Set-valued \alpha-fractal functions

TL;DR

This paper introduces set-valued -fractal functions, explores their properties, estimates approximation errors, and defines a new graph concept with bounds on fractal dimension, showing it as an attractor of an iterated function system.

Contribution

It presents a novel concept of set-valued -fractal functions, introduces a new graph definition, and establishes their properties and fractal dimension bounds.

Findings

The new graph of -fractal functions is an attractor of an iterated function system.

Provides bounds on the fractal dimension of the new graph.

Estimates the perturbation error between functions and their -fractal approximations.

Abstract

In this paper, we introduce the concept of the $\alpha$-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 $\alpha$-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 $\alpha$-fractal function is an attractor of an iterated function system.