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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 {\it J. Tabor} and {\it 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)].

39B12Iterative and composite functional equations
39B22Functional equations for real functions
26A18Iteration of functions of one real variable
Full Text: DOI EuDML
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