Theory of Differential Inequalities for Two-Point Boundary Value Problems and their Applications to Three-Point B.V.Ps Associated with n th Order Non-Linear System of Differential Equations

This paper presents a theory of differential inequalities for two-point boundary value problems (B.V.Ps) associated with the system of n th order non-linear differential equations. Using these inequalities as a tool we establish the existence and uniqueness of solutions to three-point B.V.Ps associated with the system of n th order non-linear differential equations by using the idea of matching solutions.     

Basics of Riemann Delta and Nabla Integration on Time Scales

In this paper we introduce and investigate the concepts of Riemann's delta and nabla integrals on time scales. Main theorems of the integral calculus on time scales are proved.     

Differential Equations of Fractional Order: Methods,Results and Problems. II

The article, being a continuation of the first one [A.A. Kilbas and J.J. Trujillo (2001). Differential equations of fractional order. Methods, results and problems, I. Applicable Analysis , 78 (1-2), 153-192.], deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. The methods and the results in the theory of such fractional differential equations are presented including the Dirichlet-type problem for ordinary fractional differential equations, studying such equations in spaces of generalized functions, partial fractional differential equations and more general abstract equations, and treatment of numerical methods for ordinary and partial fractional differential equations. Problems and new trends of research are discussed.     

Necessary-Sufficient Conditions for Permanence and Extinction in Lotka-Volterra System with Discrete Delays

In this article, we give the necessary and sufficient conditions for permanence and extinction of a two-species Lotka-Volterra system with different discrete delays. We improve some previous results. Moreover, we show the permanence and extinction for this system is equivalent to that of its corresponding nondelayed system.     

All General Solutions of Post Equations

In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known （Theorem 6）. As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations （Theorem 9）.     

An Andô-Douglas type theorem in Riesz spaces with a conditional expectation

In this paper we formulate and prove analogues of the Hahn-Jordan decomposition and an Andô-Douglas-Radon-Nikodým theorem in Dedekind complete Riesz spaces with a weak order unit, in the presence of a Riesz space conditional expectation operator. As a consequence we can characterize those subspaces of the Riesz space which are ranges of conditional expectation operators commuting with the given conditional expectation operators and which have a larger range space. This provides the first step towards a formulation of Markov processes on Riesz spaces.     

Basic Homogenization Results for a Biconnected ε-Periodic Structure

We present, in a bounded domain, a model of an l -periodic structure composed of two phases, both being connected but only one reaching the boundary of the domain, avoiding in this way the local type convergences of the homogenization process. In this framework we revise some basic tools of the homogenization theory in porous media: the extension and the restriction operators, the Ne ) as inequality. Moreover, we obtain some compacity properties which reduce the proof of the pressure type convergences from the homogenization of fluid flows through porous media to the expected procedure of a priori estimations and two-scale convergences. As all the properties can be proved without much technical difficulties, avoiding annoying hypotheses and the use of Kolmogorov's criterion of compacity, the present structure seems one of the most convenient realistic models of porous media that can be studied with the methods of homogenization.     

A generalization of Andô's theorem and Parrott's example

Andô's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting contractions. We generalize both of these results as follows. Any -tuple of contractions that commute according to a graph without a cycle can be dilated to an -tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.

     

Analysis of a viral infection model with immune impairment,intracellular delay and general non-linear incidence rate

In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptotically stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one.     

总人口规模变化的年龄结构MSEIR流行病模型的再生数

在总人口规模变化和疾病影响死亡率的假设下,讨论了带二次感染和接种疫苗的年龄结构MSEIR流行病模型.首先给出再生数R(ψ,λ)(这里ψ(a)是接种疫苗率,λ是总人口的增长指数)的显式表达式.其次,证明了当R(ψ,λ)<1时,系统的无病平衡态是稳定的;当R(ψ,λ)>1时,无病平衡态是不稳定的.     

Reproductivity for a nematic liquid crystal model

In this paper we prove existence of weak solution with the reproductivity in time property, for a penalized PDE’s system related to a nematic liquid crystal model. This problem is relatively explict when time-independent Dirichlet boundary conditions are imposed for the orientation of crystal molecules. Nevertheless, for the time-dependent case, the treatment of the problem is completely different. The verification of a maximum principle for weak reproductive solutions is fundamental in the argument. Finally, the relation between reproductive and periodic in time (regular) solutions will be pointed out, differenting the 2D and 3D cases. Basically, in two-dimensional domains every reproductive solution is regular and time periodic, whereas the problem remains open for three-dimensional domains.     

More on the Difference Equation y n + 1 = ( p + y n −1 )/( qy n + y n −1 )

From previous studies of the equation in the title with positive parameters p and q and positive initial conditions we know that if q h 4 p + 1 then the equilibrium is a global attractor. We also know that if q > 4 p + 1 then every solution eventually enters and remains in the interval [ p / q , 1]. In this strip there exists a "unique" prime period two solution that is locally asymptotically stable. In this paper, we provide more insight as to the behavior of solutions of the equation in the title in the strip [ p / q , 1], where a one-dimensional stable manifold lives.     

A new modified resource budget model for nonlinear dynamics in citrus production

Alternate bearing or masting is a general yield variability phenomenon in perennial tree crops. This paper first presents a theoretical modeling and simulation study of the mechanism for this dynamics in citrus, and then provides a test of the proposed models using data from a previous 16-year experiment in a citrus orchard. Our previous studies suggest that the mutual effects between vegetative and reproductive growths caused by resource allocation and budgeting in plant body might be considered as a major factor responsible for the yield oscillations in citrus. Based on the resource budget model proposed by Isagi et al. (J Theor Biol. 1997;187:231-9), we first introduce the new leaf growth as a major energy consumption component into the model. Further, we introduce a nonlinear Ricker-type equation to replace the linear relationship between costs for flowering and fruiting used in Isagi's model. Model simulations demonstrate that the proposed new models can successfully simulate the reproductive behaviors of citrus trees with different fruiting dynamics. These results may enrich the mechanical dynamics in tree crop reproductive models and help us to better understand the dynamics of vegetative-reproductive growth interactions in a real environment.     

Difference Equations for the Masses of Leptons and Quarks in TeV Quantum Gravity

It is shown here that a 3 2 4 lattice in the wall for TeV quantum gravity with n =2 extra small-scale spatial dimensions can account for the fermion masses in a strikingly accurate manner. The family index, the electromagnetic charge number coupling, and the Yukawa coupling for lepton and quark mass generation in the minimal Standard Model (with a single Higgs) are related here to t' Hooft discreteness in the wall. Discrete values for the two transverse spatial distances in the wall are viewed as geometrical correspondents of the family index and the electromagnetic charge number coupling. The mass spectrum of Dirac leptons and quarks can then be understood as a manifestation of a Yukawa coupling that depends on the transverse wall coordinates. Linear homogeneous difference equations are considered to govern the Yukawa coupling or, more appropriately, the Yukawa field on the wall lattice. The solution to the latter difference equations yields experimentally consistent pole mass values for all twelve leptons and quarks. With the Yukawa field extending through the bulk, mass elevation for the second and third families features the torus radii ratio R 2 / R 1 =41/10.     

Spectral Analysis for an Indefinite Singular Sturm-Liouville Problem

Direct and inverse problems of spectral analysis are studied for an indefinite singular boundary value problem coming from astrophysics. We establish properties of the spectrum, prove completeness and expansion theorems and investigate the inverse problem of recovering the differential equation from the given spectral characteristics.     

On the dynamics of radially symmetric granulomas

A granuloma is a collection of macrophages that contains bacteria or other foreign substances that the body?s immune response is unable to eliminate. In this paper we present a simple mathematical model of radially symmetric granuloma dynamics. The model consists of a coupled system of two semi-linear parabolic equations for the macrophage density, and the bacterial density. The boundary of the granuloma is free. This simple framework makes it possible to conduct a mathematical analysis of the system dynamics. In particular, we show that the model system has a unique solution, and that, depending on the biological parameters; the bacterial load either disappears over time or persists. We use numerical methods to establish the existence of stationary solutions and examine how a stationary solution changes with the reproductive rate of the bacteria. These simulations show that the structure of the granuloma breaks down as the reproductive rate of the bacteria increases.     

Dynamical behaviors of a discrete HIV‐1 virus model with bilinear infective rate

We establish a discrete virus dynamic model by discretizing a continuous HIV‐1 virus model with bilinear infective rate using ‘hybrid’ Euler method. We discuss not only the existence and global stability of the uninfected equilibrium but also the existence and local stability of the infected equilibrium. We prove that there exists a crucial value similar to that of the continuous HIV‐1 virus dynamics, which is called the basic reproductive ratio of the virus. If the basic reproductive ratio of the virus is less than one, the uninfected equilibrium is globally asymptotically stable. If the basic reproductive ratio of the virus is larger than one, the infected equilibrium exists and is locally stable. Moreover, we consider the permanence for such a system by constructing a Lyapunov function v_n. Copyright © 2013 John Wiley & Sons, Ltd.     

Infinite Quasimonotone Systems of Ordinary Differential Equations with Applications to Stochastic Processes

We deal with the initial value problem for countably infinite linear systems of ordinary differential equations of the form y '( t ) = A ( t ) y ( t ) where A ( t ) = ( a ij ( t ): i , j S 1) is a measurable, infinite and essentially positive matrix, i.e., a ij ( t ) S 0 for i p j . The main novelty of our approach is the systematic use of a classical comparison theorem for finite linear systems which leads easily to the existence of a nonnegative minimal solution and its properties. Application to generalized stochastic birth and death processes produces criteria for honest and dishonest probability distributions. A short proof of the Kolmogorov and Chapman-Kolmogorov equations for stochastic processes follows. The results hold for L 1 -coefficients. Our method extends to nonlinear infinite systems of quasimonotone type and can be used for numerical procedures that yield exact results; cf. the Addendum.     

Monotone Method for Second Order Periodic Boundary Value Problems and Periodic Solutions of Delay Difference Equations

In this article, we employ monotone iterative technique to study the existence of solutions for second order periodic boundary value problem and periodic solutions of delay difference equations.     

Stability of an age-structured epidemiological model for hepatitis C

This article introduces an age-structured epidemiological model for the disease transmission dynamics of hepatitis C. We first show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number ? ₀ is below one, in this case, the disease always dies out, then we prove that at least one endemic steady state exists when the reproductive number ? ₀ is above one, the stability conditions for the endemic steady states are also given.