Countable zipper fractal interpolation and some elementary aspects of the associated nonlinear zipper fractal operator

This note aims to extend the notion of affine zipper fractal interpolation function from the case of a finite data set to an infinite sequence of data points. We work with a slightly more general setting wherein the assumption of affinity on the functions involved in the construction of the zipper fractal interpolant is dropped. Invoking the iterative functional equation for the countable zipper fractal interpolant, its stability with a perturbation of data points and sensitivity to perturbations in the maps that define the zipper are examined. In the second part of this note, the countable zipper fractal interpolation is used to obtain a parameterized family of zipper fractal functions corresponding to a prescribed real-valued Lipschitz continuous function on a closed bounded interval in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}. An operator obtained by associating each Lipschitz continuous function to its fractal counterpart is approached from the standpoint of nonlinear functional analysis and perturbation theory of operators.