Set-Valued α -Fractal Functions

Abstract

In this paper, we introduce the concept of the α-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its α-fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an α-fractal function is an attractor of an iterated function system. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023.