Symmetry

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27 pages, 746 KiB

Open AccessArticle

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://doi.org/10.3390/sym14071401 - 7 Jul 2022

Cited by 5 | Viewed by 1771

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

Open AccessArticle

Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Type

by Marcelina Mocanu

Symmetry 2021, 13(11), 2072; https://doi.org/10.3390/sym13112072 - 2 Nov 2021

Cited by 3 | Viewed by 1632

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

Open AccessArticle

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://doi.org/10.3390/sym13101895 - 8 Oct 2021

Cited by 12 | Viewed by 1582

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

Open AccessArticle

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://doi.org/10.3390/sym13081313 - 21 Jul 2021

Cited by 5 | Viewed by 1652

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

Open AccessArticle

Stability of Bi-Additive Mappings and Bi-Jensen Mappings

by Jae-Hyeong Bae and Won-Gil Park

Symmetry 2021, 13(7), 1180; https://doi.org/10.3390/sym13071180 - 30 Jun 2021

Cited by 2 | Viewed by 1197

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

4 f (\frac{x + y}{2}, \frac{z + w}{2}) = 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

Open AccessArticle

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://doi.org/10.3390/sym13061095 - 21 Jun 2021

Cited by 6 | Viewed by 1779

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

Open AccessArticle

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://doi.org/10.3390/sym13060996 - 2 Jun 2021

Cited by 21 | Viewed by 2093

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

Open AccessArticle

The Stability of Isometries on Restricted Domains

by Ginkyu Choi and Soon-Mo Jung

Symmetry 2021, 13(2), 282; https://doi.org/10.3390/sym13020282 - 7 Feb 2021

Cited by 2 | Viewed by 1379

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) ∥ = ∥ x - y ∥

and

{∥ f (x) - f (y) ∥}^{2} = {∥ x - y ∥}^{2}

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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