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Fractal Fract
Special Issues
Nonlinear Fractional Differential Equation and Fixed-Point Theory
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Nonlinear Fractional Differential Equation and Fixed-Point Theory

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

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 15268

Special Issue Editors

Prof. Dr. Mujahid Abbas

E-Mail Website
Guest Editor
1. Department of Mathematics, Government College University, Lahore 54000, Pakistan
2. Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Rroad, Pretoria 0002, South Africa
Interests: nonlinear operator theory; best approximations; fuzzy logic; soft set theory; convex optimization theory; fixed-point theory and its applications
Special Issues, Collections and Topics in MDPI journals
Dr. Talat Nazir

E-Mail Website
Guest Editor
Department of Mathematical Sciences, University of South Africa, Johannesburg 0003, South Africa
Interests: functional analysis; fixed point theory and applications; optimization theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We invite you to contribute a manuscript to this Special Issue on “Nonlinear Fractional Differential Equation and Fixed-Point Theory”.

Nonlinear fractional differential equations have been of great interest for the past three decades. This is due to the intensive development of the theory of fractional calculus and its applications. Apart from diverse areas of pure mathematics, fractional differential equations can be used in the modeling of various fields of science and engineering, such as rheology, self-similar dynamical processes, porous media, fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many other branches of science. A significant characteristic of a fractional order differential operator which distinguishes it from the integer-order differential operator is that it is nonlocal in nature, that is, the future state of a dynamical system or process involving fractional derivative depends on its current state as well as its past states. In fact, this feature of fractional-order operators has contributed toward the popularity of fractional-order models, which are recognized as more realistic and practical than classical integer-order models.

Fixed-point theory provides essential tools for solving problems arising in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization problems, equilibrium problems, complementarity problems, selection and matching problems, and problems of proving the existence of solution of integral and differential equations.

The aim of this Special Issue is to gather and publish new results on fractional differential equation and fixed-point theory and their applications. We welcome papers on topics including, but not limited to the following:

  • Fixed-point theory and its application;
  • Approximation theory and its application;
  • Variational inequality and its applications;
  • Fractional differential equation;
  • Iterated function system;
  • Fractal theory;
  • Differential equations;
  • Finite difference equations;
  • Nonlinear analysis;
  • Dynamical system;
  • Nonlinear convex analysis.

Prof. Dr. Mujahid Abbas
Prof. Dr. Talat Nazir
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.

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

Published Papers (9 papers)

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Research

19 pages, 339 KiB  
Article
Lyapunov Inequalities for Systems of Tempered Fractional Differential Equations with Multi-Point Coupled Boundary Conditions via a Fix Point Approach
by Hailong Ma and Hongyu Li
Fractal Fract. 2024, 8(12), 754; https://doi.org/10.3390/fractalfract8120754 - 21 Dec 2024
Viewed by 854
Abstract
In this paper, we study a system of nonlinear tempered fractional differential equations with multi-point coupled boundary conditions. By applying the properties of Green’s function and the operator and combining the method of matrix analysis, we obtain the corresponding Lyapunov inequalities under two [...] Read more.
In this paper, we study a system of nonlinear tempered fractional differential equations with multi-point coupled boundary conditions. By applying the properties of Green’s function and the operator and combining the method of matrix analysis, we obtain the corresponding Lyapunov inequalities under two Banach spaces. And, we have compared two Lyapunov inequalities under certain conditions. An example is given to verify our results. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
29 pages, 416 KiB  
Article
New Studies for Dynamic Programming and Fractional Differential Equations in Partial Modular b-Metric Spaces
by Abdurrahman Büyükkaya, Dilek Kesik, Ülkü Yeşil and Mahpeyker Öztürk
Fractal Fract. 2024, 8(12), 724; https://doi.org/10.3390/fractalfract8120724 - 9 Dec 2024
Cited by 1 | Viewed by 721
Abstract
This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation R, proposing a novel definition for a generalized ℷC-type [...] Read more.
This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation R, proposing a novel definition for a generalized ℷC-type Suzuki R-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
22 pages, 842 KiB  
Article
Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium
by Muhammad Shaheryar, Fahim Ud Din, Aftab Hussain and Hamed Alsulami
Fractal Fract. 2024, 8(10), 609; https://doi.org/10.3390/fractalfract8100609 - 18 Oct 2024
Cited by 5 | Viewed by 1567
Abstract
We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for [...] Read more.
We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
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17 pages, 331 KiB  
Article
Matkowski-Type Functional Contractions under Locally Transitive Binary Relations and Applications to Singular Fractional Differential Equations
by Faizan Ahmad Khan, Nidal H. E. Eljaneid, Ahmed Alamer, Esmail Alshaban, Fahad Maqbul Alamrani and Adel Alatawi
Fractal Fract. 2024, 8(1), 72; https://doi.org/10.3390/fractalfract8010072 - 22 Jan 2024
Viewed by 1630
Abstract
This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally J-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the [...] Read more.
This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally J-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the help of our findings, we deal with the existence and uniqueness theorems pertaining to the solution of a variety of singular fractional differential equations. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
17 pages, 335 KiB  
Article
Advancements in Best Proximity Points: A Study in F-Metric Spaces with Applications
by Badriah Alamri
Fractal Fract. 2024, 8(1), 62; https://doi.org/10.3390/fractalfract8010062 - 16 Jan 2024
Viewed by 1513
Abstract
The purpose of this study is to explore the existence and uniqueness of the best proximity points for α,Θ-proximal contractions, a novel concept introduced in the context of F-metric spaces. Moreover, we provide an example to show the usability [...] Read more.
The purpose of this study is to explore the existence and uniqueness of the best proximity points for α,Θ-proximal contractions, a novel concept introduced in the context of F-metric spaces. Moreover, we provide an example to show the usability of the obtained results. To broaden the scope of this research area, we leverage our best proximity point results to demonstrate the existence and uniqueness of solutions for differential equations (equation of motion) and also for fractional differential equations. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
14 pages, 311 KiB  
Article
Solving a Nonlinear Fractional Differential Equation Using Fixed Point Results in Orthogonal Metric Spaces
by Afrah Ahmad Noman Abdou
Fractal Fract. 2023, 7(11), 817; https://doi.org/10.3390/fractalfract7110817 - 12 Nov 2023
Cited by 6 | Viewed by 1549
Abstract
This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal [...] Read more.
This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
19 pages, 365 KiB  
Article
Hermite–Hadamard-Type Inequalities via Caputo–Fabrizio Fractional Integral for h-Godunova–Levin and (h1, h2)-Convex Functions
by Waqar Afzal, Mujahid Abbas, Waleed Hamali, Ali M. Mahnashi and M. De la Sen
Fractal Fract. 2023, 7(9), 687; https://doi.org/10.3390/fractalfract7090687 - 15 Sep 2023
Cited by 12 | Viewed by 1346
Abstract
This note generalizes several existing results related to Hermite–Hadamard inequality using h-Godunova–Levin and (h1,h2)-convex functions using a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. This study uses a non-singular kernel and constructs some [...] Read more.
This note generalizes several existing results related to Hermite–Hadamard inequality using h-Godunova–Levin and (h1,h2)-convex functions using a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. This study uses a non-singular kernel and constructs some new theorems associated with fractional order integrals. Furthermore, we demonstrate that the obtained results are a generalization of the existing ones. To demonstrate the correctness of these results, we developed a few interesting non-trivial examples. Finally, we discuss some applications of our findings associated with special means. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
15 pages, 375 KiB  
Article
A New Technique to Achieve Torsional Anchor of Fractional Torsion Equation Using Conservation Laws
by Nematollah Kadkhoda, Elham Lashkarian, Hossein Jafari and Yasser Khalili
Fractal Fract. 2023, 7(8), 609; https://doi.org/10.3390/fractalfract7080609 - 8 Aug 2023
Cited by 2 | Viewed by 3235
Abstract
The main idea in this research is introducing another approximate method to calculate solutions of the fractional Torsion equation, which is one of the applied equations in civil engineering. Since the fractional order is closed to an integer, we convert the fractional Torsion [...] Read more.
The main idea in this research is introducing another approximate method to calculate solutions of the fractional Torsion equation, which is one of the applied equations in civil engineering. Since the fractional order is closed to an integer, we convert the fractional Torsion equation to a perturbed ordinary differential equation involving a small parameter epsilon. Then we can find the exact solutions and approximate symmetries for the alternative approximation equation. Also, with help of the definition of conserved vector and the concept of nonlinear self-adjointness, approximate conservation laws(ACL) are obtained without approximate Lagrangians by using their approximate symmetries. In order to apply the presented theory, we apply the Lie symmetry analysis (LSA) and concept of nonlinear self-adjoint Torsion equation, which are very important in mathematics and engineering sciences, especially civil engineering. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
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16 pages, 377 KiB  
Article
Relative Controllability of ψ-Caputo Fractional Neutral Delay Differential System
by Kothandapani Muthuvel, Panumart Sawangtong and Kalimuthu Kaliraj
Fractal Fract. 2023, 7(6), 437; https://doi.org/10.3390/fractalfract7060437 - 29 May 2023
Cited by 2 | Viewed by 1279
Abstract
The aim of this work is to analyze the relative controllability and Ulamn–Hyers stability of the ψ-Caputo fractional neutral delay differential system. We use neutral ψ-delayed perturbation of the Mitttag–Leffler matrix function and Banach contraction principle to examine the Ulam–Hyers stability [...] Read more.
The aim of this work is to analyze the relative controllability and Ulamn–Hyers stability of the ψ-Caputo fractional neutral delay differential system. We use neutral ψ-delayed perturbation of the Mitttag–Leffler matrix function and Banach contraction principle to examine the Ulam–Hyers stability of our considered system. We formulate the Grammian matrix to establish the controllability results of the linear fractonal differential system. Further, we employ the fixed-point technique of Krasnoselskii’s type to establish the sufficient conditions for the relative controllability of a semilinear ψ-Caputo neutral fractional system. Finally, the theoretical study is validated by providing an application. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)
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