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On a linear iterative equation. (English) Zbl 0831.39006
Consider the iterative equation \((*)\) \(\sum^k_{i = 0} a_i f^i(x) = 0\), where \(a_0, \ldots, a_k\) are reals \((a_k \neq 0)\) and its characteristic polynomial \(w (\lambda) = \sum^k_{i = 0} a_i \lambda^i\). Assume that: (i) \(w\) has a unique positive root \(r_0\), (ii) \(r_0\) is a simple root, (iii) the absolute values of other complex roots of \(w\) are greater than \(r_0\). The main result (Theorem 1) states that if either \(D \subset (- \infty, 0)\), or \(D \subset (0, + \infty)\), and \(f : D \to D\) satisfies \((*)\) then \(r_0 D \subset D\) and \(f(x) = r_0x\) for \(x \in D\).
This is a generalization of a result of the reviewer [On an equation of linear iteration. Aequationes Math. (to appear)] where instead of conditions (i)–(iii) it is assumed that \(a_0 < 0\), \(a_1, \ldots, a_k \geq 0\) and the greatest common divisor of the support of \((a_1, \ldots, a_k)\) is equal to 1.
The following lemma being a consequence of the Theorem of Kronecker serves as the main tool in the proof of Theorem 1: Let \(\varphi_1, \ldots, \varphi_q\), \(\varphi_{q + 1}, \ldots, \varphi_r \in (0,2 \pi)\), \(\alpha_1, \ldots, \alpha_r \in \mathbb{R}\) and let \(P(m) = \alpha_1 \cos (m \varphi_1) + \cdots + \alpha_q \cos (m \varphi_q) + \alpha_{q + 1} \sin (m \varphi_{q + 1}) + \cdots + \alpha_r \sin (m \varphi_r)\) for \(m \in \mathbb{Z}\). If \(\liminf_{m \to \infty} P(m) \geq 0\) then \(P(m) = 0\) for \(m \in \mathbb{Z}\).
A number of examples shows that the assumptions of Theorem 1 are essential and answers some questions asked by the reviewer [loc. cit.].

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