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On a linear iterative equation. (English) Zbl 0831.39006

Consider the iterative equation (*) i=0 k a i f i (x)=0, where a 0 ,...,a k are reals (a k 0) and its characteristic polynomial w(λ)= i=0 k a i λ 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(-,0), or D(0,+), and f:DD satisfies (*) then r 0 DD and f(x)=r 0 x for xD.

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 0 and the greatest common divisor of the support of (a 1 ,...,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 ϕ 1 ,...,ϕ q , ϕ q+1 ,...,ϕ r (0,2π), α 1 ,...,α r and let P(m)=α 1 cos(mϕ 1 )++α q cos(mϕ q )+α q+1 sin(mϕ q+1 )++α r sin(mϕ r ) for m. If lim inf m P(m)0 then P(m)=0 for m.

A number of examples shows that the assumptions of Theorem 1 are essential and answers some questions asked by the reviewer [loc. cit.].


MSC:
39B12Iterative and composite functional equations
26A18Iteration 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.