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On an equation of linear iteration. (English) Zbl 0872.39010
Given positive numbers \(a_1,\dots, a_k\) and positive integers \(n_1,\dots, n_k\) consider the equation \[ \sum_{i=1}^k a_if^{n_i}(x)=x.\tag \(*\) \] Let \(c\) be the only positive real number satisfying \(\sum_{i=1}^k a_ic^{n_i}=1\). Assuming that the greatest common divisor of \(n_1,\dots,n_k\) equals 1, \(D\subset(-\infty,0)\) or \(D\subset (0,+\infty)\), and \(f:D\to D\) is a solution of \((*)\), the author proves that \(cd\subset D\) and \(f(x)=cx\) for \(x\in D\).
Reviewer’s remark: This result was generalized by J. Tabor and J. Tabor [Result. Math. 27, No. 3-4, 412-421 (1995; Zbl 0831.39006)]. They answered also in negative three of the five questions formulated in the paper under review. Cf. also the abstracts of talks in Aequationes Math. 51, pp. 159, 163-164, 170 (1996)].
Reviewer: K.Baron (Katowice)

MSC:
39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
26A18 Iteration of real functions in one variable
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References:
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