Smooth solutions for the p-order functional equation f(φ(x)) =φ^p(f(x))

This paper deals with the p-order functional equation f(φ (x)) = φ^p(f(x)), φ (0) = 1, −1 ≤φ (x) ≤1, x ∈ [−1, 1], where p ≥ 2 is an integer, φp is the p-fold iteration of φ, and f(x) is smooth odd function on [−1, 1] and satisfies f(0) = 0, −1 f (x) 0, (x ∈ [−1, 1]). Using constructive method, the existence of unimodal-even-smooth solutions of the above equation on [−1, 1] can be proved.