On convex iterative roots of non-monotonic mappings

摘要

Let I be an interval. We consider the non-monotonic convex self-mappings such that is convex. They have the property that all iterates are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every there exists a convex iterative root of order and the sequence satisfies the condition of compatibility, that is A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions such that and . We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions.