Differentiable solutions of an equation with product of iterates

Abstract

In the previous work [2], we considered continuous solutions of an iterative equation involving the multiplication of iterates. In this paper, we continue to investigate this equation for differentiable solutions. Similar to continuous solutions until [2], there is no obtained result on differentiable solutions of such an equation on non-compact intervals of R.Although our strategy here is to use conjugation to reduce the equation to the well-known polynomial-like iterative equation as in [2], all known results on differentiable solutions of the latter are given on compact intervals. We re-explore polynomial-like iterative equation on the whole of R and prove the existence and uniqueness of differentiable solutions of our equation on R + and R − .