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Fractal Fract
Special Issues
Fixed Point Theory and Fractals
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Fixed Point Theory and Fractals

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: 31 May 2025 | Viewed by 2980

Special Issue Editors

Prof. Dr. María A. Navascués

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Guest Editor
Department of Applied Mathematics, Universidad de Zaragoza, 50018 Zaragoza, Spain
Interests: fractals; fixed point theory; approximation; iterative methods
Special Issues, Collections and Topics in MDPI journals
Dr. Bilel Selmi

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Guest Editor
Department of Mathematics, University of Monastir, Monastir 5000, Tunisia
Interests: fractal geometry; multifractal analysis; dynamical systems
Dr. Cristina Serpa

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Guest Editor
1. ISEL—Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisbon, Portugal
2. CMAFcIO—Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, 1749-016 Lisbon, Portugal
Interests: fractals; fractal regression; Hausdorff dimension; fractal calculus; systems of iterative equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent decades, fractal theory has proven to be extremely useful for the modelling of a great quantity of natural and social phenomena. Its fields of applications range from biotechnology to financial markets, for instance.

Fractal geometry builds a bridge between classical geometry and modern analysis. The static models of the old geometry are enriched with the dynamics of an infinite iterative process, where the outputs are not merely points but more sophisticated geometric objects and structures.

A fractal set can be described in very different ways, but the current mathematical research tends to define a fractal as the fixed point of an operator on the space of compact subsets of a space of metric type. Iterated function systems provide a way of constructing an operator of this kind, and a procedure for the approximation of its fixed points. Thus, the relationships between fractal and fixed-point theories are deep and increasingly intricate.

This issue is aimed at emphasizing the relationships between both fields, including their theoretical as well as their applied aspects. This volume also welcomes acute insights in any of the disciplines with potential (even if they are not developed) influences in the other.

Prof. Dr. María A. Navascués
Dr. Bilel Selmi
Dr. Cristina Serpa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fixed point theorems
  • fractal sets, functions and measures
  • fixed-point approximation and stability
  • iterative methods for the solution of differential and integral equations
  • iterated function systems
  • discrete dynamical systems
  • fractional calculus related to fixed point theory, dynamical systems or deterministic/stochastic fractals
  • applications of these fields

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Published Papers (4 papers)

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Research

24 pages, 21684 KiB  
Article
An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets
by Khairul Habib Alam, Yumnam Rohen, Anita Tomar, Naeem Saleem, Maggie Aphane and Asima Razzaque
Fractal Fract. 2025, 9(1), 40; https://doi.org/10.3390/fractalfract9010040 - 15 Jan 2025
Viewed by 616
Abstract
This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity [...] Read more.
This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function sin(um)−αu+β, where u,α,β∈C and m≥2. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractals)
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13 pages, 361 KiB  
Article
An Iterative Method for the Approximation of Common Fixed Points of Two Mappings: Application to Fractal Functions
by María A. Navascués
Fractal Fract. 2024, 8(12), 745; https://doi.org/10.3390/fractalfract8120745 - 17 Dec 2024
Viewed by 598
Abstract
This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides sufficient conditions for the strong [...] Read more.
This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides sufficient conditions for the strong convergence of the sequence generated by the iterative scheme to a common fixed point of two operators. The method is illustrated with some examples of application. The procedure is used to approach a common solution of two Fredholm integral equations of the second kind. In the second part of the article, the existence of a fractal function coming from two different Read–Bajraktarević operators is proved. Afterwards, a study of the approximation of fixed points of a fractal convolution of operators is performed, in the framework of Lebesgue or Bochner spaces. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractals)
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22 pages, 506 KiB  
Article
Topological Degree for Operators of Class (S)+ with Set-Valued Perturbations and Its New Applications
by Evgenii S. Baranovskii and Mikhail A. Artemov
Fractal Fract. 2024, 8(12), 738; https://doi.org/10.3390/fractalfract8120738 - 14 Dec 2024
Viewed by 643
Abstract
We investigate the topological degree for generalized monotone operators of class (S)+ with compact set-valued perturbations. It is assumed that perturbations can be represented as the composition of a continuous single-valued mapping and an upper semicontinuous set-valued mapping with aspheric [...] Read more.
We investigate the topological degree for generalized monotone operators of class (S)+ with compact set-valued perturbations. It is assumed that perturbations can be represented as the composition of a continuous single-valued mapping and an upper semicontinuous set-valued mapping with aspheric values. This allows us to extend the standard degree theory for convex-valued operators to set-valued mappings whose values can have complex geometry. Several theoretical aspects concerning the definition and main properties of the topological degree for such set-valued mappings are discussed. In particular, it is shown that the introduced degree has the homotopy invariance property and can be used as a convenient tool in checking the existence of solutions to corresponding operator inclusions. To illustrate the applicability of our approach to studying models of real processes, we consider an optimal feedback control problem for the steady-state internal flow of a generalized Newtonian fluid in a 3D (or 2D) bounded domain with a Lipschitz boundary. By using the proposed topological degree method, we prove the solvability of this problem in the weak formulation. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractals)
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18 pages, 340 KiB  
Article
Common Attractors of Generalized Hutchinson–Wardowski Contractive Operators
by Bilal Iqbal, Naeem Saleem, Iram Iqbal and Maggie Aphane
Fractal Fract. 2024, 8(11), 651; https://doi.org/10.3390/fractalfract8110651 - 9 Nov 2024
Viewed by 616
Abstract
The aim of this paper is to obtain a fractal set of ℑ-iterated function systems comprising generalized ℑ-contractions. For a variety of Hutchinson–Wardowski contractive operators, we prove that this kind of system admits a unique common attractor. Consequently, diverse outcomes are [...] Read more.
The aim of this paper is to obtain a fractal set of ℑ-iterated function systems comprising generalized ℑ-contractions. For a variety of Hutchinson–Wardowski contractive operators, we prove that this kind of system admits a unique common attractor. Consequently, diverse outcomes are obtained for generalized iterated function systems satisfying various generalized contractive conditions. An illustrative example is also provided. Finally, the existence results of common solutions to fractional boundary value problems are obtained. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractals)
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