Characterization of Non-Isolated Forts and Stability of an Iterative Functional Equation

Abstract

The problem of finding a solution f : X →X of the iterative functional equation f n = F for a given positive integer n ≥ 2 and a function F : X → X on a non-empty set X is known as the iterative root problem. The non-strictly monotone points (or forts) of F play an essential role in finding a continuous solution f of f n = F whenever X is an interval in the real line. In this thesis, we define the forts for any continuous function f : I →J, where I and J are arbitrary intervals in the real line R. We study the non-monotone behavior of forts under composition and characterize the sets of isolated and non-isolated forts of iterates of any continuous self-map on an arbitrary interval I to study the continuous solutions of f n = F. Consequently, we obtain an example of an uncountable measure zero dense set of non-isolated forts in the real line. We define the notions of iteratively closed set in the space of continuous self-maps and the non-monotonicity height of any continuous self-map. We prove that continuous self-maps of non-monotonicity height 1 need not be strictly monotone on its range, unlike continuous piecewise monotone functions. Also, we obtain sufficient conditions for the existence of continuous solutions of f n = F for a class of continuous functions of non-monotonicity height 1. Further, we discuss the Hyers-Ulam stability of the iterative functional equation f n = F for continuous self-maps of non-monotonicity height 0 and 1.