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Set-Valued \(\alpha \)-Fractal Functions

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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.

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Acknowledgements

This work is supported by MHRD Fellowship to the 1st author as a TA-ship at the Indian Institute of Technology (BHU), Varanasi. Some results of this paper have been presented at the conference, “ AMS Fall Western Virtual Sectional Meeting (formerly at the University of New Mexico): SS 13A - Special Session on Fractal Geometry and Dynamical Systems., October 23–24, 2021”.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India

    Megha Pandey & Tanmoy Som

  2. Department of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj, 211015, India

    Saurabh Verma

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  1. Megha Pandey
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  2. Tanmoy Som
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  3. Saurabh Verma
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Each author contributed equally to this manuscript.

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Correspondence to Megha Pandey.

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Communicated by Edward B. Saff.

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Pandey, M., Som, T. & Verma, S. Set-Valued \(\alpha \)-Fractal Functions. Constr Approx (2023). https://doi.org/10.1007/s00365-023-09652-2

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  • DOI: https://doi.org/10.1007/s00365-023-09652-2

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