On regular solutions of some simple iterative functional equations

We consider the iterative functional equation Φ(αx) = L[Φ(x)] + F(x) (x ∈ E) (2) where Φ : E → G is the unknown function, E and G are complex normed linear spaces, F : E → G is a given function, α is a root of unity of order n, I is the identical mapping in G and L : G → G is a given linear mapping satisfying Ln = I. By using simple direct methods, we obtain the general solution of (2), its continuous solutions, and, in the case E = ℂq, its differentiable solutions. The results presented here complete a previous paper dealing with the case L(x) = βx (x ∈ G) where β is some complex number.