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Fractal Fract
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Metric Spaces with Its Application to Fractional Differential Equations
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Metric Spaces with Its Application to Fractional Differential Equations

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

Deadline for manuscript submissions: 14 February 2025 | Viewed by 11691

Special Issue Editors

Prof. Dr. Khurram Shabbir

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Guest Editor
Department of Mathematics, Government College University of Lahore, 54000 Lahore, Pakistan
Interests: algebraic geometry; topology; inequalities; applied mathematics; metric spaces; iteration schemes; fractional partial differential equations
Dr. Monica-Felicia Bota

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
Interests: applied mathematics; metric spaces; nonlinear operators; fractal-fractional model; fractional derivative; FPDE; ODE; PDE
Special Issues, Collections and Topics in MDPI journals
Dr. Liliana Guran

E-Mail Website
Guest Editor
Department of Pharmaceutical Sciences, “Vasile Goldis” Western University of Arad, 310025 Arad, Romania
Interests: applied mathematics; fixed point theory; metric spaces; nonlinear operators; ODE; PDE; FDE
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Metric space and fixed point theorems in metric spaces are powerful tools in applied mathematics. Metric space has been proven to be a very interesting topic for researchers who work in fixed point theory. The existence of a solution of differential and integral fractional equations has been proved using the metric space and the fixed point techniques.

In the last century, the study of fractional differential equations became very dynamic. Fractional order operators are actually nonlinear operators but they are more practical than the ones given in the classical form. Fractional order operators have many applications in various scientific fields, such as physics, fluid mechanics, entropy theory, viscoelasticity, chemistry, biology, dynamical systems, signal processing, and so on. Keeping this in mind, virtual many real-world phenomena can become known problems of fractional differential and integral equations.

Some natural phenomena such as, the growth of bacteria, freezing water, brain waves, have been approached in recent years, using the concepts of fractals. Their mathematics has reached major scientific perspectives. Various phenomena having a pulse, rhythm or pattern, have the potential to be modelled by a fractal.

This Special Issue invites and welcomes review, expository, and original research papers addressing state-of-the-art developments in pure and applied mathematics via fractals and fractional calculus, along with their wide-ranging applications in the physical, natural, computational, environmental, engineering, and statistical sciences, all mixed with fixed points techniques. This Special Issue is dedicated, but not limited, to the following topics of interest:

  • Metric spaces;
  • Fixed points theorems;
  • Well-posedness;
  • Stability;
  • Fractional differential equations with different kernels;
  • Fractal patterns;
  • Statistical convergence;
  • Decision-making problems;
  • Numerical and computational methods;
  • Mathematical physics;
  • Mathematics in biology;
  • Intuitionistic fuzzy relations.

Prof. Dr. Khurram Shabbir
Dr. Monica-Felicia Bota
Dr. Liliana Guran
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

  • metric spaces
  • fixed points theorems
  • well-posedness
  • stability
  • fractional differential equations with different kernels
  • fractal patterns
  • statistical convergence
  • decision making problems
  • numerical and computational methods
  • mathematical physics
  • mathematics in biology
  • intuitionistic fuzzy relations

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (9 papers)

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Research

16 pages, 1066 KiB  
Article
A Novel Approach for Solving Fractional Differential Equations via a Multistage Telescoping Decomposition Method
by Fouzia Chita, Souad Ayadi, Meltem Erden Ege, Ozgur Ege and Rajagopalan Ramaswamy
Fractal Fract. 2025, 9(2), 65; https://doi.org/10.3390/fractalfract9020065 - 22 Jan 2025
Viewed by 468
Abstract
This study presents a novel algorithm for solving nonlinear fractional initial value problems, utilizing a multistage telescoping decomposition method (FTDM). By combining the MFTDM with the Elzaki transform, this method significantly improves computational efficiency and accuracy. Through a series of numerical experiments, the [...] Read more.
This study presents a novel algorithm for solving nonlinear fractional initial value problems, utilizing a multistage telescoping decomposition method (FTDM). By combining the MFTDM with the Elzaki transform, this method significantly improves computational efficiency and accuracy. Through a series of numerical experiments, the proposed approach is demonstrated to be highly practical, reliable and straightforward, offering a robust framework for solving nonlinear fractional differential equations effectively. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
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17 pages, 345 KiB  
Article
Numerical Algorithm for Coupled Fixed Points in Normed Spaces with Applications to Fractional Differential Equations and Economics
by Lifang Guo, Salha Alshaikey, Abeer Alshejari, Muhammad Din and Umar Ishtiaq
Fractal Fract. 2025, 9(1), 37; https://doi.org/10.3390/fractalfract9010037 - 14 Jan 2025
Viewed by 475
Abstract
This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This [...] Read more.
This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This class of operators expands on existing cyclic contractions, including interpolative Kannan mappings, interpolative Reich–Rus–Ćirić contractions, and other known contractions in the literature. We demonstrate the existence and uniqueness of fixed points for these operators and provide an example to illustrate our findings. Moreover, we discuss the applications of our results in solving nonlinear integral equations. Furthermore, we introduce the idea of a coupled interpolative enriched cyclic Reich–Rus–Ćirić operator and establish the existence of a strongly coupled fixed-point theorem for this contraction. Finally, we provide an application to fractional differential equations to show the validity of the main result. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
18 pages, 3475 KiB  
Article
Analysis of Caputo-Type Non-Linear Fractional Differential Equations and Their Ulam–Hyers Stability
by Ekber Girgin, Abdurrahman Büyükkaya, Neslihan Kaplan Kuru, Mudasir Younis and Mahpeyker Öztürk
Fractal Fract. 2024, 8(10), 558; https://doi.org/10.3390/fractalfract8100558 - 26 Sep 2024
Cited by 1 | Viewed by 850
Abstract
This study presents two novel frameworks, termed a quasi-modular b-metric space and a non-Archimedean quasi-modular b-metric space, and various topological properties are provided. Using comparison and simulation functions, this paper rigorously proves several fixed point theorems in the non-Archimedean quasi-modular b [...] Read more.
This study presents two novel frameworks, termed a quasi-modular b-metric space and a non-Archimedean quasi-modular b-metric space, and various topological properties are provided. Using comparison and simulation functions, this paper rigorously proves several fixed point theorems in the non-Archimedean quasi-modular b-metric space. As a useful application, it also establishes Ulam–Hyers stability for the fixed point problem. Finally, this study concludes with a unique solution to a non-linear fractional differential equation, making a substantial contribution to the discipline. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
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24 pages, 365 KiB  
Article
Solving Fractional Random Differential Equations by Using Fixed Point Methodologies under Mild Boundary Conditions
by Hasanen A. Hammad and Saleh Fahad Aljurbua
Fractal Fract. 2024, 8(7), 384; https://doi.org/10.3390/fractalfract8070384 - 28 Jun 2024
Viewed by 847
Abstract
This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we [...] Read more.
This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
10 pages, 330 KiB  
Article
Stability in the Sense of Hyers–Ulam–Rassias for the Impulsive Volterra Equation
by El-sayed El-hady, Süleyman Öğrekçi, Tania A. Lazăr and Vasile L. Lazăr
Fractal Fract. 2024, 8(1), 47; https://doi.org/10.3390/fractalfract8010047 - 12 Jan 2024
Viewed by 1471
Abstract
This article aims to use various fixed-point techniques to study the stability issue of the impulsive Volterra integral equation in the sense of Ulam–Hyers (sometimes known as Hyers–Ulam) and Hyers–Ulam–Rassias. By eliminating key assumptions, we are able to expand upon and enhance some [...] Read more.
This article aims to use various fixed-point techniques to study the stability issue of the impulsive Volterra integral equation in the sense of Ulam–Hyers (sometimes known as Hyers–Ulam) and Hyers–Ulam–Rassias. By eliminating key assumptions, we are able to expand upon and enhance some recent findings. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
17 pages, 574 KiB  
Article
A Numerical Scheme and Application to the Fractional Integro-Differential Equation Using Fixed-Point Techniques
by Arul Joseph Gnanaprakasam, Balaji Ramalingam, Gunaseelan Mani, Ozgur Ege and Reny George
Fractal Fract. 2024, 8(1), 34; https://doi.org/10.3390/fractalfract8010034 - 4 Jan 2024
Cited by 5 | Viewed by 1471
Abstract
In this paper, we introduce the notion of orthogonal α–F–convex contraction mapping and prove some fixed-point theorems for self-mapping in orthogonal complete metric spaces. The proven results generalize and extend some of the well-known results in the literature. Following the [...] Read more.
In this paper, we introduce the notion of orthogonal α–F–convex contraction mapping and prove some fixed-point theorems for self-mapping in orthogonal complete metric spaces. The proven results generalize and extend some of the well-known results in the literature. Following the derivation of these fixed-point results, we propose a solution for the fractional integro-differential equation, utilizing the fixed-point technique within the context of orthogonal complete metric spaces. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
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19 pages, 488 KiB  
Article
Novel Fuzzy Contractions and Applications to Engineering Science
by Mudasir Younis and Afrah Ahmad Noman Abdou
Fractal Fract. 2024, 8(1), 28; https://doi.org/10.3390/fractalfract8010028 - 29 Dec 2023
Cited by 15 | Viewed by 1728
Abstract
The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as “Kannan-graph-fuzzy contraction”. This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled [...] Read more.
The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as “Kannan-graph-fuzzy contraction”. This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled by nonlinear models. It is anticipated that using Kannan-graph-fuzzy contractions will offer a fresh approach to discussing solution existence in situations where these ideas overlap. This strategy might have a significant impact on a lot of real-world applications, proving how practical multidisciplinary techniques are for tackling issues in the real world originating from technology and engineering. The article includes several illustrated examples based on computer simulations, making the results more explicit. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
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19 pages, 410 KiB  
Article
A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications
by Danish Ali, Shahbaz Ali, Darab Pompei-Cosmin, Turcu Antoniu, Abdullah A. Zaagan and Ali M. Mahnashi
Fractal Fract. 2023, 7(11), 790; https://doi.org/10.3390/fractalfract7110790 - 30 Oct 2023
Cited by 2 | Viewed by 1724
Abstract
Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a [...] Read more.
Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the D** iteration technique, for approximating fixed points in generalized α-nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the D** iteration approach to the fixed points of generalized α-nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
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24 pages, 3864 KiB  
Article
Computational Study for Fiber Bragg Gratings with Dispersive Reflectivity Using Fractional Derivative
by Hira Tariq, Ghazala Akram, Maasoomah Sadaf, Maria Iftikhar and Liliana Guran
Fractal Fract. 2023, 7(8), 625; https://doi.org/10.3390/fractalfract7080625 - 16 Aug 2023
Cited by 7 | Viewed by 1248
Abstract
In this paper, the new representations of optical wave solutions to fiber Bragg gratings with cubic–quartic dispersive reflectivity having the Kerr law of nonlinear refractive index structure are retrieved with high accuracy. The residual power series technique is used to derive power series [...] Read more.
In this paper, the new representations of optical wave solutions to fiber Bragg gratings with cubic–quartic dispersive reflectivity having the Kerr law of nonlinear refractive index structure are retrieved with high accuracy. The residual power series technique is used to derive power series solutions to this model. The fractional derivative is taken in Caputo’s sense. The residual power series technique (RPST) provides the approximate solutions in truncated series form for specified initial conditions. By using three test applications, the efficiency and validity of the employed technique are demonstrated. By considering the suitable values of parameters, the power series solutions are illustrated by sketching 2D, 3D, and contour profiles. The analysis of the obtained results reveals that the RPST is a significant addition to exploring the dynamics of sustainable and smooth optical wave propagation across long distances through optical fibers. Full article
(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)
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