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Symmetry
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Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory
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Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 14418

Special Issue Editors

Dr. Garik Petrosyan

E-Mail Website
Guest Editor
Voronezh State Pedagogical University 86 Lenin St., 394043 Voronezh, Russia
Interests: differential equation; differential inclusion; controllability problem; fixed point; condensing multimap; measure of non-compactness; topological degree; fractional derivative
Dr. Alexander Zaslavski

E-Mail Website
Guest Editor
Department of Mathematics, The Technion—Israel Institute of Technology, Haifa 32000, Israel
Interests: variational and optimal control problems on unbounded domains; optimization theory and related topics; infinite products of operators and their applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The investigation of systems with symmetry and control systems is a complicated and very important part of contemporary mathematical control theory and harmonic analysis, which has numerous applications and attracts the attention of a number of researchers around the world. In turn, the development of the theory of differential inclusions is associated with the fact that they provide a convenient and natural tool for describing control systems of various classes, systems with discontinuous characteristics, which are studied in various branches of the optimal control theory, mathematical physics, radio physics, acoustics, etc. However, solving these problems within the frameworks of existing theories is often a very difficult problem, since many of them find sufficiently adequate descriptions in terms of differential equations and inclusions with fractional derivatives. The theory of differential equations of fractional order originates from the ideas of Leibniz and Euler, but only by the end of the XX century did interest in this topic increase significantly. The subject of fractional calculus has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous diverse and widespread fields of science and engineering.

The aim of this Special Issue is to show recent advances in the theory of systems with symmetry, control systems, fractional calculus and fixed point applications to scientific problems.

Dr. Garik Petrosyan
Prof. Alexander Zaslavski
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. Symmetry 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 2400 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

  • symmetry
  • control system
  • fixed point
  • fractional derivative
  • fractional differential equations
  • fractional differential inclusions
  • applications of fractional calculus

Published Papers (10 papers)

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Research

15 pages, 300 KiB  
Article
Solution of Fredholm Integral Equation via Common Fixed Point Theorem on Bicomplex Valued B-Metric Space
by Gunaseelan Mani, Arul Joseph Gnanaprakasam, Ozgur Ege, Nahid Fatima and Nabil Mlaiki
Symmetry 2023, 15(2), 297; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15020297 - 21 Jan 2023
Cited by 3 | Viewed by 1090
Abstract
The notion of symmetry is the main property of a metric function. The area of fixed point theory has a suitable structure for symmetry in mathematics. The goal of this paper is to find fixed point and common fixed point results in a [...] Read more.
The notion of symmetry is the main property of a metric function. The area of fixed point theory has a suitable structure for symmetry in mathematics. The goal of this paper is to find fixed point and common fixed point results in a bicomplex valued b-metric space for mixed type rational contractions with control functions. Some well-known literature findings were generalized in our main findings. We provide an example to strengthen and validate our main results. As an example, in the context of bicomplex-valued b-metric space, we develop fixed point and common fixed point results for the rational contraction mapping. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
16 pages, 300 KiB  
Article
New Contractive Mappings and Solutions to Boundary-Value Problems in Triple Controlled Metric Type Spaces
by Fatima M. Azmi
Symmetry 2022, 14(11), 2270; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14112270 - 29 Oct 2022
Cited by 4 | Viewed by 1018
Abstract
In this study, we utilize the notion of triple controlled metric type space that preserves the symmetry property, which is a generalization of b-metric-type spaces, to prove new fixed-point results. We introduce (α-F)-contractive mappings and Θ-contractive mappings [...] Read more.
In this study, we utilize the notion of triple controlled metric type space that preserves the symmetry property, which is a generalization of b-metric-type spaces, to prove new fixed-point results. We introduce (α-F)-contractive mappings and Θ-contractive mappings on triple controlled metric type space settings. Then, we establish the existence and uniqueness of fixed-point results on complete triple controlled metric type space. Moreover, some examples and applications to boundary-value problems of the fourth-order differential equation are presented to display the usage of the obtained result. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
19 pages, 333 KiB  
Article
Approximating Common Solution of Minimization Problems Involving Asymptotically Quasi-Nonexpansive Multivalued Mappings
by Min Wang, Umar Ishtiaq, Naeem Saleem and Imo Kalu Agwu
Symmetry 2022, 14(10), 2062; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14102062 - 03 Oct 2022
Cited by 4 | Viewed by 992
Abstract
In this paper, an iterative scheme for finding common solutions of the set of fixed points for a pair of asymptotically quasi-nonexpansive mapping and the set of minimizers for the minimization problem is constructed. Using the idea of the jointly demicloseness principle, strong [...] Read more.
In this paper, an iterative scheme for finding common solutions of the set of fixed points for a pair of asymptotically quasi-nonexpansive mapping and the set of minimizers for the minimization problem is constructed. Using the idea of the jointly demicloseness principle, strong convergence results are achieved without imposing any compactness condition on the space or the operator. Our results improve, extend and generalize many important results in the literature. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
7 pages, 203 KiB  
Article
An Algorithm Based on Unions of Nonexpansive Mappings in Metric Spaces
by Alexander J. Zaslavski
Symmetry 2022, 14(9), 1852; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091852 - 06 Sep 2022
Cited by 3 | Viewed by 857
Abstract
In this paper, we obtain two extensions of a recent result of Tam, which was proved for iterates of set-valued paracontracting operators in a finite-dimensional space. Our results are obtained for operators in an arbitrary metric space. In the first result, we study [...] Read more.
In this paper, we obtain two extensions of a recent result of Tam, which was proved for iterates of set-valued paracontracting operators in a finite-dimensional space. Our results are obtained for operators in an arbitrary metric space. In the first result, we study exact iterates of the set-valued mapping, while in the second one, we deal with its inexact iterates, taking computational errors into account. In a particular case of a Banach space, if all the operators are symmetric, then the set of all limit points of iterates is symmetric too. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
19 pages, 1079 KiB  
Article
Fractional-View Analysis of Fokker-Planck Equations by ZZ Transform with Mittag-Leffler Kernel
by Azzh Saad Alshehry, Muhammad Imran, Rasool Shah and Wajaree Weera
Symmetry 2022, 14(8), 1513; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081513 - 24 Jul 2022
Cited by 8 | Viewed by 1259
Abstract
This work combines a ZZ transformation with the Adomian decomposition method to solve the fractional-order Fokker-Planck equations. The fractional derivative is represented in the Atangana-Baleanu derivative. It is looked at with graphs that show that the accurate and estimated results are close to [...] Read more.
This work combines a ZZ transformation with the Adomian decomposition method to solve the fractional-order Fokker-Planck equations. The fractional derivative is represented in the Atangana-Baleanu derivative. It is looked at with graphs that show that the accurate and estimated results are close to each other, indicating that the method works. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series type result that quickly converges to actual answers. The acquired outcomes suggest that the novel analytical solution method is simple to use and very successful at assessing complicated equations that occur in related research and engineering fields. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
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17 pages, 326 KiB  
Article
A New Extension to the Intuitionistic Fuzzy Metric-like Spaces
by Fahim Uddin, Umar Ishtiaq, Khalil Javed, Suhad Subhi Aiadi, Muhammad Arshad, Nizar Souayah and Nabil Mlaiki
Symmetry 2022, 14(7), 1400; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071400 - 07 Jul 2022
Cited by 5 | Viewed by 1148
Abstract
In this manuscript, we introduce the concept of intuitionistic fuzzy controlled metric-like spaces via continuous t-norms and continuous t-conorms. This new metric space is an extension to intuitionistic fuzzy controlled metric-like spaces, controlled metric-like spaces and controlled fuzzy metric spaces, and intuitionistic fuzzy [...] Read more.
In this manuscript, we introduce the concept of intuitionistic fuzzy controlled metric-like spaces via continuous t-norms and continuous t-conorms. This new metric space is an extension to intuitionistic fuzzy controlled metric-like spaces, controlled metric-like spaces and controlled fuzzy metric spaces, and intuitionistic fuzzy metric spaces. We prove some fixed-point theorems and we present non-trivial examples to illustrate our results. We used different techniques based on the properties of the considered spaces notably the symmetry of the metric. Moreover, we present an application to non-linear fractional differential equations. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
16 pages, 1098 KiB  
Article
Quasi-Synchronization and Quasi-Uniform Synchronization of Caputo Fractional Variable-Parameter Neural Networks with Probabilistic Time-Varying Delays
by Renyu Ye, Chen Wang, Axiu Shu and Hai Zhang
Symmetry 2022, 14(5), 1035; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14051035 - 18 May 2022
Cited by 5 | Viewed by 1293
Abstract
Owing to the symmetry between drive–response systems, the discussions of synchronization performance are greatly significant while exploring the dynamics of neural network systems. This paper investigates the quasi-synchronization (QS) and quasi-uniform synchronization (QUS) issues between the drive–response systems on fractional-order variable-parameter neural networks [...] Read more.
Owing to the symmetry between drive–response systems, the discussions of synchronization performance are greatly significant while exploring the dynamics of neural network systems. This paper investigates the quasi-synchronization (QS) and quasi-uniform synchronization (QUS) issues between the drive–response systems on fractional-order variable-parameter neural networks (VPNNs) including probabilistic time-varying delays. The effects of system parameters, probability distributions and the order on QS and QUS are considered. By applying the Lyapunov–Krasovskii functional approach, Hölder’s inequality and Jensen’s inequality, the synchronization criteria of fractional-order VPNNs under controller designs with constant gain coefficients and time-varying gain coefficients are derived. The obtained criteria are related to the probability distributions and the order of the Caputo derivative, which can greatly avoid the situation in which the upper bound of an interval with time delay is too large yet the probability of occurrence is very small, and information such as the size of time delay and probability of occurrence is fully considered. Finally, two examples are presented to further confirm the effectiveness of the algebraic criteria under different probability distributions. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
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13 pages, 287 KiB  
Article
Existence of Attractive Solutions for Hilfer Fractional Evolution Equations with Almost Sectorial Operators
by Mian Zhou, Bashir Ahmad and Yong Zhou
Symmetry 2022, 14(2), 392; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020392 - 16 Feb 2022
Cited by 3 | Viewed by 1604
Abstract
The purpose of this paper is to investigate the existence of attractive solutions for a Cauchy problem of fractional evolution equations with Hilfer fractional derivative, which is a generalization of both the Riemann–Liuoville and Caputo fractional derivatives. Our methods are based on the [...] Read more.
The purpose of this paper is to investigate the existence of attractive solutions for a Cauchy problem of fractional evolution equations with Hilfer fractional derivative, which is a generalization of both the Riemann–Liuoville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzela theorem, Schauder’s fixed point theorem, the Wright function and Kuratowski’s measure of noncompactness. The symmetric structure of the spaces and the operators defined by us plays a crucial role in showing the existence of fixed points. We obtain the global existence and attractivity results of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
18 pages, 388 KiB  
Article
The Optimal Control Problems for Generalized Elliptic Quasivariational Inequalities
by Shih-Sen Chang, Abdullah Ali H. Ahmadini, Salahuddin, Min Liu and Jinfang Tang
Symmetry 2022, 14(2), 199; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020199 - 20 Jan 2022
Cited by 3 | Viewed by 1157
Abstract
In this article, we propose an optimal control problem for generalized elliptic quasi-variational inequality with unilateral constraints. Then, we discuss the sufficient assumptions that ensure the convergence of the solutions to the optimal control problem. The proofs depend on convergence results for generalized [...] Read more.
In this article, we propose an optimal control problem for generalized elliptic quasi-variational inequality with unilateral constraints. Then, we discuss the sufficient assumptions that ensure the convergence of the solutions to the optimal control problem. The proofs depend on convergence results for generalized elliptic quasi-variational inequalities, obtained by the arguments of compactness, lower semi-continuity, monotonicity, penalty and different estimates. As an application, we addressed the abstract convergence results in the analysis of optimal control associated with boundary value problems. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
15 pages, 316 KiB  
Article
A New Class of Coupled Systems of Nonlinear Hyperbolic Partial Fractional Differential Equations in Generalized Banach Spaces Involving the ψ–Caputo Fractional Derivative
by Zidane Baitiche, Choukri Derbazi, Mouffak Benchohra and Yong Zhou
Symmetry 2021, 13(12), 2412; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122412 - 13 Dec 2021
Cited by 7 | Viewed by 2048
Abstract
The current study is devoted to investigating the existence and uniqueness of solutions for a new class of symmetrically coupled system of nonlinear hyperbolic partial-fractional differential equations in generalized Banach spaces in the sense of ψ–Caputo partial fractional derivative. Our approach is [...] Read more.
The current study is devoted to investigating the existence and uniqueness of solutions for a new class of symmetrically coupled system of nonlinear hyperbolic partial-fractional differential equations in generalized Banach spaces in the sense of ψ–Caputo partial fractional derivative. Our approach is based on the Krasnoselskii-type fixed point theorem in generalized Banach spaces and Perov’s fixed point theorem together with the Bielecki norm, while Urs’s approach was used to prove the Ulam–Hyers stability of solutions of our system. Finally, some examples are provided in order to illustrate our theoretical results. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)
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