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Fractal Fract
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New Trends on Fixed Point Theory
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New Trends on 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 (15 November 2022) | Viewed by 26051

Special Issue Editors

Dr. Zoran D. Mitrovic

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Guest Editor
Department of Theoretical Mathematics, Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina
Interests: nonlinear analysis; fixed point theory; variational inequalities; best approximations; equilibrium theory
Dr. Reny Kunnel Chacko George

E-Mail Website
Guest Editor
Department of Mathematics, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia
Interests: nonlinear analysis; fixed point theory and applications; operator theory; generalized metric topologies
Dr. Liliana Guran

E-Mail Website
Guest Editor
Department of Hospitality Services, Faculty of Business, Babes-Bolyai University, 400174 Cluj-Napoca, 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,

The fixed point theory occupies an important place in the nonlinear functional analysis, numerical mathematics, economics as well as in applied sciences.

This Special Issue is dedicated to the modern theory of a fixed point from two aspects: metric and topological.

We plan to report new results on fixed points and their applications on different class equations and inclusions, such as the generalized classes of metric spaces obtained in the last few years such as modular metric, b-metric and F-metric spaces, bipolar metric spaces, fuzzy metric spaces, etc. In these classes of space, a number of open problems have been realized in recent years, and new possibilities for the development of the metric theory of a fixed point have been provided.

Additionally, we expect applications in fractional differential equations, fractional differential inclusions, fractional boundary value problems and fractional operators.

In this Special Issue, we hope to publish papers showing aspects of the application of topological methods in the fixed point theory, focusing primarily on Schauder-type theorems for single-valued mappings.

The final goal is to publish novel results concerning multivalued mappings (Kakutani-type theorems). This primarily refers to applications in mathematical economics (the existence of a maximal element and the equilibrium problem).

Dr. Zoran D. Mitrovic
Dr. Reny Kunnel Chacko George
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

  • existence and uniqueness of fixed points
  • fractional differential inclusions
  • fractional differential equations
  • fractional boundary value problems
  • fractional operators with applications
  • generalized metric space
  • iteration processes for fixed points or best proximity points
  • best approximation problems
  • variational inequalities
  • equilibrium problems
  • KKM mapping

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Published Papers (13 papers)

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Research

20 pages, 671 KiB  
Article
On Novel Mathematical Modeling for Studying a Class of Nonlinear Caputo-Type Fractional-Order Boundary Value Problems Emerging in CGT
by Ali Turab, Wutiphol Sintunavarat and Jong-Suk Ro
Fractal Fract. 2023, 7(2), 99; https://doi.org/10.3390/fractalfract7020099 - 17 Jan 2023
Viewed by 1762
Abstract
Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. [...] Read more.
Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C4H10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
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28 pages, 385 KiB  
Article
Fixed Point for Almost Contractions in v-Generalized b-Metric Spaces
by Anshuka Kadyan, Savita Rathee, Anil Kumar, Asha Rani and Kenan Tas
Fractal Fract. 2023, 7(1), 60; https://doi.org/10.3390/fractalfract7010060 - 4 Jan 2023
Cited by 1 | Viewed by 2131
Abstract
We present a significant example to show that the class of v-generalized b-metric spaces properly contains the class of v-generalized metric spaces as well as b-metric spaces. This is accomplished because the example provided by Došenović et al. (2020) [...] Read more.
We present a significant example to show that the class of v-generalized b-metric spaces properly contains the class of v-generalized metric spaces as well as b-metric spaces. This is accomplished because the example provided by Došenović et al. (2020) is insufficient to expose the generality of v-generalized b-metric spaces over the existing related spaces. Therefore, we establish fixed point theorems by defining generalized almost contractions of rational type and Reich type in v-generalized b-metric spaces. Moreover, we compare the proven results with the already existing fixed point theorems in this space by presenting suitable examples. As a consequence of these fixed point theorems, we further develop some common fixed-point results that ensure the existence and uniqueness of coincidence points and common fixed points for a pair of self maps. Finally, we use the outcome to check that the given Fredholm integral equation has a solution and that it is also unique. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
15 pages, 383 KiB  
Article
On the Solvability of Mixed-Type Fractional-Order Non-Linear Functional Integral Equations in the Banach Space C(I)
by Vijai Kumar Pathak, Lakshmi Narayan Mishra, Vishnu Narayan Mishra and Dumitru Baleanu
Fractal Fract. 2022, 6(12), 744; https://doi.org/10.3390/fractalfract6120744 - 16 Dec 2022
Cited by 17 | Viewed by 1546
Abstract
This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional (κ,ϕ)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space C([1,T]) arising [...] Read more.
This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional (κ,ϕ)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space C([1,T]) arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo’s fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
13 pages, 292 KiB  
Article
New Results for Weakly Compatible (WC) and R-Weakly Commuting (RWC) Mappings with an Applicationin Dynamic Programming
by Penumarthy Parvateesam Murthy, Sanjay Kumar, Rajesh Kumar, Pusplata Sahu, Zoran D. Mitrović and Reny George
Fractal Fract. 2022, 6(12), 733; https://doi.org/10.3390/fractalfract6120733 - 10 Dec 2022
Viewed by 1353
Abstract
The aim of this paper is to obtain some new results about common fixed points. Our results use weaker conditions than those previously used. We have relaxed the conditions for commutating pair mappings and compatible mappings of the type (A), [...] Read more.
The aim of this paper is to obtain some new results about common fixed points. Our results use weaker conditions than those previously used. We have relaxed the conditions for commutating pair mappings and compatible mappings of the type (A), which were introduced in 1976. The theorems are enriched by using the concept of WC and various types of weakly commuting pairs of maps in metric spaces. To discuss the existence and uniqueness of the common solutions, we have obtained an application to the functional equations in dynamic programming. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
12 pages, 315 KiB  
Article
Common Fixed Point for Meir–Keeler Type Contraction in Bipolar Metric Space
by Penumarthy Parvateesam Murthy, Chandra Prakash Dhuri, Santosh Kumar, Rajagopalan Ramaswamy, Muhannad Abdullah Saud Alaskar and Stojan Radenovi’c
Fractal Fract. 2022, 6(11), 649; https://doi.org/10.3390/fractalfract6110649 - 4 Nov 2022
Cited by 7 | Viewed by 1374
Abstract
In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using [...] Read more.
In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using the extension of Meir–Keeler contraction in the setting of bipolar metric spaces have been proved. The derived results have been supplemented with non-trivial examples. Our results extend and generalise the results established in the past. We have provided an application to find an analytical solution to an Integral Equation to supplement the derived result. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
48 pages, 421 KiB  
Article
Analysis of a Hybrid Coupled System of ψ-Caputo Fractional Derivatives with Generalized Slit-Strips-Type Integral Boundary Conditions and Impulses
by Zhiwei Lv, Ishfaq Ahmad, Jiafa Xu and Akbar Zada
Fractal Fract. 2022, 6(10), 618; https://doi.org/10.3390/fractalfract6100618 - 21 Oct 2022
Cited by 7 | Viewed by 1748
Abstract
In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving ψ-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered [...] Read more.
In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving ψ-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered system. For the existence and uniqueness of the solution, we use the Banach contraction principle. With the help of Schaefer’s fixed-point theorem and some assumptions, we also obtain at least one solution of the mentioned system. Finally, the main results are verified with an appropriate example. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
22 pages, 385 KiB  
Article
Discussion on the Approximate Controllability of Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators
by Chandrabose Sindhu Varun Bose, Ramalingam Udhayakumar, Ahmed M. Elshenhab, Marappan Sathish Kumar and Jong-Suk Ro
Fractal Fract. 2022, 6(10), 607; https://doi.org/10.3390/fractalfract6100607 - 18 Oct 2022
Cited by 25 | Viewed by 1731
Abstract
This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of [...] Read more.
This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
8 pages, 299 KiB  
Article
Positive Solutions for Perturbed Fractional p-Laplacian Problems
by Mengfei Tao and Binlin Zhang
Fractal Fract. 2022, 6(10), 571; https://doi.org/10.3390/fractalfract6100571 - 8 Oct 2022
Cited by 2 | Viewed by 1498
Abstract
In this article, we consider a class of quasilinear elliptic equations involving the fractional p-Laplacian, in which the nonlinear term satisfies subcritical or critical growth. Based on a fixed point result due to Carl and Heikkilä, we can well overcome the lack [...] Read more.
In this article, we consider a class of quasilinear elliptic equations involving the fractional p-Laplacian, in which the nonlinear term satisfies subcritical or critical growth. Based on a fixed point result due to Carl and Heikkilä, we can well overcome the lack of compactness which has been a key difficulty for elliptic equations with critical growth. Moreover, we establish the existence and boundedness of the weak solutions for the above equations. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
19 pages, 345 KiB  
Article
Best Proximity Point Theorems for the Generalized Fuzzy Interpolative Proximal Contractions
by Khalil Javed, Maha M. A. Lashin, Muhammad Nazam, Hamed H. Al Sulami, Aftab Hussain and Muhammad Arshad
Fractal Fract. 2022, 6(8), 455; https://doi.org/10.3390/fractalfract6080455 - 21 Aug 2022
Viewed by 1676
Abstract
The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy [...] Read more.
The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy complete metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing Ψ,Φ-fuzzy proximal contractions and Ψ,Φ-fuzzy proximal interpolative contractions. The obtained results improve and generalize many best proximity point theorems published earlier. Moreover, we provide many nontrivial examples to validate our best proximity point theorem. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
13 pages, 1284 KiB  
Article
Stability Analysis and Computational Interpretation of an Effective Semi Analytical Scheme for Fractional Order Non-Linear Partial Differential Equations
by Javed Iqbal, Khurram Shabbir and Liliana Guran
Fractal Fract. 2022, 6(7), 393; https://doi.org/10.3390/fractalfract6070393 - 16 Jul 2022
Cited by 7 | Viewed by 2087
Abstract
In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional [...] Read more.
In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional order partial differential equations. Since the Laplace transform cannot be applied to non-linear problems, the combination of the variational iteration technique with it will give a better and rapidly convergent sequence. Exact solutions may also exist, but we will show that the coupled technique is much better to approximate the exact solutions. The Caputo–Fabrizio fractional derivative will be used throughout the study. In addition, some possible implications of the results given here are connected with fixed point theory. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
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13 pages, 839 KiB  
Article
On the Global Well-Posedness of Rotating Magnetohydrodynamics Equations with Fractional Dissipation
by Muhammad Zainul Abidin, Muhammad Marwan, Humaira Kalsoom and Omer Abdalrhman Omer
Fractal Fract. 2022, 6(6), 340; https://doi.org/10.3390/fractalfract6060340 - 17 Jun 2022
Cited by 1 | Viewed by 1814
Abstract
This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation (−Δ)℘ for 12<℘≤1. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional [...] Read more.
This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation (−Δ)℘ for 12<℘≤1. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional rotating magnetohydrodynamics equations in a more generalized Besov spaces characterized by the time evolution semigroup related to the generalized linear Stokes–Coriolis operator. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
22 pages, 415 KiB  
Article
Neutrosophic Double Controlled Metric Spaces and Related Results with Application
by Fahim Uddin, Umar Ishtiaq, Aftab Hussain, Khalil Javed, Hamed Al Sulami and Khalil Ahmed
Fractal Fract. 2022, 6(6), 318; https://doi.org/10.3390/fractalfract6060318 - 6 Jun 2022
Cited by 9 | Viewed by 2064
Abstract
In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction [...] Read more.
In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
13 pages, 342 KiB  
Article
Fixed Point Results for Generalized F-Contractions in b-Metric-like Spaces
by Huaping Huang, Kastriot Zoto, Zoran D. Mitrović and Stojan Radenović
Fractal Fract. 2022, 6(5), 272; https://doi.org/10.3390/fractalfract6050272 - 17 May 2022
Cited by 11 | Viewed by 2751
Abstract
The purpose of this paper is to introduce several generalized F-contractions in b-metric-like spaces and establish some fixed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, [...] Read more.
The purpose of this paper is to introduce several generalized F-contractions in b-metric-like spaces and establish some fixed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, we find the existence and uniqueness of a solution to a class of integral equations in the context of b-metric-like spaces. Full article
(This article belongs to the Special Issue New Trends on Fixed Point Theory)
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