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On a linear iterative equation. (English) Zbl 0831.39006

Consider the iterative equation $\left(*\right)$ ${\sum }_{i=0}^{k}{a}_{i}{f}^{i}\left(x\right)=0$, where ${a}_{0},...,{a}_{k}$ are reals $\left({a}_{k}\ne 0\right)$ and its characteristic polynomial $w\left(\lambda \right)={\sum }_{i=0}^{k}{a}_{i}{\lambda }^{i}$. 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 $D\subset \left(-\infty ,0\right)$, or $D\subset \left(0,+\infty \right)$, and $f:D\to D$ satisfies $\left(*\right)$ then ${r}_{0}D\subset D$ and $f\left(x\right)={r}_{0}x$ 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 ${a}_{0}<0$, ${a}_{1},...,{a}_{k}\ge 0$ and the greatest common divisor of the support of $\left({a}_{1},...,{a}_{k}\right)$ 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 ${\varphi }_{1},...,{\varphi }_{q}$, ${\varphi }_{q+1},...,{\varphi }_{r}\in \left(0,2\pi \right)$, ${\alpha }_{1},...,{\alpha }_{r}\in ℝ$ and let $P\left(m\right)={\alpha }_{1}cos\left(m{\varphi }_{1}\right)+\cdots +{\alpha }_{q}cos\left(m{\varphi }_{q}\right)+{\alpha }_{q+1}sin\left(m{\varphi }_{q+1}\right)+\cdots +{\alpha }_{r}sin\left(m{\varphi }_{r}\right)$ for $m\in ℤ$. If ${lim inf}_{m\to \infty }P\left(m\right)\ge 0$ then $P\left(m\right)=0$ for $m\in ℤ$.

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 Iterative and composite functional equations 26A18 Iteration of functions of one real variable
References:
 [1] Hardy G.H. and Wright E.M., An introduction to the theory of numbers. Fourth edition, Oxford, 1971. [2] Hildebrand Francis B., Finite-difference equations and simulations, Prentice – Hall, Inc. Englewood Cliffs, New Jersey, 1968. [3] Jarczyk W., On an equation of linear iteration, Aequationes Math, (to appear). [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.