PERIODIC SOLUTIONS OF THE POINCARE FUNCTIONAL EQUATION: UNIQUENESS

Alexander Sarkovskii has shown that the only continuous periodic functions satisfying the duplication formula psi(2x) = 2 psi(x)(2) - 1 are of the form psi(x) = cos(omega x), where omega is a real constant. The aim of this paper is to generalize this result as follows: Let I be a compact interval, f : I -> I be a continuous surjection and k > 1 be an integer. Let psi : R -> I and phi : R -> I be two solutions of the Poincare functional equation psi(kx) = f (psi(x)) with psi cosine-like and phi non-constant, continuous and periodic. We will show that there exist a positive real number omega and an integer m such that phi(x) = psi(omega x + m lambda/(k - 1)) (x is an element of R), where lambda is the principal period of psi. As a tool, we employ the simultaneous Schroder-difference functional equation sigma(kx) = +/- k sigma(x), sigma(x + 1) = epsilon(x)sigma(x) + r(x), where epsilon(x) = +/- 1 and r(x) is an element of Z for all x >= 0, and determine its solution.

Automated summary

This paper generalizes the results of Alexander Sarkovskii and explores the relationship between continuous periodic functions that satisfy a specific functional equation.