Differentiable Solutions of Equations Involving Iterated Functional Series

Let f be a self-mapping on a topological space X and f denote the mth iterate of f , that is, f f ◦ fm−1, f0 id, m 1, 2, . . . . Let C X,X be the set of all continuous self-mappings on X. Equations having iteration as their main operation, that is, including iterates of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations 1–4 , because it concludes the problem of iterative roots 1, 5, 6 , that is, finding f ∈ C X,X such that f is identical to a given F ∈ C X,X . As a natural generalization of the problem of iterative roots, a class of iterative equations named as polynomial-like iterative equation