Advance in Functional Equations

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

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 11211

Special Issue Editors

Institute of Mathematics, Lodz University of Technology, al. Politechniki 8, 93-590 Łódź, Poland
Interests: functional equations; functional inequalities; fuzzy logic; Hyers-Ulam stability; classical analysis
Special Issues, Collections and Topics in MDPI journals
Department of Natural Sciences, Rzeszów University, Rzeszów, Poland
Interests: functional equations; functional inequalities; utility theory; decision making under risk
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is our pleasure to invite you to contribute to the special issue of Symmetry dedicated to functional equations. Functional equations and inequalities find their applications in many areas of science, including approximate reasoning, physics, economy and behavioral sciences. Symmetry is a fundamental attribute of several functional equations, including the famous Cauchy functional equations. Free variables appearing in many equations often enjoy several symmetry properties and many techniques rely on exploring them. Therefore, functional equations and inequalities fit into the scope of the multidisciplinary journal Symmetry.

This issue is mainly devoted to research connected with functional equations and inequalities of one and several variables, their stability in a sense of Ulam and applications in all areas of science. In particular, we seek for submissions which bring new methods and new ideas to the field and which discover connections with various branches of science. Papers dealing with applications of functional equations or inequalities are also welcome.

Prof. Włodzimierz Fechner
Prof. Jacek Chudziak
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

  • functional equation
  • functional inequality
  • stability in a sense of Ulam
  • generalized Hyers–Ulam stability of functional equation
  • hyperstability and superstability
  • applications of functional equations
  • applications of functional inequalities
  • utility theory
  • risk maeasures and their properties
  • symmetry in functional equations and inequalities

Related Special Issue

Published Papers (8 papers)

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Research

27 pages, 746 KiB  
Article
A Scaled Dai–Yuan Projection-Based Conjugate Gradient Method for Solving Monotone Equations with Applications
by Ali Althobaiti, Jamilu Sabi’u, Homan Emadifar, Prem Junsawang and Soubhagya Kumar Sahoo
Symmetry 2022, 14(7), 1401; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071401 - 07 Jul 2022
Cited by 3 | Viewed by 1264
Abstract
In this paper, we propose two scaled Dai–Yuan (DY) directions for solving constrained monotone nonlinear systems. The proposed directions satisfy the sufficient descent condition independent of the line search strategy. We also reasonably proposed two different relations for computing the scaling parameter at [...] Read more.
In this paper, we propose two scaled Dai–Yuan (DY) directions for solving constrained monotone nonlinear systems. The proposed directions satisfy the sufficient descent condition independent of the line search strategy. We also reasonably proposed two different relations for computing the scaling parameter at every iteration. The first relation is proposed by approaching the quasi-Newton direction, and the second one is by taking the advantage of the popular Barzilai–Borwein strategy. Moreover, we propose a robust projection-based algorithm for solving constrained monotone nonlinear equations with applications in signal restoration problems and reconstructing the blurred images. The global convergence of this algorithm is also provided, using some mild assumptions. Finally, a comprehensive numerical comparison with the relevant algorithms shows that the proposed algorithm is efficient. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
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21 pages, 384 KiB  
Article
Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Type
by Marcelina Mocanu
Symmetry 2021, 13(11), 2072; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112072 - 02 Nov 2021
Cited by 3 | Viewed by 1165
Abstract
We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to [...] Read more.
We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some specific function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
10 pages, 309 KiB  
Article
Analysis on ψ-Hilfer Fractional Impulsive Differential Equations
by Kulandhaivel Karthikeyan, Panjaiyan Karthikeyan, Dimplekumar N. Chalishajar, Duraisamy Senthil Raja and Ponnusamy Sundararajan
Symmetry 2021, 13(10), 1895; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101895 - 08 Oct 2021
Cited by 10 | Viewed by 1188
Abstract
In this manuscript, we establish the existence of results of fractional impulsive differential equations involving ψ-Hilfer fractional derivative and almost sectorial operators using Schauder fixed-point theorem. We discuss two cases, if the associated semigroup is compact and noncompact, respectively. We consider here [...] Read more.
In this manuscript, we establish the existence of results of fractional impulsive differential equations involving ψ-Hilfer fractional derivative and almost sectorial operators using Schauder fixed-point theorem. We discuss two cases, if the associated semigroup is compact and noncompact, respectively. We consider here the higher-dimensional system of integral equations. We present herewith new theoretical results, structural investigations, and new models and approaches. Some special cases of the results are discussed as well. Due to the nature of measurement of noncompactness theory, there exists a strong relationship between the sectorial operator and symmetry. When working on either of the concepts, it can be applied to the other one as well. Finally, a case study is presented to demonstrate the major theory. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
13 pages, 288 KiB  
Article
Existence, Uniqueness, and Stability Analysis of the Probabilistic Functional Equation Emerging in Mathematical Biology and the Theory of Learning
by Ali Turab, Won-Gil Park and Wajahat Ali
Symmetry 2021, 13(8), 1313; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081313 - 21 Jul 2021
Cited by 5 | Viewed by 1261
Abstract
Probabilistic functional equations have been used to analyze various models in computational biology and learning theory. It is worth noting that they are linked to the symmetry of a system of functional equations’ transformation. Our objective is to propose a generic probabilistic functional [...] Read more.
Probabilistic functional equations have been used to analyze various models in computational biology and learning theory. It is worth noting that they are linked to the symmetry of a system of functional equations’ transformation. Our objective is to propose a generic probabilistic functional equation that can cover most of the mathematical models addressed in the existing literature. The notable fixed-point tools are utilized to examine the existence, uniqueness, and stability of the suggested equation’s solution. Two examples are also given to emphasize the significance of our findings. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
12 pages, 275 KiB  
Article
Stability of Bi-Additive Mappings and Bi-Jensen Mappings
by Jae-Hyeong Bae and Won-Gil Park
Symmetry 2021, 13(7), 1180; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071180 - 30 Jun 2021
Cited by 2 | Viewed by 963
Abstract
Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation [...] Read more.
Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation f(x+y,z+w)=f(x,z)+f(y,w) and the bi-Jensen functional equation 4fx+y2,z+w2=f(x,z)+f(x,w)+f(y,z)+f(y,w). Full article
(This article belongs to the Special Issue Advance in Functional Equations)
12 pages, 277 KiB  
Article
New Results for Oscillation of Solutions of Odd-Order Neutral Differential Equations
by Clemente Cesarano, Osama Moaaz, Belgees Qaraad, Nawal A. Alshehri, Sayed K. Elagan and Mohammed Zakarya
Symmetry 2021, 13(6), 1095; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061095 - 21 Jun 2021
Cited by 4 | Viewed by 1339
Abstract
Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system’s past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a [...] Read more.
Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system’s past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufficient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
11 pages, 277 KiB  
Article
Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses
by Snezhana Hristova and Mohamed I. Abbas
Symmetry 2021, 13(6), 996; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13060996 - 02 Jun 2021
Cited by 19 | Viewed by 1680
Abstract
The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the [...] Read more.
The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
11 pages, 279 KiB  
Article
The Stability of Isometries on Restricted Domains
by Ginkyu Choi and Soon-Mo Jung
Symmetry 2021, 13(2), 282; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020282 - 07 Feb 2021
Cited by 2 | Viewed by 1130
Abstract
We will prove the generalized Hyers–Ulam stability of isometries, with a focus on the stability for restricted domains. More precisely, we prove the generalized Hyers–Ulam stability of the orthogonality equation and we use this result to prove the stability of the equations [...] Read more.
We will prove the generalized Hyers–Ulam stability of isometries, with a focus on the stability for restricted domains. More precisely, we prove the generalized Hyers–Ulam stability of the orthogonality equation and we use this result to prove the stability of the equations f(x)f(y)=xy and f(x)f(y)2=xy2 on the restricted domains. As we can easily see, these functional equations are symmetric in the sense that they become the same equations even if the roles of variables x and y are exchanged. Full article
(This article belongs to the Special Issue Advance in Functional Equations)
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