A Study on Iterative Root Problem

Abstract

The iterative root problem is one of the classical problem in the theory of iterative functional equations and is described as follows: Given a non-empty X, a self map F on X and a fixed positive integer n, to find another self map f on X such that fn = F. If such a function f exists, then it is called an nth iterative root of F. Existence of iterative roots for strictly monotone continuous functions are wellstudied. Among the piecewise monotone continuous (PM) functions, the existence of iterative roots of functions with height less than two is also well-studied. In this thesis, we develop the method of characteristic interval to any continuous functions and discuss the properties of non-isolated forts of any continuous functions on a compact interval. This helps us to derive the conditions on the existence of iterative roots for a class of PM functions with non-monotonicity height greater than one and a class of continuous functions with infinitely many forts. As an application we obtain a new class of functions which is dense in the space of all continuous functions from a compact interval into itself. We also provide sufficient conditions on the existence of solutions of series-like iterative functional equation for a class of PM functions. We conclude the thesis with results on the uniqueness of iterative roots of order preserving homeomorphisms by using the set of points of coincidence.