Symmetry

Research

21 pages, 301 KiB

Open AccessArticle

Degree of L_p Approximation Using Activated Singular Integrals

by George A. Anastassiou

Symmetry 2024, 16(8), 1022; https://doi.org/10.3390/sym16081022 (registering DOI) - 10 Aug 2024

In this article we present the

L_{p}

,

p \geq 1

, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions [...] Read more.

In this article we present the

L_{p}

,

p \geq 1

, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order

L_{p}

modulus of smoothness. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

13 pages, 293 KiB

Open AccessArticle

The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents

by Mostafa Bachar

Symmetry 2024, 16(8), 978; https://doi.org/10.3390/sym16080978 - 1 Aug 2024

Viewed by 448

Abstract

This study looks closely into the analysis of the variation of constants formula given by [...] Read more.

This study looks closely into the analysis of the variation of constants formula given by

Φ (t) = S (t) Φ (0) + \int_{0}^{t} S (t - σ) F (σ, Φ (σ)) d σ,

for

t \in [0, T], T > 0

, within the context of modular function spaces

L_{ρ}

. Additionally, this research explores practical applications of the variation of constants formula in variable exponent Lebesgue spaces

L^{p (\cdot)}

. Specifically, the study examines these spaces under certain conditions applied to the exponent function

p (\cdot)

and the functions

F

as well as the semigroup

S (t)

, utilizing the symmetry properties of the algebraic semigroup. This investigation sheds light on the intricate interplay between parameters and functions within these mathematical frameworks, offering valuable insights into their behavior and properties in

L^{p (\cdot)}

. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

35 pages, 458 KiB

Open AccessArticle

Stability Analysis of Some Types of Singularly Perturbed Time-Delay Differential Systems: Symmetric Matrix Riccati Equation Approach

by Valery Y. Glizer

Symmetry 2024, 16(7), 838; https://doi.org/10.3390/sym16070838 - 3 Jul 2024

Viewed by 695

Abstract

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are [...] Read more.

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

9 pages, 262 KiB

Open AccessArticle

Positive Radial Symmetric Solutions of Nonlinear Biharmonic Equations in an Annulus

by Yongxiang Li and Shengbin Yang

Symmetry 2024, 16(7), 793; https://doi.org/10.3390/sym16070793 - 24 Jun 2024

Viewed by 444

Abstract

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation

▵^{2} u = f (u, ▵ u)

on an annular domain

Ω

in

R^{N}

with the Navier boundary conditions [...] Read more.

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation

▵^{2} u = f (u, ▵ u)

on an annular domain

Ω

in

R^{N}

with the Navier boundary conditions

{u |}_{\partial Ω} = 0

and

{▵ u |}_{\partial Ω} = 0

, where

f : R^{+} \times R^{-} \to R^{+}

is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow

f (ξ, η)

to have superlinear or sublinear growth on

ξ, η

as

| (ξ, η) | \to 0

and ∞. Our discussion is mainly based on the fixed-point index theory in cones. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

16 pages, 283 KiB

Open AccessArticle

Coupled Fixed Point Theory in Subordinate Semimetric Spaces

by Areej Alharbi, Maha Noorwali and Hamed H. Alsulami

Symmetry 2024, 16(4), 499; https://doi.org/10.3390/sym16040499 - 19 Apr 2024

Viewed by 616

Abstract

The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of [...] Read more.

The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

10 pages, 299 KiB

Open AccessArticle

Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents

by Mostafa Bachar, Mohamed A. Khamsi and Osvaldo Méndez

Symmetry 2023, 15(11), 2014; https://doi.org/10.3390/sym15112014 - 2 Nov 2023

Viewed by 726

Abstract

We investigate the existence of solutions for the Fredholm integral equation

Φ (ϑ) = G (ϑ, Φ (ϑ)) + \int_{0}^{1} F (ϑ, ζ, Φ (ζ)) d ζ,

[...] Read more.

We investigate the existence of solutions for the Fredholm integral equation

Φ (ϑ) = G (ϑ, Φ (ϑ)) + \int_{0}^{1} F (ϑ, ζ, Φ (ζ)) d ζ,

for

ϑ \in [0, 1]

, in the setting of the modular function spaces

L_{ρ}

. We also derive an application of this research within the framework of variable exponent Lebesgue spaces

L^{p (\cdot)}

subject to specific conditions imposed on the exponent function

p (\cdot)

and the functions

F

and

G

. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

7 pages, 231 KiB

Open AccessArticle

Three Convergence Results for Inexact Iterates of Uniformly Locally Nonexpansive Mappings

by Simeon Reich and Alexander J. Zaslavski

Symmetry 2023, 15(5), 1084; https://doi.org/10.3390/sym15051084 - 15 May 2023

Cited by 2 | Viewed by 926

Abstract

In 2006, together with D. Butnariu, we showed that if all iterates of a nonexpansive self-mapping of a complete metric space converge, then all its inexact iterates with summable computational errors converge too. In a recent paper of ours, we have extended this [...] Read more.

In 2006, together with D. Butnariu, we showed that if all iterates of a nonexpansive self-mapping of a complete metric space converge, then all its inexact iterates with summable computational errors converge too. In a recent paper of ours, we have extended this result to uniformly locally nonexpansive self-mappings of a complete metric space. In the present paper, we establish analogous results for uniformly locally nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space. In the particular case of a Banach space, if the operator is symmetric, then the set of all limit points of its iterates is also symmetric. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

16 pages, 319 KiB

Open AccessFeature PaperArticle

Existence and Nonexistence of Positive Solutions for Perturbations of the Anisotropic Eigenvalue Problem

by Olena Andrusenko, Leszek Gasiński and Nikolaos S. Papageorgiou

Symmetry 2023, 15(2), 495; https://doi.org/10.3390/sym15020495 - 13 Feb 2023

Viewed by 948

Abstract

We consider a Dirichlet problem, which is a perturbation of the eigenvalue problem for the anisotropic p-Laplacian. We assume that the perturbation is

(p (z) - 1)

-sublinear, and we prove an existence and nonexistence theorem for positive [...] Read more.

We consider a Dirichlet problem, which is a perturbation of the eigenvalue problem for the anisotropic p-Laplacian. We assume that the perturbation is

(p (z) - 1)

-sublinear, and we prove an existence and nonexistence theorem for positive solutions as the parameter

λ

moves on the positive semiaxis. We also show the existence of a smallest positive solution and determine the monotonicity and continuity properties of the minimal solution map. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

26 pages, 646 KiB

Open AccessArticle

Improvement of Unconstrained Optimization Methods Based on Symmetry Involved in Neutrosophy

by Predrag S. Stanimirović, Branislav Ivanov, Dragiša Stanujkić, Vasilios N. Katsikis, Spyridon D. Mourtas, Lev A. Kazakovtsev and Seyyed Ahmad Edalatpanah

Symmetry 2023, 15(1), 250; https://doi.org/10.3390/sym15010250 - 16 Jan 2023

Cited by 4 | Viewed by 1746

Abstract

The influence of neutrosophy on many fields of science and technology, as well as its numerous applications, are evident. Our motivation is to apply neutrosophy for the first time in order to improve methods for solving unconstrained optimization. Particularly, in this research, we [...] Read more.

The influence of neutrosophy on many fields of science and technology, as well as its numerous applications, are evident. Our motivation is to apply neutrosophy for the first time in order to improve methods for solving unconstrained optimization. Particularly, in this research, we propose and investigate an improvement of line search methods for solving unconstrained nonlinear optimization models. The improvement is based on the application of symmetry involved in neutrosophic logic in determining appropriate step size for the class of descent direction methods. Theoretical analysis is performed to show the convergence of proposed iterations under the same conditions as for the related standard iterations. Mutual comparison and analysis of generated numerical results reveal better behavior of the suggested iterations compared with analogous available iterations considering the Dolan and Moré performance profiles and statistical ranking. Statistical comparison also reveals advantages of the neutrosophic improvements of the considered line search optimization methods. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

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18 pages, 569 KiB

Open AccessArticle

Descent Derivative-Free Method Involving Symmetric Rank-One Update for Solving Convex Constrained Nonlinear Monotone Equations and Application to Image Recovery

by Aliyu Muhammed Awwal, Adamu Ishaku, Abubakar Sani Halilu, Predrag S. Stanimirović, Nuttapol Pakkaranang and Bancha Panyanak

Symmetry 2022, 14(11), 2375; https://doi.org/10.3390/sym14112375 - 10 Nov 2022

Cited by 2 | Viewed by 1332

Abstract

Many practical applications in applied sciences such as imaging, signal processing, and motion control can be reformulated into a system of nonlinear equations with or without constraints. In this paper, a new descent projection iterative algorithm for solving a nonlinear system of equations [...] Read more.

Many practical applications in applied sciences such as imaging, signal processing, and motion control can be reformulated into a system of nonlinear equations with or without constraints. In this paper, a new descent projection iterative algorithm for solving a nonlinear system of equations with convex constraints is proposed. The new approach is based on a modified symmetric rank-one updating formula. The search direction of the proposed algorithm mimics the behavior of a spectral conjugate gradient algorithm where the spectral parameter is determined so that the direction is sufficiently descent. Based on the assumption that the underlying function satisfies monotonicity and Lipschitz continuity, the convergence result of the proposed algorithm is discussed. Subsequently, the efficiency of the new method is revealed. As an application, the proposed algorithm is successfully implemented on image deblurring problem. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry II)

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