Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros’ problem

Abstract

Abstract. We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a generalized problem of Zoltán Boros posed during the 50th ISFE.Mathematics Subject Classification. 39B12, 39A06.

A note on the order of polynomial-like iterative equations

“…. ≤ |λ p | < |r 1 | and r 1 , r 2 are real with r 1 < −1 < r 2 < 0, then a continuous surjection f : I → I satisfies equation (1) if and only if it satisfies equation…”

“…Let n be a positive integer and I ⊂ R be an interval. We are interested in so-called polynomial-like iterative equations, namely, equations of the form (1) a n f n (x) + . .…”

“…, a 1 , a 0 are given real numbers and a 0 = 0. In general, it is difficult to find all continuous functions satisfying equation (1) even in the case n = 3; a partial solution in this case was given in [7] and the complete solution for n = 2 was presented in [3,5]. One of methods for finding solutions to equation (1) involves lowering its order.…”

“…In general, it is difficult to find all continuous functions satisfying equation (1) even in the case n = 3; a partial solution in this case was given in [7] and the complete solution for n = 2 was presented in [3,5]. One of methods for finding solutions to equation (1) involves lowering its order. Such results were obtained in [1,6,9] (see Theorem 2 below); the present paper contains a new result in this spirit.…”

“…then a continuous function f : I → I satisfies equation(1) if and only if it satisfies(5) withR(b m , . .…”