Iterated Functions Systems Composed of Generalized θ-Contractions

Rajan, Pasupathi; Navascués, María A.; Chand, Arya Kumar Bedabrata

doi:10.3390/fractalfract5030069

Open AccessArticle

Iterated Functions Systems Composed of Generalized θ-Contractions

by

Pasupathi Rajan

^1,†,

María A. Navascués

^2,†

and

Arya Kumar Bedabrata Chand

^1,*,†

¹

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India

²

Departamento de Matemática Aplicada, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, 50018 Zaragoza, Spain

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Fractal Fract. 2021, 5(3), 69; https://doi.org/10.3390/fractalfract5030069

Submission received: 2 June 2021 / Revised: 9 July 2021 / Accepted: 9 July 2021 / Published: 14 July 2021

(This article belongs to the Special Issue Fractal Functions and Applications)

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Abstract

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized

θ

-contraction IFS, which is a finite collection of generalized

θ

-contraction functions

T_{1}, \dots, T_{N}

from finite Cartesian product space

X \times \dots \times X

into X, where

(X, d)

is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued

θ

-contraction IFSs are Picard. Finally, when the map

θ

is continuous, we show the relation between the code space and the attractor of

θ

-contraction IFS.

Keywords: iterated function systems; fixed point; attractor; fractal; θ-contraction; picard operator; code space

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MDPI and ACS Style

Rajan, P.; Navascués, M.A.; Chand, A.K.B. Iterated Functions Systems Composed of Generalized θ-Contractions. Fractal Fract. 2021, 5, 69. https://doi.org/10.3390/fractalfract5030069

AMA Style

Rajan P, Navascués MA, Chand AKB. Iterated Functions Systems Composed of Generalized θ-Contractions. Fractal and Fractional. 2021; 5(3):69. https://doi.org/10.3390/fractalfract5030069

Chicago/Turabian Style

Rajan, Pasupathi, María A. Navascués, and Arya Kumar Bedabrata Chand. 2021. "Iterated Functions Systems Composed of Generalized θ-Contractions" Fractal and Fractional 5, no. 3: 69. https://doi.org/10.3390/fractalfract5030069

APA Style

Rajan, P., Navascués, M. A., & Chand, A. K. B. (2021). Iterated Functions Systems Composed of Generalized θ-Contractions. Fractal and Fractional, 5(3), 69. https://doi.org/10.3390/fractalfract5030069

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Iterated Functions Systems Composed of Generalized θ-Contractions

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