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 Zoltan Boros posed during the 50th ISFE.
Highlights
Suppose I ⊂ R is a non-degenerated interval and let g : I → I be a function
The main objective of this paper is to give particular conditions under which a converse holds, i.e., we want to find some conditions guaranteeing that if a continuous function g satisfies (1.1), there is a divisor of aN rN + · · · + a1r + a0 such that g is a solution to the corresponding iterative equation of lower order
A function g satisfies Eq (1.1) if and only if for every x0 ∈ I the sequencem∈N0 given by xm = g(xm−1) for all m ∈ N satisfies the recurrence relation (1.3)
Summary
Suppose I ⊂ R is a non-degenerated interval and let g : I → I be a function. Assuming g to be continuous we are interested in lowering the order of the equation aN gN (x) + · · · + a1g(x) + a0x = 0,. One of methods for finding solutions to Eq (1.1), and to its nonhomogenous counterpart, where zero on the right-hand side is replaced by an arbitrary continuous function, is based on lowering its order The 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. The main objective of this paper is to give particular conditions under which a converse holds, i.e., we want to find some conditions guaranteeing that if a continuous function g satisfies (1.1), there is a divisor of aN rN + · · · + a1r + a0 such that g is a solution to the corresponding iterative equation of lower order
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