Positive Solutions Research Articles

Stability analysis and Hopf bifurcation of a fractional order HIV model with saturated incidence rate and time delay

In this paper, a fractional order HIV model with saturated incidence rate and time delay is proposed and analyzed. Firstly, the existence and uniqueness of positive solutions are proved. Secondly, the basic reproduction number and the sufficient conditions for the stability of two equilibriums are obtained. Thirdly, by using time delay as the bifurcation parameter, it is found that Hopf bifurcation may occur when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to verify the theoretical results. Finally, some discussions and conclusions are listed.

Oscillatory Features of Fourth-Order Emden–Fowler Differential Equations with Sublinear Neutral Terms

This article examines the oscillatory characteristics of a fourth-order Emden–Fowler differential equation, specifically when it includes a sublinear neutral term. Our methodology centers on establishing multiple theorems that introduce innovative conditions to guarantee that there are no positive solutions to the examined equation. Due to the symmetry between non-oscillatory solutions, we obtain oscillation conditions by excluding only positive solutions. We employ the Riccati technique in various ways to achieve this objective. The criteria presented in this study complement and generalize many findings published in the literature. We support the efficiency of our findings by applying them to an example.

Absence of Positive Solutions of Some Nonlinear Inequalities with Transformations of the Argument in a Half-Space

Absence of Positive Solutions of Some Nonlinear Inequalities with Transformations of the Argument in a Half-Space

A quantitative version of the Gidas-Ni-Nirenberg Theorem

A celebrated result by Gidas, Ni & Nirenberg asserts that classical positive solutions to semilinear equations −Δu=f(u) in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small perturbations of this equation and study the quantitative stability counterpart of this result.

Safety evaluation of genetically modified crops

This study focuses on the application of genetically modified (GM) crops in modern agricultural production, delving into the assessment of their safety and consumer acceptance issues, while analyzing the mechanisms through which these factors influence market dynamics. The background highlights that, despite the potential of genetic modification technology to enhance the overall performance of crops, public concerns regarding their safety significantly affect consumer acceptance and, consequently, market performance. An evaluation of existing literature on the safety evaluation methods for GM crops is first conducted, identifying shortcomings in integrating consumer acceptance, and market dynamics. To address this gap, an evaluation system that incorporates consumer acceptance into the safety evaluation of GM crops was developed, utilizing the FuzzyID3 algorithm. Furthermore, employing multi-attribute decision theory, a decision model for assessing the stances of market stakeholders towards GM crops was established. This model, through the calculation and weighting of the distances from positive and negative ideal solutions across various modules, offers a novel perspective for market analysis. The methodology employed herein provides a robust tool for the safety evaluation and market forecasting of GM crops, holding practical value for guiding policy formulation and industry development.

Neutral Emden–Fowler Differential Equation of Second Order: Oscillation Criteria of Coles Type

In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (avηxvz′v)′ + qvFxgv = 0, where v≥v0 and the corresponding function z = x + px∘h. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion.

Existence results for a class of singular p(x)-Kirchhoff equations

In the present paper, we study a class of p ( x ) -Kirchhoff-type equations involving variable singularities. Using the Ekeland's variational principle, a constrained minimization and a global minimum method, we show the existence and uniqueness of positive solutions for the weak ( γ ∈ ( 0 , 1 ) ) and strong ( γ ∈ ( 1 , ∞ ) ) variable singularities. We provide two applications to illustrate the main results.

A discussion on some new boundary value problems involving the q-derivative operator via the existence of positive solutions

A discussion on some new boundary value problems involving the q-derivative operator via the existence of positive solutions

A unified stochastic SIR model driven by Lévy noise with time-dependency

We propose a unified stochastic SIR model driven by Lévy noise. The model is structural enough to allow for time-dependency, nonlinearity, discontinuity, demography, and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.

Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy

In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines (IL2&IL12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ IL_{2} \\& IL_{12}$$\\end{document}) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCDIL2IL12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$IL_{2} IL_{12}$$\\end{document}Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system’s boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system’s global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients’ production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research.

Research and Implementation of Indoor Positioning Algorithm Based on Bluetooth 5.1 AOA and AOD.

With the addition of Bluetooth AOA/AOD direction-finding capabilities in the Bluetooth 5.1 protocol and the introduction of antenna array technology into the Bluetooth platform to further enhance positioning accuracy, Bluetooth has gradually become a research hotspot in the field of indoor positioning due to its standard protocol specifications, rich application ecosystem, and outstanding advantages such as low power consumption and low cost compared to other indoor positioning technologies. However, current indoor positioning based on Bluetooth AOA/AOD suffers from overly simplistic core algorithm implementations. When facing different application scenarios, the standalone AOA or AOD algorithms exhibit weak applicability, and they also encounter challenges such as poor positioning accuracy, insufficient real-time performance, and significant effects of multipath propagation. These existing problems and deficiencies render Bluetooth lacking an efficient implementation solution for indoor positioning. Therefore, this paper proposes a study on Bluetooth AOA and AOD indoor positioning algorithms. Through an analysis of the principles of Bluetooth's newly added direction-finding functionality and combined with research on array signal DOA estimation algorithms, the paper ultimately integrates the least squares algorithm to optimize positioning errors in terms of accuracy and incorporates an anti-multipath interference algorithm to address the impacts of multipath effects in different scenarios. Experimental testing demonstrates that the indoor positioning algorithms applicable to Bluetooth AOA and AOD can effectively mitigate accuracy errors and overcome multipath effects, exhibiting strong applicability and significant improvements in real-time performance.

Comparison Theorems for Oscillation of Higher-Order Neutral Delay Differential Equations

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons.

Positive solutions of Kirchhoff type problems with critical growth on exterior domains

Positive solutions of Kirchhoff type problems with critical growth on exterior domains

Dynamical Behaviors of a Stochastic Susceptible-Infected-Treated-Recovered-Susceptible Cholera Model with Ornstein-Uhlenbeck Process

In this study, a cholera infection model with a bilinear infection rate is developed by considering the perturbation of the infection rate by the mean-reverting process. First of all, we give the existence of a globally unique positive solution for a stochastic system at an arbitrary initial value. On this basis, the sufficient condition for the model to have an ergodic stationary distribution is given by constructing proper Lyapunov functions and tight sets. This indicates in a biological sense the long-term persistence of cholera infection. Furthermore, after transforming the stochastic model to a relevant linearized system, an accurate expression for the probability density function of the stochastic model around a quasi-endemic equilibrium is derived. Subsequently, the sufficient condition to make the disease extinct is also derived. Eventually, the theoretical findings are shown by numerical simulations. Numerical simulations show the impact of regression speed and fluctuation intensity on stochastic systems.

Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, x″(t)+f(x(t))x′(t)+φ(t)xμ(t)−1xγ(t)=e(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x''(t)+f(x(t))x'(t)+\\varphi (t)x^{\\mu}(t)-\\frac{1}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+∞)→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:(0,+\\infty )\\rightarrow R$\\end{document} is continuous, which may have a singularity at the origin, the sign of φ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varphi (t)$\\end{document}, e(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e(t)$\\end{document} is allowed to change, and μ, γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T-periodic solution when μ∈[0,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu \\in [0,+\\infty )$\\end{document}.

Impact of hunting cooperation in predator and anti‐predator behaviors in prey in a predator–prey model

In this paper, we propose a predator–prey model with hunting cooperation in predator and anti‐predator behaviors in prey. The conditions for the existence and the stability of the unique positive constant equilibrium are given. It is found that with the increasing of the birth rate of the prey, the trivial solution loses its stability, and the semi‐trivial solution emerges and also loses its stability. For the positive constant solution, we find that as the hunting cooperation in predator increases or the fear decreases, the positive constant equilibrium loses its stability, and Hopf bifurcation occurs. We also derive the existence of limit cycles by Poincaré‐Bendixson theorem. We also study a diffusive model and derive that self‐diffusion can induce Turing instability. Finally, we conduct numerical simulations to present our conclusions.

Positive Solutions of Elliptic Systems with Superlinear Terms on the Critical Hyperbola

Positive Solutions of Elliptic Systems with Superlinear Terms on the Critical Hyperbola

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Closed-Form Kinematics Solutions for Redundant Medical Robots with Joint Limits and Singularity Avoidance

Closed-form forward and inverse position, velocity and acceleration kinematics problems solutions are developed for a redundant medical robot family. Developed solutions allow one to choose a specific joint as a set one. Solutions work in constant time, suitable for real-time operation. Methods for joint limit and singularity avoidance are proposed. Simulation experiments are performed on simulated KUKA MED 14 r820 robot.

Schrödinger–Poisson systems with zero mass in the Sobolev limiting case

AbstractWe study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted ‐Laplace operator and without the mass term, and a higher‐order fractional Poisson equation. Since the system is set in , the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign‐changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case .