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A functional equation involving iterates of a bijection on the unit interval. II. (English) Zbl 0899.39005
An example is exhibited of a functional equation of the form \[ \sum^n_{k=0} c_kf^k (x)=0 \] \((c_k\in \mathbb{R}\), \(\sum^n_{k=0} c_k=0\), \(f^k\) denoting the \(k\)-th iterate of \(f)\) which has a bijective solution \(f\) (of \([0,1]\) onto itself) and such that \(f^r(x)\neq x\) for any \(r>0\). Thus the problem posed by the authors in part I [ibid. No. 7, 899-908 (1993; Zbl 0518.39005)] has been settled.

MSC:
39B12 Iteration theory, iterative and composite equations
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[1] Mukherjea, A.; Ratti, J.S., On a functional equation involving iterates of a bijection on the unit interval, Nonlinear analysis: theory, methods & applications, 7, 8, 899-908, (1983) · Zbl 0518.39005
[2] Ratti, J.S.; Lin, Y.F., A functional equation involving f and f−1, Colloquium mathematicum, LX/LXI, 519-523, (1990) · Zbl 0731.39006
[3] Weinian Zhang, The discussion on the iterated equation σi = 1naifi(x) = F(x), (1985), Inst. of Math. Sciences, Academia Sinica China
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