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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.].

MSC:
 39B12 Iteration theory, iterative and composite equations 26A18 Iteration of real functions in one variable
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References:
 [1] Hardy G.H. and Wright E.M., An introduction to the theory of numbers. Fourth edition, Oxford, 1971. · Zbl 0020.29201 [2] Hildebrand Francis B., Finite-difference equations and simulations, Prentice – Hall, Inc. Englewood Cliffs, New Jersey, 1968. · Zbl 0157.22702 [3] Jarczyk W., On an equation of linear iteration, Aequationes Math, (to appear). · Zbl 0872.39010 [4] Jarczyk W., Problem 9, Report of Meeting, The Thirty-first International Symposium on Functional Equations, August 22–28, 1993, Debrecen, Aequationes Math. 47(1994). [5] Łojasiewicz S., Introduction to complex analytic geometry, Birkhäuser Verlag, Basel – Boston – Berlin, 1991.
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