Comparison Principle Research Articles

Աշշուրի պետությունը մ.թ.ա. III հազարամյակի սկզբներից մինչև մ.թ.ա. XV դարի վերջերը

The study of the history of Assyria and the Assyrians has its important place in the study of the Civilizations of the Ancient Near East. Based on the selected chronological periods, that is, from the beginning of the 3rd Millennium BC to the End of the 15th Century BC, the physico-geographic location, origin, political and religious significance of the city of Aššur and the transitional stages of the city and then the state are discussed and analyzed. Methods and materials: The work uses a historical-comparative research method. The principle of comparison and contrast of this method was applied in the study of sources. Mostly inscriptions left by the rulers of the Ancient Near East and especially Assyria were used. Analysis During the study of the material, the stages of the history of Assyria and its principles, problems of chronology, as well as Old Assyrian and Middle Assyrian royal lists were examined in detail. The methods of formation and development of government bodies at the early stage of statehood as well as the management system of the Assyrian State are characterized.

Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source∂tu=Δum+|x|σup, posed for (x,t)∈RN×(0,T), with exponents 1<p<m and σ>0, are established. More precisely, we identify the optimal class of initial conditionsu0 for which (local in time) existence is ensured and prove non-existence of solutions for the complementary set of data. We establish then (local in time) uniqueness and a comparison principle for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem blows up in a finite timeT∈(0,∞) and finite speed of propagation holds true for t∈(0,T): if u0∈L∞(RN) is an initial condition with compact support and blow-up time T>0, then u(t) is compactly supported for t∈(0,T). We also establish in this work the absence of localization at the blow-up time T for solutions stemming from compactly supported data.

The convergence rate of p-harmonic to infinity-harmonic functions

The purpose of this paper is to prove a uniform convergence rate of the solutions of the p-Laplace equation with Dirichlet boundary conditions to the solution of the infinity-Laplace equation as . The rate scales like for general solutions of the Dirichlet problem and like for solutions with positive gradient. An explicit example shows that it cannot be better than . The proof of this result solely relies on the comparison principle with the fundamental solutions of the p-Laplace and the infinity-Laplace equation, respectively. Our argument does not use viscosity solutions, is purely metric, and is therefore generalizable to more general settings where a comparison principle with Hölder cones and Hölder regularity is available.

A New Design Technique for a High-Speed and High dV/dt Immunity Floating-Voltage Level Shifter

This paper presents a high-speed level shifter with about 500 V/ns power supply slew immunity. In this designed structure, a narrow pulse-controlled current source is adapted to accelerate the voltage conversion and reduce the power consumption. A Fast-Slewing Circuit speeds up the operation of the level shifter based on the current comparison principle. Edge detection technology is used to filter the generated voltage noise and achieve high dV/dt immunity. The proposed level shifter simulated with the 0.18 μm BCD (bipolar-CMOS-DMOS) process shows fast responses with a typical delay of 1.49 ns and 500 V/ns dV/dt immunity in the 200 V high-voltage application, which only occupies a 0.022 mm2 active area with the 0.18 μm BCD process.

Clinical practice guidelines for the treatment of primary Sjögren's syndrome by off-label drug use in China(2023 edition)

Sjögren's syndrome (SjS) is one of the most prevalent rheumatic diseases. Off-label drug use is very common in the clinical practice of the treatment of SjS. In order to establish evidence-based justifications and systematically evaluate the off-label drug use in SjS treatment, the Committee of Chinese Primary Health Care Foundation for Rheumatologists and Immunologists referred to published guidelines, consensus statements, and case series, and reevaluated the outcomes of the randomized controlled trials about the marketed immunosuppressive drugs treating SjS. The single-item breakdown of the Sjögren's tool for assessing response (STAR) which was proposed in 2022 was employed as the outcome. The committee established the corresponding clinical questions based on the population, intervention, comparison, and outcome (PICO) principle, and then carried out evidence retrieval, and synthesized meta-analysis using the grading of recommendations, assessment, development, and evaluation (GRADE) methodology. Finally, 21 recommendations addressing twelve clinical questions were formed which included two strong recommendations, fourteen weak recommendations, and five consensus-based recommendations. The overarching objective of the guideline is to provide clinicians with a rational basis for medication selection and to offer guidance to pharmacy departments for management purposes.

Matrix measure-based distributed impulsive consensus on nonlinear multi-agent systems with mixed time-varying delays

This paper concentrates on researching the global and exponential leader-following consensus issue for an array of nonlinear multi-agent systems with the system time-varying delay and the distributed time-varying delay. An innovative distributed impulsive controller is proposed to force the states of all agents to track the trajectories of leader agent. By jointly introducing the matrix measure protocol, the comparison principle for impulsive systems, and the average impulsive interval, sufficient criteria for the realization of leader-following consensus are derived. In addition, considering different functions of impulsive signal, two different convergence rates are precisely estimated by utilizing the parameter variation formula, respectively. Finally, two numerical examples are given to demonstrate the effectiveness of proposed control strategy and the rightness of theoretical analysis.

Analysis of a reaction–diffusion dengue model with vector bias on a growing domain

In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number ℜ 0 ρ . The results show that (i) if ℜ 0 ρ > 1 , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when ℜ 0 ρ < 1 . Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure.

Spreading dynamics of an impulsive reaction-diffusion model with shifting environments

This paper focuses on the effects of environmental improvement and worsening on the spread and invasion of populations with birth pulse. We propose an impulsive reaction-diffusion model with a shifting environment to describe the dynamics of species with distinct reproduction stage and dispersal stage. First, the impulsive reaction-diffusion model is reduced to a discrete-time recursive system defined by a discrete map. Next, with the aid of the appropriate test function and comparison principle, we obtain some sufficient conditions on the nonexistence and uniqueness of nontrivial fixed points of the discrete map. This, combined with the abstract theory of spatially non-translation dynamical systems, enables us to establish the existence of traveling wave solutions and the asymptotic propagation properties of the model.

Stabilization of hybrid stochastic differential equations with multiple delays via intermittent control

In this paper, we study stability of hybrid stochastic differential equations (SDEs) with multiple delays by intermittent control based on time‐varying delay observations. By constructing an auxiliary non‐delayed hybrid stochastic differential equation and using ‐matrix theory, Lyapunov method, and the comparison principle, sufficient criterions to guarantee the th moment exponential stability and almost sure exponential stability of hybrid SDEs are derived. Finally, a simulation example is shown to illustrate the validity of theoretical results.

Elastic graphs with clamped boundary and length constraints

We study two minimization problems concerning the elastic energy on curves given by graphs subject to symmetric clamped boundary conditions. In the first, the inextensible problem, we fix the length of the curves while in the second, the extensible problem, we add a term penalizing the length. This can be considered as a one-dimensional version of the Helfrich energy. In both cases, we prove existence, uniqueness and qualitative properties of the minimizers. A key ingredient in our analysis is the use of Noether identities valid for critical points of the energy and derived from the invariance of the energy functional with respect to translations. These identities allow us also to prove curvature bounds and ordering of the minimizers even though the problem is of fourth order and hence in general does not allow for comparison principles.

Error estimate of a transformed L1 scheme for a multi-term time-fractional diffusion equation by using discrete comparison principle

This work is concerned with the multi-term time-fractional diffusion equation ∑j=0JbjDtαju−pΔu+c(x,t)u=f, where Dtαj is the Caputo derivative with 1>α0>α1>⋯>αJ>0 and bj, p are positive constants. The solution of this problem usually has a weak singularity near the initial time. To handle such difficulty, a smoothing transformation t=s1/α0 is applied so that an equivalent re-scaled fractional differential equation is obtained. Then the equivalent equation is solved by the transformed L1 scheme of the Caputo derivative and the standard 3-point discretization of the spatial derivative on uniform meshes both in time and space direction. The α-robust error estimate with the temporal convergence order O(Nα0−2) is given by using the discrete comparison principle, which does not blow up as α→1−. Finally, numerical results are given to confirm our error analysis.

Convergence of the solutions of the nonlinear discounted Hamilton–Jacobi equation: The central role of Mather measures

Given a continuous Hamiltonian H:(x,p,u)↦H(x,p,u) defined on T⁎M×R, where M is a closed connected manifold, we study viscosity solutions, uλ:M→R, of discounted equations:H(x,dxuλ,λuλ(x))=cin M where λ>0 is called a discount factor and c is the critical value of H(⋅,⋅,0).When the Hamiltonian H is convex and superlinear in p and non–decreasing in u, under an additional non–degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions (uλ)λ>0 converges to a specific solution u0 ofH(x,dxu0,0)=cin M. Our non–degeneracy condition requires H to be increasing (in u) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in u.

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle.

Hopf bifurcation and patterns in a modified SIR model

Infectious diseases have constantly threatened human safety because the diffusion of the susceptible and infected may make more individuals infected and even die. In this paper, a modified SIR model with both external stimulus and diffusion is considered to illustrate the dynamical mechanism of the periodic outbreak and pattern formation. Firstly, we propose a modified SIR model based on the propagation behaviour of infectious diseases to show the effects of the different parameters and diffusion on the outbreak. The Hopf bifurcation and multiscale methods are performed to analyze the stability of this model, which explains the dynamical mechanism of the periodic outbreak. Then, the pattern formation and Turing instability are discussed through comparison principles to reveal the role of periodic disturbances and diffusion in selecting pattern formation. Also, we find rich patterns that may occur when the frequency modulation is close to the intrinsic frequency. Finally, our theoretical results are verified by numerical simulation.

Asymptotic Boundedness of Conformable Fractional Delay Differential Systems

This article focuses on the problem of asymptotic boundedness for non-autonomous conformable fractional delay differential systems. Firstly, by utilizing the definition of conformable fractional derivative and fractional comparison principle, we propose non-autonomous conformable fractional Halanay inequalities. And then, based on the Halanay inequalities, sufficient conditions for the asymptotic boundedness are constructed for the systems. Finally, a numerical example shows the applicability of the proposed technique.

Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.

New Criteria for Analyzing the Permanence, Periodic Solution, and Global Attractiveness of the Competition and Cooperation Model of Two Enterprises with Feedback Controls and Delays

This paper studies a class of the non-autonomous competition and cooperation model of two enterprises involving discrete time delays and feedback controls. The paper proposes new criteria for analyzing the permanence, periodic solution, and global attractiveness of the model. The common mathematical techniques of the Lyapunov method, the continuation theorem, and the comparison principle are used in this paper. By means of the comparison principle and inequality techniques, the concept of permanence is investigated, which refers to the long-term survival of the enterprises within the competitive and cooperative framework. Meanwhile, using the continuation theorem, we establish conditions under which the system exhibits periodic behavior. Additionally, the global attractiveness of the system is derived by constructing multiple Lyapunov functionals. Finally, an example is presented to illustrate the applicability and validity of the proposed criteria in this paper. This example serves as a demonstration that showcases the main results derived from the analysis.

Traveling waves for the Keller-Segel-FKPP equation with strong chemotaxis

We show that there exist traveling wave solutions of the Keller-Segel-FKPP equation, which models a diffusing and logistically growing population subject to chemotaxis. In contrast to previous results, our result is in the strong aggregation regime; that is, we make no smallness assumption on the parameters. The lack of a smallness condition makes L∞-estimates difficult to obtain as the comparison principle no longer gives them “for free.” Instead, our proof is based on suitable energy estimates in a carefully tailored uniformly local Lp-space. Interestingly, our uniformly local space involves a scaling parameter, the choice of which is a crux of the argument. Numerical experiments exploring the stability, qualitative properties, and speeds of these waves are presented as well.

Fourth-Order Neutral Differential Equation: A Modified Approach to Optimizing Monotonic Properties

In this article, we investigate some qualitative properties of solutions to a class of functional differential equations with multi-delay. Using a modified approach, we first derive a number of optimized relations and inequalities that relate the solution xs to its corresponding function zs and its derivatives. After classifying the positive solutions, we follow the Riccati approach and principle of comparison, where fourth-order differential equations are compared with first-order differential equations to obtain conditions that exclude the positive solutions. Then, we introduce new oscillation conditions. With regard to previous relevant results, our results are an extension and complement to them. This work has theoretical significance in that it uncovers some new relationships that aid in developing the oscillation theory of higher-order equations in addition to the applied relevance of neutral differential equations.

Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the C0,α, C1,α and C2,α regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.