arXiv subjects

Chaitanya Gopalakrishna

Publications and source records attributed to Chaitanya Gopalakrishna.

9 recordsLinked to original sources

arXiv2024

An equation in nonlinear combination of iterates

In this paper we deal with an equation in nonlinear combination of iterates. Although it can be reduced by the logarithm conjugacy to a form for application of Schauder's or Banach's fixed point theorems, a difficulty called Zero Problem is encountered for continuous solutions because the domain does not contain $0$. So we consider solutions with weaker regularity, using the Knaster-Tarski fixed point theorem for complete lattices to give order-preserving solutions. Then we give semi-continuous solutions and integrable solutions.

math.DS↗

arXiv2023

A note on iterated maps of the unit sphere

Let $\mathcal{C}(S^{m})$ denote the set of continuous maps from the unit sphere $S^{m}$ in $\mathbb{R}^{m+1}$ into itself endowed with the supremum norm. We prove that the set $\{f^n: f\in \mathcal{C}(S^{m})~\text{and}~n\ge 2\}$ of iterated maps is not dense in $\mathcal{C}(S^{m})$. This, in particular, proves that the periodic points of the iteration operator of order $n$ are not dense in $\mathcal{C}(S^m)$ for all $n\ge 2$, providing an alternative proof of the result that these operators are not Devaney chaotic on $\mathcal{C}(S^m)$ proved in [M. Veerapazham, C. Gopalakrishna, W. Zhang, Dynamics of the iteration operator on the space of continuous self-maps, Proc. Amer. Math. Soc., 149(1) (2021), 217--229].

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arXiv2022

Iterative Roots of Multifunctions

Some easily verifiable sufficient conditions for the nonexistence of iterative roots for multifunctions on arbitrary nonempty sets are presented. Typically if the graph of the multifunction has a distinguished point with a relatively large number of paths leading to it then such a multifunction does not admit any iterative root. These results can be applied to single-valued maps by considering their pullbacks as multifunctions. This has been illustrated by showing the nonexistence of iterative roots of some specified orders for certain complex polynomials.

math.DS↗

arXiv2022

The non-iterates are dense in the space of continuous self-maps

In this paper we develop a tool to identify functions which have no iterative roots of any order. Using this, we prove that when $X$ is $[0,1]^m$, $\mathbb{R}^m$ or $S^1$, every non-empty open set of the space $\mathcal{C}(X)$ of continuous self-maps on $X$ endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order $n\ge 2$. This, in particular, proves that the complement of $\{f^n: f\in \mathcal{C}(X)~\text{and}~n\ge 2\}$, the set of non-iterates, is dense in $\mathcal{C}(X)$ for these $X$.

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arXiv2022

Iterative square roots of functions

An iterative square root of a function $f$ is a function $g$ such that $g(g(\cdot))=f(\cdot)$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. This is used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^m$ and to the whole of $\mathbb{R}^m$ for every positive integer $m.$ On the other hand, we also prove that every continuous self-map of a space homeomorphic to the unit cube in $\mathbb{R}^m$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

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arXiv2021

A note on Fredholm integral equation

This note gives results on the existence of semi-continuous solutions of a Fredholm integral equation of the second kind using Tarski's fixed point theorem.

math.AP↗

arXiv2021

Differentiable solutions of an equation with product of iterates

In the previous work [2] (i.e., arXiv:2105.03385), we considered continuous solutions of an iterative equation involving the multiplication of iterates. In this paper, we continue to investigate this equation for differentiable solutions. Similar to continuous solutions until [2], there is no obtained result on differentiable solutions of such an equation on non-compact intervals of $\mathbb{R}$. Although our strategy here is to use conjugation to reduce the equation to the well-known polynomial-like iterative equation as in [2], all known results on differentiable solutions of the latter are given on compact intervals. We re-explore polynomial-like iterative equation on the whole of R and prove the existence and uniqueness of differentiable solutions of our equation on $\mathbb{R}_+$ and $\mathbb{R}_-$.

math.DS↗

arXiv2021

Iteration and iterative equation on lattices

In this paper we investigate iteration of maps on lattices and the corresponding polynomial-like iterative equation. Since a lattice need not have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using Tarski's fixed point theorem, we prove the existence of order-preserving solutions on convex complete sublattices of Riesz spaces. Further, in $\mathbb{R}^n$ and $\mathbb{R}$, special cases of Riesz space, we discuss upper semi-continuous solutions and integrable solutions respectively. Finally, we indicate more special cases of Riesz space for discussion on the iterative equation.

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arXiv2021

Continuous solutions of an iterative equation with multiplication

Iterative equation is an equality with an unknown function and its iterates. There were not found a result on iterative equations with multiplication of iterates of the unknown function on $\mathbb{R}$. In this paper we use an exponential function to reduce the equation in conjugation to the well-known form of polynomial-like iterative equation, but we encountered two difficulties: the reduction restricts our discussion of the equation to $\mathbb{R}_+$; the reduced polynomial-like iterative equation is defined on the whole $\mathbb{R}$ but known results were given on comapct intervals. We revisit the polynomial-like iterative equation on the whole $\mathbb{R}$ and give existence, uniqueness, stability and construction of continuous solutions of our equation on $\mathbb{R}_+$. Then we technically extend our solutions from $\mathbb{R}_+$ to $\mathbb{R}_-$.

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