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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
##### Keywords:
iterative functional equation; iterates; general solution
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##### References:
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