A functional equation arising from an asymptotic formula for iterates

Abstract

We consider the real solutions of the functional equation (*) ilj"'(i) = -i15(mi), i,3(0) = 0, m where m € N and ipm denotes the m-th iterate of the unknown function ip. We will handle this functional equation for a fixed m, but also for all naturals m, and give a representation of all C'2-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.