2. Thesis and Dissertations
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Item Characterization of Non-Isolated Forts and Stability of an Iterative Functional Equation(National Institute of Technology Karnataka, Surathkal, 2021) Palanivel, R.; Murugan, V.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.Item A Study on Iterative Root Problem(National Institute of Technology Karnataka, Surathkal, 2018) Kumar M, Suresh; Murugan, VThe 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.