Gronau, Detlef; Sablik, Maciej A functional equation arising from an asymptotic formula for iterates. (English) Zbl 0818.39005 Pr. Nauk. Uniw. Śląsk. Katowicach 1444, Ann. Math. Silesianae 8, 173-187 (1994). 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) Cited in 1 Review MSC: 39B12 Iteration theory, iterative and composite equations 26A18 Iteration of real functions in one variable Keywords:asymptotic formula; functional equation; iterate; real continuous solution PDFBibTeX XMLCite \textit{D. Gronau} and \textit{M. Sablik}, Pr. Nauk. Uniw. Śląsk. Katowicach, Ann. Math. Silesianae 1444(8), 173--187 (1994; Zbl 0818.39005)