Positive Solutions Research Articles

Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms

Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms

SIR epidemic model with non-Lipschitz stochastic perturbations

This paper investigates a stochastic SIR epidemic model with non-Lipschitz coefficients in the diffusion term. Firstly, we study the existence and uniqueness of the global positive solution of this model. Then, we prove that the stochastic process has a stationary distribution, and we also establish an exact expression of the density function of the stationary distribution under certain parameter constraints. Finally, we discuss the extinction of the epidemic and the asymptotic properties around the equilibrium points of the deterministic SIR model.

Existence and Multiplicity Results for an Elliptic Problem Involving Mixed Local and Nonlocal Operator

In this work, we consider the following mixed local-nonlocal quasilinear ellipic problem$$(P_\lambda)\left\{\begin{array}{rcll} -\Delta_p u+(-\Delta)_p^s u = \lambda f(u) \text{in}\Omega,\\ u 0 \text{in }\Omega,\\ u = 0 \text{in }\mathbb{R}^N \setminus\Omega,\end{array}\right.$$where Ω⊂RN is a bounded regular domain in RN with 0s1pN and f: R→R is a continuous function, that have a finite number of zeroes, changing sign between them. The main goal of this paper is to prove the existence and multiplicity of positive solutions for such problems by using variational methods.

On singular φ−Laplacian BVPs of nonlinear fractional differential equation

This paper investigates the existence of multiple positive solutions for a class of φ−Laplacian boundary value problem with a nonlinear fractional differential equation and fractional boundary conditions. Multiple solutions are proved under slight conditions on a possibly degenerating source term. Approximation techniques together with the fixed-point index theory a on cone of a Banach space are employed. Some illustrating examples of are also supplied. Mathematics Subject Classification (2010): 34A08, 34B15, 34B18, 47H10. Received 27 July; Accepted 13 December 2021

GERMANY’S SECURITY MANAGEMENT IN THE LIGHT OF THE INTEGRATED SECURITY – CURRENT STATE AND PROSPECTS

In June 2023, the Federal Government of Germany presented the first national security strategy (NSS). The NSS promotes Integrated Security as a new foreign and security policy. However, research has shown that the idea of Integrated Security is not new and that the integration of all resources of national power has been a goal for Germany for 30 years. Research has purpose to give an answer to following questions: Can the current organization of the German national security system support the idea of Integrated Security as a new security policy and what are the key requirements related to the organization, functioning and management of the German national security system that should be fulfilled so that Integrated Security can be implemented? The basic hypothesis of this research is that the national security system of Germany is not organized and governed in a way that can enable the implementation of Integrated Security, primarily because, on the strategic level, Germany does not have a national security council which could serve as a focal point and impetus for integration, coordination and government of a security system. In order to provide answers to the above stated questions in the first part of paper we explained Integrated Security and the national security objectives of the NSS, then we described organization of the three ministries and security services as the most important actors in the implementation of the NSS objectives at the operational level and at the end we did the capability analysis of the strategic level of management to implement Integrated Security. The research showed that the organizational solutions have not been aligned with the determinations stated in the NSS. The result of this research are proposed solutions for organization and position of the Federal Security Council which should be provided with competencies that will enable the integration of elements from all levels into a single structure which aims to achieve the national security objectives.

An Up-View Visual-Based Indoor Positioning Method via Deep Learning

Indoor positioning plays a crucial role in various domains. It is employed in various applications, such as navigation, asset tracking, and location-based services (LBS), in Global Navigation Satellite System (GNSS) denied or degraded areas. The visual-based positioning technique is a promising solution for high-accuracy indoor positioning. However, most visual positioning research uses the side-view perspective, which is susceptible to interferences and may cause concerns about privacy and public security. Therefore, this paper innovatively proposes an up-view visual-based indoor positioning algorithm. It uses the up-view images to realize indoor positioning. Firstly, we utilize a well-trained YOLO V7 model to realize landmark detection and gross extraction. Then, we use edge detection operators to realize the precision landmark extraction, obtaining the landmark pixel size. The target position is calculated based on the landmark detection and extraction results and the pre-labeled landmark sequence via the Similar Triangle Principle. Additionally, we also propose an inertial navigation system (INS)-based landmark matching method to match the landmark within an up-view image with a landmark in the pre-labeled landmark sequence. This is necessary for kinematic indoor positioning. Finally, we conduct static and kinematic experiments to verify the feasibility and performance of the up-view-based indoor positioning method. The results demonstrate that the up-view visual-based positioning is prospective and worthy of research.

Selecting Optimal Charcoal Company using Multi-Criteria Decision Making Methodology

A popular technique used in many industries to assist decision-makers in evaluating and selecting the best alternatives from a variety of accessible possibilities is multi-criteria decision-making (MCDM). This study proposed an MCDM methodology for selecting optimal charcoal from various options and criteria. Various criteria influence the selection problem, such as financial, safety, labour, power supply, production, and transportation. This study used the Combinative Distance Assessment (CODAS) method as an MCDM method to rank the alternatives and use the best one. This study used nine criteria and twenty alternatives. The requirements are divided into positive and negative criteria to compete for the positive and negative ideal solution using the CODAS method. The criteria weights are computed. The rank of alternatives is checked by using the sensitivity analysis. The results show the rank of other options is stable in different cases.

Role of ART and PrEP treatments in a stochastic HIV/AIDS epidemic model

In this paper, a stochastic HIV/AIDS epidemic model is presented to study the synthetic effect of ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments among MSM ( men who have sex with men). Firstly, we give the global stability of disease-free equilibrium and the endemic equilibrium in terms of basic reproduction number R0 for deterministic model. And then the existence of global positive solutions and the existence of unique ergodic stationary distribution under R0S>1 for stochastic model are given. Further, the long-time stochastic dynamic of the model is investigated, including the criteria on the extinction and persistence in mean for the stochastic model. Finally, we give some numerical simulations to illustrate our theoretical results, and the sensitive analysis shows that ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments can effectively control the spread of AIDS among MSM population.

Stochastic near-optimal controls for treatment and vaccination in a COVID-19 model with transmission incorporating Lévy jumps

The COVID-19 pandemic has triggered a groundbreaking reliance on mathematical modelling as an important tool for studying and managing the spread of the virus since its emergence. Public health preventive measures such as vaccination and therapeutics can effectively reduce or eradicate an infectious disease. This work investigates these two strategies for controlling the COVID-19 epidemic through a stochastic epidemiological modelling approach. The existence and uniqueness of a positive solution of the stochastic system is studied. A priori estimates of the vaccination and treatment controls are established. Sufficient and necessary conditions are obtained for the near-optimal control problem of the stochastic model using the maximum condition of the Hamiltonian function and the Ekeland principle. Finally, to support our theoretical results, numerical simulations for a combination of optimized vaccination and treatment strategies were presented to understand the challenges posed by COVID-19 in Brazil.

Positive solutions of parameter-dependent nonlocal differential equations with convolution coefficients

In this paper we investigate a parameter-dependent nonlocal differential equations with convolution coefficients. Using the Birkhoff–Kellogg type theorem, existence of positive solutions is established. Under additional growth conditions, we obtain upper and lower bounds for the parameter. An example is also given to illustrate the main results.

Liouville type theorems involving fractional order systems

Abstract In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).

Dynamic Analysis of a Ratio-Dependent Food Chain Model with Prey-Taxis

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results.

Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment

In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C0—semigroup theory, we prove that the system has a unique positive time-dependent solution (TDS) that satisfies the probability condition. As our second result, we prove that the TDS of the system strongly converges to its steady-state solution (SSS) if the service rates of the servers are constants. For this asymptotic behavior, we analyze the spectrum of the system operator associated with the system. Additionally, the stability of the semigroup generated by the system operator is also discussed.

Calculation and Analysis of Equilibrium Position of Aerostatic Bearings Based on Bivariate Interpolation Method

The solution of equilibrium positions is a critical component in the calculation of the dynamic characteristic coefficients of aerostatic bearings. The movement of the rotor in one direction leads to bidirectional variations in the air film force, resulting in low efficiency when using conventional calculation methods. It can even lead to iterative divergence if the initial value is improperly selected. This study concentrates on the orifice throttling aerostatic bearings and proposes a novel method called the bivariate interpolation method (BIM) to calculate the equilibrium position. The equilibrium equation for the rotor under the combined influence of air film forces, gravity, and external loads is established. A calculation program based on the finite difference method is developed to determine the equilibrium position. The process of solving the equilibrium position and the convergence is compared with the secant method and the search method. Furthermore, the variation trend of the equilibrium position and stiffness when the external loads changes are studied based on the BIM. Finally, the correctness of the BIM to solve the equilibrium position is proved by comparing it with the experiment results. The calculation results indicate that the BIM successfully resolves the problem of initial value selection and exhibits superior computational efficiency and accuracy. The equilibrium position initially moves away from the direction of the external load as the load increases, and then this gradually approaches the load direction. The main stiffness increases with increases in the external load, while the variation in cross stiffness depends on the direction of the external load.

An extension of fuzzy TOPSIS approach using integral values for banking performance evaluation

Banks play a crucial role as financial intermediaries in advancing economies. Increasing market share is a top priority for all banks in today's fiercely competitive market. To achieve this, banks are expected to enhance their efficiency, which boosts their competitive capacity and helps decision-makers identify areas for development. This paper introduces an extended fuzzy TOPSIS approach that incorporates integral values to assess banking performance. Our method includes the averaging and normalization of decision-makers assigned ratings and weights, ensuring a standardized scale for comparison. Subsequently, we create membership functions for the ultimate fuzzy evaluation values of alternatives using interval arithmetic for fuzzy numbers. We employ a novel fuzzy number ranking method based on integral values to streamline the aggregation of fuzzy numbers to defuzzify the weighted ratings. Ultimately, we introduce a closeness coefficient for determining the ranking order of alternatives by computing their distances to both the positive and negative ideal solutions. To illustrate the effectiveness and applicability of our proposed method, we present an application example.

Dynamics of a stochastic delay predator-prey model with fear effect and diffusion for prey

We establish a stochastic delay predator-prey model with fear effect and prey diffusion, and investigate the dynamic behavior of the model. Initially, we use Itô's formula to prove the existence and uniqueness of a global positive solution and the stochastic ultimate boundedness of the system. Subsequently, we provide sufficient conditions for the extinction and persistence of prey and predator, and discuss three possible scenarios. Beside, we construct a suitable Lyapunov function and obtain the conditions for the existence of an ergodic stationary distribution. Finally, we present some numerical simulations to verify the analytical results.

Multiplicity and concentration of positive solutions to the double phase Kirchhoff type problems with critical growth

The aim of this paper is to study the multiplicity and concentration of positive solutions to the $(p,q)$ Kirchhoff-type problems involving a positive potential and a continuous nonlinearity with critical growth at infinity. Applying penalization techniques, truncation methods and the Lusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its minimum values.

Existence of positive solutions in the space of Hölder functions for a class of nonlinear fractional differential equations with integral boundary conditions

We study the existence of positive solutions for a fractional differential equation with integral boundary conditions. Our solutions are placed in the space of Hölder functions and the main tools used in the proof of the results are the classical Schauder fixed point theorem and a sufficient condition about the relative compactness in Hölder spaces. Moreover, some examples are shown illustrating the results.

Global Multiplicity of Positive Solutions for Nonlinear Robin Problems with an Indefinite Potential Term

We consider a Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction is parametric and exhibits the competing effects of a concave (sublinear) and of a convex (superlinear) terms (“concave-convex” problem). The parameter multiplies the convex term. We prove an existence and multiplicity theorem which is global in parameter.

Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section.