Mathematics

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21 pages, 330 KiB

Open AccessArticle

On the Evolution Operators of a Class of Linear Time-Delay Systems

by Manuel De la Sen

Mathematics 2024, 12(22), 3575; https://doi.org/10.3390/math12223575 - 15 Nov 2024

Viewed by 237

This paper studies the properties of the evolution operators of a class of time-delay systems with linear delayed dynamics. The considered delayed dynamics may, in general, be time-varying and associated with a finite set of finite constant point delays. Three evolution operators are [...] Read more.

This paper studies the properties of the evolution operators of a class of time-delay systems with linear delayed dynamics. The considered delayed dynamics may, in general, be time-varying and associated with a finite set of finite constant point delays. Three evolution operators are defined and characterized. The basic evolution operator is the so-called point delay operator, which generates the solution trajectory under point initial conditions at

t_{0} = 0

. Furthermore, this paper also considers the whole evolution operator and the delay strip evolution operator, which define the solution trajectory, respectively, at any time instant and along a strip of time whose size is that of the maximum delay. These operators are defined for any given bounded piecewise continuous function of initial conditions on an initialization time interval of measure being identical to the maximum delay. It is seen that the semigroup property of the time-invariant undelayed dynamics, which is generated by a

C_{0}

-semigroup, becomes lost by the above evolution operators in the presence of the delayed dynamics. This fact means that the point evolution operator is not a strongly and uniformly continuous one-parameter semigroup, even if its undelayed part has a time-invariant associated dynamics. The boundedness and the stability properties of the time-delay system, as well as the strong and uniform continuity properties of the evolution operators, are also discussed. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

17 pages, 1951 KiB

Open AccessArticle

Double Tseng’s Algorithm with Inertial Terms for Inclusion Problems and Applications in Image Deblurring

by Purit Thammasiri, Vasile Berinde, Narin Petrot and Kasamsuk Ungchittrakool

Mathematics 2024, 12(19), 3138; https://doi.org/10.3390/math12193138 - 7 Oct 2024

Viewed by 779

Abstract

In this research paper, we present a novel theoretical technique, referred to as the double Tseng’s algorithm with inertial terms, for finding a common solution to two monotone inclusion problems. Developing the double Tseng’s algorithm in this manner not only comprehensively expands theoretical [...] Read more.

In this research paper, we present a novel theoretical technique, referred to as the double Tseng’s algorithm with inertial terms, for finding a common solution to two monotone inclusion problems. Developing the double Tseng’s algorithm in this manner not only comprehensively expands theoretical knowledge in this field but also provides advantages in terms of step-size parameters, which are beneficial for tuning applications and positively impact the numerical results. This new technique can be effectively applied to solve the problem of image deblurring and offers numerical advantages compared to some previously related results. By utilizing certain properties of a Lipschitz monotone operator and a maximally monotone operator, along with the identity associated with the convexity of the quadratic norm in the framework of Hilbert spaces, and by imposing some constraints on the scalar control conditions, we can achieve weak convergence to a common zero point of the sum of two monotone operators. To demonstrate the benefits and advantages of this newly proposed algorithm, we performed numerical experiments to measure the improvement in the signal–to–noise ratio (ISNR) and the structural similarity index measure (SSIM). The results of both numerical experiments (ISNR and SSIM) demonstrate that our new algorithm is more efficient and has a significant advantage over the relevant preceding algorithms. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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13 pages, 1268 KiB

Open AccessArticle

A New Variant of the Conjugate Descent Method for Solving Unconstrained Optimization Problems and Applications

by Aliyu Muhammed Awwal, Mahmoud Muhammad Yahaya, Nuttapol Pakkaranang and Nattawut Pholasa

Mathematics 2024, 12(15), 2430; https://doi.org/10.3390/math12152430 - 5 Aug 2024

Cited by 2 | Viewed by 700

Abstract

Unconstrained optimization problems have a long history in computational mathematics and have been identified as being among the crucial problems in the fields of applied sciences, engineering, and management sciences. In this paper, a new variant of the conjugate descent method for solving [...] Read more.

Unconstrained optimization problems have a long history in computational mathematics and have been identified as being among the crucial problems in the fields of applied sciences, engineering, and management sciences. In this paper, a new variant of the conjugate descent method for solving unconstrained optimization problems is introduced. The proposed algorithm can be seen as a modification of the popular conjugate descent (CD) algorithm of Fletcher. The algorithm of the proposed method is well-defined, and the sequence of the directions of search is shown to be sufficiently descending. The convergence result of the proposed method is discussed under the common standard conditions. The proposed algorithm together with some existing ones in the literature is implemented to solve a collection of benchmark test problems. Numerical experiments conducted show the performance of the proposed method is very encouraging. Furthermore, an additional efficiency evaluation is carried out on problems arising from signal processing and it works well. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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18 pages, 1401 KiB

Open AccessArticle

Existence and Sensitivity Analysis of a Caputo Fractional-Order Diphtheria Epidemic Model

by Idris Ahmed, Chanakarn Kiataramkul, Mubarak Muhammad and Jessada Tariboon

Mathematics 2024, 12(13), 2033; https://doi.org/10.3390/math12132033 - 29 Jun 2024

Cited by 3 | Viewed by 902

Abstract

Diphtheria, a potentially life-threatening infectious disease, is primarily caused by the bacterium Corynebacterium diphtheriae. This pathogen induces a range of severe symptoms, including respiratory distress, cardiac arrhythmias, and, in extreme cases, fatal outcomes. This paper aim to unravel the transmission dynamics of [...] Read more.

Diphtheria, a potentially life-threatening infectious disease, is primarily caused by the bacterium Corynebacterium diphtheriae. This pathogen induces a range of severe symptoms, including respiratory distress, cardiac arrhythmias, and, in extreme cases, fatal outcomes. This paper aim to unravel the transmission dynamics of diphtheria infection within the Caputo fractional derivatives framework, establishing the solutions’ existence and uniqueness. Through forward normalized sensitivity analysis, we scrutinize the key parameters influencing the basic reproduction number, a pivotal metric in understanding and controlling the spread of the disease. The results indicate that reducing the values of the interaction rate, transmission rate, and birth rate plays a key role in curtailing diphtheria transmission. Furthermore, employing an effective numerical tool, we present graphical representations that delineate the influence of various crucial model parameters on infection dynamics. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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14 pages, 434 KiB

Open AccessArticle

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

by Sunil Panday, Shubham Kumar Mittal, Carmen Elena Stoenoiu and Lorentz Jäntschi

Mathematics 2024, 12(12), 1809; https://doi.org/10.3390/math12121809 - 11 Jun 2024

Cited by 1 | Viewed by 841

Abstract

In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order [...] Read more.

In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to

10.5208

. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from

1.5157

to

1.6011

. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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11 pages, 1181 KiB

Open AccessArticle

Passive Stabilization of Static Output Feedback of Disturbed Nonlinear Stochastic System

by Ping-Tzan Huang, Chein-Chung Sun, Cheung-Chieh Ku and Yun-Chen Yeh

Mathematics 2023, 11(21), 4435; https://doi.org/10.3390/math11214435 - 26 Oct 2023

Viewed by 865

Abstract

This paper investigates the Static Output (SO) control issue of the disturbed nonlinear stochastic system, which achieves passivity. Through the application of fuzzy sets and the stochastic differential equation, a Takagi–Sugeno (T-S) fuzzy model with the terms of multiplicative noise and external disturbance [...] Read more.

This paper investigates the Static Output (SO) control issue of the disturbed nonlinear stochastic system, which achieves passivity. Through the application of fuzzy sets and the stochastic differential equation, a Takagi–Sugeno (T-S) fuzzy model with the terms of multiplicative noise and external disturbance can be constructed to describe the considered systems. Furthermore, the Parallel Distributed Compensation (PDC) concept is used to design a fuzzy controller exhibiting an SO feedback scheme structure. To attenuate the effect of external disturbance, the PDC-based SO fuzzy controller is designed to exhibit passivity. During the derivation of some sufficient conditions, a line-integral Lyapunov function is utilized to avoid the conservative term produced using the derivative membership function. Using converting technologies, a stability criterion belonging to Linear Matrix Inequality (LMI) forms is proposed such that the derived conditions are convex hull problems and are solved through an optimization algorithm. Then, the proposed criterion is used to discuss the problem of SO controller design of ship fin stabilizing systems with added disturbance and noise. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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12 pages, 256 KiB

Open AccessFeature PaperArticle

On an Extension of a Spare Regularization Model

by Abdellatif Moudafi

Mathematics 2023, 11(20), 4285; https://doi.org/10.3390/math11204285 - 14 Oct 2023

Cited by 1 | Viewed by 956

Abstract

In this paper, we would first like to promote an interesting idea for identifying the local minimizer of a non-convex optimization problem with the global minimizer of a convex optimization one. Secondly, to give an extension of their sparse regularization model for inverting [...] Read more.

In this paper, we would first like to promote an interesting idea for identifying the local minimizer of a non-convex optimization problem with the global minimizer of a convex optimization one. Secondly, to give an extension of their sparse regularization model for inverting incomplete Fourier transforms introduced. Thirdly, following the same lines, to develop convergence guaranteed efficient iteration algorithm for solving the resulting nonsmooth and nonconvex optimization problem but here using applied nonlinear analysis tools. These both lead to a simplification of the proofs and to make a connection with classical works in this filed through a startling comment. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

11 pages, 300 KiB

Open AccessArticle

Existence and Uniqueness of Positive Solutions for the Fractional Differential Equation Involving the ρ(τ)-Laplacian Operator and Nonlocal Integral Condition

by Piyachat Borisut and Supak Phiangsungnoen

Mathematics 2023, 11(16), 3525; https://doi.org/10.3390/math11163525 - 15 Aug 2023

Cited by 2 | Viewed by 1096

Abstract

This paper aims to investigate the Caputo fractional differential equation involving the

ρ (τ)

Laplacian operator and nonlocal multi-point of Riemann–Liouville’s fractional integral. We also prove the uniqueness of the positive solutions for Boyd and Wong’s nonlinear contraction via the Guo–Krasnoselskii [...] Read more.

This paper aims to investigate the Caputo fractional differential equation involving the

ρ (τ)

Laplacian operator and nonlocal multi-point of Riemann–Liouville’s fractional integral. We also prove the uniqueness of the positive solutions for Boyd and Wong’s nonlinear contraction via the Guo–Krasnoselskii fixed-point theorem in Banach spaces. Finally, we illustrate the theoretical results and show that by solving the nonlocal problems, it is possible to obtain accurate approximations of the solutions. An example is also provided to illustrate the applications of our theorem. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

11 pages, 301 KiB

Open AccessArticle

Fixed Point Results in Soft Fuzzy Metric Spaces

by Sonam, Ramakant Bhardwaj and Satyendra Narayan

Mathematics 2023, 11(14), 3189; https://doi.org/10.3390/math11143189 - 20 Jul 2023

Cited by 7 | Viewed by 1753

Abstract

The primary objective of the paper is to present the Banach contraction theorem in soft fuzzy metric spaces while taking into consideration a restriction on the soft fuzzy metric between the soft points of the absolute soft set. A new altering distance function, [...] Read more.

The primary objective of the paper is to present the Banach contraction theorem in soft fuzzy metric spaces while taking into consideration a restriction on the soft fuzzy metric between the soft points of the absolute soft set. A new altering distance function, namely the

Ψ

-contraction function, is introduced on soft fuzzy metric spaces, and some fixed point results are proven by considering soft mappings that comprise

Ψ

-contraction with the continuity of soft t-norm. In addition to that, some illustrations are supplied for the support of the established soft fuzzy Banach contraction theorem and fixed point results over

Ψ

-contraction mappings. The obtained results generalize and extend some well-known results present in the literature on fixed point theory. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

16 pages, 10116 KiB

Open AccessArticle

An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application

by Kamonrat Sombut, Kanokwan Sitthithakerngkiet, Areerat Arunchai and Thidaporn Seangwattana

Mathematics 2023, 11(9), 2107; https://doi.org/10.3390/math11092107 - 28 Apr 2023

Cited by 1 | Viewed by 1237

Abstract

In this paper, we propose an inertial forward–backward splitting method for solving the modified variational inclusion problem. The concept of the proposed method is based on Cholamjiak’s method. and Khuangsatung and Kangtunyakarn’s method. Cholamjiak’s inertial technique is utilized in the proposed method for [...] Read more.

In this paper, we propose an inertial forward–backward splitting method for solving the modified variational inclusion problem. The concept of the proposed method is based on Cholamjiak’s method. and Khuangsatung and Kangtunyakarn’s method. Cholamjiak’s inertial technique is utilized in the proposed method for increased acceleration. Moreover, we demonstrate that the proposed method strongly converges under appropriate conditions and apply the proposed method to solve the image restoration problem where the images have been subjected to various obscuring processes. In our example, we use the proposed method and Khuangsatung and Kangtunyakarn’s method to restore two medical images. To compare image quality, we also evaluate the signal-to-noise ratio (SNR) of the proposed method to that of Khuangsatung and Kangtunyakarn’s method. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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6 pages, 265 KiB

Open AccessArticle

Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability L^p(·)

by Mohamed A. Khamsi and Osvaldo D. Méndez

Mathematics 2023, 11(1), 157; https://doi.org/10.3390/math11010157 - 28 Dec 2022

Viewed by 1127

Abstract

In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in

L^{p (\cdot)} (Ω)

obtained under the assumptions

p_{+} < \infty

and the property

(R)

satisfied by

ρ

will [...] Read more.

In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in

L^{p (\cdot)} (Ω)

obtained under the assumptions

p_{+} < \infty

and the property

(R)

satisfied by

ρ

will force

p_{-} > 1

. Therefore, the conclusion is well known. In this note, we establish said conclusion without the assumption

p_{+} < \infty

. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

20 pages, 508 KiB

Open AccessArticle

Approximating Common Fixed Points of Nonexpansive Mappings on Hadamard Manifolds with Applications

by Konrawut Khammahawong, Parin Chaipunya and Kamonrat Sombut

Mathematics 2022, 10(21), 4080; https://doi.org/10.3390/math10214080 - 2 Nov 2022

Viewed by 1404

Abstract

The point of this research is to present a new iterative procedure for approximating common fixed points of nonexpansive mappings in Hadamard manifolds. The convergence theorem of the proposed method is discussed under certain conditions. For the sake of clarity, we provide some [...] Read more.

The point of this research is to present a new iterative procedure for approximating common fixed points of nonexpansive mappings in Hadamard manifolds. The convergence theorem of the proposed method is discussed under certain conditions. For the sake of clarity, we provide some numerical examples to support our results. Furthermore, we apply the suggested approach to solve inclusion problems and convex feasibility problems. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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