Faculty Publications
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Item Series-Like Iterative Functional Equation for PM Functions(IOP Publishing Ltd, 2021) Suresh Kumar, M.; Murugan, V.Given a non-empty subset X of the real line and a self map G on X, the functional equation representing G as an infinite linear combination of iterations of a self map g on X is known as the series-like functional equation. The solutions of the series-like functional equation have been studied only for the class of continuous strictly monotone functions. In this paper, we prove the existence of solutions of series-like functional equations for the class of continuous non-monotone functions using characteristic interval. © Published under licence by IOP Publishing Ltd.Item Existence of continuous solutions for an iterative functional series equation with variable coefficients(2009) Murugan, V.; Subrahmanyam, P.V.We obtain theorems on the existence and uniqueness of the solution for iterative functional equations of the type where Hi's and F are given functions and ?i's are nonnegative functions such that on [a, b]. Stability of the solution is also discussed. © Birkhäuser Verlag, Basel, 2009.Item Large Time Asymptotics with Error Estimates to Solutions of a Forced Burgers Equation(2017) Satyanarayana, E.; Mohd, M.; Murugan, V.This article deals with a forced Burgers equation (FBE) subject to the initial function, which is continuous and summable on R. Large time asymptotic behavior of solutions to the FBE is determined with precise error estimates. To achieve this, we construct solutions for the FBE with a different initial class of functions using the method of separation of variables and Cole–Hopf like transformation. These solutions are constructed in terms of Hermite polynomials with the help of similarity variables. The constructed solutions would help us to pick up an asymptotic approximation and to show that the magnitude of the difference function of the true and approximate solutions decays algebraically to 0 for large time. © 2016 Wiley Periodicals, Inc., A Wiley CompanyItem Subcommuting and comparable iterative roots of order preserving homeomorphisms(Emerald Group Holdings Ltd., 2019) Murugan, V.; Suresh Kumar, M.It is known that the iterative roots of continuous functions are not necessarily unique, if it exist. In this note, by introducing the set of points of coincidence, we study the iterative roots of order preserving homeomorphisms. In particular, we prove a characterization of identical iterative roots of an order preserving homeomorphism using the points of coincidence of functions. © 2019, Veerapazham Murugan and Murugan Suresh Kumar.Item Relation between kneading matrices of a map and its iterates(Korean Mathematical Society kms@kms.or.kr, 2020) Gopalakrishna, C.; Murugan, V.It is known that the kneading matrix associated with a continuous piecewise monotone self-map of an interval contains crucial combinatorial information of the map and all its iterates, however for every iterate of such a map we can associate its kneading matrix. In this paper, we describe the relation between kneading matrices of maps and their iterates for a family of chaotic maps. We also give a new definition for the kneading matrix and describe the relationship between the corresponding determinant and the usual kneading determinant of such maps. ©2020 Korean Mathematical Society.Item Iterative roots of continuous functions with non-isolated forts(Springer Science and Business Media B.V., 2020) Murugan, V.; Suresh Kumar, M.S.Iterative root problem is one of the classical problem in analysis and isdescribed as follows: given a topological space X, a continuous self map F on X and a fixed positive integer n, to find another self map f on X such that fn= F. This problem is solved only for a particular class of continuous functions and is still unsolved for any continuous function. In this paper, we present results on non-existence of iterative roots of continuous functions having finitely manynon-isolated forts. © 2017, Forum D'Analystes, Chennai.Item NON-ISOLATED, NON-STRICTLY MONOTONE POINTS OF ITERATES OF CONTINUOUS FUNCTIONS(Michigan State University Press, 2021) Murugan, V.; Palanivel, R.There are continuous functions with complicated yet interesting sets of non-isolated non-strictly monotone points. This paper aims to characterize the sets of isolated and non-isolated non-strictly monotone points of the composition of continuous functions. Consequently, an uncountable dense set of measure zero in the real line and whose complement is also uncountable and dense is obtained. © 2021 Michigan State University Press. All rights reserved.Item Iterative roots of continuous functions and Hyers–Ulam stability(Birkhauser, 2021) Murugan, V.; Palanivel, R.In this paper, we prove that continuous non-PM functions with non-monotonicity height equal to 1 need not be strictly monotone on its range, unlike PM functions. An existence theorem is obtained for the iterative roots of such functions. We also discuss the Hyers–Ulam stability for the functional equation of the iterative root problem. © 2020, Springer Nature Switzerland AG.Item Hyers-Ulam stability of an iterative equation for strictly increasing continuous functions(Birkhauser, 2023) Palanivel, R.; Murugan, V.The Hyers-Ulam stability of the iterative equation fn= F for continuous functions F was studied under the assumptions that F is a homeomorphism on its range, and the equation has stability on its range. It is important to study the stability of the equation for homeomorphisms on intervals. In this paper, theorems on stability are obtained using the properties of monotonic approximate solutions. The method is based on the stability of two derived iterative equations. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.Item Iterative roots of PM functions extended from the characteristic interval(Birkhauser, 2023) Murugan, V.; Suresh Kumar, M.S.; Jarczyk, J.; Jarczyk, W.In this paper we consider the problem of finding iterative roots of order less than the number of forts of continuous piecewise monotone(PM) functions with nonmonotonicity height greater than one. We present sufficient conditions to extend iterative roots of PM functions from the characteristic interval which determines the behavior of the function under iteration. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.