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On an equation of linear iteration. (English) Zbl 0872.39010

Given positive numbers a 1 ,,a k and positive integers n 1 ,,n k consider the equation

i=1 k a i f n i (x)=x·(*)

Let c be the only positive real number satisfying i=1 k a i c n i =1. Assuming that the greatest common divisor of n 1 ,,n k equals 1, D(-,0) or D(0,+), and f:DD is a solution of (*), the author proves that cdD and f(x)=cx for xD.

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)].


MSC:
39B12Iterative and composite functional equations
39B22Functional equations for real functions
26A18Iteration of functions of one real variable
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