) · Zbl 0899.39005 |
| [6] | Urabe, H., On entire solutions of the iterative functional equation \(g(g(z))=F(z)\), Bull. Kyoto Univ. (Ser. B), 74, 13-26 (1989) · Zbl 0689.30018 |
| [7] | Weinian, Z., Discussion on the iterated equation ∑\(_{i=1}^{n\) · Zbl 0639.39006 |
| [8] | Weinian, Z., Discussion on the differentiable solutions of the iterated equation ∑\(_{i=1}^{n\) · Zbl 0717.39005 |
| [9] | Mai, Jiehua; Liu, Xinhe, Existence, uniqueness and stability of (( C^m\) solutions of iterative functional equations, Science in China (Series A), 43, 9, 897-913 (2000) · Zbl 0999.39020 |
| [10] | Mai, Jiehua; Liu, Xinhe, Existence and uniqueness of analytic solutions of iterative functional equations, (Dynamical Systems—Proceeding of the International Conference in Honor of Professor Liao Shantao (1999), World Scientific), 213-222 · Zbl 0999.39020 |
| [11] | Gonzalez, L. B., Linear independence of iterates of entire functions, Proc. Amer. Math. Soc., 112, 4, 1033-1036 (1991) · Zbl 0728.30020 |
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