A functional equation arising from an asymptotic formula for iterates - Gronau, Detlef; Sablik, Maciej (Annales Mathematicae Silesianae, 1994)
We consider the real solutions of the functional equation (⚹) ϕm(x) = 1/m ϕ(mx), ϕ(0) = 0, where m∈ℕ and ϕm denotes the m-th iterate of the unknown function ϕ. We will handle this functional equation for a fixed m, but also for all naturals m, and give a representation of all C2-solutions (even weaker, see Theorem 2.1) of (⚹), but also treat the case of other solutions of this equation. In the introduction we will show the origin of this equation.