Research

15 pages, 255 KiB

Open AccessArticle

Fixed Point Theorems for Set-Valued Contractions in Metric Spaces

by Seong-Hoon Cho

Axioms 2024, 13(2), 86; https://doi.org/10.3390/axioms13020086 - 27 Jan 2024

Viewed by 744

In this paper, the concepts of Wardowski-type set-valued contractions and Işik-type set-valued contractions are introduced and fixed point theorems for such contractions are established. A positive answer to the open Question is given. Examples to support main theorems and an application to integral [...] Read more.

In this paper, the concepts of Wardowski-type set-valued contractions and Işik-type set-valued contractions are introduced and fixed point theorems for such contractions are established. A positive answer to the open Question is given. Examples to support main theorems and an application to integral inclusion are given. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

22 pages, 392 KiB

Open AccessArticle

On the Monotonic and Asymptotic Properties of Positive Solutions to Third-Order Neutral Differential Equations and Their Effect on Oscillation Criteria

by Amira Essam, Osama Moaaz, Moutaz Ramadan, Ghada AlNemer and Ibrahim M. Hanafy

Axioms 2023, 12(12), 1086; https://doi.org/10.3390/axioms12121086 - 28 Nov 2023

Viewed by 815

Abstract

The monotonic properties of positive solutions to functional differential equations of the third order are examined in this paper. It is generally known that by optimizing the relationships between a solution and its corresponding function, as well as its derivatives, one can improve [...] Read more.

The monotonic properties of positive solutions to functional differential equations of the third order are examined in this paper. It is generally known that by optimizing the relationships between a solution and its corresponding function, as well as its derivatives, one can improve the oscillation criterion for neutral differential equations. Based on this, we obtain new relationships and inequalities and test their effect on the oscillation parameters of the studied equation. To obtain the oscillation parameters, we used Riccati techniques and comparison with lower-order equations. Finally, the progress achieved in oscillation theory for third-order equations was measured by comparing our results with previous relevant results. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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19 pages, 1334 KiB

Open AccessArticle

New Two-Level Time-Mesh Difference Scheme for the Symmetric Regularized Long Wave Equation

by Jingying Gao, Qingmei Bai, Siriguleng He and Eerdun Buhe

Axioms 2023, 12(11), 1057; https://doi.org/10.3390/axioms12111057 - 17 Nov 2023

Viewed by 772

Abstract

The paper introduces a new two-level time-mesh difference scheme for solving the symmetric regularized long wave equation. The scheme consists of three steps. A coarse time-mesh and a fine time-mesh are defined, and the equation is solved using an existing nonlinear scheme on [...] Read more.

The paper introduces a new two-level time-mesh difference scheme for solving the symmetric regularized long wave equation. The scheme consists of three steps. A coarse time-mesh and a fine time-mesh are defined, and the equation is solved using an existing nonlinear scheme on the coarse time-mesh. Lagrange’s linear interpolation formula is employed to obtain all preliminary solutions on the fine time-mesh. Based on the preliminary solutions, Taylor’s formula is utilized to construct a linear system for the equation on the fine time-mesh. The convergence and stability of the scheme is analyzed, providing the convergence rates of

O (τ_{F}^{2} + τ_{C}^{4} + h^{4})

in the discrete

L_{\infty}

-norm for

u (x, t)

and in the discrete

L_{2}

-norm for

ρ (x, t)

. Numerical simulation results show that the proposed scheme achieves equivalent error levels and convergence rates to the nonlinear scheme, while also reducing CPU time by over half, which indicates that the new method is more efficient. Furthermore, compared to the earlier time two-mesh method developed by the authors, the proposed scheme significantly reduces the error between the numerical and exact solutions, which means that the proposed scheme is more accurate. Additionally, the effectiveness of the new scheme is discussed in terms of the corresponding conservation laws and long-time simulations. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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11 pages, 283 KiB

Open AccessArticle

Investigation of the Oscillatory Properties of Solutions of Differential Equations Using Kneser-Type Criteria

by Yousef Alnafisah and Osama Moaaz

Axioms 2023, 12(9), 876; https://doi.org/10.3390/axioms12090876 - 13 Sep 2023

Viewed by 655

Abstract

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study’s methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using [...] Read more.

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study’s methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using a comparison technique to connect the oscillation of the studied equation with some second-order equations, the second aspect takes advantage of the significant progress made in the study of the oscillation of second-order equations. Numerous applications of functional differential equations of the neutral type served as the inspiration for the study of a subclass of these equations. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

15 pages, 1385 KiB

Open AccessArticle

Numerical Solution of Time-Fractional Schrödinger Equation by Using FDM

by Moldir Serik, Rena Eskar and Pengzhan Huang

Axioms 2023, 12(9), 816; https://doi.org/10.3390/axioms12090816 - 25 Aug 2023

Viewed by 717

Abstract

In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo derivative, and the fourth-order compact finite difference scheme [...] Read more.

In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo derivative, and the fourth-order compact finite difference scheme is utilized for discretizing the spatial term. The unconditional stability and convergence of the scheme in the maximum norm are proved. Finally, we verified the theoretical result with a numerical test. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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15 pages, 400 KiB

Open AccessArticle

Resolvent-Free Method for Solving Monotone Inclusions

by Yan Tang and Aviv Gibali

Axioms 2023, 12(6), 557; https://doi.org/10.3390/axioms12060557 - 05 Jun 2023

Viewed by 1054

Abstract

In this work, we consider the monotone inclusion problem in real Hilbert spaces and propose a simple inertial method that does not include any evaluations of the associated resolvent and projection. Under suitable assumptions, we establish the strong convergence of the method to [...] Read more.

In this work, we consider the monotone inclusion problem in real Hilbert spaces and propose a simple inertial method that does not include any evaluations of the associated resolvent and projection. Under suitable assumptions, we establish the strong convergence of the method to a minimal norm solution. Saddle points of minimax problems and critical points problems are considered as the applications. Numerical examples in finite- and infinite-dimensional spaces illustrate the performances of our scheme. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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15 pages, 337 KiB

Open AccessArticle

Recent Results on Expansive-Type Evolution and Difference Equations: A Survey

by Behzad Djafari Rouhani and Mohsen Rahimi Piranfar

Axioms 2023, 12(4), 373; https://doi.org/10.3390/axioms12040373 - 13 Apr 2023

Viewed by 671

Abstract

In this survey, we review some old and new results initiated with the study of expansive mappings. From a variational perspective, we study the convergence analysis of expansive and almost-expansive curves and sequences governed by an evolution equation of the monotone or non-monotone [...] Read more.

In this survey, we review some old and new results initiated with the study of expansive mappings. From a variational perspective, we study the convergence analysis of expansive and almost-expansive curves and sequences governed by an evolution equation of the monotone or non-monotone type. Finally, we propose two well-defined algorithms to remedy the shortcomings concerning the ill-posedness of expansive-type evolution systems. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

26 pages, 3354 KiB

Open AccessArticle

A Mathematical Model for Zika Virus Infection and Microcephaly Risk Considering Sexual and Vertical Transmission

by Mahmoud A. Ibrahim and Attila Dénes

Axioms 2023, 12(3), 263; https://doi.org/10.3390/axioms12030263 - 03 Mar 2023

Cited by 5 | Viewed by 1467

Abstract

We establish a compartmental model for Zika virus disease transmission, with particular attention paid to microcephaly, the main threat of the disease. To this end, we consider separate microcephaly-related compartments for affected infants, as well as the role of asymptomatic carriers, the influence [...] Read more.

We establish a compartmental model for Zika virus disease transmission, with particular attention paid to microcephaly, the main threat of the disease. To this end, we consider separate microcephaly-related compartments for affected infants, as well as the role of asymptomatic carriers, the influence of seasonality and transmission through sexual contact. We determine the basic reproduction number of the corresponding time-dependent model and time-constant model and study the dependence of this value on the mosquito-related parameters. In addition, we demonstrate the global stability of the disease-free periodic solution if

R_{0} < 1

, whereas the disease persists when

R_{0} > 1

. We fit our model to data from Colombia between 2015 and 2017 as a case study. The fitting is used to figure out how sexual transmission affects the number of cases among women as well as the number of microcephaly cases. Our sensitivity analyses conclude that the most effective ways to prevent Zika-related microcephaly cases are preventing mosquito bites and controlling mosquito populations, as well as providing protection during sexual contact. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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20 pages, 534 KiB

Open AccessArticle

An Inertial Subgradient Extragradient Method for Approximating Solutions to Equilibrium Problems in Hadamard Manifolds

by Olawale Kazeem Oyewole and Simeon Reich

Axioms 2023, 12(3), 256; https://doi.org/10.3390/axioms12030256 - 01 Mar 2023

Cited by 4 | Viewed by 1146

Abstract

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to [...] Read more.

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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12 pages, 1374 KiB

Open AccessArticle

Parameter Estimation Analysis in a Model of Honey Production

by Atanas Z. Atanasov, Slavi G. Georgiev and Lubin G. Vulkov

Axioms 2023, 12(2), 214; https://doi.org/10.3390/axioms12020214 - 17 Feb 2023

Cited by 2 | Viewed by 1484

Abstract

Honeybee losses are an extensive global problem. In this study, a new compartment model of honeybee population that mainly concerns honey production is developed. The model describes the interaction of the food stock with the brood (immature bees), adult bees and produced honey. [...] Read more.

Honeybee losses are an extensive global problem. In this study, a new compartment model of honeybee population that mainly concerns honey production is developed. The model describes the interaction of the food stock with the brood (immature bees), adult bees and produced honey. In the present paper, the issue of an adequate model recovery is addressed and the parameter identification inverse problem is solved. An adjoint equation procedure to obtain the unknown parameter values by minimizing the functional error during a period of time is proposed. Numerical simulations with realistic data are discussed. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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9 pages, 227 KiB

Open AccessArticle

Two Convergence Results for Inexact Infinite Products of Non-Expansive Mappings

by Alexander J. Zaslavski

Axioms 2023, 12(1), 88; https://doi.org/10.3390/axioms12010088 - 14 Jan 2023

Viewed by 1099

Abstract

We analyze the asymptotic behavior of infinite products of non-linear operators which take a non-empty, closed subset of a complete metric space into the space, taking into account summable computational errors. Our results can be applied in methods for solving convex feasibility and [...] Read more.

We analyze the asymptotic behavior of infinite products of non-linear operators which take a non-empty, closed subset of a complete metric space into the space, taking into account summable computational errors. Our results can be applied in methods for solving convex feasibility and optimization problems. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

20 pages, 299 KiB

Open AccessArticle

Fixed-Point Theorems for ℒ_γ Contractions in Branciari Distance Spaces

by Seong-Hoon Cho

Axioms 2022, 11(9), 479; https://doi.org/10.3390/axioms11090479 - 18 Sep 2022

Cited by 2 | Viewed by 1000

Abstract

In this paper, the concepts of Suzuki-type ℒ_γ contractions and Suzuki–Berinde-type ℒ_γ contractions are introduced, and new fixed-point theorems for these two contractions are established. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

20 pages, 662 KiB

Open AccessArticle

Deterministic and Stochastic Prey–Predator Model for Three Predators and a Single Prey

by Yousef Alnafisah and Moustafa El-Shahed

Axioms 2022, 11(4), 156; https://doi.org/10.3390/axioms11040156 - 28 Mar 2022

Cited by 4 | Viewed by 2183

Abstract

In this paper, a deterministic prey–predator model is proposed and analyzed. The interaction between three predators and a single prey was investigated. The impact of harvesting on the three predators was studied, and we concluded that the dynamics of the population can be [...] Read more.

In this paper, a deterministic prey–predator model is proposed and analyzed. The interaction between three predators and a single prey was investigated. The impact of harvesting on the three predators was studied, and we concluded that the dynamics of the population can be controlled by harvesting. Some sufficient conditions were obtained to ensure the local and global stability of equilibrium points. The transcritical bifurcation was investigated using Sotomayor’s theorem. We performed a stochastic extension of the deterministic model to study the fluctuation environmental factors. The existence of a unique global positive solution for the stochastic model was investigated. The exponential–mean–squared stability of the resulting stochastic differential equation model was examined, and it was found to be dependent on the harvesting effort. Theoretical results are illustrated using numerical simulations. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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