A note on the order of polynomial-like iterative equations

Abstract

We show that, under reasonable assumptions, two negative roots can be eliminated from the characteristic equation of a polynomial-like iterative equation. This result gives a new case where we may lower the order of such an equation. 2010 Mathematics Subject Classification. 39B12.

Means of iterates

“…First such results on the whole real line were obtained in [8] in the case where all roots of the characteristic equation are real and satisfy some special conditions. Further research in this direction was obtained in [1,2,12,17] and some of them will be crucial tools in this paper.…”

“…Assume that J ⊂ R is an interval, ϕ : I → J is a bijection onto J and F : I → I. Then F satisfies (1) with M given by (2) on I if and only if…”

“…Consider integers n 2, 0 k n and a non-trivial interval I ⊂ R. The following problem arises when studying polynomial-like iterative equations, means of functions or convergence of means of iterates: Determine all continuous functions F : I → I with the k-fold iterate F k being a mean of all its subsequent iterates up to F n ; more precisely, find all self-mappings F ∈ C(I) such that (1) F k (x) = M (x, F (x), F 2 (x), . .…”

“…In this paper we concentrate on equation (1) in the case where M is a quasiarithmetic mean, i.e., with M being of the form (2) M (x 0 , ..., x n ) = ϕ −1 1 n + 1 n i=0 ϕ(x i ) for all x 0 , . .…”

“…. , x n ∈ I, where ϕ is a continuous bijection form I onto an interval J ⊂ R. Observe that it is enough to consider equation (1) with M being the arithmetic mean which follows from the following remark. It is easy to prove that any self-mapping satisfying (3) is injective (see, e.g.…”