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A functional equation arising from an asymptotic formula for iterates. (English) Zbl 0818.39005

Denoting the \(m\)-th iterate of \(\varphi\) by \(\varphi^ m\), let \(\varphi\) be a real continuous solution of \(\varphi^ m (x) = \varphi (mx)/m\), \(m > 1\), in an open interval containing 0, let \(\varphi (0) = 0\), and let \(\varphi'' (0) = 2b\) exist. If \(\varphi' (0) = 1\) then \(\varphi (x) = x/(1 - bx)\). If \(m\) is odd and \(\varphi' (0) = - 1\) then \(\varphi (x) = - x\). Further solutions are also constructed.
Reviewer: L.Berg (Rostock)

MSC:

39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
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