Axioms

19 pages, 311 KiB

Open AccessArticle

An Extension of Left Radau Type Inequalities to Fractal Spaces and Applications

by Bandar Bin-Mohsin, Abdelghani Lakhdari, Nour El Islem Karabadji, Muhammad Uzair Awan, Abdellatif Ben Makhlouf, Badreddine Meftah and Silvestru Sever Dragomir

Axioms 2024, 13(9), 653; https://doi.org/10.3390/axioms13090653 (registering DOI) - 23 Sep 2024

Abstract

In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained [...] Read more.

In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained results not only represent an extension of certain previously established findings to fractal sets but also a refinement of these when the fractal dimension

μ

is equal to one. Finally, to support our findings, we present a practical application to demonstrate the effectiveness of our results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

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15 pages, 299 KiB

Open AccessArticle

Investigation of the Oscillatory Properties of Fourth-Order Delay Differential Equations Using a Comparison Approach with First- and Second-Order Equations

by Osama Moaaz, Shaimaa Elsaeed, Asma Al-Jaser, Samia Ibrahim and Amira Essam

Axioms 2024, 13(9), 652; https://doi.org/10.3390/axioms13090652 (registering DOI) - 23 Sep 2024

Abstract

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed [...] Read more.

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method’s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

3 pages, 145 KiB

Open AccessEditorial

Advances in Difference Equations

by Azhar Ali Zafar

Axioms 2024, 13(9), 651; https://doi.org/10.3390/axioms13090651 (registering DOI) - 22 Sep 2024

Abstract

This editorial concerns the Special Issue of Axioms entitled “Advances in Difference Equations” [...] Full article

(This article belongs to the Special Issue Advances in Difference Equations)

16 pages, 314 KiB

Open AccessArticle

Fuzzy Fixed Point Theorems in

S

-Metric Spaces: Applications to Navigation and Control Systems

by Maryam Iqbal, Afshan Batool, Aftab Hussain and Hamed Alsulami

Axioms 2024, 13(9), 650; https://doi.org/10.3390/axioms13090650 (registering DOI) - 22 Sep 2024

Abstract

This manuscript examines fuzzy fixed point results using the concepts of

S

-metric space. We introduce two contractive maps,

γ

- and

γ

-weak contractions, within the context of

S

-metric spaces. These contractive maps form the cornerstone of our research, offering a [...] Read more.

This manuscript examines fuzzy fixed point results using the concepts of

S

-metric space. We introduce two contractive maps,

γ

- and

γ

-weak contractions, within the context of

S

-metric spaces. These contractive maps form the cornerstone of our research, offering a novel approach to solving mathematical problems. We explore fixed point results derived from the application of these maps, showcasing their utility in finding solutions in diverse mathematical scenarios. Furthermore, we provide concrete examples that illustrate the practical relevance and versatility of our theorems, emphasizing their potential applications across a wide range of scientific and engineering domains. This manuscript presents the novel concepts of

γ

- and

γ

-weak contractions and establishes their importance in mathematical research. By demonstrating their effectiveness in solving real-world problems and offering illustrative examples, our work contributes valuable tools and insights to the broader scientific community, enhancing our understanding of contractive maps and their applications. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

38 pages, 3307 KiB

Open AccessArticle

A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

by Theodore E. Simos

Axioms 2024, 13(9), 649; https://doi.org/10.3390/axioms13090649 (registering DOI) - 22 Sep 2024

Abstract

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and [...] Read more.

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches. Full article

(This article belongs to the Special Issue The Numerical Analysis and Its Application)

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15 pages, 1616 KiB

Open AccessArticle

Pretest Estimation for the Common Mean of Several Normal Distributions: In Meta-Analysis Context

by Peter M. Mphekgwana, Yehenew G. Kifle and Chioneso S. Marange

Axioms 2024, 13(9), 648; https://doi.org/10.3390/axioms13090648 (registering DOI) - 22 Sep 2024

Abstract

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary [...] Read more.

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary test estimator for the common mean

(μ)

with unknown and unequal variances. When there exists prior information regarding the population mean with consideration that

μ

might be equal to the reference value for the population mean, a hypothesis test can be conducted:

H_{0} : μ = μ_{0}

versus

H_{1} : μ \neq μ_{0}

. The initial sample is used to test

H_{0}

, and if

H_{0}

is not rejected, we become more confident in using our prior information (after the test) to estimate

μ

. However, if

H_{0}

is rejected, the prior information is discarded. Our simulations indicate that the proposed preliminary test estimator significantly decreases the mean squared error (MSE) values compared to unbiased estimators such as the Garybill-Deal (GD) estimator, particularly when

μ

closely aligns with the hypothesized mean (

μ_{0}

). Furthermore, our analysis indicates that the proposed test estimator outperforms the existing method, particularly in cases with minimal sample sizes. We advocate for its adoption to improve the accuracy of common mean estimation. Our findings suggest that through careful application to real meta-analyses, the proposed test estimator shows promising potential. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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10 pages, 360 KiB

Open AccessArticle

A Note on Some Novel Laplace and Stieltjes Transforms Associated with the Relaxation Modulus of the Andrade Model

by Juan Luis González-Santander and Alexander Apelblat

Axioms 2024, 13(9), 647; https://doi.org/10.3390/axioms13090647 (registering DOI) - 20 Sep 2024

Abstract

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a [...] Read more.

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function

F_{α, β} (s)

by using two different approaches: the Bromwich integral and the decomposition of

F_{α, β} (s)

in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

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15 pages, 291 KiB

Open AccessArticle

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

by Said Mesloub, Eman Alhazzani and Hassan Eltayeb Gadain

Axioms 2024, 13(9), 646; https://doi.org/10.3390/axioms13090646 (registering DOI) - 20 Sep 2024

Abstract

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove [...] Read more.

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments. Full article

(This article belongs to the Special Issue Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods)

31 pages, 591 KiB

Open AccessArticle

Solutionsof Fuzzy Goursat Problems with Generalized Hukuhara (gH)-Differentiability Concept

by Noor Jamal, Muhammad Sarwar, Kamaleldin Abodayeh, Manel Hleili, Saowaluck Chasreechai and Thanin Sitthiwirattham

Axioms 2024, 13(9), 645; https://doi.org/10.3390/axioms13090645 (registering DOI) - 20 Sep 2024

Abstract

In this manuscript, we will discuss the solutions of Goursat problems with fuzzy boundary conditions involving gH-differentiability. The solutions to these problems face two main challenges. The first challenge is to deal with the two types of fuzzy gH-differentiability:

(i)

-differentiability [...] Read more.

In this manuscript, we will discuss the solutions of Goursat problems with fuzzy boundary conditions involving gH-differentiability. The solutions to these problems face two main challenges. The first challenge is to deal with the two types of fuzzy gH-differentiability:

(i)

-differentiability and

(i i)

-differentiability. The sign of coefficients in Goursat problems and gH-differentiability produces sixteen possible cases. The existing literature does not afford a solution method that addresses all the possible cases of this problem. The second challenge is the mixed derivative term in Goursat problems with fuzzy boundary conditions. Therefore, we propose to discuss the solutions of fuzzy Goursat problems with gH-differentiability. We will discuss the solutions of fuzzy Goursat problems in series form with natural transform and Adomian decompositions. To demonstrate the usability of the established solution methods, we will provide some numerical examples. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)

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33 pages, 53062 KiB

Open AccessArticle

An Improved MOEA/D with an Auction-Based Matching Mechanism

by Guangjian Li, Mingfa Zheng, Guangjun He, Yu Mei, Gaoji Sun and Haitao Zhong

Axioms 2024, 13(9), 644; https://doi.org/10.3390/axioms13090644 (registering DOI) - 20 Sep 2024

Abstract

Multi-objective optimization problems (MOPs) constitute a vital component in the field of mathematical optimization and operations research. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) decomposes a MOP into a set of single-objective subproblems and approximates the true Pareto front (PF) by optimizing [...] Read more.

Multi-objective optimization problems (MOPs) constitute a vital component in the field of mathematical optimization and operations research. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) decomposes a MOP into a set of single-objective subproblems and approximates the true Pareto front (PF) by optimizing these subproblems in a collaborative manner. However, most existing MOEA/Ds maintain population diversity by limiting the replacement region or scale, which come at the cost of decreasing convergence. To better balance convergence and diversity, we introduce auction theory into algorithm design and propose an auction-based matching (ABM) mechanism to coordinate the replacement procedure in MOEA/D. In the ABM mechanism, each subproblem can be associated with its preferred individual in a competitive manner by simulating the auction process in economic activities. The integration of ABM into MOEA/D forms the proposed MOEA/D-ABM. Furthermore, to make the appropriate distribution of weight vectors, a modified adjustment strategy is utilized to adaptively adjust the weight vectors during the evolution process, where the trigger timing is determined by the convergence activity of the population. Finally, MOEA/D-ABM is compared with six state-of-the-art multi-objective evolutionary algorithms (MOEAs) on some benchmark problems with two to ten objectives. The experimental results show the competitiveness of MOEA/D-ABM in the performance of diversity and convergence. They also demonstrate that the use of the ABM mechanism can greatly improve the convergence rate of the algorithm. Full article

(This article belongs to the Special Issue Mathematical Optimizations and Operations Research)

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17 pages, 1026 KiB

Open AccessArticle

Research on Change Point Detection during Periods of Sharp Fluctuations in Stock Prices–Based on Bayes Method β-ARCH Models

by Fenglin Tian, Yong Wang, Qi Qin and Boping Tian

Axioms 2024, 13(9), 643; https://doi.org/10.3390/axioms13090643 - 19 Sep 2024

Abstract

In periods of dramatic stock price volatility, the identification of change points in stock price time series is important for analyzing the structural changes in financial market data, as well as for risk prevention and control in the financial market. As their residuals [...] Read more.

In periods of dramatic stock price volatility, the identification of change points in stock price time series is important for analyzing the structural changes in financial market data, as well as for risk prevention and control in the financial market. As their residuals follow a generalized error distribution, the problem of estimating the change point parameters of the

β

-ARCH model is solved by combining the Kalman filtering method and the Bayes method innovatively, and we give a method for parameter estimation of the Bayes factors for the occurrences of change points, the expected values of the change point positions, and the variance of the change points. By detecting the change points of the price of eight stocks with a high number of limit up and limit down changes occurring in the observation period, the following conclusions are obtained: (1) Change point detection using the

β

-ARCH model based on the Bayes method is effective. (2) For different values of

β

, this research study finds that based on the classical ARCH model (i.e.,

β = 1

) of the change point parameter, the results are relatively optimal. (3) The accuracy of change point detection can be improved by correcting stock short-term effects by using the Kalman filtering method. Full article

(This article belongs to the Special Issue Applications of Bayesian Methods in Statistical Analysis)

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25 pages, 2125 KiB

Open AccessArticle

Dynamic Analysis and Optimal Control of a Fractional Order Fishery Model with Refuge and Protected Area

by Wenjun Gao, Xiu Jia and Ruiqing Shi

Axioms 2024, 13(9), 642; https://doi.org/10.3390/axioms13090642 - 19 Sep 2024

Abstract

In this paper, a mathematical analysis of fractional order fishery model with stage structure for predator is carried out under the background of prey refuge and protected area. First, it is demonstrated that the solution exists and is unique. The paper aims to [...] Read more.

In this paper, a mathematical analysis of fractional order fishery model with stage structure for predator is carried out under the background of prey refuge and protected area. First, it is demonstrated that the solution exists and is unique. The paper aims to analyze predator-prey dynamics in a fishery model through the application of fractional derivatives. It is worth emphasizing that we explicitly examine how fractional derivatives affect the dynamics of the model. The existence of each equilibrium point and the stability of the system at the equilibrium point are proved. The theoretical results are proved by numerical simulation. Alternatively, allocate harvesting efforts within an improved model aimed at maximizing economic benefits and ecologically sustainable development. The ideal solution is obtained by applying Pontryagin’s optimal control principle. A large number of numerical simulations show that the optimal control scheme can realize the sustainable development of the ecosystem. Full article

(This article belongs to the Special Issue Mathematical Modeling, Simulations and Applications)

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13 pages, 2478 KiB

Open AccessArticle

The Geometry of Dynamic Time-Dependent Best–Worst Choice Pairs

by Sasanka Adikari, Norou Diawara and Haim Bar

Axioms 2024, 13(9), 641; https://doi.org/10.3390/axioms13090641 - 19 Sep 2024

Abstract

There has been increasing interest in best–worst discrete choice experiments (BWDCEs) in health economics, transportation research, and other fields over the last few years. BWDCEs have distinct advantages compared to other measurement approaches in discrete choice experiments (DCEs). A systematic study of best–worst [...] Read more.

There has been increasing interest in best–worst discrete choice experiments (BWDCEs) in health economics, transportation research, and other fields over the last few years. BWDCEs have distinct advantages compared to other measurement approaches in discrete choice experiments (DCEs). A systematic study of best–worst (BW) choice pairs can be traced back to the 1990s. Recently, new ideas have been introduced to the subject. Calculating utility helps measure the attractiveness of BW choices. The goal of this paper is twofold. First, we extend the idea of the BW choice pair to include dynamic, time-dependent transition probability and capture utility at each time and for each choice pair. Second, we used the geometry of BW choice pairs to capture the correlations among them and to characterize and clarify the BW choice pairs in the network, where properties can be derived within each class. This paper discusses BWDCEs, the probability transition matrix of choices over time, and the utility function. The proposed network classification for BW choice pairs is laid out. A detailed simulated example is presented, and the results are compared with the classical

K

-means classification. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics)

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14 pages, 249 KiB

Open AccessArticle

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

by Şahap Çetin, Yalçın Yılmaz and Coşkun Yakar

Axioms 2024, 13(9), 640; https://doi.org/10.3390/axioms13090640 - 19 Sep 2024

Abstract

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. [...] Read more.

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative

(Δ γ)

instead of the classical derivative

(d γ)

in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained. Full article

(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)

11 pages, 305 KiB

Open AccessArticle

A Framework for I*-Statistical Convergence of Fuzzy Numbers

by Tanushri, Ayaz Ahmad and Ayhan Esi

Axioms 2024, 13(9), 639; https://doi.org/10.3390/axioms13090639 - 18 Sep 2024

Abstract

In this study, we investigate the concept of

I^{*}

-statistical convergence for sequences of fuzzy numbers. We establish several theorems that provide a comprehensive understanding of this notion, including the uniqueness of limits, the relationship between

I^{*}

-statistical convergence and classical [...] Read more.

In this study, we investigate the concept of

I^{*}

-statistical convergence for sequences of fuzzy numbers. We establish several theorems that provide a comprehensive understanding of this notion, including the uniqueness of limits, the relationship between

I^{*}

-statistical convergence and classical convergence, and the algebraic properties of

I^{*}

-statistically convergent sequences. We also introduce the concept of

I^{*}

-statistical pre-Cauchy and

I^{*}

-statistical Cauchy sequences and explore its connection to

I^{*}

-statistical convergence. Our results show that every

I^{*}

-statistically convergent sequence is

I^{*}

-statistically pre-Cauchy, but the converse is not necessarily true. Furthermore, we provide a sufficient condition for an

I^{*}

-statistically pre-Cauchy sequence to be

I^{*}

-statistically convergent, which involves the concept of

I^{*} - l i m i n f

. Full article

(This article belongs to the Special Issue Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking and Their Applications)

17 pages, 311 KiB

Open AccessArticle

Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space

by Dimple Singh, Priya Goel, Ramandeep Behl and Iñigo Sarría

Axioms 2024, 13(9), 638; https://doi.org/10.3390/axioms13090638 - 18 Sep 2024

Abstract

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type [...] Read more.

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type

(ψ - α)

-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones. Full article

(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)

18 pages, 1055 KiB

Open AccessArticle

Mathematical Model for the Control of Red Palm Weevil

by Zuhur Alqahtani, Areej Almuneef and Moustafa El-Shahed

Axioms 2024, 13(9), 637; https://doi.org/10.3390/axioms13090637 - 18 Sep 2024

Abstract

The red palm weevil (Rhynchophorus ferrugineus) is a highly destructive pest, causing severe damage to palm trees and significantly reducing their productivity. This paper aims to develop and analyze a mathematical model that captures the interactions between palm trees, Rhynchophorus ferrugineus [...] Read more.

The red palm weevil (Rhynchophorus ferrugineus) is a highly destructive pest, causing severe damage to palm trees and significantly reducing their productivity. This paper aims to develop and analyze a mathematical model that captures the interactions between palm trees, Rhynchophorus ferrugineus, and entomopathogenic nematodes as a means of integrated control. We identify the equilibrium points of the system and perform a stability analysis to assess the system’s behavior. Additionally, we design a linear quadratic regulator (LQR) to limit the spread of the red palm weevil within a locally linearized framework. The feedback control law, which is both straightforward and immediately implementable, is employed to avoid the need for complex cost calculations, thus simplifying the solution to the optimal control problem. Numerical simulations demonstrate that the proposed control strategy is effective in reducing the number of infected palm trees. The results indicate that increasing the population of entomopathogenic nematodes can significantly decrease the red palm weevil population, offering a promising approach to mitigating this pest’s impact. Full article

(This article belongs to the Topic Mathematical Modeling)

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17 pages, 296 KiB

Open AccessArticle

On Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

by Norah Alessa and Mohammed Guediri

Axioms 2024, 13(9), 636; https://doi.org/10.3390/axioms13090636 - 17 Sep 2024

Abstract

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form

(\bar{M}, \bar{g})

, where

\bar{M} = R \times_{f} M

and [...] Read more.

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form

(\bar{M}, \bar{g})

, where

\bar{M} = R \times_{f} M

and

\bar{g} = ϵ d t^{2} + f^{2} (t) g_{M}

. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where

(\bar{M}, \bar{g})

is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field

\partial_{t}

and the so-called support function

θ

to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)

11 pages, 614 KiB

Open AccessArticle

Necessary and Sufficient Criteria for a Four-Weight Weak-Type Maximal Inequality in the Orlicz Class

by Erxin Zhang

Axioms 2024, 13(9), 635; https://doi.org/10.3390/axioms13090635 - 17 Sep 2024

Abstract

Let

Φ_{i} (i = 1, 2)

be two N-functions, f be a

μ

-measurable function, and

ω_{i} (i = 1, 2, 3, 4)

be four weight functions. This study presents necessary and [...] Read more.

Let

Φ_{i} (i = 1, 2)

be two N-functions, f be a

μ

-measurable function, and

ω_{i} (i = 1, 2, 3, 4)

be four weight functions. This study presents necessary and sufficient conditions for weight functions

(ω_{1}, ω_{2}, ω_{3}, ω_{4})

such that the inequality

\int_{{x : M f (x) > λ}} Φ_{1} (λ ω_{1} (x)) ω_{2} (x) d μ (x) \leq c_{1} \int_{X} Φ_{2} (c_{1} | f (x) | ω_{3} (x))

ω_{4} (x) d μ (x)

holds, which extends several established results. Full article

(This article belongs to the Special Issue Theory of Functions and Applications II)

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